WilsonLoopExpectationsforNon-Abelian ...

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arXiv:2111.07540v1 [math-ph] 15 Nov 2021 Wilson Loop Expectations for Non-Abelian Gauge Fields Coupled to a Higgs Boson at Low and High Disorder Arka Adhikari Stanford University [email protected] November 16, 2021 Abstract We consider computations of Wilson loop expectations to leading or- der at large β in the case where a non-abelian gauge field interacts with a Higgs boson. By identifying the main order contributions from mini- mal vortices, we can express the Wilson loop expectations via an explicit Poisson random variable. This paper treats multiple cases of interests, including the Higgs boson at low and high disorder, and finds efficient polymer expansion like computations for each of these regimes. Contents 1 Introduction 2 1.1 Background and History ....................... 2 1.2 Organization of the Paper ...................... 3 1.3 Preliminary Notation and Discussion ................ 4 1.3.1 Previous Work: Wilson Loops in Abelian Lattice Higgs Model ............................. 6 1.4 Extension of the Analysis of Wilson Loop Expectation ...... 6 1.4.1 The Excitations of the Higgs field .............. 7 1.5 The Low Disorder Regime: High κ ................. 8 1.6 The High Disorder Regime: κ Low ................. 9 1.6.1 Informal Discussion ..................... 9 2 The Low Disorder Regime: A Toy Model 12 2.1 Introduction and Informal Discussion ................ 12 2.1.1 Higgs boson Configuration .................. 12 2.1.2 Wilson Loop Action ..................... 14 2.2 Rigorous Definitions ......................... 14 2.3 Analysis of the Wilson Loop Action ................ 19 1

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Wilson Loop Expectations for Non-Abelian

Gauge Fields Coupled to a Higgs Boson at Low

and High Disorder

Arka Adhikari

Stanford University

[email protected]

November 16, 2021

Abstract

We consider computations of Wilson loop expectations to leading or-

der at large β in the case where a non-abelian gauge field interacts with

a Higgs boson. By identifying the main order contributions from mini-

mal vortices, we can express the Wilson loop expectations via an explicit

Poisson random variable. This paper treats multiple cases of interests,

including the Higgs boson at low and high disorder, and finds efficient

polymer expansion like computations for each of these regimes.

Contents

1 Introduction 2

1.1 Background and History . . . . . . . . . . . . . . . . . . . . . . . 21.2 Organization of the Paper . . . . . . . . . . . . . . . . . . . . . . 31.3 Preliminary Notation and Discussion . . . . . . . . . . . . . . . . 4

1.3.1 Previous Work: Wilson Loops in Abelian Lattice HiggsModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Extension of the Analysis of Wilson Loop Expectation . . . . . . 61.4.1 The Excitations of the Higgs field . . . . . . . . . . . . . . 7

1.5 The Low Disorder Regime: High κ . . . . . . . . . . . . . . . . . 81.6 The High Disorder Regime: κ Low . . . . . . . . . . . . . . . . . 9

1.6.1 Informal Discussion . . . . . . . . . . . . . . . . . . . . . 9

2 The Low Disorder Regime: A Toy Model 12

2.1 Introduction and Informal Discussion . . . . . . . . . . . . . . . . 122.1.1 Higgs boson Configuration . . . . . . . . . . . . . . . . . . 122.1.2 Wilson Loop Action . . . . . . . . . . . . . . . . . . . . . 14

2.2 Rigorous Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Analysis of the Wilson Loop Action . . . . . . . . . . . . . . . . 19

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2.3.1 Trivial Configurations . . . . . . . . . . . . . . . . . . . . 202.3.2 The main term contribution . . . . . . . . . . . . . . . . . 222.3.3 Bounding the probability of large vortices . . . . . . . . . 23

2.4 Approximating Mγ as a Poisson Random Variable . . . . . . . . 27

3 The Non-Abelian Case 31

3.1 Model and Some Preliminary Discussion . . . . . . . . . . . . . . 313.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.2 Removing Trivial Gauge Invariances . . . . . . . . . . . . 32

3.2 The case of Low Disorder in the Higgs Field . . . . . . . . . . . . 34

4 The High Disorder Regime: Small κ 36

4.1 Expansion in Random Currents . . . . . . . . . . . . . . . . . . . 364.2 Knot Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Identifying the Main Order Excitations . . . . . . . . . . . . . . . 444.4 Approximation by a Poisson Random Variable . . . . . . . . . . 49

5 Decorrelation Estimates and a More Precise Expansion 53

5.1 Minimal Vortices and Decorrelation . . . . . . . . . . . . . . . . . 585.2 Approximation by Poisson Random Variables . . . . . . . . . . . 74

A Details on Vertex Decompositions 81

1 Introduction

1.1 Background and History

Quantum field theory and the Standard Model are one of the greatest successesof modern physics; they are able to compute the behavior of the smallest par-ticles to remarkable accuracy. However, these physical theories have yet to begiven a rigorous formulation due to multiple difficulties in trying to define theHamiltonians used on a continuous space. Glimm and Jaffe [11] attempted todefine quantum field theories by first defining a Euclidean lattice gauge theoryand then applying a Wick rotation to the proper complex space. Seiler [14]pointed out that there were issues when trying to follow this path in general;Seiler instead proposed that the basic object one should study are appropriaterandom functions on an appropriate space of closed curves.

One promising strategy to do this was via the method of lattice gauge the-ories. Inspired by ideas from statistical physics, Wilson [17] proposed latticegauge theories as a means of computing some quantities of interest in latticegauge theory; more specifically, he wanted to explain quark confinement. Lat-tice gauge theories are essentially quantum field theories defined on a discretelattice; on this discrete space, the Hamiltonians can be well-defined. The ulti-mate hope is that one could take the limit of quantities defined on the latticeas the lattice size of the lattice goes to ∞ and the lattice spacing goes to 0 todefine a continuous theory.

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This approach is promising, but yet there is still no proof that in the inter-esting physical dimension of 4 there is a way to take the lattice spacing to 0 andstill obtain obtain a meaningful probability distribution in the end. However,it is still a very interesting question to determine whether there are importantphysical quantities of interest such that there is a meaningful limit when com-puted on lattice gauge theories as the size of the lattice goes to ∞. The mostimportant quantity of interest are Wilson loop observables. These Wilson loopexpectation will be formally defined at the end of subsection 1.3, but we candescribe the physical meaning of these values here. In the Standard Model,many larger subatomic particles, called baryons, are theorized to be composedof smaller constituents called quarks. An unusual fact about quarks is thatindividual quarks are not found alone in nature. It is argued that in real worldscenarios, the quarks are tightly bound to each other. Wilson in [17] arguesthat if Wilson loop expectations satisfy appropriate decay conditions, namelythe Area Law, then this would be sufficient to establish quark binding.

Though the most physically relevant case of lattice gauge theories occur whenstudying the group SU(4), there is substantial literature in both the math andphysics literature studying lattice gauge theories on finite groups [2, 3, 7, 12, 13,15, 16]. In more recent history, there were multiple works computing the valuesof Wilson loop expectation for various discrete groups. In particular the papers[5] and [4] compute Wilson loop expectations in the case of pure gauge field.The paper [10] computes Wilson loop expectations for a pure U(1) gauge group.However, in the Standard Model, it is expected that the gauge field interactswith multiple other particles of interest; the interaction with other particlesmakes the analysis substantially more completed. This model considers thecomputation of Wilson loop expectations when our Hamiltonian includes aninteraction with a Higgs boson. Under certain conditions, we would expectthat the analysis in this work could be adapted to the case of multiparticleinteractions without too much difficulty. We remark here that the work [9]treats the problem of computing Wilson loop expectations with a Higgs boson,but only in the case that the gauge group is abelian and there is low disorderin the Higgs field. This work substantially generalizes the analysis to non-abelian groups and to low disorder as well, amongst other generalizations. Inthe proceeding subsections, we give further mathematical background on theproblem of computing Wilson loop expectations as well as our main results.

1.2 Organization of the Paper

In this short subsection, we describe the organization of the paper. In the restof the introduction, we present the notation we will use as well as further detailsabout the problem of computing Wilson loop expectations on models containinga Higgs boson. We discuss some of the history behind the problem and introduceour two major new regimes, the low disorder and the high disorder regime.

These two regimes have different polymer expansions an each have their ownsection. Section 2 analyzes the low disorder regime with a toy model, where Gand H are both abelian Z2 groups. Within this section, we introduce the model

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in Subsection 2.1, introduce our polymer expansion in subsection 2.2, performour identification of the main order contributions in Subsection 2.3, and finallycompare our model to a computation involving a Poisson random variable inSubsection 2.4.

Wondrously, there are no substantial differences in the low disorder regimebetween the abelian and non-abelian cases. Thus, Section 3 has the simulta-neous job of introducing the notation of the non-abelian Higgs boson model insubsection 3.1 and perform the brief modifications necessary to analyze the lowdisorder regime in Subsection 3.2.

In Section 4, we give the more involved analysis of the lower disorder regime.The first two Subsections 4.1 and 4.2 introduce elements of our polymer likeexpansion. Subsection 4.3 identifies the main order contribution and Subsection4.4 performs a comparison to a Poisson random variable.

In Section 5, we perform a far more delicate analysis of the error terms inSection 4 involving particular properties of the minimal vortices that form thecrux of our main-order contribution in Section 4. There are delicate decorrela-tion estimates established in Subsection 5.1 which are helpful in improving theerror term and then improving the approximation by Poisson random variablesin Subsection 5.2.

1.3 Preliminary Notation and Discussion

Lattice gauge theories, as their name implies, are studied in some lattice ΛN :=[−N,N ]d which is a sublattice of Zd. As a graph, the structure is inheritedas a subgraph of Zd. We let VN denote the set of vertices of ΛN and we letEN be the set of oriented edges of ΛN ; the orientation distinguishes the edgee = (x, y) from −e = (y, x). Also important in lattice gauge theories are the setof plaquettes PN . If we let x1, x2, x3, x4 be a set of four vertices such that xi+1

is adjacent to xi (with x5 = x1), then the square bounded by these four verticeswill be called a plaquette. Plaquettes have a natural orientated inherited fromthe order in which we traverse the boundary vertices. (Thus, if we let p be theplaquette with boundary vertices traversed in the order x1, x2, x3 and then x4,then −p is the plaquette with boundary vertices traversed in the order x4, x3, x2and then x1.)

A gauge configuration σ with gauge group G on a lattice, ΛN ⊂ Zd is anassignment σ : EN → G such that σ−e = (σe)

−1. The gauge group G alsocomes with a unitary representation ρ : G→ UD to the appropriate space of Dby D unitary matrices.

A Higgs boson configuration is the assignment φ : VN → C of a complexvalued function at each vertex. Later in this introduction, we will provide somerestrictions on the values that φ could take. Associated to the Higgs boson is acovariant derivative that mixes the action of Higgs boson and the gauge field.Let e be the edge that connects the vertex x to the vertex y. Then, we have,

Deφ = φxTr[ρ(σe)]− φy . (1)

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With this, we have the following interaction term between the Higgs bosonand the gauge field,

SD(σ, φ) :=1

2

e∈EN

|Deφ|2 = 2d∑

x∈VN

|φx|2 −∑

e=(x,y)∈EN

φx Tr[ρ(σe)]φy. (2)

A gauge field configuration also has a self-energy, corresponding to the cur-vature form, or Wilson action functional. Associated to any oriented plaquettep, whose boundary vertices are x1, x2, x3 and x4 ( with edges ei = (xi, xi+1)), wecan define the action ψp along the plaquette to be Tr[ρ(σe1 )ρ(σe2 )ρ(σe3 )ρ(σe4 )].Note that due to the cyclic property of the trace, it is indeed the case that thevalue of ψp does not depend in particular on which vertex we start the orderingof the boundary vertices( though the actions will differ if we consider ψp orψ−p). The Wilson action function will be the sum of the actions ψp over alloriented plaquettes ψp.

SW (σ) := −∑

p∈PN

ψp(σ). (3)

For an abelian group and a 1-dimensional representation ρ, we can representthis in a much more direct form as follows:

SW (σ) := −∑

p∈PN

ρ((dσ)p). (4)

If we consider the oriented plaquette p with oriented boundary edges e1,e2,e3and e4, then (dσ)p is the well-defined element

i σei . We needed to introducemore general notation for the non-abelian case since we are no longer able toassign a canonical element to each plaquette.

The Higgs boson also has a self- interaction given by the following form,

V (φ) :=∑

x∈VN

(|φx|2 − 1)2. (5)

Consider the following interaction Hamiltonian between the Higgs boson anda gauge field,

HN,β,κ,ζ(σ, φ) := −βSW (σ)− κSD(σ, φ) − ζV (φ). (6)

We consider the formal limit when we take ζ → ∞. In this limit, the valuesof |φx| are confined to 1, so φx can only take values along the unit circle. Wewill let φx take values in some finite subgroup, H , of the multiplicative group ofthe unit circle. The group H is isomorphic to Zk( the additive group of integers

mod k) for some k; namely, φx can only take values of the form e2πijk for some

0 ≤ j < k. We also remark that after fixing |φ| = 1, we have that φ = φ−1.This replacement will be used many times when expressing the Hamiltonian.

We consider the following measure of gauge and Higgs boson configurationson the lattice ΛN .

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µ(σ, φ) =1

Zexp[HN,β,κ(σ, φ)], (7)

where Z is the partition function coming from the sum of exp[HN,β,κ(σ, φ)] overall configurations of σ and φ.

On loop γ, the Wilson loop action Wγ(σ, φ) is the product Tr[∏

e∈γ ρ(σe)]of the σ’s along γ. As before, the cyclic property of the trace ensures that thisquantity is well-defined.

1.3.1 Previous Work: Wilson Loops in Abelian Lattice Higgs Model

In this previous work, the authors considered the case that G was abelian, ρwas a one-dimensional representation, and H is contained inside the image ofρ(G)(which will be a subgroup of the unit circle). This simplification allows oneto effectively gauge out the effect of the Higgs boson.

Before describing this construction, we remark that we can always assumethat the only element g ∈ G such that ρ(g) = 1 is the identity element. Ifthis were not true, then we can remove a trivial gauge invariance that does notchange the Hamiltonian action, but modifies the group G so that there is onlythe only element with ρ(g) = 1 is the identity. The construction is as follows:first let G′ be the subgroup of elements g′ in G consisting of those elementswith ρ(g′) = G. Since multiplying a configuration σe by some element g′ ∈ G′

at any edge does not change the Hamiltonian, we see that this is a trivial gaugeinvariance. We might as well consider the quotient group G/G′ as our gaugegroup. This G/G′ has the property that the only element g with ρ(g) = 1 is theidentity element.

Now let us return to our simplification of the Higgs boson action. We canmake the following map that preserves the action, but simplifies the Higgs bosonterm. Let ηx be the term such that ρ(ηx) = φx (this is unique since ρ(g) = 1 onlyif g = 1), then we see that under the map (φx, σe) → (1, η−1

x σeηy) where (x, y)are the endpoints corresponding to the edge e. We let the map σe → η−1

x σeηybe Fφ. For any choice of φ, this Fφ is a bijection between the set of gauge fieldconfigurations σe and itself.

This argument shows that when considering the expectation of a gauge in-variant function, such as a Wilson loop expectation, one can merely considerthe expectation with respect to the net Hamiltonian,

HN,β,κ = −κ∑

e∈EN

ρ(σe)− β∑

p∈PN

ρ((dσ)p). (8)

This reduces the question of the Higgs boson to a slightly modified questionabout pure gauge configurations. To some extent, the paper [9] only dealt withcase under the assumption that κ is sufficiently large.

1.4 Extension of the Analysis of Wilson Loop Expectation

One can imagine multiple changes to the model that make the model proposedin (8) substantially more difficult to analyze.

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One question that one could ask is the following:What happens if the group H for the Higgs boson is not contained in the

image ρ(G) of the gauge group?The above question is well-defined for abelian groups with a 1-dimensional

representation. For more general groups and representations, the analogue isfollows. Let G be the group of elements whose image under the representationρ is a multiple of the identity, i.e. g ∈ G if ρ(g) = cI, where c is some constant.We let X be the group of these coefficients c from elements of G; since ρ is aunitary representation, X is a subgroup of the unit circle. The analogue wehave to consider is that H is not a subgroup of these elements X .

If H were not contained in X , one immediately sees that one cannot imme-diately gauge out the effects of the Higgs boson. If one now has to consider anon-trivial Higgs boson group, one must also need to consider the excitationsof the Higgs field.

For example, if the Higgs field is constant, one automatically knows thatthe lowest energy states are those that set σe = 1 and any other configura-tion on edges can be reasonably understood to be an exponentially suppressedexcitation.

However, if we do not start with a constant Higgs field, it may be possiblethat we could get a lower energy configuration by setting σe 6= 1 instead ofσe = 1.

1.4.1 The Excitations of the Higgs field

To describe this phenomenon, we will first give a simple calculation. Considerthe case that G = Z3 while H = {e2πik9 } for k between 0 and 9 and β = 0 (orat least β ≪ κ). Observe that H is isomorphic to Z9.

In this case, one can apply a gauge transformation similar to one applied inthe previous subsection. However, ρ(σe) can only take values {1, e2πi39 , e2πi 69 }rather than all of H . Thus, the furthest we are able to simplify the values of φxunder a gauge transformation are to {1, e2πi 19 , e2πi 29 }. Let θx = {1, e2πi19 , e2πi 29 }be the simplified value of φx under the gauge transformation.

Now consider an edge e = (x, y) such that θx = e2πi29 and θy = 1. We

see that the maximum value of exp[θxρ(σe)θy + θxρ(σ−1e )θy] is exp[2 cos(2π

89 )]

which occurs when σe = e2πi69 . Observe that the maximum is not exp[2], as one

would get if θx = θy.What this illustrates is the fact that if one fixes the configuration of the Higgs

boson at every lattice point, one could possibly have substantial changes to thelowest energy configuration. Thus, in order to analyze the Higgs boson model ingeneral, one would necessarily need a method to analyze the excitations of theHiggs boson beyond just reducing it to a case of a modified pure gauge field. Wewill show that various scales of κ will change the behavior of the excitations ofthe Higgs boson. One will have to treat these different regimes through differenttypes of analysis.

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1.5 The Low Disorder Regime: High κ

In the case that κ is large, much like the case dealt with in [9], one finds that thelarge value of κ prevents excitations of σe 6= 1 for many edges. They can showthat in this regime, one can reduce the computation of Wilson loop expectationsto the computation of the contribution from minimal vortices, e.g. excitationsthat consist of single edge with σe 6= 1 with all other edges in a neighborhoodaround it set to σe = 1.

When we introduce a Higgs boson field, we can assert a slightly similarstatement. Though it is not the case that, given a fixed Higgs boson fieldconfiguration, the lowest energy configuration fixes all values of σ to σe = 1, wedo see that an edge e = (x, y) with σe 6= 1 and arbitrary Higgs boson valuesφx,φy at the boundary vertices will certainly have less energy at that edge thana configuration that sets φx = φy and σe = 1. Thus, if one is willing to modifythe Higgs field in addition to the gauge field configuration, one may be able totreat edges with σe 6= 1 as an excitation.

This is exactly what is done in Section 2. In this section, beyond just gener-alizing the analysis of [9] to a more general Higgs boson group, we simplify theanalysis of [9] by providing a true polymer expansion. One further consequenceof this polymer expansion is that it helps express the Wilson loop expectation asa computation involving an explicit Poisson random variable. Roughly speak-ing, if we define the support of a configuration (σ, φ) of a joint gauge field andHiggs field configuration as those plaquettes that bound excited edges e withσe 6= 1 or φx 6= φy, then we derive the desired properties behind a true polymerexpansion. The details of the polymer expansion are given in Definition 1. Themain principle behind a polymer expansion is that it is easy to analyze prob-abilities if one can always perform the following procedure. If a configurationhas support P1 ∪ P2, where P1 and P2 are disjoint, then configuration can bebijectively split into a configuration supported on P1 and another supported onP2.

There is a rather straightforward way to do this splitting for the gauge fieldconfigurations σ, but one has to exert effort in order to split the Higgs fieldconfigurations into the disjoint supports. Section 2 performs this analysis in atoy case that G = Z2 and H = Z2, where the splitting is easier to describe. Theanalysis for more general gauge group G and Higgs boson group H is similar,but more cumbersome in the description of the division.

We informally describe our main theorem in this section as follows.

Informal Theorem 1 (Informal Version of Theorems 1 and 3 of Section 2). For sufficiently large β and κ, the main order contributions to the Wilsonloop observables are by the number of minimal vortices (roughly, those plaquetteexciations centered around a single edge with σe 6= 1) that are centered aroundedges of γ. Furthermore, the number of minimal vortex excitations along theedges of γ can be treated as roughly a Poisson random variable with parameterO(|γ|).

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Define the quantity

Aβ,κ :=

g 6=1 ρ(g) exp[12βRe[(ρ(g)− ρ(1))]] exp[2κRe[ρ(g)− ρ(1)]]∑

g 6=1 exp[12βRe[(ρ(g)− ρ(1))]] exp[2κRe[ρ(g)− ρ(1)]]

E[Wγ ] ≈ E[AXβ,κ], (9)

where X is a Poisson random variable with expectation |γ| exp[12βRe[ρ(g) −ρ(1)] exp[2κRe[ρ(g)− ρ(1)]].

Wondrously, our polymer expansion is robust enough to treat the case ofnon-abelian G with few changes to the proof.

1.6 The High Disorder Regime: κ Low

A difficulty in the case of small κ is one can no longer consider edges with σe 6= 1as excitations of the configuration (σ, φ). Instead, the predominant suppressionto the probability is caused by excited plaquettes with ρ((dσ)p) 6= 0. Due tothis fact, we see that it would be better to separate the gauge field excitationsas follows. We first create an auxiliary field η : VN → G satisfying the propertythat σe = ηxσeη

−1y has as ‘few’ nontrivial edges with σe 6= 1 as possible. The

effects of the Higgs boson interaction on the gauge field configuration can beunderstood as a fluctuation of the auxiliary field ηx. What follows is an informaldiscussion of the procedure being described.

1.6.1 Informal Discussion

To do this, we introduce a spanning tree T with base point b of the lattice Λ. Ourgoal is to find the auxiliary field η : VN → G such that the field σe = η−1

x σeηyfor e = (x, y) satisfies some particular properties. The property we try to ensureis the following: for each edge e ∈ T we assign the value σe = 1.

The auxiliary field η can be constructed inductively as follows. We firstset ηb = 1. Now, for any other vertex v on the lattice ΛN , let p = (b =v0), v1, v2, . . . , (vn = v) be the path on the spanning tree T connecting b to v.Assume that we have already assigned the values ηv0 to ηvn−1 . Then, we chooseηvn := σ−1

e ηvn−1 , where e is the edge (vn−1, vn). Clearly, with this choice ofηvm , we have σe = 1 for e = (vn−1, vn). Since T is a spanning tree, this fixes allvalues of η on VN . Finally, we can define σ as σe = η−1

x σeηy for e = (x, y). Weremark that in the course of the proof, T will not be fixed a-priori and will bechosen as is most appropriate for that part of the proof.

Let us give some heuristics for the partition function and other associatedquantities for this new Hamiltonian with the auxiliary field η. In the computa-tion of the partition function to highest order in β (for β sufficiently large andregardless of the specific value of κ), we can ignore the fluctuation of σe and setσe = 1 for all edges e. The partition function can be treated as a sum over allpossible configurations of ηv and φv for the Higgs boson.

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Namely, Z ≈ ∑

ηv ,φvexp[

e=(v,w) κφvTr[ρ(ηvη−1w )]φ−1

w ]. The configurationof the values of σe are fixed to 1, since these excitations would be lower orderin β.

When computing the probability of seeing a single excited minimal vortexP (e) in the Wilson action, the probability can be computed as a fraction withsome numerator and Z as the denominator. To leading order, we see that thenumerator can roughly be expressed as a sum over configurations with σ′

e = 1for e′ not the center of the single excited vortex and σe 6= 1 for e = (a, b)the center of the single excited vortex. We see that it would be important tospecifically consider the values of ηa, ηb, φa, φb.

We see that one way to write the numerator as,

σe 6=1

exp[12β(Tr[ρ(σe)]− Tr[ρ(1)])]

ηa,ηb,φa,φb

exp[κRe[φaTr[ρ(ηaσeη−1b )]φ−1

b − φaTr[ρ(ηaη−1b )]φ−1

b ]]

R

exp[∑

e=(v,w)

κφvTr[ρ(ηwη−1w )]φ−1

w ],

(10)

where∑

R denotes the sum over all elements not specified by ηa, ηb, φa, φb.Additionally, for the quantity inside the second sum, we will use the conventionφa = φa and so on. The reason we introduce this notation is to specificallyindicate that we fix the values of φa to φa and so on.

Now, the ratio

R exp[∑

e=(v,w) κφvTr[ρ(ηwη−1w )]φ−1

w ]∑

ηv ,φvexp[

e=(v,w) κφvTr[ρ(ηvη−1w )]φ−1

w ](11)

is the probability that under the Ising type interaction φvTr[ρ(ηvη−1w )]φ−1

w on

edges, we will get ηa = ηa, ηb = ηb, φa = φa, φb = φb for some specified values ofφa, φb, ηa, ηb. These can be considered to be some type of magnetization.

We see at low κ, we see that understanding the behavior of the η’s and the φfluctuations are non-trivial questions. Furthermore, we see that the introductionof the η and φ fields can lead to highly non-trivial correlations.

From the example above, we can imagine what would happen if we considera more general family of excited plaquettes. For example, let us consider twoexcited minimal vortices centered at edge e1 = (a1, b1) and e2 = (a2, b2). To dothis, we have to adjust the computation of the numerator to specifically fix thevalues of ηai,bi and φai,bi .

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Namely, we see that we can better express the numerator as,

σei

i

exp[12β(Re[ρ(σei )]− Re[ρ(1)])]

ηai,bi,φai,bi

i

exp[κRe[φaiTr[ρ(ηai

σei η−1bi

)]φbi − φaiTr[ρ(ηai

η−1bi

)]φbi ]]

R

exp[∑

e=(v,w)

κφvTr[ρ(ηwη−1w )]φw ],

(12)

where as before∑

R is the sum over variables that are not ηai,bi , φai,bi while

φaiis set equal to φv in the second sum.The important ratio we have to consider in this example is,

R exp[∑

e=(v,w) κφvTr[ρ(ηwη−1w )]φw ]

φv ,ηvexp[

e=(v,w) κφvTr[ρ(ηwη−1w )]φw]

. (13)

We see that the computation involves understanding the correlation betweenηai, ηbi and φai

, φbi at different sites. Though at small κ we would expectexponential decay of correlations, it is the very presence of these correlationsthat makes it difficult to define a polymer expansion in the case of small κ purelyfrom knowing the values of σ and φ, even in the abelian case.

For simplicity of notation, let us consider the abelian case. The main issuecan be observed by looking at the following example. Consider a gauge fieldconfiguration σ1 such that the set of excited plaquettes , those with d(σ)p 6= 0 isa small set with size k, but the set of edges with σe with σe 6= 0 is much larger,say of O(k4). This is an obstruction to a polymer expansion.

Naively, what this means is that if we find another configuration σ2 whosesupport contains one of these O(k4) edges, then there will be correlationsbetween observing both σ1 and σ2. Namely, P(σ1σ2) is far different fromP(σ1)P(σ2). This is in contrast to the case of a pure abelian gauge field, whereconfigurations can easily be split as the set of excited plaquettes associated toσ1 and σ2 are disjoint.

For small κ, we found an auxiliary Hamiltonian based on the random currentsrepresentation of the Ising model [8] that exactly allows us to characterize whenthe effects of κ cause two disjoint configurations to be correlated with each other.The random current expansion introduces a new field I(e) for each edge e andcouples our original Hamiltonian, HN (σ, φ) to a new Hamiltonian H(σ, φ, I) ,equation (72), in the new variables I. One can define a polymer expansion usingthese new variables, as in Definition 7. Unfortunately, even in the abelian case,the introduction of the Higgs Field leads to knotting problems, as in the non-abelian pure gauge field case [4]. Through this new polymer expansion, we showthat the Wilson loop expectation can be reduced to understanding the effectsof minimal vortices lying along the edges of γ in section 4. There is a simpleargument with a worse error rate showing that the number of minimal vorticesalong γ can be treated as a Poisson random variable in the end of the same

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section 4. However, by carefully understanding the decorrelation between thecontribution of the Wilson loop action from different minimal vortices, we canimprove our error analysis. Roughly speaking, this involves relating quantitiesof the form (13) to products of those of the form (11). This rather delicate taskwas performed in Section 5.

We have the following informal version of our main result,

Informal Theorem 2 (Based on Theorems 4 and 5 of Section 4 and Theorem9 of Section 5). For β sufficiently large and κ sufficiently small, the main ordercontribution to Wilson loop expectations come from minimal vortices centeredaround the edges of γ .

One can compute the Wilson loop expectation as,

E[Wγ ] ≈ E[Tr[DXβ,κ]], (14)

where Dβ,κ is a matrix defined in Theorem 7 and X is a Poisson RandomVariable whose expecation is O(|γ|).

2 The Low Disorder Regime: A Toy Model

2.1 Introduction and Informal Discussion

As we have discussed in the introduction, when the gauge group is not the sameas the symmetry group of the Higgs field, the fluctuations of the Higgs bosonwill affect the lowest energy configurations of the gauge field. To illustrate themain ideas in the low disorder regime(large κ), we consider the following slightlysimplified model.

On the lattice ΛN = [−N,N ]4, we allow the Higgs boson φp at each latticepoint p ∈ ΛN to take 2 values, either +1 or −1 (which we will later refer to+ and −, respectively, and will be called charges). The gauge group G of thegauge field σ is Z2; as such, we will talk about setting edges e to σe = 1 orσe = −1. We will also let ρ be a 1-dimensional representation of Z2.

We remark that since for all elements g ∈ Z2, we have g = g−1, we do notneed to concern ourselves with the orientations of edges or of plaquettes. Thus,we do not need to take into account the orientation of edges. Furthermore,since Higgs boson values φ are assigned to vertices and gauge field values σare assigned to edges, we may use language as ‘assign + to a vertex v’ tounambigously mean assigning the Higgs boson value φv = + at the vertex v and‘assign -1 to an edge e’ to mean assigning the gauge field value σe, σ−e = −1 toeither orientation of the edge e.

2.1.1 Higgs boson Configuration

We start by describing, in words, the type of interaction between the gauge groupand the Higgs field. Consider an edge e ∈ EN of the lattice. If it connects twopoints of the same charge, then assigning {+1} to e will give energy E1 andassigning {−1} to e will give energy E2 < E1. If instead, the edge e connects

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two vertices of opposite charges, then assigning {−1} as the gauge field to edgee will give energy E3 and assigning {1} to the edge e will give energy E4 < E3.The energies are ordered as follows

E1 > E3, E2, E4.

We will have a standard Wilson loop energy of the form βρ((dσ)p), where forthe abelian group Z2 we can understand (dσ) as the sum of all the gauge groupelements along the edges that bound p.

After subtracting an appropriate constant, we can formally write our Hamil-tonian as follows,

HN,β,κ(φ, σ) =∑

e=(v,w)∈EN

κ[f(σe, φvφ−1w )− f(1, 1)] + β

p∈PN

[ρ((dσ)p)− ρ(1)].

(15)where f(1, 1) = E1, f(1,−1) = E2, f(−1, 1) = E3, f(−1,−1) = E4 and ourmeasure on the lattice is

µN,β,κ(σ, φ) =1

ZΛ,βexp[HN,β,κ(σ, φ)]. (16)

Here ZΛ,β is the partition function,

ZΛ,β =∏

v∈VN

φv∈{−1,1}

e∈EN

σe∈Z2

exp[HN,β,κ(σ,φ)]. (17)

The action considered above is rather similar to the action in (6) if the gaugegroup G were chosen to be Z2 and the Higgs boson group H is Z4. In this case,one would only be able to gauge out the Higgs boson field to take only twovalues ( + or − ) here; + corresponds to φx = 1 or −1 while − correspondsto φx = i or φx = −i. In this case, if there is an edge e = (v, w) such thatφv and φw are assigned the same value, then the lowest energy configurationwould assign {+1} to σe. By contrast, if there is an edge e = (v, w) such thatφv and φw are assigned different values, then assigning σe to be {−1} and {+}would give the same energy to the edge; this energy E would be less than themaximum energy assigned when φv and φw are the same sign and σe is assignedthe value {+1}.

With the Hamiltonian in (15) and ignoring the effect of the Wilson actionfor now, we expect the following behavior. Each of the sites would have thesame sign (either + or -), while we would expect that each of the gauge fieldswould be assigned the value {+1}. The fact that we would expect nearly allvertices to be assigned the same charge is due to the constraints E3, E4 < E1

and a Peierl’s argument.Consider an assignment of Higgs boson charges in which the φ’s are not con-

stant. WLOG, we can consider the case that there is a connected neighborhoodN of vertices assigned a negative charge, but this neighborhood N is surroundedby an ocean of positive charge. We can let E(N) be the edges that connect N

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to its complement N c. We now argue that we can obtain a less excited configu-ration by flipping the charges of the vertices in N and changing the assignmentof all the edges in E(N) to 1. Clearly, the change in the energy of this configu-ration is given by at least κ|E(N)|(E1 −max(E3, E4)), where |E(N)| is the sizeof the set E(N). Thus, we expect that deviations of the Higgs field from theconstant will be suppressed exponentially for large enough κ.

The previous discussion shows one important concept. When dealing witha pure gauge field, one only needs to considering fluctuations of the edges σefrom the identity. This leads to an understanding of Wilson loops expectationsvia the associated plaquette computations. Here, the fluctuations comes in twoways; as before, we must still consider fluctuations of edges from the low energyconfigurations, but we must further consider the fluctuations of the Higgs fieldconfigurations from the identity.

2.1.2 Wilson Loop Action

As is standard, we want to compute the expected value of the Wilson loop actionas the size of the lattice goes to infinity. We consider a closed non-intersectingloop γ in ΛN consisting of edges e1, e2, . . . , em.

We can define the Wilson loop action as on a configuration C = (φv, σe) as

Wγ(C) =m∑

i=1

ρ(σe) = ρ(

m∑

i=1

σe). (18)

We use the notation 〈γ, σe〉 =∑m

i=1 σe.Recall that on abelian groups we can consider σe to be a one-form supported

on the edges, EN ; for reference, see the discussion in Section 3 of [4]. Thus,we can interpret 〈γ, σ〉 as an integral on the set of 1-forms. More importantly,we can apply Stokes’ theorem and can find a surface q such that δq = γ and(q)p ∈ {−1, 0, 1} for all plaquettes p ∈ PN such that

〈γ, σ〉 =∑

p∈PN

qp(dσ)p. (19)

When we write the Wilson loop action in terms of the integration of a two-form over a surface, we see that we are able directly see the effect of nontrivialplaquettes those with (dσ)p 6= 1; later, we will see that this decomposition willallow us to determine whether some excitation of the Higgs and gauge field Cwould be independent of the Wilson action on the loop.

In the next section, we will start rigorously describing some notions we canuse to characterize fluctuations of the Higgs field and the gauge field.

2.2 Rigorous Definitions

There is a trivial symmetry that preserves the Hamiltonian; we flip the signs ofall the Higgs boson configurations ( ’+’ to ’-’ and ’-’ to ’+’), while the gaugefield configurations σe are unchanged. For some simplicity in the proof, we may

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assume that N is odd. When N is odd, we can always apply a global flip ofthe Higgs boson charges to get a unique configuration satisfying the conditionthat

v∈VNφv > 0. (All this condition means is that the majority of charges

are + instead of −.) At this point, we can start giving definitions of the typesof excitations we will consider so that we can consider a cluster expansion.

Definition 1. [Support of Configurations]We define Ce, our excited edges, as

Ce = {e = (v, w) ∈ EN : φv 6= φw} ∪ {e ∈ EN : σe = −1}.

The support of our configuration is

supp(C) = {p ∈ PN : ∃e ∈ Ce s.t. e ∈ δp}. (20)

Remark 1. Without the Higgs field action, an edge with σe 6= 1 is not neces-sarily excited. For example, one could find a plaquette whose boundary edgesare all −1. This plaquette would not be of lower energy in the Hamiltonian andwe would not consider this plaquette to be excited. By contrast, with the Higgsaction, a single edge with σe = −1 drives it to lower energy.

Lemma 1. Consider a configuration C = (σe, φv) If p is a plaquette such that(dσ)p 6= 1, then p ∈ supp(C).

Proof. Clearly, if (dσ)p 6= 1, then there clearly is some edge e in the boundary ofp such that σe 6= 1. Thus, the only non-trivial computations to the probabilityvalue from our clusters come from what we define to be the support of ourdistribution.

We will now define the function Φ(P ) where P is a set of plaquettes in PN ;this is a crucial part of our cluster expansion.

Definition 2. We let E(P ) denote the set of edges e such that there exists aplaquette p ∈ P with e ∈ δp. Similarly, we let V (P ) denote the set of verticessuch that there is a plaquette p ∈ P such that v is a boundary vertex of p.

Φ(P ) =∑

C=(σe,φv)supp(C)=P∑

v φv>0

p∈supp(C)exp[β(ρ((dσ)p)− ρ(1))]

e∈E(P )e=(v,w)

exp[κ(f(σe, φvφ−1w )− f(1, 1))].

(21)

We also have a similar cluster expansion formula incorporating the effects

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of the Wilson action.

ΦW (P ) =∑

C=(σe,φv)supp(C)=P∑

v φv>0

p∈supp(C)exp[β(ρ((dσ)p)− ρ(1))]

×∏

e∈E(P )e=(v,w)

exp[κ(f(σe, φvφ−1w )− f(1, 1))]

e∈E(P )∩γ

ρ(σe).

(22)

Remark 2. Though after this section, we will not use this specific definitionof Φ(P ), we will use V (P ) and E(P ) frequently throughout the course of thispaper.

Our second definition gives us criterion to determine whether our clusterswould interact with each other, or could be considered to split from each other.

Definition 3. We define the following graph G2 with vertex set PN as follows.We say that there is an edge between two plaquettes p1 and p2 if there exists a3-cell such that both p1 and p2 are on the boundary of said 3-cell.

We call a set of plaquettes V a vortex if the plaquettes of V form a connectedset in the graph G2.

Furthermore, we say that two sets of plaquettes P1 and P2 are compatible ifthe set P1 and P2 are disconnected in G2. Otherwise, we say that P1 and P2

are incompatible.The decomposition of some plaquette set P into its maximal connected sub-

components P = V1 ∪ V2 . . . ∪ VN is called a vortex decomposition of V . It isclear to see that Vi and Vj are compatible for any distinct pair i and j.

We have the following Lemma that allows us compute the function Φ(P ) interms of Φ evaluated on the elements of its vortex decomposition.

Lemma 2. Let P1 and P2 be two compatible sets of plaquettes. Then,

Φ(P1 ∪ P2) = Φ(P1)Φ(P2). (23)

As a consequence, if V1 ∪ V2 . . . ∪ VN is a vortex decomposition of P , then wehave,

Φ(V1 ∪ V2 . . . ∪ VN ) = Φ(V1)Φ(V2) . . .Φ(VN ). (24)

The function ΦW would satisfy a similar property; namely,

ΦW (V1 ∪ V2 . . . ∪ VN ) = ΦW (V1)ΦW (V2) . . .ΦW (VN ). (25)

Proof. Our goal is to find a bijection C = (σe, φv) → (C1 = (σ1e , φ

1v), C2 =

(σ2e , φ

2v)) where C is a configuration whose support is P1 ∪P2 and Ci is a config-

uration whose support is Pi that satisfies the following equation.

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p∈Cexp[β(ρ((dσ)p − ρ(1))]

e∈E(P1∪P2)e=(v,w)

exp[κ(f(σe, φvφ−1w )− f(1, 1))] =

p1∈C1

exp[β(ρ((dσ)p1 − ρ(1))]∏

e∈E(P1)e1=(v1,w1)

exp[κ(f(σe1 , φv1φ−1w1

)− f(1, 1))]

×∏

p2∈C2

exp[β(ρ((dσ)p2)− ρ(1))]∏

e2∈E(P2)e2=(v2,w2)

exp[κ(f(σe2 , φv2φ−1w2

)− f(1, 1))].

(26)We will assume that P2 is a vortex for simplicity.Recall that V (P1) and V (P2)(resp. E(P1) and E(P2)) denote the set of

vertices(resp. edges) that form a boundary vertex(resp. edge) of some plaquettein P1 and P2.

Consider a configuration C whose support is P1 ∪ P2. Let e be an edge inthe complement of E(P1 ∪ P2). Then, σe = 1 for this configuration; otherwise,if σe 6= 1, then a plaquette that contains e as a boundary edge would be in thesupport of C by our definition of support. This would imply e ∈ E(P1 ∪ P2),which is a contradiction.

Now consider an edge e ∈ E(P1) ∩ E(P2). If σe 6= 1, then all the plaquettesp that contain e as a boundary edge are in supp(C) and are connected to eachother in G2. However, since e is in E(P1), at least one of these plaquettesmust be in P1. For the same reason, one of these plaquettes must also be inP2. However, this implies that P1 and P2 are connected to each other; this is acontradiction.

Thus, there is a simple way to describe the gauge field configurations σe forthe configurations C1 and C2. For edges e in E(P1), we will set σ

1e = σe, σ

2e = 1.

for edges e in E(P2), we will set σ1e = 1 and σ2

e = σe. For edges e not in eitherE(P1) or E(P2), we merely set σe = 1. Note that this is well defined since foredges e ∈ E(P1) ∩ E(P2), we know that σe = 1.

It is more complicated to describe the map on the Higgs field. Just as in theIsing model, one can expect to see islands of charges that include each other.Our procedure for mapping the Higgs field configurations from C to C1 and C2does not merely involve applying a restriction map as we have done for theedges. Instead, one must apply appropriate charge flips in order to ensure thegauge constraints

φ1v ≥ 0,∑

φ2v ≥ 0 are satisfied.We will describe the problem formally as follows. Divide V (P1 ∪ P2)

c intoconnected components as B1 ∪ B2 ∪ . . . ∪ BN and, similarly, V (Pi)

c as Bi1 ∪

Bi2 ∪ . . . ∪ Bi

N1. We see from Lemma 19 in the Appendix that each set Bi is

monocharged. If we propose an charge assignment C1 with support P1, then wetoo must make sure that each of the sets B1

i are monocharged.Note that each set B1

i can be decomposed as follows B1i ⊂ V (P2) ∪ Bi1 ∪

. . . Bimi, where the union is minimal in |mi|. If it were the case that all the

sets Bik were assigned the same charge, e.g. cil = ci1 for all l ∈ {1, . . . ,mi1},

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then we would be able to safely assign the charge ci1 to all vertices in B1i in the

configuration C1.The problem occurs when there are two components Bik1

and Bik2in the

decomposition of B1i that are assigned two different charges cik1 6= cik2 . For

simplicity of notation, WLOG, we are considering the set B11 with two subcom-

ponents B1 and B2 of different charges. For simplicity of later description, letus also assume that B1 and B2 do not contain a boundary vertex of VN .

In the remainder of this proof, we will frequently use notation from Lemma20 in the Appendix. Let V(B1) and V(B2) be as in Lemma 20, the set of verticesconnected to B1 and B2 respectively, having the same charge. From the sameLemma, the exterior boundaries EB(V(B1)) and EB(V(B2)) are connected sets.Thus, EB(V(Bi)) must be entirely contained in either P1 or P2. If both of theseexternal boundaries were in P1, then this would imply that B1 and B2 wouldbe in different connected components in the splitting of B1

i . Therefore, it mustbe the case that one of the external boundaries, say EB(V(B2)) was in P2.

In this case, what one can do is the following. We can embed our lat-tice inside Zd. In the full lattice Zd, find the set S containing infinity whoseunique(internal) boundary is EB(V2). Now consider the complement of S in Zd.We see that V(B2) is a subset of Sc. Now, what one can do is to completely flipthe signs of all vertices in Sc. Under this transformation, the only edges whoseHamiltonian action value changes are those edges in E(Sc). However, all of theseedges must belong to E(P2) rather than E(P1). Thus, this flip applied to C willnot change the Hamiltonian action when restricted to edges of E(P2)

c ∪E(P1).As a consequence of this, if we consider an edge connecting a vertex of V (P1)to its complement, then this edge will still have its adjacent vertices assignedthe same charge even after this flip. Furthermore, this flip changes the chargesso that the charges of B2 will match those of its closest neighbors in Sc.

Through a careful ordering, one can come up with a series of charge flipsthat do not affect the Hamiltonian action’s value on E(P2)

c ∪ E(P1) and willensure that for each component B1

i ⊂ V (P )∪Bi1 ∪Bi2 ∪ . . .∪Biim the chargesassociated to the Bil ’s are the same. Intuitively, this procedure involves flippingthe outermost island, and then further applying flips to correct the internalislands. At this point, one can assign this charge to all the vertices in B1

i in C1.When restricted to edges in E(P2)

c ∪E(P1), the Hamiltonian actions of C1 andC are the same. If after this procedure

φv < 0, then one can perform a globalflip of the Higgs boson charge.

Through this procedure, we get a configuration C1 whose Hamiltonian actionon the set of edges E(P2)

c∪E(P1) are the same as those of C. We can similarlyconstruct C2. Furthermore, if we consider an edge in (E(P1)

c∪E(P2))∩(E(P2)c∪

E(P1)), then the action of this edge is trivial; the vertices attached to it areassigned the same color and σe = 1.

One can see for this map C → (C1, C2), the desired relationship (26) holds.In addition, given two different configurations (C1, C2) supported on P1 and P2

respectively, one can reverse the construction here to get a configuration C whosesupport is P1 ∪P2. This procedure is fairly tedious, but is still based on finding

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the appropriate sets to flip iteratively until one gets the desired configuration.The construction of this bijection completes the proof of Lemma 2.

In summary, the previous lemma has shown that the cluster expansion splitsbased on a vortex decomposition. We will now present the link between ourcluster expansion function Φ and probabilities under our Hamiltonian.

Lemma 3. Let ZΛ,β be the partition function associated to the Hamiltonian(15) on the lattice Λ. We have the following relations,

ZΛ,β,κ = 2∑

P⊂PN

Φ(P ), (27)

P (supp(C) = P ) =Φ(P )

P⊂PNΦ(P )

. (28)

Proof. In the sum that appears in the definition of ZΛ,β,κ, one can first sum overall subsets P that could possibly be a support of the configuration, and thenover all configurations C that have P as its support. After possibly applying atrivial global flip to ensure that

v∈VNφv > 0, the first equation is merely the

definition of Φ(P ). The global flip is the reason for the existence of the factorof 2 on the right hand side of equation (27). The second equation comes fromthe definition of the probability distribution and the previous equation (27) onZΛ,β,κ. Note that we have implicitly canceled a factor of 2 that comes from theglobal flip symmetry.

2.3 Analysis of the Wilson Loop Action

Our ultimate goal in this section is to argue that the main contribution to theWilson loop expectations come from minimal vortices, P (e).

Definition 4 (Minimal Vortices). Minimal vortices P (e) have the followingstructure: they consist of the edge e and the 2(d−1) plaquettes that use the edgee as a boundary edge. This should be thought of as the smallest excitations, asthe easiest way to create a minimal vortex is to excite the single edge σe 6= 1and to set all other surrounding edges, e′, to σe′ = 1.

Remark 3. Minimal vortices will be a common structure appearing in the pre-vious works [4] and [9] and will appear in later sections as well. We remarkthat in this section, that a configuration C that has a minimal vortex P (e) in itssupport must assign all the vertices of V (P (e)) the same Higgs boson charge.Otherwise, one would be able to find at least 2 edges in V (P (e)) connecting oppo-site charges. One of these edges will be e′ 6= e. Under our definition of support,all of the plaquettes that use e′ as a boundary vertex will be in the support of C.This means that P (e) cannot be a vortex in the support of C. Furthermore, alledges e′ in E(P (e)) aside from e itself must be assigned σe′ = 1.

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We let Wγ represent our Wilson loop action on the loop γ. In addition, welet Sγ be a surface whose boundary is γ.

The analysis in this section depends on describing which plaquette configu-rations form the main contribution to the Wilson loop equation and boundingaway the contributions of rare configurations by the function Φ constructedearlier.

2.3.1 Trivial Configurations

One difference from the case of a pure gauge field is that there are excited con-figuration C whose support would intersect the loop γ, but would not have anycontribution to the Wilson loop action. In this subsection, we find some we givesome conditions on the configuration which will ensure that the configurationwill not have any contribution to the Wilson loop action. By ‘removing’ thecontribution of these trivial configurations, we can improve our error analysislater.

Our first definition describes a geometric condition for a vortex V to haveno contribution to the Wilson loop action.

Definition 5. We call a vortex V non-contributing to the Wilson loop actionγ if it satisfies one of the following two properties:

• There is no edge e ∈ γ that is simultaneously a boundary edge of someplaquette p ∈ V .

• It is a minimal vortex not centered around an edge e ∈ γ.

We will show later that if a vortex is non-contributing then, as its namestates, it will not have a contribution to the Wilson loop action.

To appropriately account for the Wilson loop contribution in orders of e−β,we need to describe other configurations C that are not immediately removeddue to the geometric condition above. In order to do this, we start by firstdescribing a modification of φ that only takes into account the ’nontrivial’ pla-quette excitations with respect to the Wilson loop functional.

Definition 6. We call a configuration C = (φv , σe) Wilson loop non-trivial ifthere is a plaquette p in supp C such that there is a boundary edge in p thatintersects the loop γ and there is another plaquette p′ in supp C with (dσ)p′ 6= 1.A configuration that is not Wilson loop non-trivial is Wilson loop trivial.

We define the functional φNT (V ) which takes values on sets of plaquettes asfollows,

ΦNT (P ) =∑

C=(σe,φv)supp(C)=P∑

v φv>0C is Wilson Loop Nontrivial

p∈supp(C)exp[β(ρ((dσ)p)− ρ(1))]

×∏

e∈E(P )e=(v,w)

exp[κ(f(σe, φvφ−1w )− f(1, 1))].

(29)

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The purpose of introducing the above Definition 6 is to separate out thoseconfigurations which will have a non-trivial effect on the Wilson loop actionWγ .Namely, there needs to be some plaquette with (dσ)p 6= 1 in order to ensurethat Wγ 6= 1.

Remark 4. We remark that if there is a configuration C with support P1 ∪ P2

with P1, P2 compatible such that C is Wilson loop non-trivial when restricted toP2, then after the splitting construction in Lemma 2, then the configuration C2is Wilson loop non-trivial on P2. This is due to the fact that our constructionis merely a restriction map applied to the gauge field configuration σ. Thisallows us to establish multiplicative analogues of our result in 2 with ΦNT . Forexample, the sum of exp[HN,β,κ(σ, φ)] over configurations with support P1 ∪ P2

and are Wilson loop non-trivial when restricted to P2 is Φ(P1)ΦNT (P2).

The following lemma takes into account both the sources of trivialities tothe Wilson loop action.

Lemma 4. Consider a vortex V whose associated edge set E(V ) has no edgein common with γ. Then,

E[Wγ |supp C = V ] = 1. (30)

For similar reasons, we also have the following statement.

E[Wγ |C is Wilson Loop Trivial] = 1. (31)

Proof. Due to our definition of support, it is clear that if V is a vortex whoseedge set E(V ) shares no edge with γ, then σe = 1 for all edges e ∈ γ. Otherwise,if σe 6= 1 for an edge in γ, then the plaquettes surrounding that edge would bein the support of our configuration. This would imply that E(V ) contains anedge of γ. Finally, it is clear that if σe = 1 for all edges of γ, then the Wilsonloop action Wγ is clearly 1. This deals with the first case of non-contributingvortex from Definition 5.

The second case is that the non-contributing vortex is minimal P (e), butthe central edge e is not in γ. From our discussion in 3, we know that all edgese′ ∈ E(P (e)) that are not e itself are set to 1. Again, this would imply thatσe is 1 along all edges of γ and, furthermore, the Wilson loop action is 1. Thisdeals with the case of (30).

Now, we deal with (31). Note that if a configuration C is Wilson looptrivial then either E(supp C) will not contain an edge of γ or (dσ)p = 1 for allplaquettes p ∈ PN . We have already dealt with the former possibility in thelast paragraph. We now deal with the second possibility.

A benefit of the abelian case is that we can rewrite our Wilson loop expec-tation as the integral of γ with respect to a 1-from σ. As we have mentionedearlier, Stokes’ theorem allows us to express this Wilson loop integral as theintegral of a surface Sγ whose boundary is γ with respect to the 2-from (dσ).Now, if (dσ)p = 1 for all p ∈ PN , then clearly the integral 〈Sγ , dσ〉 = 1. Thus,the Wilson loop action is still 1. This completes the proof of equation (31).

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2.3.2 The main term contribution

The following lemma describes the contribution to the Wilson loop action fromthe minimal vortices. These should be the simplest excitations.

Lemma 5. Consider an excitation C and let supp C = V1 ∪ V2 ∪ . . . VN be thevortex decomposition of supp C.

We say that C belongs to the event E if satisfies certain properties regardingthe vortices in the support. To be in the set E, at least one of the following holdsfor each i:

• The vortex Vi is minimal and is centered around an edge e in γ .

• Vi is non-contributing to the Wilson loop action.

• Vi is not non-contributing to the Wilson loop action, but C restricted to Viis Wilson loop trivial.

Restricted to the event E, we have

E[Wγ |E] = ρ(−1)Mγ , (32)

where Mγ is the number of minimal vortices centered along an edge of γ.

Proof. By definition,

E[Wγ |supp(C) = P ] =ΦW (P )

Φ(P ),

.Consider a decomposition into compatible sets of plaquettes of the form,

V1 ∪ V2 ∪ V3, where each Vi has the vortex decomposition V1 = V 11 ∪ . . . ∪ V 1

a ,V2 = V 2

1 ∪ . . . ∪ V 2b , V3 = V 3

1 ∪ . . . ∪ V 3c .

Define EV1,V2,V3 to be the set of configurations C on which V1 is non-contributing V2 are minimal and centered on some edge of γ, and V3 are notnon-contributing, but C is trivial on V3. Assume that EV1,V2,V3 is non-empty.

We explicitly see that,

E[Wγ |C ∈ EV1,V2,V3 ] =

∏ai=1 Φ(V

1i )∏b

j=1 ΦW (V 2j )∏c

k=1[Φ(V3k )− ΦNT (V

3k )]

∏ai=1 Φ(V

1i )∏b

j=1 Φ(V2j )∏c

k=1[Φ(V3k )− ΦNT (V 3

k )]

= ρ(−1)|V2|.(33)

Noting that E = ∪ V1,V2,V3

all compatibleEV1,V2,V3 , a disjoint union, this gives the result

(32) after a removal of the conditioning.

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2.3.3 Bounding the probability of large vortices

The following lemma bounds the probability of observing a vortex V in thesupport by the functions Φ we have defined earlier.

Lemma 6. Let V be a vortex of cardinality |V | = m, where the cardinalitycounts the number of plaquettes in V . Then, we have the following equation,

P(V is a vortex ∈ supp C) ≤ Φ(V ). (34)

Similarly,

P(V is a vortex ∈ supp C, C is non-trivial on V ) ≤ ΦNT (V ). (35)

Proof. Let PV be the set of possible supports of configurations C that include Vin the vortex decomposition. If P is a vortex decomposition PV , we use P \V todenote the remaining vortices. We remark that P \V is still a legitimate vortexdecomposition corresponding to some configuration C. By applying Lemma 2,we see that,

P∈PV

Φ(P ) =∑

P∈PV

Φ(V )Φ(P \ V ). (36)

Since each vortex decomposition p \ V can be constructed to be the support ofsome configuration Z, we see that we immediately have the inequality,

Z ≥ 2∑

P∈PV

Φ(P \ V ). (37)

Thus,

P(V is a vortex of supp C) =2∑

P∈PVΦ(P )

Z

≤ Φ(V )

P∈PVΦ(P \ V )

P∈PVΦ(P \ V )

= Φ(V ),

(38)

as desired.For the second part of the theorem, one can see that we can count the

contribution of excitations whose support contains V but is non-trivial on V as

2ΦNT (V )∑

P∈PV

Φ(P \ V ). (39)

This is due to the fact that the construction in Lemma 2 merely projects thevalue of σe to its appropriate subset. Thus, if (dσ)p 6= 1 for some plaquette, itwill hold true for the appropriate projection. Since we assumed C was nontrivialon V , the same must hold true for the projection to V .

Our next lemma uses our previous lemma to prove that large vortices areexponentially suppressed in probability.

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Lemma 7. Define the constant C1 as

C1 := 28 max

(

exp[−2βRe(ρ(1)− ρ(−1))],

exp[− κ

(d− 1)(Ref(1, 1)− max

(a,b) 6=(1,1)Ref(a, b))]]

)

.

(40)

Let V be a vortex with 2k oriented plaquettes (alternatively, k pairs of pla-quettes p,−p), then we have the following bound.

φ(V ) ≤ (C1)k, (41)

As a consequence of this bound and the previous Lemma 6, the probabilitythat there is a configuration C whose support contains a vortex of size k that isnot non-contributing to the Wilson loop action is bounded above by

2(d− 1)|γ|(C(d)C1)k, (42)

where C(d) is a constant that only depends on the dimension d.In the case that,

2β(ρ(1)− ρ(−1)) >κ

(d− 1)[Ref(1, 1)− max

(a,b) 6=(1,1)Ref(a, b)],

then we have a better estimate when we consider Wilson loop non-trivial con-figurations.

ΦNT (V ) ≤ exp[−4(d− 1)βRe[ρ(1)− ρ(−1)]](C1)k−2(d−1). (43)

Similarly, the probability that there is a configuration C such that there is avortex V of size 2k that is not non-contributing to the Wilson loop and suchthat C is nontrivial when restricted to V is bounded above by

2(d− 1)|γ|(C(d)C1)k−2(d−1)(28C(d) exp[−2βRe(ρ(1)− ρ(−1))])2(d−1). (44)

Proof. Let C be a configuration with support V = {P1, P2, . . . , Pk}.If 2k1 of these plaquettes p satisfy the property that (dσ)p 6= 1 (here, we

implicitly include a plaquette and its negative), then the probability of thisconfiguration is immediately reduced by exp[−k12βRe(ρ(1)− ρ(−1))].

Now, if the plaquette p does not satisfy (dσ)p = 1, then necessarily, eitherone of the edges on its boundary satisfies σe = −1 or there is an edge e = (v, w)on its boundary with φv 6= φw. In any case, the presence of such an edge (andthus also its negative) reduces the probability of the configuration by a factorof,

exp[−2κ(Ref(1, 1)− max(a,b) 6=(1,1)

Ref(a, b))]. (45)

Now a single edge can be the cause of the excitation of at most 4(d − 1)

oriented plaquettes. Thus, there must be at least 2(k−k1)4(d−1) excited oriented edges

pairs e and −e.

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Thus, the probability of seeing this configuration is at most

exp[−k12βRe(ρ(1)− ρ(−1))− k − k1(d− 1)

κ(Ref(1, 1)− max(a,b) 6=(1,1)

Ref(a, b))]. (46)

At this point, we now need to calculate the number of configurations withsupport V . Note that there are at most 4k vertices that are boundary verticesof plaquettes in V . These are the only vertices that could be assigned the Higgsboson charge −1. There will be at most 24k ways to assign the Higgs bosoncharge. Similarly, there are 24k ways to assign the value of σe at the edges thatbound some plaquette in the support.

Finally, we must find a way to bound the number of ways to get a connectedset of k plaquettes that would intersect one of the edges of γ.

Lemma 3.4.2 of [4] gives a useful structure theorem. In dimension 4, thereare at most (20e)k vortices of size 2k that could contain any given plaquette p.A modification to general dimension will show that there is some constant C(d)such that the number of vortices of size 2k that contain any given plaquette is(C(d))k. Here, C(d)e−1 is the number of plaquettes that are adjacent to anygiven plaquette in the adjacency graph of plaquette G2 from Definition 3.

The logic applied is similar. Each vortex is a connected subgraph of G2 andthus has a connected spanning tree. The problem of counting the number ofvortices of size 2k can now be divided into two parts. The first part is to countthe number of non-isomorphic rooted spanning trees of size k. Now given aspanning tree T , the second part is to count the number of ways to embed thisspanning tree in G2 with the root of the spanning three the same as our specialplaquette p. Since no two vortices of size k can have the same spanning treewhen embedded in G2, our counting procedure will clearly be an upper boundon the number of vortices containing p.

The number of non-isomorphic spanning trees of size k can be bounded byk(k!)−1kk−2. To get this, we divide the number of labeled spanning trees fromCayley’s Theorem by the number of permutations of the labels. We also have kchoices for the root. This is less than ek and is good enough for our purposes.

Now to embed the graph in G2, we use the following graph. Assume that wehave already embedded the vertex v in G2. Let w be a neighbor of v that wehave not yet fixed. There are C(d)e−1 ways to choose to embed the vertex w;it has to be one of the C(d)e−1 neighbors of v in G2. If we inductively performthis procedure starting from the root, we see that for every tree T , there are atmost (C(d)e−1)k ways to embed the spanning tree in G2. Combining our twoestimates show that there are at most (C(d))k different vortices of size k thatcould contain any given plaquette p.

Now, if we consider the 2(d− 1)|γ| possible plaquettes that are adjacent toedges of γ, then we see that there are at most 2(d− 1)|γ|(C(d))k vortices of sizek that are not-noncontributing to the Wilson loop action.

By a union bound, the total probability of seeing one of these excitations isbounded from above by,

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2(d− 1)|γ|(28C(d))k maxk1

exp

[

− k12βRe(ρ(1)− ρ(−1))

− k − k1(d− 1)

κ(Ref(1, 1)− max(a,b) 6=(1,1)

Ref(a, b))

]

.

(47)

If we consider the case that

βRe(ρ(1)− ρ(−1)) <κ(Ref(1, 1)−max(a,b) 6=(1,1)Ref(a, b))

2(d− 1), (48)

then the maximum is located at k1 = k. Otherwise, the maximum is located atk1 = 0.

By considering only configurations that are Wilson loop non-trivial, we canget further cancellations in the case detailed in (48).

A non-trivial configurations necessarily has k1 > 0. In fact, k1 ≥ 2(d−1) forany configuration σ such that there exists a plaquette p with (dσ)p 6= 0. Thisis Lemma 3.4.6 of [4] generalized to higher dimensions.

Thus, we could say that the probability of seeing a configuration whosesupport contains a vortex V of size k that is not non-contributing and is avortex of size k and, simultaneously, is non-trivial when restricted to this V isat most,

2(d− 1)|γ|(28C(d))k exp[

− 4(d− 1)βRe(ρ(1)− ρ(−1))

− k − 2(d− 1)

(d− 1)κ(Ref(1, 1)− max

(a,b) 6=(1,1)Ref(a, b))

]

.

(49)

As a consequence, we can now state the main result of our section. Thistheorem shows that the main contribution to the Wilson loop expectation comesfrom the minimal vortices we considers.

Theorem 1. Recall the constants C(d) and C1 from our previous Lemma 7.Assume that C(d)C1 < 1 and define the constant D as

D := max

[

e−2βRe(ρ(1)−ρ(−1))

1− 28C(d)e−2βRe(ρ(1)−ρ(−1)),

e−κ

(d−1)(Ref(1,1)−max(a,b)6=(1,1) Ref(a,b))

1− 28C(d)e−κ

(d−1)(Ref(1,1)−max(a,b)6=(1,1) Ref(a,b))

]

.

(50)

We have the following bounds on the expectation of the Wilson loop action,

|E[Wγ ]− E[ρ(−1)Mγ ]| ≤2(d− 1)|γ|(28C(d))2(d−1)+1 exp[−4(d− 1)βRe(ρ(1)− ρ(−1)]D,

(51)

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where Mγ is the number of minimal vortices that are centered around an edge eof γ.

Proof. Recall the event E from Lemma 5.The result of Lemma 7 shows that the probability of the event Ec is bounded

by the union of the events described by probability (44) for k varying between7 and ∞. Adding all the terms together and performing the union bound givesthe desired result.

2.4 Approximating Mγ as a Poisson Random Variable

We can write Mγ as the sum of random events,

Mγ :=∑

e∈γ

1[FP (e)], (52)

where FP (e) is the event that e is the center of a minimal vortex P (e) in thesupport of our configuration.

To showMγ behaves approximately as a Poisson random variable, we wouldwant to show that for e 6= e′, the events FP (e) and FP (e′) are approximatelyindependent from each other. For each edge e, we let Be to be the set ofedges e′ in γ such that a minimal vortex centered around an edge e′ connect tothe minimal vortex centered around e based on adjacency in the graph G2 ofplaquettes from Definition 3.

The following Theorem from [6] details exactly the degree of approximationof some some of random variables to the Poisson distribution.

Theorem 2. Consider the following constants.

b1 :=∑

e∈γ

e′∈Be

P(FP (e))P(FP (e′)),

b2 :=∑

e∈γ

e′∈Be\eP(1(FP (e))1(FP (e′))),

b3 :=∑

γ

E[

|E[1(FP (e))|1(FP (e′))e′ 6∈ Be]− P(FP (e))|

]

.

(53)

Let L(Mγ) denote the law of Mγ and let λ = E[Mγ ]. Then

dTV (L(Mγ),Poisson(λ)) ≤ min(1, λ−1)(b1 + b2) + min(1, 1.4λ−1/2)b3. (54)

We start with the following corollary of the proof of Lemma 6.

Corollary 1 (of Lemma 6). Let E1 and E2 be two set of edges. Let M(E1, E2)be the event that our configuration has a minimal vortex centered on each edgeof E1 and no minimal vortices centered on any edge of E2. Assume thatM(E1, E2) > 0 (this amounts to assuming that the minimal vortices centeredaround E1 do not intersect each other). Now consider some other union of vor-tices U . The probability that some union of vortices U appears in the support ofsome configuration, conditionally on being in the event M(E1, E2) is less thanΦ(U).

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Proof. Let (σ, φ) be a configuration with support U⋃

ei∈E1P (ei) ∪ R where

P (ei) is the minimal vortex centered around ei. In addition, R is some union ofvortices that do not intersect U or P (ei) and does not contain a minimal vortexcentered at an edge of E2.

The sum of exp[HN,β,κ(σ, φ)] of all configurations with this support is givenby 2Φ(U)

ei∈E1Φ(ei)Φ(R).

Thus, the sum of exp[HN,β,κ(σ, φ)] over all configurations such that thesupport contains U will be

2Φ(U)∏

ei∈E1

Φ(ei)∑

R

Φ(R), (55)

where the sum in R goes over all unions of vortices that do not intersect U , theP (ei)’s and do not contain a minimal vortex centered around an edge of e2.

All configurations with support⋃

ei∈E1P (ei) ∪R will be in M(E1, E2).

Thus, the sum of exp[HN,β,κ] for all configurations found in M(E1, E2)(we will call this Z(M(E1, E2)) the partition function restricted to the eventM(E1, E2)) will be satisfy the relation

Z(M(E1, E2)) ≥ 2∏

ei∈E1

Φ(P (e1))∑

R

Φ(R).

The probability that restricted to M(E1, E2) that U will be in the supportis the ratio of the quantity in (55) with the partition function Z(M(E1, E2)).This ratio is clearly less than Φ(U).

This is one of our major tools in computing E[1[FP (e)]|1[FP (e′ ], e′ 6∈ Be].

Lemma 8. Let γ be a loop that has no self-intersection. In addition, recall thedimension dependent constant C(D) and the constant D from Theorem 1. If2(d− 1)C(d)D < 1, we have the following bounds on the constants b1, b2 and b3and λ.

b1 ≤ 8(d− 1)|γ|Φ(P (e))2,b2 = 0,

b3 ≤2|γ|Φ(P (e))(d− 1)C(d)D

1− 2(d− 1)C(d)D,

|λ− |γ|Φ(P (e))| ≤ 2|γ|Φ(P (e))(d− 1)C(d)D

1− 2(d− 1)C(d)D.

(56)

As a consequence of Theorem 2, we see that

dTV (L(Mγ),Poisson(|γ|Φ(P (e))))

≤ 8(d− 1)|γ|(Φ(P (e)))2 + 4|γ|Φ(P (e))(d− 1)C(d)D

1− 2(d− 1)C(d)D.

(57)

Remark 5. The function Φ evaluated on a minimal plaquette P (e) does notdepend on the minimal plaquette chosen. Thus, in a minor abuse of notation,we treat it like a constant. Explicitly, we may always substitute,

Φ(P (e)) = exp[−4(d−1)Re(ρ(1)−ρ(−1))] exp[−2κRe(f(1, 1)−f(−1, 1))] (58)

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Proof. We start with the most difficult part: computing a bound on b3. Fix anedge e. We now attempt to compute

E[

|E[1(FP (e))|1(FP (e′))e′ 6∈ Be]− P(FP (e))|

]

,

for our fixed edge e. Recall the notation M(E1, E2) and Z(M(E1, E2)) fromthe proof of Corollary 1. Let E1 be some set of edges in γ \ Be and E2 be theremaining edges in γ \Be that are not contained in E1.

We know compute E[1[FP (e)]|M(E1, E2)]. Events in M(E1, E2) can be di-vided into two parts,

1. G(E1, E2): These are configuration (σ, I) inM(E1, E2) whose support hasa vortex decomposition V1 ∪ . . . ∪ Vm such that P (e) does not intersectany Vi. Thus, P (e)

⋃mi=1 Vi is a valid vortex decomposition.

2. B(E1, E2): These are configurations (σ, φ) in M(E1, E2) whose vortexdecomposition V1 . . . Vm does contain a vortex Vj such that Vj containsa plaquette that is in P (e). Thus, P (e)

⋃mi=1 Vi is not a valid vortex

decomposition.

If we let Z(G(E1, E2)) and Z(B(E1, E2)) be the sum of exp[HN,β,κ] forconfigurations in them ( the partition functions), then we see that the probabilityof seeing P (ei) as a plaquette in the support conditional on the eventM(E1, E2)is given by,

Φ(V )Z(G(E1, E2))

Z(G(E1, E2)) + Z(B(E1, E2)). (59)

If we show that Z(B(E1, E2)) ≤ cZ(M(E1, E2)) for some constant c < 1,then we see that the probability of seeing P (ei) in the support conditionalon M(E1, E2) is greater than (1− c)Φ(P (e)) and less than Φ(P (e)).

By applying Corollary 1, we can bound Z(B(E1,E2))Z(M(E1,E2))

by the sum of Φ(V )

over all configurations V that contain a plaquette of P (ei) in its support. Wehave performed a version of his sum when trying to compute the probability ofthe event Ec from 5. Recall the bound (41) on the probability of observing avortex excitation of size 2k. Notice that a minimal vortex has 2(d−1) plaquettesattached to it. We can provide a union bound over all of these 2(d− 1) vorticesand all vortices of size 2k that intersect these plaquettes. Recall that there areat most (C(d))k vortices of size 2k that contain any given plaquette.

We see that the probability of the event B(E1, E2) conditioned on G(E1, E2)is less than,

2(d− 1)∑

k=1

Ck1 (C(d))

k ≤ 2(d− 1)C(d)D, (60)

provided C(d)C1 < 1.We thus see that the expectation E[1[FP (e)]|M(E1, E2)] is greater thanΦ(P (e))

1−2(d−1)C(d)D , provided 2(d− 1)C(d)D < 1.

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Through the same logic, we can show that P(FP (e)) ≥ Φ(P (e))1−2(d−1)C(d)D . Re-

calling our previous upper bound of Φ(P (e)) on both of these quantities, we seethat,

|E[1[FP (e)]|M(E1, E2)]− E[FP (e)]| ≤2Φ(P (e))(d− 1)C(d)D

1− 2(d− 1)C(d)D. (61)

We can now integrate over our conditioning on M(E1, E2) to show

E[

|E[1(FP (e))|1(FP (e′))e′ 6∈ Be]− P(FP (e))|

]

<2Φ(P (e))(d− 1)C(d)D

1− 2(d− 1)C(d)D.

Since this bound did not depend on the specific choice of edge e, we see thatour bound on b3 is,

b3 ≤ 2|γ|Φ(P (e))(d− 1)C(d)D

1− 2(d− 1)C(d)D. (62)

b2 is trivially 0 since by definition, a support of a configuration cannot havesupport on both P (e) and P (e′) when e′ is in Be.

Now, to estimate b1, we use the bound P(FP (e)) ≤ Φ(P (e)) for all edges.Now for any edge e, there are at most 8(d− 1) other edges e′ in Be that lie in γ.Here, we use the assumption that γ has no self-intersection. We can now apply2 to assert that,

dTV (L(Mγ),Poisson(λ)) ≤ 8(d− 1)|γ|(Φ(P (e)))2 + 2|γ|Φ(P (e))(d− 1)C(d)D

1− 2(d− 1)C(d)D.

(63)Finally, an immediate consequence of the fact that

Φ(P (e))

1− 2(d− 1)C(d)D≤ P(FP (e)) ≤ Φ(P (e))

shows that the |λ− |γ|Φ(P (e))| ≤ 2|γ|Φ(P (e))(d−1)C(d)D1−2(d−1)C(d)D .

Corollary 3.1 of [1] shows that dTV (Poisson(λ),Poisson(|γ|Φ(P (e))) ≤ |λ −|γ|Φ(P (e))|. We can combine this with our earlier bound on the total variationto show the desired equation (57).

Combining the results of Lemma 8 and Theorem 1 will give us our followingmain result.

Theorem 3. Assume that the conditions of Theorem 1 and 8 hold. Let X be aPoisson random variable with expectation parameter |γ|Φ(P (e)) = exp[−4(d −1)βRe(ρ(1)− ρ(−1))] exp[−2κRe(f(1, 1)− f(−1, 1))]. (Recall that the only wayto excite a minimal vortex is to excite the center of the edge that is on its center.)Then we have that,

|E[Wγ ]− E[ρ(−1)X ]| ≤ 8(d− 1)|γ|(Φ(P (e)))2 + 4|γ|Φ(P (e))(d− 1)C(d)D

1− 2(d− 1)C(d)D

+ 2(d− 1)|γ|(28C(d))2(d−1)+1 exp[−4(d− 1)βRe(ρ(1)− ρ(−1)]D.(64)

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Proof. This is a simple triangle inequality, provided we characterize the valueof the difference of E[ρ(−1)X ] and E[ρ(−1)Mγ ]. We can choose a coupling ofX and Mγ such that P(X 6= Mγ) = dTV (L(Mγ),Poisson(|γ|φ(P (e))). Now,|ρ(−1)Mγ ,X | ≤ 1 for all values of X or Mγ . Thus, we see that, under thiscoupling

|E[ρ(−1)X ]− E[ρ(−1)Mγ ]| ≤ P(X 6=Mγ) ≤ dTV (L(Mγ),Poisson(|γ|φ(P (e))).

We can apply the triangle inequality with our estimate on |E[Wγ ]−E[ρ(−1)Mγ ]|and complete the proof.

3 The Non-Abelian Case

3.1 Model and Some Preliminary Discussion

3.1.1 Definitions

To treat the non-abelian case, we need significantly different notation. Thispreliminary section will discuss many of our new conventions. The Hamiltonianwe will consider in the case of a non-abelian gauge field G will be as follows:

HN,β,κ(σ, φ) =∑

p∈PN

β(ψp(σ) − ψp(1)) +∑

e∈EUN

κ[ge(σ, φ) − gu(1, 1)].(65)

σe is still a map from the set of oriented edges EN → G, a non-abeliangroup with σ−e = σ−1

e , ρ(·) is a D dimensional unitary representation of G.Finally, φx will be represented as a field VN → H taking values in H , a finitemultiplicative subgroup of the unit circle. In some sense, we can interpret φ asa scalar field.

There are a few technical differences from the presentation of the abeliangauge model. First of all, we need to be more careful when defining the ‘current’around a plaquette p. If a plaquette p has boundary vertices v, w, x and y andhas an oriented boundary consisting of edges e1 = (v, w), e2 = (w, x), e3 = (x, y)and e4 = (y, v), then the ‘current’ defined around the oriented plaquette p wouldbe the product σe1σe2σe3σe4 and we set ψp(σ) = Tr[ρ(σe1σe2σe3σe4 )], where ρis a D dimensional unitary representation of the group G. We remark that evenif one chose a different start edge for the boundary( e2, e3, e4 and e1 in thatorder, for example), the ‘current’ might change, but the Hamiltonian will notchange due to the multiplicative property ρ and the cyclic property of trace.Thus, ψp(σ) will be well defined regardless of how we choose the starting edgeof the boundary of p.

Recall also that PN is a set of oriented plaquettes, so PN would also contain−p whose boundary is −e4,−e3,−e2 and −e1 in that order. We see that our‘currents’ satisfy (σe1σe2σe3σe4 )

−1 = σ−e4σ−e3σ−e2σ−e1 .With respect to these‘currents’, it is obvious ψp(σ) = ψ−p(σ). By our earlier remarks, this relation

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does not depend on the specific choice of start point. We thus see that this partof the Hamiltonian will ultimately take real values.

We introduce the notion of unoriented edgesEUN . In contrast to EN , for every

adjacent pair of vertices (x, y) in ΛN , EUN includes only a single undirected edge

between x and y rather than both e = (x, y) and e− = (y, x). The functiongue (σ, φ) will consist of the sum φxTr[ρ(σe)]φ

−1y + φyTr[ρ(σ−e)]φ

−1x where e is

the oriented edge (x, y) and −e is the oriented edge (y, x). Namely, it is the sumof the Higgs boson action φxTr[ρ(σe)]φ

−1y over the oriented edge pair {e,−e} in

EN . The reason we introduce g instead of using our previous sum over EN isthat we will eventually have to perform an expansion of the Hamiltonian withrespect to these unoriented edges. One can see that the function ge is manifestlyreal.

Remark 6. Cao [4] introduces many notions from algebraic topology to describenon-abelian gauge field configurations. Though such concepts would be useful inplaces, it is not essential to understanding the logic of the proof in this section.Whenever we need to refer to some of these topological notions, I would refer tothe appropriate location in [4].

3.1.2 Removing Trivial Gauge Invariances

We discuss some gauge invariances that simplify our analysis. First of all, wecan assume that there is no element σ ∈ G such that ρ(σ) = I, the identitymatrix. The set of all elements g′ ∈ G such that ρ(g′) = I forms a normalsubgroup G′ of G. To understand our Hamiltonian, it suffices to interpret σeas a map from EN to the quotient group G/G′ instead. From a map fromEN → G/G′, one can obtain all maps from EN → G by freely multiplying eachedge with some member of the group G′. As ρ(g) = I for all members in G′,this will not change the Hamiltonian, nor Wilson loop values. Essentially, thereduction from G to G/G′ is the removal of a trivial gauge invariance.

Furthermore, after this reduction, we know that if Ψp(σ) = Ψp(1), thenwe can say in a well-defined sense that the ‘current’ around the plaquette p isexactly 1. The condition Ψp(σ) = Ψp(1) for p surrounded by boundary edgese1,e2,e3 and e4 in that order asserts that the product σe1σe2σe3σ4

is an elementof G whose image under ρ is the identity matrix. Our reduction shows the onlysuch element is 1, so σe1σe2σe3σe4 = 1. By multiplying by σ−1

e1 on the left andσe1 on the right, we can derive that σe2σe3σe4σe1 = 1 and similar for otherchoices of start vertex.

Now, |Tr(ρ(σe))| < D unless ρ(σe) = e2πijn I for some integers j and n.

Observe that since we are considering a unitary representation, all eigenvaluesof ρ(g) have absolute value less than 1. Thus, we know that |φxTr[ρ(σe)]φ−1

y +

φxTr[ρ(σe)]φ−1y | ≤ 2D for all possible choices of φx, σe and φy. Equality can

only occur if ρ(σe) = e2πijn I for some integer j and if e

2πijn ∈ H .

Let X be the set of values e2πijn such that there exists an element g of G such

that ρ(g) = e2πijn I. We see that X is a subgroup of the multiplicative group

of the unit circle, since if ρ(g1) = c1I and ρ(g2) = c2I, where X1 and X2 are

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roots of unity, then ρ(g1g2) = c1c2I. This implies that if c1 ∈ X and c2 ∈ X ,then the product c1c2 is also in X . Furthermore, ρ(g) = ρ(g)−1, so if ρ(g) = cI,then ρ(g−1) = c−1I. This shows that X contains inverses. Manifestly X hasthe identity, so X must be a subgroup of the unit circle.

As we have done in the abelian case, we can consider the Higgs field to takevalues in H/X instead of H . Observe that the Wilson loop expecation can becomputed as the ratio,

〈Wγ〉 =

σe∈Ge∈EN

φx∈Hx∈VN

Wγ(σ) exp[HN,β,κ(σ, φ)]

σe∈Ge∈EN

φx∈Hx∈VN

exp[HN,β,κ(σ, φ)]. (66)

Fix some representative hi ∈ H for each coset class Ci in H/X . Now, fix aparticular map φx = VN → H ; we will denote this map by M . For every vertexchoose an element ζMv and ηMv (depending on M) such that φv(ζ

Mv )−1 = hi

for some i and ρ(ηMv ) = ζMv I . Notice that the transformation ({φv}, {σe}) →({φv(ζMv )−1}, {ηMv σe(η

Mw )−1} for e = (v, w) does not change the Wilson loop

action nor the value of the Hamiltonian. Define φMv := φv(ζMv )−1 and σM

e =ηMv σe(η

Mw )−1.

Another way to represent the Wilson loop expectation is as,

〈Wγ〉 =∑

M=φx:VN→H

σe:EN→GWγ(σM ) exp[HN,β,κ(σ

M , φM )]∑

M=φx:VN→H

σe:EN→G exp[HN,β,κ(σM , φM )]. (67)

In both the numerator and denominator, when considering the internal sumover σ, the map σ → σM can be treated as a change of parameters for thesummation. That is, for fixed M , if σe is uniform over all maps from EN → G,then σM

e is uniform for all maps from EN → G.We can reparameterize the sum as,

〈Wγ〉 =∑

M=φx:VN→H

σe:EN→GWγ(σ) exp[HN,β,κ(σ, φM )]

M=φx:VN→H

σe:EN→G exp[HN,β,κ(σ, φM )]. (68)

Now, the map φM is a set of maps from VN → H/X . The map M → φM isonto and the preimage of any φM is of size |X ||VN |. This shows that we canreduce our problem to computing Wilson loop expectations with a Higgs fieldtaking values in H/X . After these reductions, we see that |φxTr[ρ(σe)]φ−1

y +

φxTr[ρ(σe)]φ−1y | < 2d unless φx = φy and ρ(σe) = 1. Thus, we may consider

edges e = (x, y) with σe 6= 1 or φx 6= φy as real excitations of the Hamiltonian.

Remark 7. If instead of choosing φx to be a scalar, we do the following steps

1. We let φx take values in a general group H.

2. We assume H has a unitary representation ρφ with the same dimensionas the unitary representation on G, ρ and the intersection of the groupsρφ(H) ∩ ρ(G) is a normal subgroup of the image ρφ(H).

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3. We change the Higgs boson interaction to,

e=(x,y)∈EuN

Tr[ρφ(φx)ρ(σe)ρφ(φy)] + Tr[ρφ(φx)ρ(σe)ρφ(φ−1y )]. (69)

Then, after similar transformations as we have detailed above, we can show wehave an excitation with exponential suppression exp[−O(κ)] whenever we havean edge e = (x, y) that satisfies σe 6= 1 or φx 6= φy.

To show this, not that since ρφ(φx)ρ(σe)ρφ(φ−1y ) is a unitary matrix, so the

absolute value of its trace is at most D, the dimension of the representation.Furthermore, the trace plus its conjugate is less than 2D unless the unitarymatrix considered is the identity.

If ρφ(φx)ρ(σe)ρφ(φ−1y ) is the identity, then ρ(σe) = ρφ(φyφ

−1x ), and there is

a common element in the image of ρφ(H) and ρ(G). We can quotient out ρφ(H)by its intersection with ρ(G) through the same gauge transformation procedurewe outlined previously if ρφ(H)∩ρ(G) is a normal subgroup of ρφ(H). If we alsoquotient out G and H by the elements such that ρ(g) = I and ρφ(h) = I, respec-tively, then we will have removed all possibilities for ρφ(φx)ρ(σe)ρφ(φ

−1y ) = I.

3.2 The case of Low Disorder in the Higgs Field

Before we analyze Wilson loop expectations under this Hamiltonian, we start bydiscussing the general difficulty of introducing non-abelian gauge interactions.

Our key tool in probabalistic computation was based on cluster expansions;thus, our first step in any problem was to first find an appropriate definitionof cluster that would satisfy nice properties. In the case of a pure gauge field,the natural choice of clusters would be to find those plaquettes such that thecurrent around it is 0, or, in other words, ψp(σ) 6= ψp(1). These are the naturalexcitations that suppress probability.

In the case of an abelian gauge field, we are actually able to use the setof excited plaquettes p with ψp(σ) 6= ψp(1) as the basis of a legitimate clusterexpansion. This due to the fact that a configuration σ whose support consiststwo compatible sets P1 ∪ P2 , according to Definition 3, can be split into twoconfigurations σ1 and σ2 whose supports are P1 and P2 respectively.

In the case of a non-abelian gauge field, it is no longer possible in general tosplit a configuration σ whose support consists of two compatible sets P1∪P2 intotwo configurations σ1 and σ2 with support P1 and P2. As shown in the papers[4] and [15] , there are topological restrictions that prevent such a splitting.To get around this difficulty, one must consider a more sophisticated conditionto determine whether we can split a configuration or not. This ‘sophisticatedcondition’ is related the knotting properties of the vortices in the support. Thecombinatorial analysis of the knotting properties of vortices was done by Caoin [4].

Now, let us return to our Higgs field model. As we have seen, a basicdifficulty is that our cluster expansion must consider both the excitations of theHiggs field and the gauge field. We have observed already in the abelian case

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with low disorder (large κ) that considering the Higgs field makes defining thesplitting substantially more complicated.

However, in the large κ case, our construction becomes more robust whenconsidering non-abelian groups. In fact, the cluster expansion we proposed inSection 2 works very well when considering non-abelian groups. We try nowto explain intuitively why we are able to more easily perform a splitting in thenon-abelian case with low κ than the pure gauge field case.

In the case of a pure gauge field, there is a large gauge symmetry. Namely,if we let η : VN → G be a map from the vertices VN to the group G, thenσe → ηxσeη

−1y does not change the value of the Hamiltonian. Thus, the basic

object in the pure gauge field case is not a configuration, but a gauge equivalenceclass of configurations. A gauge equivalence class containing the configurationσ will also contain the configurations ηxση

−1y for any map η : VN → G.

When G is abelian, these gauge equivalence classes of a configuration σ canbe understood as a 2-form (dσ). These 2-forms can split on disjoint supports byrestriction. However, for non-abelian groups, the gauge equivalence classes arehomomorphisms to the fundamental group; for details, one can refer to Section4 of [4]. These homomorphisms cannot split disjoint supports in general; thesupport components P1 and P2 cannot be knotted with each other if one wantsto split the support between P1 and P2.

By contrast, the introduction of the Higgs field and the κ term ensures thatthere is a strong breaking of the gauge equivalence symmetry created by themaps η : VN → G. For large κ, our basic objects are indeed configurations(σ, φ) rather than gauge equivalence classes of configurations. The basic objectof study in the large κ regime of the Higgs boson stays the same, whether we areconsider abelian or non-abelian groups. For this reason, the analysis in Section2 is robust to the introduction of a non-abelian group G.

In the analysis of Section 2, we found that any edge with σe 6= 1 automat-ically suppresses the probability by a factor of exp[−O(κ)]. This allowed us toconsider plaquettes that contain an edge with σe 6= 1 as part of our support.Immediately, if the support of our configuration is P1 ∪ P2 where P1 and P2

are compatible, then we can split the gauge field part σ of our configuration byrestriction to P1 and P2.

The assignment of Higgs bosons to vertices is similar to what is done inLemma 2. Lemma 2 treated the case when H/(ρ(G) ∩H) is isomorphic to Z2.The key point of the argument was to identify the phase boundary and performappropriate flips to correct phase boundaries when we separate the supports.The only key fact that we used about the group is that if we switch φx, φy to−φx,−φy where vertices x and y bound an edge e, then the Hamiltonian forthat edge e does not change. In particular, if we apply this flip for all verticesinside a phase boundary, it does not change the Hamiltonian except on the phaseboundary. For a more general group H , we still see that the Hamiltonian doesnot change on an edge if we apply the transformation φx → hφx, φy → hφy foran arbitrary group element in H/(H ∩ ρ(G)). This will allow the argument togo through when we consider Higgs boson groups larger than Z2. Combiningthese two facts together, we see that all of the arguments outlined in Section 2

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apply verbatim when κ is sufficiently large.

Remark 8. Even if we consider the case outline in Remark 7, with the Higgsboson taking non-abelian values, we can still apply the vertex assignment proce-dure outline in Lemma 2. This is again due to the fact that the transformationφx → hφx and φy → hφy does not change the Hamiltonian.

Tr[ρφ(hφx)ρ(σe)ρφ((hφx)−1)] = Tr[ρφ(h)ρφ(φx)ρ(σe)ρφ(φ

−1y )ρφ(h

−1)]

= Tr[ρφ(h−1)ρφ(h)ρφ(φx)ρ(σe)ρφ(φ

−1y )] = Tr[ρφ(φx)ρ(σe)ρφ(φ

−1x )].

(70)

The first equality used the fact that ρφ is a representation. The second inequalityused that Tr[AB] = Tr[BA] for general matrices A and B. The final inequalityagain used the fact that ρφ is a representation and ρφ(h)ρφ(h

−1) = ρφ(1) = I.

4 The High Disorder Regime: Small κ

4.1 Expansion in Random Currents

In contrast to the low disorder regime, we cannot simply consider edges withσe 6= 1 to be excitations blindly. Unfortunately, this means we would have toconsider a fundamentally new definition of cluster. We will observe later that inour new definition of cluster, the knotting problem is a serious difficulty whentrying to split the configuration. Interestingly, this difficulty even appears inthe case of an abelian gauge group with low κ. This means we have to treatthe Higgs field model with non-abelian and abelian gauge field with small κin the same way. To avoid presenting some long combinatorial estimates, wewill restrict to the important case d = 4 where we can cite these combinatorialestimates from previous results and focus on the new ideas.

Intuitively, one can imagine in the small κ case, there will be large fluctua-tions in the Higgs field which will decorrelate very quickly along large distances.To quantify this intuition in a cluster expansion, we introduce the notion of arandom cluster expansion. First, let c be some positive constant such that forany σe φx and φy, we have that,

ge(σ, θ) = φxTr[ρ(σe)]φ−1y + φxTr[ρ(σe)]φ

−1y > −c, c ≥ 0. (71)

We see that the measure generated by the Hamiltonian would be the samewhether we considered HN,β,κ or the following Hamiltonian,

HN,β,κ : =∑

p∈PN

β[ψp(σ)− ψp(1)] +∑

e∈EuN

κ[ge(σ, φ) + c]. (72)

We can relate the above model to a random current model. This has threesets of random variables, the σ’s, the φ’s, and a new set of edge activations I(e).The marginal distributions of σ’s and φ’s are those given by the Hamiltonian

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HN,β,κ. Provided one has a configuration φx and σe, the distribution of I(e) isgiven as follows:

P(I(e) = k) =[κ(ge(σ, φ) + c)]k

k! exp[κ(ge(σ, φ) + c)], (73)

where k can be any non-negative integer. The benefit of the random currentexpansion is that it is equivalent to finding the measure associated with thefollowing Hamiltonian.

HN,β,κ(φ, σ, I) :=∑

p∈PN

β(ψp(σ) − ψp(1))

+∑

e∈EN

I(e)[κ(ge(σ, φ) + c)]− log I(e)!.(74)

I(e) is only allowed to take non-negative integer values. Essentially, the idea isthat summing over I(e) returns the exponential.

With this in hand, we can start writing up our definition of cluster.

Definition 7. Given an configuration of the form C := (φ, σ, I), we can definethe set of excited vertices as

V := {v ∈ VN : ∃e s.t. v ∈ δ(e) and I(e) 6= 0}. (75)

We can define the support of the configuration as,

supp C := {p ∈ PN :ψp(σ) 6= ψp(1) or

∃v ∈ V s.t. v is a boundary vertex of p}. (76)

In words, a plaquette is excited if either the plaquette has a non-trivial currentrunning through it or it has a vertex on its boundary that is adjacent to an edgethat is excited with non-zero I(e) value.

We still have the same notion of compatibility of plaquette sets P1 and P2

from Definition 3. We also remark that the condition ψp(σ) 6= ψp(1) here is thesame as the condition σp 6= 1 in Section 4 of [4]. In addition, the support of aconfiguration only depends on the values of σ and I, not on φ.

Remark 9. We can show that, with regards to this definition, the excitationswith smallest support are still minimal vortices P (e) centered around an edge e.We remark that if there is an excited edge e with I(e) 6= 0, then the support willcontain all plaquettes that share a boundary vertex with this plaquette. This iscertainly more than 12 oriented plaquettes ( 6 plaquette pairs {p,−p}). Further-more, Lemma 4.3.3 of [4] asserts that the smallest vortex of the form supp(σ, 0)(i.e. all I(e) = 0) has 12 excited oriented plaquettes( 6 plaquette pairs {p,−p}).

Furthermore, in order to avoid dealing with tedious edge cases, we will as-sume a boundary condition on which we will not find excitation whose supportcontains a plaquette on the boundary of ΛN . One can show that for loops suffi-ciently far away from the boundary, the effect of excitations on the boundary isof size e−N , where N is the size of our lattice.

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We now define the analogue of the function Φ as ,

Φ(P ) =1

|G||VN |−1|H/X ||VN |

supp(σ,I)=P

p∈PN

exp[β(ψp(σ)− ψp(1))]

φ

e=(x,y)∈EN

(κ(ge(σ, φ) + c))I(e)

I(e)!.

(77)

The normalization is introduced in order to derive some multiplicativityrelations in the future. Because we are dealing with non-abelian gauge groups,we have a more complicated condition that ensures that if the support canbe decomposed into two compatible plaquette sets P1 and P2 satisfying thiscomplicated condition, then Φ(P1 ∪ P2) = Φ(P1)Φ(P2). A sufficient conditionfor this splitting is the well-separated condition of Cao [4][Lemma 4.1.21].

Definition 8 (Well-Separated). This definition is derived from the one used in[4]. We modify some of the notation used in [4] to follow the conventions of thismanuscript.

We say that two plaquette sets P1 and P2 are well-separated if we can findsome rectangle R in ΛN that satisfies the following properties: P1 is containedin P (R), the plaquettes inside R, while P2 is contained in P (R)c, the plaquettesoutside R. Furthermore, P1 and P2 do not contain a plaquette belonging to theboundary of R.

Wit this definition in hand, we can prove the following lemma.

Lemma 9. Consider two plaquette sets P1 and P2 that are well-separated ac-cording to Definition 8. Then, we can make the following assertion,

Φ(P1 ∪ P2) = Φ(P1)Φ(P2). (78)

Proof. Part 1: Splitting the configuration σLet P1 and P2 be separated by the rectangle R. Let T be a spanning tree

of ΛN that is simultaneously a spanning tree of R, the complement Rc, and theboundary of R with some basepoint b.

Notice by how we defined the support of our configuration P1 ∪ P2, all pla-quettes p such that ψp(σ) 6= ψp(1) are contained in P1 or in P2.

We will say that any gauge field configuration σ is gauged with respect tothe tree T if σe = 1 for all edges e ∈ T . We can apply Lemma 4.1.21 of [4] inorder to find a unique gauge transformation on the vertices ησ : VN → G suchthat σe := ησxσe(η

σy )

−1 with ηb = 1 at the root and σ is gauged with respect tothe tree T .

Once defining σe, we see that σ can be split into a product σ1 and σ2 suchthat σe = (σ1)e(σ2)e for all edges e = (x, y) such that σ1 and σ2 satisfy thefollowing properties: (σ1)e = σe for edges e inside R and (σ1)e = 1 for edges eoutside of R. In addition, (σ2)e = σe for edges e outside of R and (σ2)e = 1for edges e inside of R. This treats how we would divide σ into two disjointsupports. Now, let us see how we would deal with activated edges.

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Using the activated edges of I(e), one can generate a subgraph I of the latticeΛN whose edges consist of the activated edges with I(e) 6= 0. One can considerthe connected clusters C1, C2, . . . , Cm. We will observe later in the last line of(79) that our function Φ will not share variables between the different clusters;this will allow us to split Φ appropriately as a product. We also remark thatgiven a cluster Ci, all of its vertices are either boundary vertices of plaquettes ofP1 or of P2 exclusively. Otherwise, there exists some vertices v and w adjacentin the cluster Ci such that v is the boundary vertex of some plaquette p1 in P1

and w is the boundary vertex of some plaquette p2 in P2. However, the edge(v, w) is activated; thus all plaquettes that either have v or w as a boundaryvertex are in the support of the configuration. In the connectivity graph ofplaquettes G2 from Definition 3, this would imply that p1 and p2 are connected.This contradicts our assumption that P1 and P2 are disconnected componentsof the support.

We can let V1 be the vertices that bound excited edges and are boundaryvertices of some plaquette in P1 and we let Ia1 be the activated edges whoseboundary vertices are both in V1. We define V2 and Ia2 similarly. From ourearlier discussion, Ia1 ∪ Ia2 is a disjoint union covering all activated edges and V1is disjoint from V2.

For an activation I, we define I1(e) to be the activation restricted to edgesof Ia1 and I2(e) to be the activation restricted to the edges of Ia2 .

We now claim that the map (σ, I) → (σ1, I1), (σ2, I2) is a bijection from thoseconfigurations (σ, I) whose support is P1 ∪ P2 and σ is gauged with respect tothe tree T and pairs of configuration (σ1, I1), (σ2, I2) whose supports are P1 andP2 respectively and σ1, σ2 are gauged with respect to the tree T . To show thatthis is a map between the proposed spaces, it suffices to show that the support of(σ1, I1) is P1 exactly. By our construction, (σ1, I1) has support contained in P1.Now, let p be an arbitrary plaquette in P1. Since p was in the support of (σ, P ),we know that either ψp(σ) 6= ψp(1) or there is a vertex v on the boundary of psuch that v is adjacent to an excited edge, e with I(e) 6= 0.

Consider the case that ψp(σ) 6= ψp(1). We know that for p ∈ P1 that ψp(σ) =ψp(σ1) by construction. Thus, in this case we have that p ∈ supp(σ1, I1). Nowconsider the case that there is a vertex v in the boundary of p1 such that v isadjacent to an activated edge e in I. Our earlier discussion shows that v mustbe in V1 and e must be in Ia1 . Thus, v is still adjacent to an activated edge inI1.

Since all of our maps have been defined by restriction, it is easy to showthe proposed map is a bijection. The splitting is clearly unique and one cancombine an arbitrary split by multiplying the σ’s and combining the I’s. Thisproves our claim on the properties of the map (σ, I) → (σ1, I1), (σ2, I2).

Part 2: The Multiplicativity of ΦWe rewrite our sum in Φ with respect to a base representative σe gauged

with respect to the spanning tree T and a separate field η : VN → G, which

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re-introduces the gauge invariance. The sum in Φ(P1 ∪ P2) can be written as,

Φ(P1 ∪ P2) =1

|G||VN ||H/X ||VN |∑

supp(σ,I)=P1∪P2

σ gauged with T

p∈P1∪P2

exp[β(ψp(σ)− ψp(1))]

η,φ

e=(x,y)∈{e:I(e) 6=0}

(κ(g(η(σ), φ) + c))I(e)

I(e)!.

(79)The new variables η : VN → G act as follows: it takes a gauge field config-

uration σ to the configuration η(σ) that takes values η(σ)e = ηxσeη−1y for the

edge e = (x, y). This reintroduces the gauge invariance we removed when defin-ing σ. We remark that we get an extra factor of 1

|G| due to the removal of the

gauge fixing ηb = 1 for the root. This can be done by a global transformationmultiplying each element η by the same element g ∈ G.

Recall our map (σ, I) → (σ1, I1), (σ2, I2) from earlier. We see that it cansplit the product as follows.

Φ(P1 ∪ P2) =(|G||H/X |)|VN |−|V1|−|V2|

(|G|||H/X |)|VN |∑

supp(σ1,I1)=P1

p∈P1

exp[β(ψp(σ1)− ψp(1))]

×∑

ηv ,φv

vinV1

e=(x,y)I1(e) 6=0

(κ(g(η(σ1), φ) + c))I1(e)

I1(e)!

supp(σ2,I2)=P2

p∈P2

exp[β(ψp(σ2)− ψp(1))]

×∑

ηv ,φv

vinV2

e=(x,y)I2(e) 6=0

(κ(g(η(σ2), φ) + c))I2(e)

I2(e)!.

(80)

We used the following facts:

1. Firstly, the mapping (σ, I) → (σ1, I1), (σ2, I2) is a bijection.

2. Secondly, the product of the exp[βψp] factors split between the disjointsupports P1 and P2.

3. Thirdly, the products (κ(g(η(σ, φ)+ c))I1(e) split between the disjoint setsE1 and E2 where the only variable values of η and φ affecting the valuesof the product over I1( I2 resp.) are those in V1 (resp. V2).

4. Fourthly, we used the fact that the restriction of σ to E1(resp. E2) isσ1(resp. σ2).

5. Finally, the sum over all remaining variables φx and ηx belonging to vari-ables not in V1 or V2 is (|G||H/X |)|V |−|V1|−|V2|.

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Now, in the second line, we can reintroduce a dummy summation over newvariables η1v, φ

1v for vertices v 6∈ V1 and a new dummy summation over variables

η2v, φ2v for vertices v 6∈ V2 in the third line. We will also relabel the variables

ηv, φv in the second line as η1v , φ1v to simplify notation; similarly, we relabel

ηv, φv as η2v, φ2v in the third line.

Compensating for the extra multiplicative factor of (|G||H/X |)|VN |−|V1| inthe second line due to the new variables we added, as with (|G||H/X |)|VN |−|V2|

in the second line, we see that we can write the above expression as expression

Φ(P1 ∪ P2) =

1

(|G||H/X |)|VN |∑

supp(σ1,I1)=P1

p∈P1

exp[β(ψp(σ1)− ψp(1))]

×∑

η1v ,φ

1v

e=(x,y)I1(e) 6=0

(κ(g(η1(σ1), φ1) + c))I1(e)

I1(e)!

1

(|G||H/X |)|VN |

supp(σ2,I2)=P2

p∈P2

exp[β(ψp(σ2)− ψp(1))]

×∑

η2v ,φ

2v

e=(x,y)I2(e) 6=0

(κ((η2(σ2), φ2) + c))I2(e)

I2(e)!.

(81)

But, this is just the product Φ(P1)Φ(P2).

4.2 Knot Expansion

The above Lemma 9 is most useful when paired with the concept of knot ex-pansions. A knot expansion serves as a more careful way to split the support ofa configuration into disjoint components that would be better for analysis.

Definition 9. Let V1 ∪ V2 . . . ∪ Vm be a vortex decomposition. A partition ofthese vortices into knots K1∪K2∪ . . .∪Kn will be called a knot decomposition iffor each j, there is a box Rj that well-separates Kj from Kj+1∪Kj+2∪ . . .∪Kn.As a consequence of Lemma 9, we would have that Φ(K1 ∪ K2 ∪ . . . ∪ Kn) =∏n

i=1 Φ(Ki).A knot decomposition will be maximal if there is no further way to partition

any Kj = K1j ∪K2

j into non-empty components such that K1j and K2

j are well-separated by a cube in ΛN .

Remark 10. Due to Lemma 4.1.10 of [4], if one of the vortices in the supportof a configuration is a minimal vortex P (e) centered around some edge e, thenone would always be able to separate out this minimal vortex in the knot decom-position; though this slightly abuses the notion of separating box. We will usuallychoose these minimal vortices to be the first knots in the knot decomposition.This is exactly the convention of [4].

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With the notion of knot decomposition and Lemma 9 in hand, we can rathereasily prove the following statement. This is essentially Lemma 4.3.10 of [4].

Lemma 10. The probability that, under the Hamiltonian HN,β,κ, we will ob-serve a configuration (σ, φ, I) whose support under the knot decomposition con-tains K would be less than Φ(K).

Proof. Consider a knot decomposition of the form K1 ∪ . . . ∪Kj ∪ . . .Km thatincludes K = Kj. |G||VN |−1|H/X ||VN |Φ(K1 ∪ . . . ∪ Kj ∪ . . . ∪ Km) will bethe sum of exp[HN,β,κ] among all configurations that have knot decompositionK1 ∪ . . . ∪Kj ∪ . . . ∪Km.

We see that,

K∈supp(σ,φ,I)

exp[HN,β,κ(σ, φ, I)] = |G||VN |−1|H/X ||VN | ∑

K∈K1∪K2...Km

m∏

i=1

Φ(Ki).

(82)Where we abuse notation to say that K ∈ K1 ∪ . . . ∪Km means that K is

one of the knots in the decomposition.We see that if we remove K = Kj from K1 ∪ . . . ∪ Km, then , though

K1 ∪ . . . ∪ Kj−1 ∪ Kj+1 . . . ∪ Km may split further in a knot decomposition,we would still satisfy a well-separatedness condition to assert that Φ(K1 ∪ . . .∪Kj−1 ∪Kj+1 ∪Km) =

∏ni=1i6=j

Φ(Ki).

By considering those configurations whose support would be contained inK1 ∪Kj−1 ∪Kj+1 ∪Km for all knot decompositions K1 ∪ . . .∪Km that containK = Kj, we see the partition function can be bounded below by,

ZHN,β,κ≥ |G||VN |−1|H/X ||VN | ∑

K∈K1∪K2...∪Km

m∏

i=1i6=j

Φ(Ki). (83)

Taking the ratio of the term in (82) with our lower bound on the partitionfunction gives us that the probability of seeing a configuration whose supportcontains K is bounded by Φ(K).

We also have the following quantitative bound on Φ(K).

Lemma 11. Let K be some union of vortices V1∪V2∪ . . .∪VN with 2k orientedplaquettes ( k pairs of plaquettes {p,−p}) in the support. Define the constant,

αβ,κ := 24|G|max(exp[2β( maxa 6=1∈G

Re[ψp(a)− ψp(1)])], (κd exp[κd])

1

2(82) ),

where d is the maximum value of ge(σ, φ) + c for all possible values σ and φ.Note that this maximum does not depend on the edge e.

Then, we have the following bound on Φ(K).

Φ(K) ≤ αkβ,κ. (84)

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Proof. Choose some spanning tree T . We see that we can express Φ(K) withrespect to configurations gauged with respect to T as,

Φ(K) =1

|G||VN ||H/X ||VN |

supp(σ,I)=K

p∈PN

exp[β(ψp(σ)− ψp(1)]

ηv ,φv

e=(x,y):I(e) 6=0

(κ(g(η(σ), φ) + c))I(e)

I(e)!.

(85)

Notice that the support of σ, I only depends on the set AI of activated edgesof I, rather than the particular values of I. Given a set of edges A we will letsupp(σ, A) to be the support of a configuration (σ, I) whose set of activatededges from I is A. We remark that the definition would not depend on thechoice I.

What we do now is replace the first sum over sets σ, A where A is now justa set of activated edges. Later, just before we take the product over all edgesin the second line, we sum values of I(a) from 1 to ∞ where a varies over allactivated edges in A.

Namely, we write,

Φ(K) =1

|G|VN |H/X ||VN |

supp(σ,A)=K

p∈K

exp[β(ψp(σ)− ψp(1)]

ηv ,φv

e=(x,y):e∈A

∞∑

I(e)=1

(κ(ge(η(σ), φ) + c))I(e)

I(e)!.

(86)

The sum∑∞

I(e)=1(κ(g(η1(σ),φ1)+c))I(e)

I(e)! can be bounded by κd exp[κdd], where

d is the maximum of ge(η(σ), φ) + c over all choices of η,σ and φ. Recall thatsince ge is a local function, there is clearly a maximum value.

The last line of (86) can be bounded by |G||VN ||H/X ||VN |(κd exp[κd])|A|, byperforming a trivial sum over all ηv and φv variables. Fortunately, this prefactorcancels out.

Now, we bound the product∏

p∈PNexp[β(ψp(σ)−ψp(1))](κd exp[κd])

|A| forany configuration with supp(σ, A) = P . Let Cp be the set of plaquettes inK such that ψp(σ) 6= ψp(1). For plaquettes in K \ Cp, we would know that,instead, they must be activated due to an edge e ∈ A that shares a vertex witha plaquette. A single vertex can be part of at most 2

(

82

)

oriented plaquettes,

so a single edge can excite at most 4(

82

)

oriented plaquettes. Thus, we see that

|A| ≥ |K|−|Cp|4(82)

.

For any configuration such that supp(σ, A) = K, we see that we can bound∏

p∈PNexp[β(ψp(σ)− ψp(1))](κd exp[κd])

|A| by

max(exp[2β( maxa 6=1∈G

Re[ψp(a)− ψp(1)])], (κd exp[κd])

1

2(82) )k.

Now, all that we have left is to count the number of configurations σ, A withsupport K. For each edge e that is a boundary edge of a plaquette p ∈ K, we

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have at most 2 choices: to activate the edge or not. Thus, the number of wayto choose A is at most 24k with 4k being the maximum number of edges thatcould possibly be boundary edges of plaquettes in K. Now, we finally have tocount the number of σ with support P . In [4][Lemma 4.3.7], this can be reducedto finding the number of homomorphisms from the fundamental group of the2-skeleton of the complement of K. From [4][Lemma B.2], this is bounded by|G|k. Multiplying all of these constant together will give us the desired boundon Φ(K).

Remark 11. Provided we know that β is sufficiently large and κ is sufficientlysmall, from this point, we could follow the proof of Corollary 1.2.1 in [4] nearlyword for word by using a sufficiently small value of the constant αβ,κ. The proofcan follow using the same steps; namely, finding a good event E, bounding theprobability of Ec, and computing the value of the Wilson loop conditioned on

Ec. If one chooses κ

1

2(82) to be of the order exp[−β], then one could expect thisto be essentially optimal.

For the remainder of this section, we will give simpler arguments derivingthe leading order behavior of the Wilson loop expectation for the convenience ofthe reader. One can express some corrections to the leading order in κ througha more delicate argument. For interested readers, this procedure is performed inSection 5 through a careful analysis of decorrelation properties of an associatedHamiltonian.

4.3 Identifying the Main Order Excitations

As we have mentioned in Remark 11, the properties of Φ we have establishedin the previous subsections are sufficient to prove that the main order terms ofthe Wilson loop expectation are given by minimal vortices.

When exp[−β] ≪ κ, we can more precisely describe the value of the errorterm with a little bit more effort and a more careful identification of the mainorder terms.

Definition 10. We call a configuration (σ, φ, I) Wilson loop non-trivial if thefollowing holds: there exists some plaquette p such that ψp(σ) 6= ψp(1). We candefine the function ΦNT by summing over exp[HN,β,κ(σ, I, φ)] over all configu-rations that are Wilson loop nontrivial. Namely, we have that, when P is someunion of vortices,

ΦNT (P ) =∑

supp(σ,φ,I)=P(σ,φ,I) W.L. Nontrivial

exp[HN,β,κ(σ, φ, I)]. (87)

A configuration (σ, φ, I) that is not Wilson loop-nontrivial is Wilson looptrivial. A consequence of being Wilson loop trivial is that there is a way tochoose a field η : VN → G such that the modified field σe = ηxσeη

−1y = 1.

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If a configuration (σ, φ, I) has support P1 ∪ P2 where P1 and P2 are com-patible, we can say that (σ, φ, I) is Wilson loop nontrivial on P1 if there is aplaquette p ∈ P1 such that ψp(σ) 6= ψp(1).

We now describe some properties of ΦNT that are reminiscent of propertiesof Φ.

If we have a configuration supported on a union of well-separated plaquettesets P1 ∪ P2, then we can say that a configuration (σ, φ, I) is Wilson loop non-trivial when restricted to P1 if there is a plaquette p ∈ P1 such that ψp(σ) 6=ψp(1).

We can follow the construction in the proof of Lemma 9 to make the followingassertion: we have the identity

|G|(|G||H/X |)−|VN | ∑

supp(σ,φ,I)=P1∪P2

(σ,φ,I) W.L. N.T. on P1

exp[HN,β,κ(σ, φ, I)] = ΦNT (P1)Φ(P2).

This is due to the fact that under the restriction map on the gauged version ofσ, e.g. the map σ → (σ1, σ2), we see we must have that σ1 must be Wilson loopnon-trivial on P1 if P was Wilson loop non-trivial on P .

Due this multiplicative property, we can follow the proof of Lemma 10 toassert that the probability that we have a configuration whose support has aknot decomposition that contains K and is Wilson loop non-trivial on K isbounded from above by ΦNT (K). Finally, since we know that since a minimalvortex has size at least 12 oriented plaquettes from Remark 9, the support ofall Wilson loop nontrivial configurations has at least 14 oriented plaquettes (7plaquette pairs {p,−p}). Following of the proof of Lemma 11, we can assertthat that ΦNT (K) ≤ (24|G|)6 exp[−12β(maxg 6=1 Re[ψp(g)− ψp(1)])]α

k−6β,κ .

We are now at the stage where we can define our good event, E, wherewe have a characterization of the Wilson loop expectation in terms of minimalvortices.

Definition 11. We will define a set E of ‘good’ configurations by applying con-ditions for a configuration to be in the complement. We say that a configuration(σ, φ, I) is in the complement, Ec, if there is a knot K in the knot expansioncorresponding to (σ, φ, I) such that K satisfies the following properties:

1. K is not a minimal vortex.

2. K is Wilson loop nontrivial.

3. There does not exist a cube BK that separates K from the plaquettes Pγ :{p ∈ Pn : ∃e ∈ γ ∩ δp}, e.g. the set of plaquettes that bound an edge of γ.

We first show that restricted to the event E, we have a rather easy compu-tation of the Wilson loop expectation, much like in Lemma 5

Lemma 12. Define the matrix Aβ,κ as follows.

Aβ,κ :=

g 6=1 ρ(g) exp[12βRe(Tr[ρ(g)]− Tr[ρ(1)])]∑

g 6=1 exp[12βRe(Tr[ρ(g)]− Tr[ρ(1)])]. (88)

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Conditioned on the event E from Definition 11, we see that we have,

E[Wγ |E] = E[AMγ

β,kappa|E], (89)

where Mγ is the number of minimal vortices in the support of the configurationthat are centered on an edge of the loop γ.

Proof. Let V1 ∪ V2 ∪ V3 be a valid knot decomposition, i.e. corresponding tosome configuration (σ, φ, I), where the following properties hold.

1. V1 consists of a union of minimal vortices P (e1) ∪ . . . ∪ P (ek) centeredaround edges ei ∈ γ.

2. V2 consists of knots that are well separated from the plaquettes that boundan edge of γ. Namely, for each knot K in V2, there exists a cube BK thatseparates K from the plaquettes Pγ .

3. V3 consists of knots K such that (σ) restricted to K is Wilson loop trivial.

Let E(V1,V2,V3) be the event that the configuration (σ, φ, I) has exactly thisknot decomposition.

Our goal is to provide an expression of

Φγ(V1 ∪ V2 ∪ V3) :=|G|

(|G||H/X |)|VN |∑

supp(σ,φ,I)=V1∪V2∪V3

Wγ(σ) exp[HN,β,κ(σ, φ, I)],

(90)based on this knot decomposition. Say our knot decomposition is K1∪ . . .∪Km.By our convention, Km can belong to V3 or V2.

Consider the case that Km belongs to V3. Let B(Km) be a cube that sep-arates Km from the other knots in the knot decomposition and T (Km) a jointspanning tree of B(Km), the boundary and the complement.

Recall that since our knotKm is in V3, we must necessarily have that ψp(σ) =ψp(1) for all plaquettes p ∈ Km. By gauging our configuration(σ, φ, I) withrespect to T (Km), we see the gauged version (σ, φ, I) satisfies σe = 1 for alledges inside B(Km).

By the calculations similar to those performed in the proof of Lemma 9, wesee that we have the following expression for Φγ(V1 ∪ V2 ∪ V3).

Φγ(V1 ∪ V2 ∪ V3) =1

(|G||H/X |)|B(Km)c|∑

supp(σ,I1)=⋃m−1

i=1 Ki

σ gauged with T (Km)

Wγ(σ)

×∏

p∈V2

exp[β(ψp(σ)− ψp(1)]∑

ηv ,φv

v∈B(Km)c

e=(x,y)I1(e) 6=0

(κ(g(η(σ), φ) + c))I1(e)

I1(e)!

× 1

(|G||H/X |)|B(Km)|

supp(1,I2)=Km

ηv ,φv

v∈B(Km)

e=(x,y)I2(e) 6=0

(κ(g(η(1), φ)) + c)I2(e)

I2(e)!.

(91)

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Again, the main fact we used is that when we gauged σ with respect toT (Km), the fact that the configuration was Wilson loop trivial on Km impliedthat σ = 1 on all edges of B(Km) after the gauging. Thus, the Wilson loopaction only depends on the component σ on the edges of B(Km)c. In addition,the splitting divides activated edges inside B(Km)c in the first component andactivated edges inside B(Km) in the second component. These activated edgesdo not share any vertices on the boundary.

By adding appropriate auxiliary variables∑

η1v, φ1v in the first line and

η2v , φ2v in the second line and normalizing, we see that we have Φγ(V1 ∪ V2 ∪

V3) = Φ(V1 ∪ V2 ∪ V3 \Km)[Φ(Km)− ΦNT (Km)].If instead, the knot Km belonged to V2, we can follow a similar argument,

with only one observation we have to make. When we choose the cube B(Km)that well-separates Km from the other knots, our condition on V2 also ensuresthat this cube well-separates Km from the plaquettes Pγ along γ. This ensureswhen we gauge our transformation according to T (Km) and get a splitting σ1inside B(KM )c and σ2 in B(Km), we have that (σ2)e = 1 for all edges in γ.Thus, we see again that Φγ(V1 ∪V2 ∪V3) = Φ(V1 ∪V2 ∪V3 \Km)Φ(Km) holds.

By fully iterating this procedure, we see that ultimately we can remove allknots that are not minimal vortices centered around edges of γ and see that,

Φγ(V1 ∪ V2 ∪ V3) = Φγ(V1)Φ(V2)∏

Kj∈V3

[Φ(Kj)− ΦNT (Kj)]. (92)

Now, we finally evaluate Φγ(V1). We remark that there are no activatededges with I(e) 6= 0. If there were such an edge, our convention would suggestthat all plaquettes that share a boundary vertex with this edge would be excited.This would be larger than a minimal vortex. To evaluate Φγ(V1), we applyCorollary 4.1.16 of [4] and choose a spanning tree of ΛN that avoids usingthe edges ei that are centers of the minimal vortices in P . When we gaugeconfigurations with respect to this spanning tree, the only non-trivial edgeswith σe 6= 1 are those edges that for the centers of the minimal vortices.

If we define the matrix

Aβ,κ :=∑

g 6=1

ρ(g) exp[12βRe(Tr[ρ(g)]− Tr[ρ(1)])],

we see that

Φγ(V1) =∏

ei∈V1

gi 6=1

exp[12βRe[Tr[ρ(gi)]− Tr[ρ(1)]]]Tr[ρ(∏

ei∈V1

ρ(ei))]. (93)

We abuse notation slightly when we write ei ∈ V1. We use this to mean thatei is the clockwise ordering of the minimal vortices that compose V1 around theloop γ.

A similar decomposition can evaluate Φ(V1 ∪ V2 ∪ V3) (we abuse notationslighlty here; we only restrict to configurations that are trivial on V3 rather than

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consider all configurations supported on V1 ∪ V2 ∪ V3) as,

Φ(V1 ∪ V2 ∪ V3) =∏

ei∈V1

gi 6=1

exp[12βRe(Tr[ρ(gi)]− Tr[ρ(1)])]Φ(V2)∏

K∈V3

[Φ(K)− ΦNT (K)]. (94)

We can take the ratios of the quantities in equations (93) and (94) to showthat conditioned on the event E(V1,V2,V3), we see that E[Wγ |E(V1,V2,V3)] =Tr[Ak

β,κ]. Since the event E is the disjoint union of E(V1,V2,V3) over all possibleknot decompositions that could appear for configurations in E, we see that

E[Wγ |E] = E[Tr[AMγ

β,κ]|E] where Mγ is the number of minimal vortices thatintersect the loop γ.

As we have done previously, our goal now is to bound the probability of theevent E. Once this is done, we can combine this statement with the result ofthe previous Lemma 12 to estimate the leading order contribution of E[Wγ ] interms of the number of minimal vortices excited along γ.

As we have done previously, we will bound from above the probability ofthe event E. It will combine the results of Lemmas 10 and 11 combined with acombinatorial count of the number of ways to knot.

Theorem 4. Provided 1024αβ,κ < 1, with αβ,κ from Lemma 11, there is someuniversal constant not depending on β, κ,G, H or γ such that, the probabilityof the complement of the event Ec from Definition 11 is bounded by

P(Ec) ≤ O

(

|G||γ| exp[12β( maxa 6=1∈G

Re[ψp(a)− ψp(1)])]

(

1

1− 1024αβ,κ

)5)

.

(95)As an immediate consequence of this estimate and the previous Lemma 12,

we have that, for the same universal constant as in equation (95) above,

|E[Wγ ]− E[AMγ

β,κ]

≤ O

(

D|G||γ| exp[12β( maxa 6=1∈G

Re[ψp(a)− ψp(1)])]

(

1

1− 1024αβ,κ

)5)

,(96)

where D was the dimension of the unitary representation ρ of G.

Proof. If we have some configuration (σ, φ, I) that is in Ec, then this meansthat the support of the configuration (σ, φ, I) contains some knot K of size mthat cannot be separated from the plaquettes in Pγ by some cube.

Lemma 4.3.5 of [4] shows that any knot of size m can be contained in a cubeof size 3m. We will give a brief sketch of this fact here. The covering cubewill be constructed inductively based on the following principle: a cube C1 ofsize 3m and a cube C2 of size 3n that intersect can be covered by a cube C3

of size 3(m + n). This large cube C3 can be constructed manually. If we let

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ximin,j and ximax,j be the smallest and larges coordinates, respectively, in the

ith dimension of cube Cj , then we may set ximin,3 = min(ximin,1, ximin,2) and

ximax,3 = max(ximax,1, ximax,2). Since the cubes C1 and C2 intersect, we must

have that |ximax,3 − ximin,3| ≤ m+ n. This is a manual construction of C3. Nowa single connected vortex of size m can be contained in a cube of size 3m. Sincea knot is union of vortices whose containing cubes intersect each other, we caniteratively apply our covering construction on cubes and cover our knot of sizem by a cube of size 3m.

If we let Sγ3m be the set of plaquettes p such that the cube of size 3m centered

around p intersects γ, we see that a knot of size m that cannot be separated byPγ by some cube must intersect a plaquette p in Sγ

3m. Notice that the size ofSγ3m = O(|γ|m4).Now, Lemma 4.3.4 of [4] asserts that the number of knots K of size m that

can contains any given plaquette p ∈ ΛN is less than (1024)m.By applying Lemma 11, we can bound the contribution that there is a knot

that is Wilson loop nontrivial of some size m that intersects a plaquette in Sγ3m.

This will be an upper bound of P(Ec).We have,

P(Ec) ≤ (24|G|)6 exp[12β( maxa 6=1∈G

Re[ψp(a)− ψp(1)])

∞∑

m=7

O(m4|γ|)(1024)mαm−6β,κ

= O

(

|G||γ| exp[12β( maxa 6=1∈G

Re[ψp(a)− ψp(1)])]

(

1

1− 1024αβ,κ

)5)

.

(97)Finally, to derive the final consequence (96), we observe the following. For

any g ∈ G, we have that ρ(g) is a D by D unitary matrix and has trace inabsolute value less than D. Thus, the Wilson loop functional Wγ(σ, φ, I) =Tr[ρ(

e∈γ σe)] necessarily has absolute value less than D.

Also, for any k we see that Akβ,κ can be represented by

g ρ(g)pk(g) wherepk is some probability distribution over the group G. This is the probabilitydistribution of a random walk of k steps on the group G starting from theidentity and with movement probability

P (h→ gh) =exp[12βRe(Tr[ρ(g)]− Tr[ρ(1)])]

g 6=1 exp[12βRe(Tr[ρ(g)]− Tr[ρ(1)])].

We see that Tr[ANγ

β,κ] ≤ D. This allows us to bound |E[Wγ ]| and |E[Tr[ANγ

β,κ]] byD on Ec. This derives our final result.

4.4 Approximation by a Poisson Random Variable

AS we have done previously, our final goal in this section is to approximate thevariable Mγ :=

e∈γ 1[FP (e)] as a Poisson random variable. In fact, the proofwe have used in Section 2.4 can hold almost word for word in this section. This

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is due to the fact that the proofs in Section 2.4 did not use the full power of thepolymer expansion, but only applied these properties of the polymer expansionto a minimal vortex. These properties of the polymer expansion restricted tominimal vortices still hold here due to Lemma 9. Our Poisson approximationTheorem is the same as Theorem 2, so we will not reproduce this here.

We start with the following Corollary, which is our analogue of Corollary 1.As before, we see that this will be a consequence of Lemma 9.

Corollary 2 (of Lemma 9). Let E1 and E2 be two sets of edges. Let E(E1, E2)be the event that the support of the configuration (σ, φ, I) has a minimal vortexcentered at every edge of E1 and there is no minimal vortex at any edge of E2.Assume that E(E1, E2) is non-empty. Then, the probability that the knot de-composition of (σ, φ, I) contains the knot K conditional on the event E(E1, E2)is less than Φ(K).

Proof. Let (σ, φ, I) be a configuration in E(E1, E2) with knot decompositionK1 ∪ K2 ∪ . . . ∪ Km with Kj = K for some j. From Remark 10, we see thateach minimal vortex, P (ei) for ei ∈ E1, is some knot Ki in the knot expansion.

We know that

1

|G||VN |−1|H/X ||VN |∑

supp(σ,φ,I)=K1∪K2...∪Km

exp[HN,β,κ(σ, φ, I)] =m∏

i=1

Φ(Ki).

Thus, we see that the sum of 1|G||VN |−1|H/X|VN

exp[HN,β,κ(σ, φ, I)] over all

configurations that contain K in the knot decomposition is

1

|G||VN |−1|H/X |VN

(σ,φ,I)∈E(E1,E2)K∈supp(σ,φ,I))

exp[HN,β,κ(σ, φ, I)] =

⋃ei∈γ P (ei)∪K∈KP (e) 6∈K,e∈E2

Ki∈KΦ(Ki).

(98)

Here, the sum is over all valid knot decompositions K( corresponding to someconfiguration) that contain P (ei) for each ei in E1 as well as K, while notcontaining any minimal vortex from E2.

If K is a valid knot decomposition for some element in E(E1, E2) containingK, then we see that K \K is a valid knot decomposition for some element inE(E1, E2). In addition, Φ(K \K) = Φ(K)Φ(K)−1.

Thus, we can bound the partition function Z(E(E1, E2)) from below as,

1

|G||VN |−1|H/Z||VN |Z(E(E1, E2)) ≥∑

⋃ei∈γ P (ei)∪K∈KP (e) 6∈K,e∈E2

Ki∈KΦ(Ki)Φ(K)−1. (99)

Taking the ratio of (98) and (99) shows that the probability when condi-tioned on E(E1, E2) of seeing K in the knot expansion is less than Φ(K).

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Using the above Corollary, we can derive the following estimates on thequantities b1, b2 and b3 from Theorem 2.

Lemma 13. Assume we satisfy the conditions of Theorem 4.Define the constant c as,

c := 6

∞∑

m=6

(1024)mαmβ,κ = 6(1024αβ,κ)

6 1

1− 1024αβ,κ. (100)

For c < 1, e have the following estimates on b1, b2, b3 and λ = E[Mγ ].

b1 ≤ |γ|Φ(P (e))2,b2 = 0,

b3 ≤ c|γ|Φ(P (e)),|λ− |γ|Φ(P (e))| ≤ c|γ|Φ(P (e)).

(101)

As a consequence of these estimates, we have the following bound on the totalvariation distance betweenMγ and a Poisson random variable X with parameter|γ|Φ(P (e)).

dTV (L[Mγ ],Poisson(|γ|Φ(P (e))) ≤ |γ|(Φ(P (e)))2 + 2c|γ|Φ(P (e)). (102)

Again, we remark that Φ(P (e)) does not depend on the edge e. It is a short-hand for

g 6=1 exp[12βRe(Tr[ρ(g)]− Tr[ρ(1)])].

Proof. By definition b2 = 0. We can also bound b1 by |γ|Φ(P (e))2 recalling thatΦ(P (e)) does not depend on the value of the specific minimal vortices.

As always, the main difficulty is to estimate the value of b3. Fix some edgee. Let E1 be some choice of edges in γ \ Be and E2 be the remaining edges inγ \Be not including the edges of E1. We condition on the event E(E1, E2) fromCorollary 2.

Now, E(E1, E2) can be divided into two types of events,

1. G(E1, E2): These are configuration (σ, φ, I) in E(E1, E2) whose supporthas a knot decompositionK1∪K2 . . .∪Km such that no knotKi intersectsa plaquette in the minimal vortex P (e).

2. B(E1, E2): These are events in E(E1, E2) that are not in G(E1, E2). Ifa configuration is in B(E1, E2), then its support contains a knot thatintersects some plaquette in P (e).

If we let Z(G(E1, E2)) and Z(B(E1, E2)) be the partition functions corre-

sponding to the events G(E1, E2) and B(E1, E2), we see that Z(E(E1∪{e},E2))Z(E(E1,E2)

is the probability that P (e) is a minimal vortex in the support of the con-figuration conditional on the event E(E1, E2). In addition, Z(E(E1, E2)) =Z(G(E1, E2)) + Z(B(E1, E2)) and Z(G(E1, E2)) = Φ(P (e))Z(E(E1 ∪ {e}, E2).

Once we show that Z(B(E1, E2)) is small relative to Z(E(E1, E2)), we willbe done. This involves bounding the probability that there is a knot in the

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knot decomposition that intersects the minimal vortex P (e). We have donesomething similar in the course of the proof of Theorem 4.

There are at most (1024)m knots of size m that could contain any given pla-quette, and from Lemma 11 and Corollary 2, the probability of a knot excitationof size m is at most αm

β,κ. Summing this over all knots of size m, we see thatthe probability that there is a knot that intersects one of the plaquettes of P (e)is less than c from (100).

Thus, we see that Z(B(E1, E2)) ≤ cZ(E(E1, E2)). Provided that c < 1, wesee that,

Z(E(E1 ∪ {e}, E2))

Z(E(E1, E2))=

Z(E(E1 ∪ {e}, E2))

Z(G(E1, E2)) + Z(B(E1, E2))

≥ (1− c)Z(E(E1 ∪ {e}, E2))

Z(G(E1, E2))≥ (1− c)Φ(P (e)).

(103)

This proves the lower bound of (1− c)Φ(P (e)) of E[1[FP (e)|E(E1, E2)]. Re-calling the previous lower bound of Φ(P (e)) for E[1[FP (e)|E(E1, E2)] we seethat E[1[FP (e)|E(E1, E2)]− P(FP (e))| ≤ cΦ(P (e)).

Since this is true for all sets E(E1, E2), we can remove the conditioning andsee that our estimate on b3 is,

b3 ≤ |γ|cΦ(P (e)). (104)

Observe that P(FP (e)) also satisfies the same upper and lower bounds ofΦ(P (e)) and (1 − c)Φ(P (e)) we have shown earlier. Thus, we also have that,

Similarly, this shows that,

|λ− |γ|Φ(P (e))| = |γ||P(FP (e))− Φ(P (e))| ≤ c|γ|Φ(P (e)). (105)

We can now apply Theorem 2 to assert that,

dTV (L[Mγ ],Poisson(λ)) ≤ |γ|(Φ(P (e)))2 + c|γ|Φ(P (e)). (106)

Since the total variation distance of two Poisson random variables is boundedby the difference of their expectations, we also have,

dTV (L[Mγ ],Poisson(|γ|Φ(P (e)))≤ |γ|(Φ(P (e)))2 + c|γ|Φ(P (e)) + |λ− |γ|Φ(P (e))|≤ |γ|(Φ(P (e)))2 + 2c|γ|Φ(P (e)),

(107)

as desired.

Since we can choose a coupling such that P(X 6=Mγ) = dTV (X,Mγ), whereX is a Poisson random variable with parameter |γ|Φ(P (e)), we automaticallyhave our main result by combining Theorem 4 and Lemma 13.

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Theorem 5. Assume that the conditions of Theorem 4 and Lemma 13 hold,then we have that,

|E[Wγ ]− E[Tr[AXβ,κ]]|

≤ O

(

|G||γ| exp[12β( maxa 6=1∈G

Re[ψp(a)− ψp(1)])]

(

1

1− 1024αβ,κ

)5)

+ |γ|(Φ(P (e)))2 + 2c|γ|Φ(P (e)).

(108)

Here, X is a Poisson random variable with parameter E[X ] = |γ|Φ(P (e)) for aminimal vortex P (e).

5 Decorrelation Estimates and a More Precise

Expansion

As promised, this section will give a more precise expansion of the main orderterms in the Wilson loop. In order to reduce technicalities and focus on theinnovations, we will assume that our group G is abelian for now.

One technical issue with the argument in section 4 is that the presenceof the Higgs boson could cause long range correlations. In order to boundthese contributions, we had to introduce the random current expansion andtreat current excitations like knotted plaquette expansions. Due to the knottingproperty, it is hard to split a configuration σ into disjoint parts in general.However, we observe that there is more power in controlling this splitting whenwe are dealing with minimal vortices.

In this section, we will reperform the analysis of section 4 specifically takingadvantage of the presence of the minimal vortices along the loop γ. We canexpress our Hamiltonian as

HN,β,κ(σ, φ) :=∑

p∈PN

β(ρ((dσ)p)− ρ(1)) +∑

e=(v,w)∈EN

κ(f(σe, φvφ−1w )− f(1, 1)),

(109)where f is our shorthand for the action of the Higgs boson and ρ is a 1-dimensional representation.

To proceed along our analysis, we first need to redefine our notion of sup-port of configurations without immediately appealing to the random currentexpansion.

Definition 12. Given a configuration {σ} on the set of edges EN , the supportof our plaquette is

supp({σe}) = {p ∈ PN : (dσ)p 6= 1}. (110)

We now use the notation S to denote configurations of gauge fields {σe}.We treat configurations of the Higgs boson in a matter differently from the gaugefields, so we do not need notation to include the Higgs boson.

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The support of any configuration S can be split into a disjoint union V1 ∪V2 ∪ . . . ∪ VN where each Vi is a maximally connected set of plaquettes(called avortex).

We see also that the smallest possible size of a vortex is 12 oriented plaquettes(6 plaquette pairs {p,−p}). One of the simplest ways to generate this vortexconsists of a single excited edge with σe 6= 1, and all other edges are set toσe = 1.

This definition of the support is the same as one corresponding to a puregauge field. Since only exciting the plaquettes, e.g. setting (dσ)p 6= 1 has aneffect on the Wilson loop action, we would expect this definition of support tobe a more natural description of the changes in the Wilson loop action.

We now define our good set in the following definition.

Definition 13. Fix a Wilson loop γ and some value K, whose value will bespecified later. We let BK be the set of all plaquettes that are at distance atmost K from some edge e in γ. (More formally, for each edge e in γ, letBe

K be the cube of side length 2K that is centered around the edge e. ThenBK = ∪e∈γB

eK). We see that |BK | ≤ |γ|(2K)d .

E will be an event whose complement we will show to be rare. E is defined asthe complement of the union of two events Ec = E1∪ E2, which we will describeas follows.

E1 is the event that the support of the configuration contains a knot K inthe knot decomposition of size ≥ 7 that cannot be separated from the plaquettesof BK by some cube C(K).

E2 is the event that the vortex decomposition of S has at least two minimalvortices M1 (centered around e1) and M2 (centered around e2) such that e1 isan edge in γ and e2 is a boundary edge of some plaquette in Be

K . If two suchminimal vortices are of distance greater than K from each other, then we willcall them K-separated.

In this section, K will not refer to a knot, but instead our decorrelationdistance. We will instead use K for the few times we refer to a knot. We seethat in the complement of the event E, either there are no vortices that intersectγ or, if there are intersections of vortices with γ, these must be minimal vorticesthat are well-spaced (at least distance K) from each other.

On the event E, we will assert that the main contribution to the Wilsonloops will be from the minimal vortices that are centered around edges of γ,much as in the cases we have considered previously. The only technicality isthat we have to ensure that the excitations have sufficient distance from eachother in order to get some independence behavior.

Unfortunately, we do not see any direct way to bound the probability of theevent Ec without appealing to the random current expansion of section 4. Wesee that that if the configuration (σ, φ) is in the event E1 from Definition 13, wesee that (σ, φ, I) must contain a knot K that intersects the region BK(γ) for anychoice of I. Thus, we can bound the event E1 by an analog of the event Ec from

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Definition 11. Since we now deal with a neighborhood of size K around γ, asimple union bound shows that PHN

(E1) can be no more than O(K4PHN(Ec)).

The event E2 is bounded in a different way. This specifically takes advantageof the fact that we are dealing with minimal vortices. Before we do this, wedefine the notion of ΦUB(V ) for a general vortex V .

Definition 14. Let V be a vortex contained in a cube C(V ). We define ΦUB(V )as follows,

ΦUB(V ) =∑

supp(σ,φ)=V

p∈V

exp[β(ρ((dσ)p)− ρ(1))]

e∈C(V )

exp[κ(max(a,b)

Re[f(a, b)]−min(c,d)

Re[f(c, d)])].(111)

The product e ∈ C(V ) is over all oriented edges found in the cube C(V ).

We see that ΦUB(P ) forms an upper bound in a very weak sense.

Lemma 14. Under the Hamiltonian (109) we are considering, the probabilitythat we will see a configuration whose support contains a given vortex V andthis V is separated by a cube C(V ) containing V from all other vortices in thesupport is less than ΦUB(V ).

Proof. Let S be a configuration whose support S contains V and V is separatedfrom all other vortices in the support by a cube C(V ) containing V . The supportof S its support can be split as supp(S) = V ∪X , where X is a plaquette set inC(V )c.

Fix a simultaneous spanning tree T of C(V ), C(V )c and the boundary ofC(V ).

The configuration σ can be uniquely gauged with respect to T such that thegauged configuration σ can be split as σ = σ1σ2 where σ1 = 1 for all edges inC(V )c and σ2 = 1 for all edges in C(V ).

With this division, we can express,∑

supp(σ,φ)=V ∪X exp[HN,β,κ(σ, φ)] as fol-lows:

Z(X ∪ V ) :=∑

supp(σ,φ)=V ∪X

exp[HN,β,κ(σ, φ)]

=1

|G|∑

supp(σ)=V ∪Xσ gauged w.r.t. T

p∈X∪V

exp[β(ρ(dσ)− ρ(1))]

η,φ

e∈EN

exp[κf(ηvσeη−1w , φvφ

−1w )].

(112)

What we have essentially done is, for every configuration, σ, find the versionthat is gauged fixed with respect to T and reintroduce the removed gauge fixingas a sum with respect to a new field η : VN → G which operates on σ as

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σe → ηxσeη−1y . However, instead of fixing some value of ηb to be 1 at a basepoint,

we introduce a global gauge transformation that multiplies each η by some groupelement in G.

Now, we can use the fact that σe splits into (σ1)e and (σ2)e bijectively withrespect to the spanning tree T .

We see that we have the expression,

Z(X ∪ V ) =1

|G|∑

supp(σ1)=Vσ1 gauged to T

supp(σ2)=Xσ2 gauged to T

ηv ,φv

e∈EN

exp[κf(ηvη−1w , φvφ

−1w )]

×∏

p∈X

exp[β(ρ((dσ2)p)− ρ(1))]

×∏

e=(v,w)∈C(V )c

exp[κ(f(ηv(σ2)eη−1w , φvφ

−1w )− f(ηvη

−1w , φvφ

−1w ))]

×∏

p∈V

exp[β(ρ((dσ1)p)− ρ(1))]

×∏

e=(v,w)∈C(V )\δC(V )

exp[κ(f(ηv(σ1)eη−1w , φvφ

−1w )− f(ηvη

−1w , φvφ

−1w ))]

×∏

e=(v,w)∈δC(V )

exp[κ(f(ηv(σ1σ2)eη−1w , φvφ

−1w )− f(ηv(σ2)eη

−1w , φvφ

−1w ))].

(113)In the last two lines, we upper bound the exponential involving κ and f

by exp[κ(maxa,b Ref(a, b) − minc,dRef(c, d))] (observing that that product inC(V ) will include the product over both e and −e as oriented edges if any edgee were in C(V )). After this replacement, the sum over σ1 splits from all othersums. This uses the fact that the resulting sum over σ1 does not depend onthe choice of the spanning tree T . We can replace the last two lines above byΦUB(V ).

We see that

Z(X ∪ V )

≤ 1

|G|ΦUB(V )∑

X disjoint fromV

supp(σ2)=X

ηv ,φv

e∈EN

exp[κf(ηvη−1w , φvφ

−1w )]

×∏

p∈X

exp[β(ρ((dσ2)p)− ρ(1))]

e=(v,w)∈E(X)

exp[κ(f(ηv(σ2)eη−1w , φvφ

−1w )− f(ηvη

−1w , φvφ

−1w ))]

≤ ΦUB(V )Z(X).(114)

We see that the probability of finding V in the support, while V is separated

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by C(V ) from the rest of the support, is bounded by,

X is separated from C(V ) Z(X ∪ V )∑

X is separated from C(V ) Z(X)≤ ΦUB(V ), (115)

as desired.

Just as we can always split away a minimal vortex in the knot decomposition,we can always apply the above lemma to a minimal vortex. In fact, since wecan control the number of edges in C(P (e)) by the number of plaquettes inour minimal vortex, ΦUB(P (e)) is not a very bad bound when only consideringminimal vortices.

A consequence of the above lemma is that the probability that our configu-ration has P (e) and P (e′), where P (e) and P (e′) are minimal vortices that arecompatible with each other is ΦUB(P (e))

2.We can bound the probability of E2 by performing a union bound over pairs

of minimal vortices P (e) and P (e′) where e is an edge of γ and e′ is an edgein the box of size K centered around the edge e. We see that PHN

(E2) ≤O(K4|γ|ΦUB(P (e))

2), where O is a universal constant not depending on pa-rameters β, κ, γ or G.

We can combine our discussion into the following lemma,

Lemma 15. Assume the conditions of Theorem 4. We have that,

P(Ec) ≤ O

(

K4|G||γ| exp[12β( maxa 6=1∈G

Re[ρ(a)− ρ(1)])]

(

1

1− 1024αβ,κ

)5)

+O(K4|γ|ΦUB(P (e))2).

(116)As mentioned earlier, the implied constants in O do not depend on |G|, |H |, β, κor γ.

Furthermore, an easy bound on ΦUB(P (e)) is B6 where B is defined as,

B := |G|4 exp[2β(maxa 6=1

Re[ρ(a)−ρ(1)])] exp[8κ(maxa,b

Re[f(a, b)]−minc,d

Re[f(c, d)])].

(117)This is due to the fact that we can restrict our product over e ∈ C(V ) in the

definition of ΦUB from Definition 14 to those edges that bound a plaquette in theminimal vortex. This will be at most 4 unoriented edges for every unorientedplaquette in the minimal vortex. Finally, recall that our minimal vortex contains6 unoriented plaquettes.

At this point, we can now discuss the computation of Wilson expectationswhen conditioned on the set E, but we must first make an aside to the decorre-lation estimates that we will use for our calculation.

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5.1 Minimal Vortices and Decorrelation

As we have seen in the previous sections, to see the effect of disorder in the gaugefield, it was important to introduce to auxiliary field η : VN → G. The maindifficulty is that the presence of the two fields η and φ can create correlationsover large distances. However, for small κ, we can expect we can establishdecorrelation estimates that would allows us to express Wilson loop espectationsas a product over nearly independent minimal vortices. This section quantifiesthese desired decorrelation estimates.

To this end, we start by giving some definitions.

Definition 15. Consider the Hamiltonian

KN :=∑

e=(v,w)∈EN

κf(ηvη−1w , φvφ

−1w ). (118)

We say that the Hamiltonian KN satisfies decorrelation estimates if the fol-lowing is true. Let V be some set of vertices in VN Let BK(V) be set consistingof all vertices of distance K from V. Let S1 and S2 be two configurations ofηv, φw that differ on a single boundary point on BK(V). Then, for any configu-

ration ηv, φv for v the vertices on V,there exists some constants c, K0 such thatfor K ≥ K0, we have the inequality,

|PKN(ηv = ηv, φv = φv|S1)− PKN

(ηv = ηv, φv = φv|S2)| ≤ |V|e−cK . (119)

Let us first discuss some consequences of the decorrelation estimate.

Lemma 16. Assume that the Hamiltonian KN satisfies the decorrelation esti-mate.

Let V1 be some set of vertices and V2 be some set of vertices outside BK(V1).

Let η1v, φ1v be some choice of configurations for the vertices in V1 and η2v, φ

2v be

some choice of configurations for vertices in V2. Then, we have the followingestimate,

|PKN(ηv = η1v, φv = φ1v, v ∈ V1, ηw = η2w, φw = φ2w , w ∈ V2)

− PKN(ηv = η1v , φv = φ1v, v ∈ V1)PKN

(ηw = η2w, φw = φ2w, w ∈ V2)| ≤2|δBK(V1)||V1|e−cK

PKN(ηw = η2w, φw = φ2w, w ∈ V2).

(120)

Here, |δBK(V1)| is the size of the boundary of the set BK(V1)

Proof. Let Vc be the set of vertices outside BK(V1) and not including V2.We see that summation, we have that,

PKN(ηv = η1v, φv = φ1v, v ∈ V1, ηw = η2w, φw = φ2w, w ∈ V2)

=∑

ηa,φa,a∈Vc

PKN(ηv = ηiv, φv = φiv, i = 1, 2, ηa = ηa, φa = φa, a ∈ Vc). (121)

The quantity on the last line can be computed as a conditional expectation.

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PKN(ηv, φv = η1v , φ

1v, v ∈ V1, ηw, φw = η2w, φ

2w, w ∈ V2, ηa, φa = ηa, φa, a ∈ Vc)

= PKN(ηv, φv = η1v, φ

1v, v ∈ V1|ηw, φw = η2w, φ

2w, w ∈ V2, ηa, φa = ηa, φa, a ∈ Vc)

× PKN(ηw, φw = η2w, φ

2w, w ∈ V2, ηa, φa = ηa, φa, a ∈ Vc).

(122)Because our Hamiltonian KN acts on nearest neighbors, the conditional

expectation on the second time only depends on the values at the boundary|δBK(V1)|.

From the inequality (119), one can show that there is a constant E such thatfor any any boundary condition B of the form,

B = {ηw, φv : ηw, φw = η2w, φ2w, w ∈ V2, ηa, φa = ηa, φa, a ∈ Vc}, (123)

we have

|PKN(ηv, φv = η1v , φ

1v, v ∈ V1|B)− E| ≤ |V1||δBK(V1)|e−cK .

To see the derivation, one can first fix some arbitrary boundary condition B.From any other boundary condition, B, one needs to change at most |δBK(V1)|terms. Thus, by applying the triangle inequality at most |δBK(V1)| times, wecan derive the last line.

Since we have the relation,

B

PKN(ηv = η1v, φv = φ1v, v ∈ V1|B)PKN

(B) = PKN(ηv = η1v , φv = φ1v), (124)

where B is a sum over all possible boundary conditions, we must necessarilyhave that |E − PKN

(ηv = η1v , φv = φ1v)| ≤ 2|V1||δBK(V1)|e−cK .Substituting back our relation on the conditional probability in equation

(122), we see that we can upper and lower bound the conditional probability

with PKN(ηv = η1v, φv = φ1v) ± 2|δBK(V1)|e−cK . Then, we can resum the

expression over Vc. We see we derive the desired inequality,

|PKN(ηv = η1v , φv = φv

1, v ∈ V1, ηw = η2w, φw = φ2w, w ∈ V2)

− PKN(ηv = η1v, φv = φv

1, v ∈ V1)PKN

(ηw = η2w, φw = φ2w, w ∈ V2)| ≤2|V1||δBK(V1)|e−cK

PKN(ηw = η2w, φw = φ2w, w ∈ V2).

(125)

As a corollary of this lemma, we can get decorrelation estimates over aproduct of different sites, provided the sites are sufficiently distant from eachother.

Corollary 3. Suppose S1, . . . , Sm are events supported on vertices V1, . . . , Vmsuch that each Vj for j 6= i lies outside the block BK(Vi) for each i. Fix some

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constant C and assume that that each boundary satisfies |δBK(Vi)| ≤ CKd−1

and |Vi| ≤ C.Assume that KN satisfies the decorrelation estimates. For some constant

c′ < c and K sufficiently large depending on c′ and C , we have the estimate,

|PKN(S1, S2, . . . , Sm)−

m∏

i=1

PKN(Si)| ≤ m

e−c′K

mini PKN(Si)

m∏

i=1

(PKN(Si) + e−c′K).

(126)

Remark 12. This corollary will essentially only be applied to the case whereVi’s are minimal vortices. Thus, we do not need to really worry about the valueof the constant C and the dependence of K on C.

Proof. The proof of this lemma involves performing some induction.We let Fk := PKN

(S1, . . . , Sk) and Ek = |PKN(S1, . . . , Sk)−

∏ki=1 PKN

(Si)|.By splitting the events {S1, . . . , Sk−1} from Sk, we see that applying equation(120) will show that

Fk ≤ Fk−1(PKN(Sk) + e−c′K), (127)

for sufficiently large K. (This allows the exponential factor to decay faster thanany polynomial factor of K that appears from |δBK(Vi)| and the constants thatappear in |Vi|.) Thus, we can derive,

Fk ≤k∏

i=1

(PKN(Si) + e−c′K), (128)

for all k.To determine Ek, we also split the events {S1, . . . , Sk−1} from Sk. We first

write Ek as

|P(S1, . . . , Sk)− P(S1, . . . , Sk−1)P(Sk) + P(S1, . . . , Sk−1)P(Sk)−k∏

i=1

P(Si)|

we see that by applying the triangle inequality, that

Ek ≤ |Fk − Fk−1PKN(Sk)|+ PKN

(Sk)Ek−1. (129)

The decorrelation estimate applied to |Fk−Fk−1PKN(Sk)| gives that this quan-

tity is less than Fk−1e−c′K . We then apply (128) to estimate Fk−1 and apply

the induction hypothesis to Ek−1. We see that,

Ek ≤e−c′Kk−1∏

i=1

(PKN(Si) + e−c′K)

+ (k − 1)e−c′K

mini∈[1,k−1] PKN(Si)

k−1∏

i=1

(PKN(Si) + e−c′K)PKN

(Sk).

(130)

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e−c′K∏k−1

i=1 (PKN(Si)+ e−c′K) can be bounded by e−c′K

PKN(Sk)

∏ki=1(PKN

(Si)+

e−c′K). We can replace the probability in the denominator by the minimum ofthe probability over all i, mini∈[1,k] PKN

(Si).

In addition, e−c′K

mini∈[1,k−1] PKN(Si)

∏k−1i=1 (PKN

(Si) + e−c′K)PKN(Sk) is bounded

by e−c′K

mini∈[1,k] PKN(Si)

∏ki=1(PKN

(Si) + e−c′K). Adding up these terms completes

the proof.

Now we can show through a percolation type argument that for sufficientlysmall κ, we can prove decorrelation estimates for the Hamiltonian KN .

Theorem 6. For κ sufficiently small, we have the decorrelation estimates asdetailed in Definition 15 for the Hamiltonian KN

Proof. This is essentially a percolation argument based on the random currentrepresentation for the Ising model.

Consider the Hamiltonian KN :=∑

e=(v,w) κf(ηvη−1w , φvφ

−1w ). Let c be a

constant such that 2Ref(ηvη−1w , φvφ

−1w ) + c > 0 for any choice of ηv,ηw,φv and

φw at any given edge e = (v, w). Considering the new Hamiltonian,

KN :=∑

e=(v,w)∈EUN

κ[f(ηvη−1w , φvφ

−1w ) + f(ηvη

−1w , φvφ

−1w ) + c], (131)

where we recall the notion of unoriented edges. The Hamiltonian KN pairs upthe edges e and −e in order to get a real number. Thus, the measure generatedby KN is the same as the measure generated by KN .

With this in hand, we now define a new probability model based on therandom current representation of the Ising model. On this probability space, wehave two sets of random variables. The first are the same (ηv, φv) configurationvariables on the vertices. The marginal distribution of these variables are givenby the Hamiltonian KN .

The second is a set of activations on each edge Ie for e ∈ EN . I(e) is avariable that takes non-negative integer values with the following probabilitydistribution. Given a configuration S = (ηv, φv) for all vertices v, we have thefollowing distribution

P (I(e) = k|S) = (κ(2Re f(ηvη−1w , φvφ

−1w ) + c)k

k! exp[κ(2Ref(ηvη−1w , φvφ

−1w ) + c)]

. (132)

What is interesting about this representation is that the joint distributioncan be represented as a new Hamiltonian as follows,

KN (η, φ, I) :=∑

e=(v,w)∈EN

I(e)κ(2Ref(ηvη−1w , φvφ

−1w ) + c)− log I(e)!. (133)

The point is, one can marginalize by summing over I(e). The summationover I(e) returns exp[κ(2Ref(ηvη

−1w , φvφ

−1w ) + c)] as desired.

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Now, let V be a set of vertices with K-boundary BK(V) surrounding it. Tounderstand the effect of boundary conditions applied to BK(V) on vertices inV , we see it will be better to understand the effect upon conditioning on thevalue I(e).

For a fixed configuration of values I(e), we let A be the set of activated edgesdefined as A := {e ∈ EN : I(e) 6= 0}. With the set of activated edges definedabove, for every vertex v, we can define a cluster Cl(v) to be the set of verticesconnected to v using only edges of A.

Now, we claim that, upon conditioning on the values of I(e). Configurationsof (ηv, φv) that lie on different clusters are independent of each other. This canbe seen by explicitly writing out this conditional probability as a summation,

P (η, φ|I) =∏

e=(v,w)∈EN

(κ(2Ref(ηvη−1w ,φvφ

−1w )+c))I(e)

I(e)!∑

η,φ

e=(v,w)∈EN

(κ(2Ref(ηvη−1w ,φvφ

−1w )+c))I(e)

I(e)!

. (134)

The partition function in the denominator can split as a product over differentclusters. Thus, different clusters once conditioned on I(e) are independent ofeach other.

Thus, if v is a vertex in V , then v can only be affected by the boundary ifthere is a path using activated edges in A connecting v to the boundary.

Namely, we see that,

|PKN(ηv = ηv, φv = φv|S1)− PKN

(ηv = ηv, φv = φv|S2)|≤∑

v∈VPKN

(v ∼A BK(v)), (135)

where we use the notation ∼A to denote connection using edges in A. Noticethat the left hand side is 0 if v were not connected to the boundary Bk(v) usingedges of A and is bounded by 1 otherwise.

Now, our goal is to derive exponential decay bounds for the percolation typeestimate PKN

(v ∼A BK(v)). To compute this probability, we condition on theconfiguration (ηv, φv) and then compute the resulting percolation process.

For any given configuration (ηv, φw), we see that the probability of any givenedge e being activated is at most, 1 − e−κ(2maxa,b Ref(a,b)+c). By a monotonecoupling on percolation processes, we see that the probability that v is connectedto the boundary BK(v) through edges in A is bounded by the probability that vis connected to the boundary BK(v) throught the edge-bond percolation processwhose probability of edge activation is 1− e−κ(2maxa,b Ref(a,b)+c). At this point,there are standard arguments to show that the probability that v is connectedto its boundary is bounded from percolation theory. For example, we can applyTheorem 1.1 Part 2 of [8].

Theorem 7. Recall the Hamiltonian KN from Definition 15 and assume it sat-isfies said decorrelation estimates. Let m(φv, φw, ηv, ηw) be the probability underlimN→∞〈·〉KN

(i.e., the limiting probability distribution on the infinite lattice)

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that on on some edge e = (v, w), we will see the configuration φv = φv and soon.

We first define the quantity,

X(g) :=∑

φ1,φ2,η1,η2

m(φ1, φ2, η1, η2) exp[−2κRe[f(η1gη−12 , φ1φ

−12 )− f(η1η

−12 , φ1φ

−12 )]],

(136)which represents the relative change of the Higgs boson action due to settingσe = g. We then define the quantity,

Dβ,γ :=

g 6=1 ρ(g) exp[−12βRe[ρ(g)− ρ(1)]]X(g)∑

g 6=1 exp[−12βRe[ρ(g)− ρ(1)]]X(g). (137)

In addition, define L to be the following constant.

L := |G|24|H |24 exp[96κ(maxa,b

Ref(a, b)−minc,d

Ref(c, d))]. (138)

(138)Provided N is large, the loop γ is sufficiently far away from the boundary,

and K is sufficiently large, we have,

|E[Wγ −DMγ

β,κ |E]| ≤ P(Ec)−1|γ|2B6LK3e−cK exp[B6|γ|L]

+ P(Ec)−1 |γ|B6[(1 + e−c′K)L]e−c′K

min ηv ,φv

v∈ minimal vortex

PKN(ηv, φv)

exp[|γ|B6[(1 + e−c′K)L]].

(139)Here Mγ is the number of edges on γ that form the center of minimal vortex

excitations. We also recall the constant B from Lemma 15 and the constant cfrom the decorrlation estimates Definition 15.

Similarly, we have,

|E[Wγ −DMγ

β,κ ]| ≤ P(Ec) + |γ|2B6LK3e−cK exp[B6|γ|L]

+|γ|B6[(1 + e−c′K)L]e−c′K

min ηv ,φv

v∈ minimal vortex

PKN(ηv, φv)

exp[|γ|B6[(1 + e−c′K)L]].

(140)

Remark 13. For finite N , there are marginal differences between the magne-tizations m(φ, η) calculated with respect to the Hamiltonian KN and its infinitelimit. However, provided we consider loops far away from the boundary (say

on order√N). We would expect such differences to decay on the order e−

√N .

In the course of the proofs that follow, we ignore such marginal differences be-tween the magnetizations computed with respect to KN and those computed withrespect to the infinite limit to simplify the presentation of the core ideas.

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Remark 14. The main benefit of the estimate (139) is that when K is large,we can supress the factor of |γ| in the third and fourth terms of the upper bound,even when |γ| = O(exp[dβ]) for example. Even in this case, we see that we couldset K = O(β), so we can suppress the factor of Kd with B when β is sufficientlylarge. This ensures that with an appropriate choice of K, the right hand side ofequation (139) is sufficiently small.

Proof of Theorem 7. On the event Ec, both W (γ) and Dβ,γ are bounded by 1.

We now consider what happens on the high probability event E.Consider a set V1 ∪V2 . . .∪VN that would form the support of an excitation

that would belong in E. Since this event belongs to E, there are two disjointpossibilities:

1. G1:The only excitations in BK are minimal vortices that intersect theloop γ and these minimal vortices are spaced at least distance K fromeach other.

2. G2 :The support of the excitation does not intersect the loop γ and weare not in G1.

When conditioned on G2, we see that 〈Wγ〉 is 1 and ρ(−1)Mγ is also 1 and,so, the difference is 0.

Now, let us comment on what we should do when we consider G1. We candecompose G1 as follows. First fix some set K1 := V1 ∪ V2 ∪ . . . ∪ Vm of K-separated minimal vortices Vi centered on edges of γ. We let the event G1

V1,...,Vm

be the event that the support of the configuration consists of minimal vorticesV1, . . . , Vm and possibly some set K2 that lies on the exterior of BK(γ).

Lemma 17. On the event G1V1,...,Vm

, we have that for sufficiently large K andvalues of κ such that KN satisfies the decorrelation estimates from Definition15 that

|E[Wγ −DMγ

β,κ |G1V1,...,Vm

]| ≤ |γ|K3e−cKLm

+me−cK

min ηv ,φv

v∈ minimal vortex

PKN(ηv, φv)

[(1 + e−cK)L]m.

(141)

Proof. We can choose a spanning tree T (V1, V2, . . . , Vm) that is a simultaneousspanning tree of the boxes BK(Vi) of size k centered around each minimal vortexVi and the complement of these boxes.

By gauging our configuration with respect to this spanning tree, we see thatany configuration σ with support V1 ∪ V2 . . . ∪ Vm ∪ P can be split into m + 1parts σ1, . . . σm and σrest such that σi has its only nontrivial edges, with σi 6= 1,at the center ei of the minimal vortex P (ei) forming Vi. Finally, σrest has itsonly non-trivial edges in the complement of the boxes BK(Vi) surrounding theminimal vortex. Furthermore, since we are dealing with an abelian group, thecontribution Wilson loop action of σrest can separate from the contribution fromthe σi’s.

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Part 1: Splitting K1 from K2

For a general group G, this splitting gives us a good way to write downthe expression of Wγ conditional on the event E(V1, . . . , Vm) is as a ratio N

D ,where we now write out the definitions of the numerator and the denominator.At this point in the proof, we are only concerned with splitting the supportof a configuration into two parts: the part supported on V1 ∪ V2 . . . ∪ Vm andpart supported on the outside of BK(γ). Throughout this part We will use thenotation σ1 to represent the part supported on the boxes BK(K1) := BK(V1)∪BK(V2) . . .∪BK(Vm) and σ2 to represent the part supported outside of the unionof the aforementioned boxes. As mentioned before, in the following expressions,each configuration is represented by its unique gauge fixed configuration

We can write out the denominator as,

D =∑

K2

η1v ,φ

1v

v∈V (K1)

η2v ,φ

2v

v∈BK(K1)c

ηv ,φv

v 6∈V (K1)∪BK(K1)c

e=(v,w)

exp[κ(f(ηvη−1w , φvφ

−1w )− f(1, 1))]

supp(σ1)=K1

p∈K1

exp[β(ρ(d(σ1)p)− ρ(1))]

×∏

e∈E(K1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

supp(σ2)=K2

p∈K2

exp[β(ρ(d(σ2)p)− ρ(1))]

×∏

e∈BK(K1)c

exp[κ(f(η2v σ2e(η

2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))].

(142)We slightly abuse notation here, in the first line, the sum v ∈ BK(K1)

c isa sum over the vertices in BK(K1)

c while the sum e ∈ BK(K1)c is a sum over

the edges in BK(K1)c. We hope this distinction is always clear in context by

the use of e or v and the variables we associate to it. Note that this is the sumof exp[HN,β,κ(σ, φ)] for configurations found in E1, i.e. the partition functionfor E1. In the above decomposition, we used the fact that, after the gauging,σ1 only takes non-trivial values on the edges that are on the boundary of someminimal vortex. In addition, we used the fact that σ2 can only take non-trivialvalues on the edges that are in the complement of BK(K1).

In the product on the first line, any appearance of ηv(resp. φv) for v a vertexin K1 would be replaced with η1v (resp. φ1v). Similar things happen with ηv forv a vertex in BK(K1)

c. We also remark that in the last line, we know that σ2takes trivial values on BK(K1), not necessarily on all of E(K2)

c. This is alsowhy we have to sum up η2v, φ

2v over all all vertices in BK(K1)

c

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We have a similar expansion for the numerator.

N =∑

K2

η1v ,φ

1v

v∈V (K1)

η2v ,φ

2v

v∈BK(K1)c

ηv ,φv

v 6∈V (K1)∪BK(K1)c

e=(v,w)

exp[κ(f(ηvη−1w , φvφ

−1w )− f(1, 1))]

supp(σ1)=K1

〈σ1, γ〉∏

p∈K1

exp[β(ρ(d(σ1)p)− ρ(1))]

×∏

e∈E(K1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

supp(σ2)=K2

〈γ, σ2〉∏

p∈K2

exp[β(ρ(d(σ2)p)− ρ(1))]

×∏

e∈BK(K1)c

exp[κ(f(η2v σ2e(η

2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))].

(143)We remark here that, in the end, we can assert that the Wilson loop expec-

tation only depends on the values of σ1 rather than on σ2. This uses the factthat K2 has support that is separated by a rectangle from γ, this σ2 could notpossibly contribute to the Wilson action.

If we define ZKNto be the partition function associated to the Hamiltonian

from Definition 15 , KN (φ, η) :=∑

e∈ENκ(f(ηvη

−1w , φvφ

−1w )−f(1, 1)), and PKN

to be the probability distribution corresponding the Hamiltonian KN , then wesee that D

ZKN

has the following expression.

DZKN

=∑

K2

η1v ,φ

1v

v1∈V (K1)

η2v ,φ

2v

v2∈BK(K1)c

PKN(η1v , φ

1v, η

2v, φ

2v)

×∑

supp(σ1)=K1

p∈K1

exp[β(ρ(d(σ1)p)− ρ(1))]

×∏

e∈E(K1)

exp[κ(f(η1v σ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

×∑

supp(σ2)=K2

p∈K2

exp[β(ρ(d(σ2)p)− ρ(1))]

×∏

e∈BK(K1)c

exp[κ(f(η2vσ2e(η

2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))],

(144)

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Similarly, we have this following expression for the numerator,

NZKN

:=∑

K1

K2

η1v ,φ

1v

v1∈V (K1)

η2v,φ

2v

v2∈BK(K1)c

PKN(η1v , φ

1v, η

2v , φ

2v)

×∑

supp(σ1)=K1

〈γ, σ1〉∏

p∈K1

exp[β(ρ(d(σ1)p)− ρ(1))]

×∏

e∈E(K1)

exp[κ(f(η1v σ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

×∑

supp(σ2)=K2

〈γ, σ2〉∏

p∈K2

exp[β(ρ(d(σ2)p)− ρ(1))]

×∏

e∈BK(K1)c

exp[κ(f(η2vσ2e(η

2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))].

(145)By introducing the probability terms PKN

, we can use decorrelation esti-mates from Lemma 16 for small κ.

|PKN(η1v, φ

1v, η

2v , φ

2v)− PKN

(η1v , φ1v)PKN

(η2v , φ2v)| ≤ |γ|K3e−cK

PKN(η2v , φ

2v),(146)

for some constant c.Let us now define the quantities

N :=∑

η1v ,φ

1v

v1∈V (K1)

PKN(η1v , φ

1v)

supp(σ1)=K1

〈γ, σ1〉∏

p∈K1

exp[β(ρ(d(σ1)p)− ρ(1))]

×∏

e∈E(K1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

K2

η2v ,φ

2v

PKN(η2v , φ

2v)

supp(σ2)=K2

〈γ, σ2〉∏

p∈K2

exp[β(ρ(d(σ2)p)− ρ(1))]

×∏

e∈BK(K1)c

exp[κ(f(η2v σ2e(η

2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))],

(147)

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and

D :=∑

η1v ,φ

1v

v1∈V (K1)

PKN(η1v , φ

1v)

supp(σ1)=K1

p∈K1

exp[β(ρ(d(σ1)p)− ρ(1))]

×∏

e∈E(K1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

K2

η2v ,φ

2v

PKN(η2v , φ

2v)

supp(σ2)=K2

p∈K2

exp[β(ρ(d(σ2)p)− ρ(1))]

×∏

e∈BK(K1)c

exp[κ(f(η2v σ2e(η

2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))].

(148)

We see that we can write the difference between ND and N

D . We have,

ND − N

D

=

N/ZKN

D/ZKN

− ND

≤∣

(N/ZKN− N )D/ZKN

DD/ZKN

+

N/ZKN(D/ZKN

− D)

DD/ZKN

.

(149)

Since we have the absolute value bound that |〈γ, σi〉| ≤ 1 and all otherremaining quantities in the expressions of N and N are positive, we see that we

have the boundN/ZKN

D/ZKN

≤ 1.

If we define,

D :=∑

η1v ,φ

1v

v1∈V (K1)

supp(σ1)=K1

p∈K1

exp[β(ρ(d(σ1)p)− ρ(1))]

×∏

e∈E(K1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

K2

η2v ,φ

2v

PKN(η2v , φ

2v)

supp(σ2)=K2

p∈K2

exp[β(ρ(d(σ2)p)− ρ(1))]

×∏

e∈BK(K1)c

exp[κ(f(η2v σ2e(η

2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))],

(150)

then we see that we can bound N/ZN−ND by |γ|K3e−cK D

D .

When computing the ratio between D and D, we can cancel out the ratio ofthe terms involving K2. We see that we only need to consider the ratio

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DD

=

η1v ,φ

1v

v1∈V (K1)

∑supp(σ1)=K1

∏p∈K1

exp[β(ρ(d(σ1)p)−ρ(1))]

×∏e∈E(K1) exp[κ(f(η

1vσ

1e(η

1w)−1,φ1

v(φ1w)−1)−f(η1

v(η1w)−1,φ1

v(φ1w)−1))]

η1v ,φ

1v

v1∈V (K1)

PKN(η1

v ,φ1v)

∑supp(σ1)=K1

∏p∈K1

exp[β(ρ(d(σ1)p)−ρ(1))]

×∏e∈E(K1) exp[κ(f(η

1vσ

1e(η

1w)−1,φ1

v(φ1w)−1)−f(η1

v(η1w)−1,φ1

v(φ1w)−1))]

. (151)

We can bound exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1) − f(η1v(η1w)

−1, φ1v(φ1w)

−1))]from above by exp[2κ(maxa,b Ref(a, b) − minc,dRef(c, d))] in the numerator,and lower bound it by exp[−2κ(maxa,b Ref(a, b)−minc,dRef(c, d))] in the de-nominator.

This allows us to bound the ratio by

DD

≤∏

e∈E(K1)

exp[4κ(maxa,b

Ref(a, b)−minc,d

Ref(c, d))]

η1v ,φ

1v

v1∈V (K1)

supp(σ1)=K1

p∈K1exp[β(ρ((dσ1)p)− ρ(1))]

η1v ,φ

1v

v1∈V (K1)

PKN(η1v , φ

1v)∑

supp(σ1)=K1

p∈K1exp[β(ρ((dσ1)p)− ρ(1))]

(152)In the fraction above, in both the numerators and denominators, the sum overη1v and φ1v splits from the sum over σ1. We can cancel out the sum over σ1. Wecan cancel out this sum over σ1 in both the numerator and denominator. The re-maining sum over η1v , φ

1v is equal to 1 in the denominator and |G||V (K1)|H ||V (K1)|

in the numerator. Considering that these are all minimal vortices that are wellseparated, we can explicitly compute |V (K1)| and |E(K1)|.

Recall the definition of L from equation 138. Our ultimate bound on DD is

Lm.Therefore, we can bound |ND − N

D | ≤ |γ|K3e−cKLm.Part 2: Decorrelating the Minimal Vortices The importance of the expression

ND is that one is able to completely cancel out the effect of excited plaquettesin K2.

Recall the spanning tree T (V1, V2, . . . , Vm) and the splitting of the excitationσ into σ1, . . . , σm between them. The spanning tree can be chosen so that theonly excited edge with σi

e 6= 1 on Vi is the edge that Vi shares with γ.

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Consider the quantity,

D :=∑

η1v ,φ

1v,...,η

mv ,φm

v

PKN(η1v , φ

1v, . . . , η

mv , φ

mv )

×∑

supp(σ1)=V1

exp[−12βRe(ρ(1)− ρ((dσ1))]

e=(v,w)∈E(V1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

× . . .

×∑

supp(σm)=Vm

exp[−12βRe(ρ(1)− ρ((dσm))]

e=(v,w)∈E(Vm)

exp[κ(f(ηmv σme (ηmw )−1, φmv (φmw )−1)− f(ηmv (ηmw )−1, φmv (φmw )−1))]

(153)and

N :=∑

η1v ,φ

1v,...,η

mv ,φm

v

PKN(η1v , φ

1v, . . . , η

mv , φ

mv )

×∑

supp(σ1)=V1

〈δγ, dσ1〉 exp[−12βRe(ρ(1)− ρ((dσ1))]

e=(v,w)∈E(V1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

× . . .

×∑

supp(σm)=V1

〈δγ, dσm〉 exp[−12βRe(ρ(1)− ρ((dσm))]

e=(v,w)∈E(Vm)

exp[κ(f(ηmv σme (ηmw )−1, φmv (φmw )−1)− f(ηmv (ηmw )−1, φmv (φmw )−1))].

(154)

We see that ND = N

D. In this equality, we implicitly used the fact that

the Vi’s are minimal vortices so (dσi) is constant on its support of 6 unorientedplaquettes. Since our vortices are separated by distanceK, we have the followinginequality from Corollary 3,

|PKN(η1v , φ

1v, . . . , η

mv , φ

mv )−

m∏

i=1

PKN(ηiv, φ

iv)|

≤ me−c′K

minj PKN(ηjv, φ

jv)

m∏

i=1

[PKN(ηiv, φ

iv) + e−cK ].

(155)

Provided γ is sufficiently far away from the boundary, one can quite readilyshow that there is constant B such that PKN

(ηjv, φjv) ≥ B for any configuration

of ηjv, φjv associated to a minimal vortex.

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Remark 15. We will give here a rough sketch of this fact. This is essentiallydue to the fact that we are considering a finite range interaction on a fixed finitesubsystem with no assignment explicitly prohbited by the Hamiltonian. Considertwo configurations (η1v, φ

1v) and (η1v , φ

1v) on some minimal vortex V1. Consider

a gauge field assignment on the lattice I = {ηv, φv} such that restricted to thevertices on K we have that ηv = η1v. Now consider the configuration I which is

equal to I on {ηv, φv} outside of V1 and equal to {η1v, φ1v} on V1. There is somefinite bound B on the shift of energy between I and I. Thus, the configuration{η1v, φ1v} on K1 is no more than B times more likely than the configuration{η1v, φ1v}. We can apply this logic for all the finitely many configurations whichshows that PKN

(·) is bounded below for all configurations on the vertices of K1.

We now define the quantities N and D as,

N :=∑

η1v ,φ

1v

v∈V (V1)

PKN(η1v , φ

1v)

supp(σ1)=V1

〈δγ, dσ1〉 exp[−12βRe(ρ(1)− ρ((dσ1))]

e=(v,w)∈E(V1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

× . . .

×∑

ηmv ,φm

v

v∈V (Vm)

PKN(ηmv , φ

mv )

supp(σm)=Vm

〈δγ, dσm〉 exp[−12βRe(ρ(1)− ρ((dσm))]

e=(v,w)∈E(Vm)

exp[κ(f(ηmv σme (ηmw )−1, φmv (φmw )−1)− f(ηmv (ηmw )−1, φmv (φmw )−1))],

(156)and

D :=∑

η1v ,φ

1v

v∈V (V1)

PKN(η1v , φ

1v)

supp(σ1)=V1

exp[−12βRe(ρ(1)− ρ((dσ1))]

e=(v,w)∈E(V1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

× . . .

×∑

ηmv ,φm

v

v∈V (Vm)

PKN(ηmv , φ

mv )

supp(σm)=Vm

exp[−12βRe(ρ(1)− ρ((dσm))]

e=(v,w)∈E(Vm)

exp[κ(f(ηmv σme (ηmw )−1, φmv (φmw )−1)− f(ηmv (ηmw )−1, φmv (φmw )−1))],

(157)As before, we see that,

N

D− N

D

≤∣

N(D− D)

DD

+

D(N− N)

DD

. (158)

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As before, we can bound∣

N

D

∣ ≤ 1, and we can relate both |D− D| and |N− N|to a third quantity, which we bound. We define,

D :=∑

η1v ,φ

1v

v∈V (V1)

[PKN(η1v , φ

1v) + e−c′K ]

supp(σ1)=V1

exp[−12βRe(ρ(1)− ρ((dσ1))]

e=(v,w)∈E(V1)

exp[κ(f(η1vσ1e(η

1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

× . . .

×∑

ηmv ,φm

v

vinV (Vm)

[PKN(ηmv , φ

mv ) + e−c′K ]

supp(σm)=Vm

exp[−12βRe(ρ(1)− ρ((dσm))]

e=(v,w)∈E(Vm)

exp[κ(f(ηmv σme (ηmw )−1, φmv (φmw )−1)− f(ηmv (ηmw )−1, φmv (φmw )−1))].

(159)

That me−c′K

minη1v,φ1

vPKN

(η1v ,φ

1v)D is a bound on |D− D| and |N− N| is a consequence

of the inequality in (155), where the minimum in the denominator is taken overall configuration of η and φ on the vertices of any minimal vortex.

Computing the ratio of D

Dis very similar to computing the ratio of D

D . Wecan always bound

e=(v,w)∈E(Vm)

exp[κ(f(ηmv σme (ηmw )−1, φmv (φmw )−1)− f(ηmv (ηmw )−1, φmv (φmw )−1))]

by exp[48κ(maxa,b Ref(a, b) − minc,dRef(c, d))] from above in the numeratorand exp[−48κ(maxa,b Ref(a, b) −minc,dRef(c, d))] from below in the denomi-nator. Once this is done, the sum over σi and ηi, φi split and we can divide thenumerator and denominator by this sum.

We bound each PKN(η1v , φ

1v) + e−c′K by 1 + e−c′K . Thus, in the numerator,

the sum over η1 and φ1 is bounded by (1+ e−c′K)(|G||H |)24 since there are |G|choices for every η and |H | choices for every φ. The sum over φ1 and η1 in thedenominator is exactly 1.

Recalling L from (138), we see that our bound we can bound D as,

D

D≤ [(1 + e−c′K)L]m, (160)

and we get the bound∣

N

D− N

D

≤ |γ|K3e−cKLm +me−c′K

min ηv ,φv

v∈ minimal vortexPKN

(ηv, φv)[(1 + e−c′K)L]m

(161)

The ratio N

Dis the desired expression Dm

β,κ. Combining all error terms, thiscompletes the proof of the Lemma.

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Return to Proof of Theorem 7.With the previous Lemma 17 in hand, we can compute the error bounds by

removing the conditioning.We see that,

|E[Wγ −DMγ

β,κ |E]| ≤ P(G2|E)[E[Wγ −DMγ

β,κ |G2]

+∑

m

V1,...,Vm

K−SeparatedMinimal

P(G1(V1, . . . , Vm)|E)|[E[Wγ −DMγ

β,κ |G1(V1, . . . , Vm)]|. (162)

One the event G2, we know that Wγ = 1 = DMγ

β,κ . We can also bound

P(G1(V1, . . . , Vm)|E) ≤ P (E)−1B6m. Notice that P(E)P(G1(V1, . . . , Vm)|E)is bounded by the probability that our configuration has a support contain-ing the K-separated minimal vortices V1, . . . Vm. The probability that we seea configuration whose support contains the minimal vortices V1 . . . Vm is less

than∏m

i=1 ΦUB(Vi) ≤ B6m by Lemma 14. Now, there are |γ|mm! ways to choose

V1, . . . , Vm such that they are centered on edges of γ.Combining these estimates, we see that we can bound the last line of (162)

by performing the summation over the last term.

m

V1,...,Vm

K−SeparatedMinimal

P(G1(V1, . . . , Vm)|E)|[E[Wγ −DMγ

β,κ |G1(V1, . . . , Vm)]|

≤ P(E)−1∞∑

m=1

B6m |γ|mm!

|γ|K3e−cKLm

+ P(E)−1∞∑

m=1

B6m |γ|mm!

me−c′K

min ηv ,φv

v∈ minimal vortexPKN

(ηv, φv)[(1 + e−c′K)L]m

≤ P(E)−1|γ|2B6LK3e−cK exp[B6|γ|L]

+ P(E)−1 |γ|B6[(1 + e−c′K)L]e−c′K

min ηv ,φv

v∈ minimal vortexPKN

(ηv, φv)exp[|γ|B6[(1 + e−c′K)L]].

(163)Notice that we only accrue error from m = 1 onwards, not at m = 0.

Remark 16. As a remark, a slight modification of the proof shows that we canbound P(G1(V1, . . . , Vm)|E) by B6m and we do not actually need to include theprefactor P(E)−1. This is actually established later in Corollary 4. To simplifynotation later, we will actually apply the above estimate without the prefactorP(E)−1.

Similarly, we can also bound |E[Wγ − DMγ

β,κ ]| without the restriction to E.

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We see that,

|E[Wγ −DMγ

β,κ ]| = |P(Ec)[E[Wγ −DMγ

β,κ |Ec] + P(G2)[E[Wγ −DMγ

β,κ |G2]

+∑

m

V1,...,Vm

K−SeparatedMinimal

P(G1(V1, . . . , Vm))[E[Wγ −DMγ

β,κ |G1(V1, . . . , Vm)]|

≤ P(Ec)|[E[Wγ −DMγ

β,κ |Ec]|+ P(G2)|[E[Wγ −DMγ

β,κ |G2]|+∑

m

V1,...,Vm

K−SeparatedMinimal

P(G1(V1, . . . , Vm))|[E[Wγ −DMγ

β,κ |G1(V1, . . . , Vm)]|.

(164)

The only difference we need to mention from our previous analysis is thatwe can bound the first term in the last inequality by P(E)c.

5.2 Approximation by Poisson Random Variables

We would again expect that with good decorrelation estimates, that the ex-citation of two minimal vortices would roughly be independent of each other.Again, our goal is to use a version of Theorem 2 in order to show that Mγ isroughly distributed according to a Poisson random variable. We will do thiswhen conditioning on the high probability event E.

Again, we write,

Mγ :=∑

e∈γ

1[FP (e)], (165)

where FP (e) is the event that there is a minimal vortex centered at an edge ofγ. Let BK(e) be the set of edges e′ such that the minimal vortex P (e′) wouldbe found in a box BK(e) of size 2K centered around the edge e.

We use PE to denote the probability distribution when restricted to the

event E. In the discussion that follows in this subsection, we will restrict ouranalysis to events that belong in E.

The version of Theorem 2 we would use in this case is as follows,

Theorem 8. Consider the following constants.

b1 :=∑

e∈γ

e′∈BK(e)

PE(FP (e))PE(FP (e′)),

b2 :=∑

e∈γ

e′∈BK(e)\ePE(1(FP (e))1(FP (e′))),

b3 :=∑

γ

EE

[

|EE [1(FP (e))|1(FP (e′))e′ 6∈ BK(e)]− PE(FP (e))|

]

.

(166)

Let L(Mγ) denote the law of Mγ and let λ = EE [Mγ ]. Then

dTV (LE(Mγ),Poisson(λ)) ≤ min(1, λ−1)(b1 + b2) + min(1, 1.4λ−1/2)b3. (167)

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LE(Mγ) is law of Mγ conditioned on the event E.

Again, a small adaptation of the proof of Lemma 14 shows the followingCorollary.

Corollary 4. [of Lemma 14] Let E1 and E2 be two sets of edges. LetM(E1, E2)(inside E) be the event that there is a minimal vortex centered around eachedge of E1 and there is no minimal vortex centered around any edge of E2.Assume that M(E1, E2) has positive probability. Let V be some minimal vortexthat is not centered around an edge in E1 or in E2. The probability that wehave a configuration (σ, φ) whose support contains V conditioned on the eventM(E1, E2) is less than ΦUB(V ).

Proof. The same adaptations that allowed Corollary 1 to follow from Lemma 6allow this Corollary to follow from Lemma 14.

We see that if a configuration (σ, φ) in M(E1, E2) has support that containsV , then the support is of the form V

e∈E1P (e) ∪ R, where R is a union of

vortices that is disjoint from V⋃

e∈E1P (e) and does not contain any minimal

vortex centered around an edge of E2.We have already computed the sum of exp[HN,β,κ(σ, φ)] for all configurations

whose support is V⋃

e∈E1P (e)∪R. This is the expression Z(V

e∈E1P (e)∪R)

from equation (112).We can follow the steps in the proof of Lemma 14 by choosing a spanning

tree T (V ) that is a spanning tree of C(V ), a cube surrounding V , and itscomplement. If we gauge the configuration σ with respect to the spanning treeT (V ) and split accordingly, we can ultimately derive that Z(V

e∈E1P (e)∪R)

is less than ΦUB(V )Z(⋃

e∈E1P (e) ∪R). This exactly uses the fact that V was

a minimal vortex and can be separated from the other excitations. We cannow assert that the sum of exp[HN,β,κ(σ, φ)] for all configurations in M(E1, E2)whose support contains V is bounded as follows,

V ∈supp((σ,φ))(σ,φ)∈M(E1,E2)

exp[HN,β,κ(σ, φ)] =∑

R

Z(V⋃

e∈E1

P (e) ∪R)

≤ ΦUB(V )∑

R

Z(⋃

e∈E1

P (e) ∪R).(168)

Here, R runs over all all vortices that are found in the vortex expansion of someconfiguration in M(E1, E2) that contains V in its support.

Z(M(E1, E2)) ≥∑

R

Z(⋃

e∈E1

P (e) ∪R).

Taking the ratio of equation (168) with our partition function Z(M(E1, E2))shows that the probability of observing the a configuration (σ, φ) whose supportcontains U when conditioned on the event Z(M(E1, E2)) is less than ΦUB(V ).

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As before, with this lemma in hand, we can start to bound the quantitiesb1, b2, b3

Lemma 18. Assume that the conditions of Theorem 7. Recall the decorrelationestimates from Lemma 16, the constant L from equation (138), and the constantB from Lemma 15. Define the constant

c := 12K4B6.

Assume that c < 1.We have the following bounds on b1, b2, b3 and λ = E[Mγ ].

b1 ≤ |γ|12K4B12,

b2 = 0

b3 ≤ |γ|∣

∣(1− c)[Dβ,κ −K3e−cKL]− [Dβ,κ +K3e−cKL]∣

∣ ,

|λ− |γ|Dβ,κ| ≤ |γ|∣

∣(1 − c)[Dβ,κ −K3e−cKL]− [Dβ,κ +K3e−cKL]∣

∣ .

(169)

As a consequence of these estimates, we have from Theorem 8 that,

dTV (LE(Mγ),Poisson(|γ|Dβ,κ)) ≤ 12|γ|K4B12

+ 2|γ|∣

∣(1− c)[Dβ,κ −K3e−cKL]− [Dβ,κ +K3e−cKL]∣

(170)

Proof. As before, the most difficult part is to bound b3. The proof is largelysimilar to the proof of Lemma 8, except for the fact that we must now deal withsome technicalities of decorrelation estimates.

As before, let E1 be some subset of edges in γ \ BK(e) and E2 be all re-maining edges in γ \BK(e). Recall the notation M(E1, E2) and Z(M(E1, E2))from the proof of Corollary 4. We also define sets G(E1, E2) and B(E1, E2)similarly to the proof of Lemma 8, but there are minor differences related tothe decorrelation.

1. G(E1, E2): This is the set of all configurations (σ, φ) in M(E1, E2) suchthat the support of the configuration contains no vortices in the boxBK(e).

2. B(E1, E2): These are all configurations in M(E1, E2) that do not belongto G(E1, E2). They are characterized by having a vortex in the supportthat intersects the box BK(e). In fact, this vortex is minimal since inwe assume M(E1, E2) ∈ E and E has no non-minimal vortices that arewithin a K neighborhood of γ.

3. TG(E1, E2): These are all configurations (σ, φ) in M(E1 ∪ {e}, E2) whosesupport does not contain any element in BK(e) aside from the minimalvortex at e. This is in fact all of M(E1 ∪{e}, E2) since we are consideringevents in the set E.

We remark thatM(E1∪{e}, E2) is the set of all configurations inM(E1, E2)that have the minimal vortex P (e) in the support.

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Our goal is to show that Z(B(E1, E2)) is small relative to Z(M(E1, E2))and that the sum of all configurations in Z(TG(E1, E2)) is a simple productDβ,κ of Z(G(E1, E2)). The former is the same as what we have done in similarto what we have done earlier. The latter requires some decorrelation estimates.

By a union bound, the ratio of Z(B(E1, E2)) to Z(M(E1, E2)) is boundedfrom above by

e′∈BK(e) PE(P (e′) ∈ supp(σ, φ)|M(E1, E2)) since all events in

Z(B(E1, E2)) necessarily have some minimal vortex excitation in BK(e). Eachindividual term PE(P (e

′) ∈ supp(σ, φ)|M(E1, E2)) is bounded from above byΦUB(P (e)) ≤ B6 from Corollary 4. Thus, we see that the ratio of Z(B(E1, E2))and Z(M(E1, E2)) is less than c := 12K4B6 by the union bound above.

Like in the proof of Lemma 8, we would like to say that Z(TG(E1,E2)Z(G(E1,E2))

is

the constant Dβ,κ. The only thing preventing this is that the activation ofthe minimal P (e) does not separate as a product from the remaining activa-tions. However, we specifically use the fact the configurations in TG(E1, E2)and G(E1, E2) have no excitation in the box BK(e) aside from the minimalvortex in P (e) for TG(E1, E2).

Proof of a Decorrelation Inequality:Let U be the support of some configuration in G(E1, E2). Recalling the

partition function Z from equation, we will show that the following is true(112)

Z(U ∪ P (e))Z(U)

−Dβ,κ

≤ K3e−cKL, (171)

where L is the constant from equation (138) and our values of K and c satisfythe decorrelation estimate from Lemma 16. Since Z(G(E1, E2)) =

U Z(U)

and Z(TG(E1, E2)) =∑

U Z(U ∪P (e)), this shows that|Z(TG(E1,E2))Z(G(E1,E2))

−Dβ,κ| ≤K3e−cKL.

The proof of (171) is very similar to the proof of Part 1 of Lemma 17. LetBK(e) be the box of size K centered around the minimal vortex P (e). Westart by choosing a spanning tree T (e) that is a spanning tree of BK(e) andits complement. As before, when we gauge out a configuration σ with respectto T (e) to σ, this allows us to split the configuration into those with supportσ1 in BK(e) and σ2 in BK(e)c. Furthermore, the only non-trivial edge of σ1would be the edge e that is the center of the minimal vortex P (e). We can thenreintroduce all other configuration by introducing a new field η : VN → G.

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We have,

Z(P (e) ∪ U) =∑

ηv ,φv

e′=(v,w) 6∈E(P (e))∪BK(e)c

exp[κ(f(ηvη−1w , φvφ

−1w )− f(1, 1))]

×∑

σ1 gauged with T (e)

p∈P (e)

exp[β(ρ((dσ1)p)− ρ(1))]

×∏

e′=(v,w)e′∈E(P (e))

exp[κ(f(η1v(σ1)e′(η1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

×∑

σ2 gauged with T (e)

p∈U

exp[β(ρ((dσ2)p)− ρ(1))]

×∏

e′=(v,w)e′∈BK(e)c

exp[κ(f(η2v(σ2)e′ (η2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))],

(172)where η1,2, φ1.2 denote the values of ηv and φv for vertices that belong to V (P (e))or BK(e)c. We can divide by the partition function ZKN

to write the productas,

Z(P (e) ∪ U)

ZKN

=∑

η1v ,φ

1v,η

2v,φ

2v

PKN(η1v, φ

1v, η

2v, φ

2v)

×∑

σ1 gauged with T (e)

p∈P (e)

exp[β(ρ((dσ1)p)− ρ(1))]

×∏

e′=(v,w)e′∈E(P (e))

exp[κ(f(η1v(σ1)e′ (η1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

×∑

σ2 gauged with T (e)

p∈U

exp[β(ρ((dσ2)p)− ρ(1))]

×∏

e′=(v,w)e′∈BK(e)c

exp[κ(f(η2v(σ2)e′(η2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))].

(173)We would like to split PKN

(η1v, φ1v, η

2v, φ

2v) into PKN

(η1v , φ1v)PKN

(η2v , φ2v). If

we had equality in this splitting, then we would immediately get the product

Dβ,κZ(P (e)∪U)

ZKN

.

We have the error

|PKN(η1v , φ

1v, η

2v, φ

2v)− PKN

(η1v, φ1v)PKN

(η2v , φ2v)| ≤ K3e−cK

PKN(η2v , φ

2v)

from Lemma 16. Combined with our observation in the previous paragraph, we

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see that

|Z(P (e) ∪ U)

ZKN

−Dβ,κZ(U)

ZKN

|

≤∑

η1v ,φ

1v,η

2v ,φ

2v

|PKN(η1v , φ

1v, η

2v, φ

2v)− PKN

(η1v , φ1v)PKN

(η2v , φ2v)|

×∑

σ1 gauged with T (e)

p∈P (e)

exp[β(ρ((dσ1)p)− ρ(1))]

×∏

e′=(v,w)e′∈E(P (e))

exp[κ(f(η1v(σ1)e′ (η1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

×∑

σ2 gauged with T (e)

p∈U

exp[β(ρ((dσ2)p)− ρ(1))]

×∏

e′=(v,w)e′∈BK(e)c

exp[κ(f(η2v(σ2)e′(η2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))]

≤ K3e−cK∑

σ1 gauged with T (e)

p∈P (e)

exp[β(ρ((dσ1)p)− ρ(1))]

×∏

e′=(v,w)e′∈E(P (e))

exp[κ(f(η1v(σ1)e′ (η1w)

−1, φ1v(φ1w)

−1)− f(η1v(η1w)

−1, φ1v(φ1w)

−1))]

×∑

η2v ,φ

2v

PKN(η2v , φ

2v)

σ2 gauged with T (e)

p∈U

exp[β(ρ((dσ2)p)− ρ(1))]

×∏

e′=(v,w)e′∈BK(e)c

exp[κ(f(η2v(σ2)e′(η2w)

−1, φ2v(φ2w)

−1)− f(η2v(η2w)

−1, φ2v(φ2w)

−1))]

≤ K3e−cKLZ(U)ZKN

,

(174)where L is the constant from equation (138). We can now divide out byZ(U)Z−1

KNon both sides. This proves the desired intermediary inequality (171).

Return to bounding b3Recall c := 12K4B6. Provided c is less than 1, we can derive the follow-

ing. Since we have Z(B(E1,E2))Z(M(E1,E))

= Z(B(E1,E2))Z(G(E1,E2))+Z(B(E1,E2))

≤ c, we see that

Z(B(E1, E2)) ≤ c

1−cZ(G(E1, E2)).

For a lower bound, we see that we have,

Z(M(E1 ∪ {e}, E2)

Z(M(E1, E2))=

Z(TG(E1, E2))

Z(G(E1, E2)) + Z(B(E1, E2))

≥ (1− c)Z(TG(E1, E2)

Z(G(E1, E2))≥ (1 − c)[Dβ,κ −K3e−cKL].

(175)

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For an upper bound, we have that,

Z(M(E1 ∪ {e}, E2)

Z(M(E1, E2))≤ Z(TG(E1, E2))

Z(G(E1, E2)

≤ [Dβ,κ +K3e−cKL].

(176)

These are lower and upper bounds of EE [1[FP (e)]|M(E1, E2)] regardless ofthe sets E1 and E2. We remark that these must be lower and upper boundson PE(FP (e)). By removing the conditioning on M(E1, E2) and summing upoverall edges in γ, we can derive the following bound on b3,

b3 ≤ |γ|∣

∣(1− c)[Dβ,κ −K3e−cKL]− [Dβ,κ +K3e−cKL]∣

∣ . (177)

This is also a bound on |λ−|γ|Dβ,κ|, referring to our earlier lower and upperbounds on PE(FP (e)).

Our bounds on b1 and b2 are much simpler. By simple summation, we seethat |γ|12K4B12 suffices as an upper bound for both b1. This merely uses thatthe probability of excitation of a single minimal vortex is less than ΦUB(P (e))and the probability of excitation of two minimal vortices that do not intersectis less than ΦUB(P (e)∪ P (e)′) = ΦUB(P (e))

2 ≤ B12. We can then perform thesummation over all e ∈ γ and e′ ∈ BK(e).

Upon reaching here, the calculation of dTV (LE(Mγ),Poisson(|γ|Dβ,κ)) isthe same as it was in the previous Lemma 8. We use Theorem 8 to computedTV (LE(Mγ),Poisson(λ)) and apply dTV (Poisson(λ),Poisson(|γ|Dβ,κ) ≤ |λ −|γ|Dβ,κ|. We get the desired comparision in (170) by a triangle inequality.

By combining this error estimate with the error estimate from Theorem 7,we will be able to prove the following main result. The proof is exactly the sameas short proof of Theorem 3 and we will omit the proof.

Theorem 9. Assume that the conditions of Theorem 7 hold and Lemma 18hold. Let X be a Poisson random variable with parameter |γ|Dβ,κ. Then, wehave the following estimate on the Wilson loop expectation,

|EE [Wγ ]− E[DXβ,κ]| ≤ 12|γ|K4B12

+ 2|γ|∣

∣(1− c)[Dβ,κ −K3e−cKL]− [Dβ,κ +K3e−cKL]∣

+ |γ|2B6LK3e−cK exp[B6|γ|L]

+|γ|B6[(1 + e−c′K)L]e−c′K

min ηv,φv

v∈ minimal vortex

PKN(ηv, φv)

exp[|γ|B6[(1 + e−c′K)L]].

(178)

As a consequence, we also have that,

|E[Wγ ]− E[DXβ,κ]| ≤ 12|γ|K4B12 + P(Ec)

+ 2|γ|∣

∣(1− c)[Dβ,κ −K3e−cKL]− [Dβ,κ +K3e−cKL]∣

+ |γ|2B6LK3e−cK exp[B6|γ|L]

+|γ|B6[(1 + e−c′K)L]e−c′K

min ηv,φv

v∈ minimal vortex

PKN(ηv, φv)

exp[|γ|B6[(1 + e−c′K)L]].

(179)

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Remark 17. Just as in Remark 14, the main point of the introduction of K isthat when K is large enough, we can use e−cK to suppress factors of |γ| evenwhen |γ| is large. In terms that do not have e−cK, we only have a polynomialpower of K and can suppress this polynomial power of K with exp[−β] for βrelatively large. Just as in the aforementioned remark, we will choose K = O(β).

A Details on Vertex Decompositions

In this section, we will describe assorted facts on Higgs boson vertex configura-tions that would be useful for our decomposition theorem.

The following Lemma details the behaviors of the Higgs boson charges onthe complement of the support of a distribution.

Lemma 19. Let P be the support of a configuration C of Higgs boson and gaugefields. As previously, we use V (P ) and E(P ) to denote the set of vertices andedges that are associated to P . Let the complement V (P ) in VN be divided intoseparate connected components V (P )c = B1∪B2∪ . . .∪BN . Then the followingstatements hold true,

• There is a single charge ci such that each vertex vi in Bi is assigned theHiggs boson charge ci, e.g. φvi = ci, ∀vi ∈ Bi.

• If v is a vertex in V (P ) that is connected by an edge in EN to a vertexvi ∈ Bi, then φv = ci, where ci is the common color of each vertex in vi.

Proof. We start with the proof of the first item.If Bi is not monocharged, then there are two vertices vi, wi in Bi that are

not the same color as well as a path pi consisting entirely of vertices in Bi

connecting vi and wi. At least one of the edges,ei , on this path p must haveopposite charges on its neighboring vertices. This would imply that ei wouldbe in supp(C); furthermore, it would imply that the vertices of this edge wouldbelong in V (P ) rather than the complement. This is a contradiction. Thisproves the first item.

Now, we prove the second item. If vi is a vertex in Bi and w is a vertexin V (P ) adjacent to v that does not have the same charge ci, then the edgee = (vi, w) w has the same charge as v would be in supp(C). Thus, the vertexv must be in V (P ). This is a contradiction.

The behavior of the Higgs boson configurations is much like the Ising model.One can imagine regions of monocharged components containing regions of othermonocharged components recursively. It is important to understand the rela-tionship between these monocharged components and the support of the con-figuration. The following lemma details these relationships.

Lemma 20. Let (φ, σ) be a configuration C of Higgs boson and gauge fields.Let V be a some connected monocharged set(e.g. all the vertices are assignedthe same Higgs boson charge) in VN and define the set V(V ) as follows,

V(V ) := {w ∈ VN : ∃ path p(v → w) s.t. ∀ vertices a ∈ p, φa = φw}. (180)

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Namely, V(V ) is the collection of vertices in VN (V ) that can be connected to Vwith vertices of the same Higgs boson charge as V .

We define E(V ) as the set of vertices connecting vertices in V to its comple-ment in VN .

E(V ) := {e = (v, w) ∈ EN : v ∈ V(V ), w ∈ V(V )c}. (181)

We finally define P(V ) to be the set of plaquettes in PN that have one of theedges in E as a boundary edge. Namely,

P(V ) := {p ∈ PN : ∃e ∈ E(V ) s.t. e ∈ δp}. (182)

Then, P(V ) is a subset of supp(C). We can call P(V ) the boundary of V(V ).Now assume further that there is no vertex in V(V ) that is a boundary vertex

of PN . We can make the following statements on the decomposition of P(V )into connected components.

There is a unique connected component, which we will call the externalboundary of V(V ); we will denote this by EB(V(V )) satisfying the followingproperties.

• EB(V(V )) is connected.

• Embed the subset V(V ) into the full lattice Zd. Consider the connectedcomponents of V(V )

cin Z

d into connected components V(V )c= B1 ∪

B2∪ . . .∪BN . Let B1 be the unique non-compact component, so it extendsto ∞. If we define E(B1) to be the set of edges connecting B1 to itscomplement and P(B1) to be the set of plaquettes having at least one edgefrom E(B1) on its boundary. Then, EB(V(V )) = P(B1).

Remark 18. The statement that P as we have defined it above is in suppC is bydefinition. The second part of the above lemma is to formally state the intuitionthat a set that is monocharged but connected and compact is separated from theoutside vertices that do not share its charge by a single connected boundary. Allother boundaries of the set V are internal and separate it from the islands ofcharge that are internal to V.

Acknowledgements: The author would like to thank Sourav Chatterjeefor suggesting this problem as well as for useful advice and Sky Cao for usefuldiscussions.

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