Limit Topology for Dimer AlgebraRandom R-Trees

(Based on Joint Discussion with Nicolai Reshetikhin)Matthew Bernard

May 2018

Abstract

We prove inverse partition correlation of Z projective limit topology by KasteleynPfaffian Pf(ΣK)∈Quot(K[D]) Grassmann field framework of bipartite (Z+)d dimer‘‘tree-like object” equipped with dual lattice height. We obtain the asymptoticscritical points of real and complex discriminants for the Grassmann-integral FermionKernel of special operators, and formulate conjecture for correlation function on largedeviation functional of height function fluctuation φ, in asymptotic behavior of largerandom spanning tree and tree-valued measure.

Keywords: Random-Dimer, Kasteleyn, Grassmann, Fermionic-Correlation

Grateful to all the conference organizers/participants

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1 Characterizations

1.1 Basic definitions and observations

Graph Γ:={i∈N |1≤ i≤n}⊂ (Z+)d |d=2,3,. . . in topological surfaceMg

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is partition [σ] if, ∀ ℓ :={i= i1· · · id, j=j1· · · jd}∈D, ∃ perfect matching:• Dimers (ℓ)ℓ∈D do not overlap, ∀D:= set of all configurations D of Γ• All vertices (i)i∈Γ :=∂D are covered: |∂D|=2

∑ℓ σD(ℓ), σD :={0, 1}

∀ |[σ]|≤|D| partitions [σ].Remark. Γ:= closed, connected; n := even; dimer is undefined ∀ i=j loop.

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Symmetric difference D∆D′ :=D∪D′\D∩D′ ⊇ (Cα | α∈N, 1≤α≤m).

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One or more circuit-coverings ⇐= transition cycles := superposition cycles:

• Disjoint even-length (simple closed curves) cycles Cα | α∈N, 1≤α≤m.• Alternating sequence of configuration-pair by ordered cyclic sequence:

elementary circuit := (ℓ1, ℓ2, . . . , ℓr) of (ℓ1, ℓ3, . . . , ℓr−3, ℓr−1) ∈D, resp.(ℓ2, ℓ4, . . . , ℓr−2, ℓr) ∈D′, for sequence i1, ℓ1, i2, ℓ2, . . . , ℓr joining vertexi1 after finite edges, starting at i1. Another elementary circuit starting atvertex ir+1 at end of edge ℓr+1∈D for Γ of vertices > r, etc.

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Proposition. Aut(D) :=subset of automorphism group Aut(D) is given by|Aut(D)|=

(n2

)! 2

n2 , where |Aut(D)|≤|Aut(Γ)|=n! .

Proof. Preserving contiguity (adjacency+distribution), ∀ [σ]∼σ∈Aut(D) :

[σ] :Γ→Γ{ σ(1, . . . , n) = (σ(1),σ(2), . . . ,σ(n−1),σ(n))

s.t. σ(2ℓ−1)=ℓ, ℓ

The local observables :=D-D correlation functions (conditional probabilities)⟨ ∏ℓ∈D

σD(ℓ)⟩

def== Prob(ℓ1 ∈ D, . . . , ℓn ∈ D) := E

[ n∏i=1

σD(ℓi)]

=∑D∈D

n∏i=1

σD(ℓi)× Prob(D) =

∑D∈D

∏ℓ∈D

εℓ ωℓn∏

i=1

σD(ℓi)∑D∈D

∏ℓ∈D

εℓ ωℓ

where εℓ∈{−1,+1}, and σD : Γ→ {0, 1} by

σD(ℓ) :={1 if ℓ ∈ D0 if ℓ ̸∈ D

∀D ∈ D

is the indicator function of ℓ on the support D for all dimer weights ω(·)and,

the numerator =∑

D∋ ℓ1, ..., ℓn

εD ωD∣∣∣ εD∈{−1,+1} ∼ ∏

ℓ∈D

εℓ .

Remark. If any ℓξ ̸= ℓη | ξ,η ∈ {1, . . . , n} have a common vertex, then⟨σD(ℓ1) · · ·σD(ℓn)⟩=0; more so, σD(ℓ)σD(ℓ)=σD(ℓ); so, ℓξ ̸=ℓη, ∀ ξ ̸=η.

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That is, in the σ-finite energy Ξ(·) measure δ(·) ∈ Ξ :E(Γ)→R, ℓ 7→ Ξℓ,probability exists by⟨∏

ℓ∈D

σD(ℓ)⟩:=

∏ℓ∈D

εℓ ωℓ∑D∈D

∏ℓ∈D

εℓ ωℓ=

εD ωDZ(Γ;ω)

=:Prob(D)∣∣∣∣ωℓ=e− Ξℓk T , εD∼∏

ℓ∈D

εℓ

=1

Zexp(−ΞDk T

) ∣∣∣ ΞD=∑ℓ∈D

Ξℓ , Z=∑D∈D

exp(−ΞDk T

)where Z := strict-sense positive continuous partition function on the objects:

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• Domains in hexagonal lattice.

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• Domains in square lattice.

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The bipartite graph (of no black-black nor white-white edge) is disjoint unionV (Γ) = V•(Γ) ⊔ V◦(Γ).

Instance.

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5 6

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15 17 19 .

Non-instance. 1 2 3

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(no bipartite structureon triangular lattices

).

Remark.D := 1-chain in cell complex of GF2 field: (ℓ)ℓ∈D∈H1

(Mg;GF2

)∼=(Z2)2g.∂D := 0-chain in cell complex of GF2 field: (i)i∈Γ∈H0

(Mg;GF2

)∼=Z2.8

1.2 Combinatorial equivalence

Proposition. Two families, respectively, on space of dimers and space oftilings, are combinatorially equivalent; that is, ∃ bijective correspondence

Dimers ←→ T ilings.Proof. Let all (v)v ∈Γ⊂Mg := closed traversal (walk); Γ finite, connected,and planar (non-intersecting edges). The set of all Γ spanning trees defines:I. The 2D cell complex Γ:

vertices, edges, faces := 0-cells, 1-cells, 2-cells, resp.1

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Disjoint interiors.

dim(∂Γn) = (n−1) /GF (2)∂Γn := boundary between

two n-cells.

Remark. Γ⊂Mg := embedding by 1-skeleton CW-complex (resp. cellulardecomposition of an oriented closed connected surface).

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II. The dual cell complex Γ∗:0-cells, 1-cells, 2-cells := ‘‘centers” of 2-cells, 1-cells, 0-cells of Γ, resp.

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Γ∗ := dualcell complexto Γ.

III. Dimer Unique pair of 2-cells on Γ∗on Γ: share a dimer:

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IV. Therefore, the global bijection:

(Dimers on Γ) ←→(

Tilings of Γ∗ byunique pair of 2-cells

).

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In particular, on the dual cell complex of bipartite graph:1

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(two ‘‘colorings” of the same tile).

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Limit topology := stacked3D boxes; projection of2D rhombus tiling:

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1.3 Space of height functions

Definition. Let Γ⊂Mg := planar, bipartite, hexagonal, such thatDimers ←−[ Discrete Surfaces.

The space of height functions is the partially ordered by overlap, given by:

HΓdef== {π : faces (Γ)→ Z}

π+ 13

π+ 239 5

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16 14π

π+ 13 π+23

π+ 139 5

13 10

16 14π

π + 23

with respect to normalization π(f0)=0 of the reference face f0.

Proposition.(i). Let ∂Γ:= boundary faces. Then, πD

∣∣∂Γ

does not depend on D.(ii). πD1D2 = πD1−πD2.

Proof. ♡.

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Theorem. The dimer configuration probability is given by

Prob(D)= 1Z

∏f

qπD(f)f

∣∣∣∣Z=∑π ∈HΓ

∏f

qπ(f)f , qf =

∏ℓ∈ ∂f

ωεℓℓ , π∈HΓ, D∼=π.

Proof. By the combinatorial equivalence,{Dimers on Γ

}∼=

bijection

{height functions

}.

The bijection then gives the measure. □

Remark. Prob(D) :=“gauge” invariant: ω(ℓ) 7−→s(ℓ+)ω(ℓ) s(ℓ−). More so,qf := invariant (‘‘essential” parameters).

Particular cases.

(i).4

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a

bc

acb

=⇒ the uniform distribution:q = a−1b c−1a b−1c = 1.

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(ii). qf :=qt,Prob(π) ∝

∏tqπ(t)t .1

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Remark.0

1.4 Kasteleyn orientation and matrix

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+ −

−+

+−

−+

−−

−+

++

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−−

−+ .

Definition. Let Γ⊂ (Z+)d :=CW-complex 1-skeleton (resp. genus g compactorientable surface cell-decomposition) closed connected embedding, endowedwith induced (counterclockwise) orientation ε(∂F) on every face boundary;Γ is Kasteleyn if ∀{i, j} edges ℓ, i ̸=j, arbitrary i-to-j orientations εK(iℓjℓ),∏

ℓ∈ ∂F

εKiℓjℓ = −1, ∀F ∈Γ∣∣ εKiℓjℓ :={−1 if ε(∂F) ∈ counter-εK(iℓjℓ)+1 if ε(∂F) ∈ εK(iℓjℓ).

Let Γ:=Kasteleyn, weighted ω>0, ∀{i, j} edges ℓ, i ̸=j,ΣKij =

∑ℓ

εKiℓjℓ ωiℓjℓ∣∣ ΣKij :=0 for i, j not adjoined and i=j loop.

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Proposition (existence). Kasteleyn orientation exists.Proof. ♡.Construction (bipartite case). Following Γ∗ spanning tree:

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−−− +++

−−−

+++

−−− +++

+++ −−−−−−+++

+++−−−

−−− +++

−−−+++

−−−+++ −−−

+++

−−−+++

−−− +++

+++−−−

−−−+++

−−− +++

+++−−−

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+++−−− +++

−−−

−−−+++

−−− +++

+++ −−−

−−−+++

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−−−+++

−−− +++

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−−−+++

−−− +++

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−−−+++

+++

+++ −−−−−−

−−− +++

+++−−−−−−+++

−−− +++

−−−

+++

−−−+++ −−−

+++

+++−−−

+++−−−

−−−+++

+++ −−−

−−− +++

−−−+++

+++−−−

−−− +++

−−−+++

−−−+++

+++

−−−

+++

−−−

−−−

+++

+++−−−

+++

−−−

Reduce Γ to small N×N→ exp(αN 2); choose ε(∂F); follow Γ∗ spanningtree, rooted outside Γ, from root; make εK(F) with each Γ∗ edge deletion.

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Definition. Two orientations are equivalent, if each is obtainable from theother through a sequence of orientation-reversing maps:

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−→148

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−−−

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−−−

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−−−+++

−−−

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+++

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−−−

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−−−

−−−+++

Theorem. All Kasteleyn orientations of Γ planar are equivalent.Proof. Induction, ∀n

Remark. Deleting a vertex changes orientation to non-Kasteleyn at ‘‘holes”:

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A

B

C

≡

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−1

−1

−1

hAhBhC=(−1)3=−1

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66 hABC=1, non-Kasteleyn orientation.

That is, hAhBhC=−hABC.

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Remark. To convert the non-Kasteleyn orientation back to Kasteleyn:

hAhBhC=1.

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Derivation. Given Γ⊂ (Z+)d := cycle-graph in plane, ∀ i, j=1, . . . , n

ΣKij =

{−ωij = ωji for ε(∂F) ∈ counter-εKij , ∀ i i j j, i i j jωij = −ωji for ε(∂F) ∈ counter-εKij , ∀ i i j j, i i j j0 for i, j not adjoined and loop i=j.

Theorem (Kasteleyn). Every ±|detΣK|12 =: Pf(ΣK) ∈ Quot(K[D]),Γ ⊂ (Z+)d, non-vanishing monomial corresponds 1-1 to [σ], ∀D on Nνedges of class Cν | ν = 1, . . . , d, with respect to sgn(σ) := (−1)t(σ) wheret(σ) :=(σ(i∈N) |1≤ i≤n)−→ (i∈N |1≤ i≤n), ∀σ∈Aut(D), such that:

(i) |Pf(ΣK)|=∑D∈D

∏ℓ∈D

εℓ ωℓ∣∣∣Pf(ΣK)=∑

σ= [σ]

(−1)t(σ)n2∏

ℓ=1

ΣKσ(2ℓ−1)σ(2ℓ).

(ii) Pf(ΣK)= 12n2

1(n2

)!

∑σ

(−1)t(σ)n2∏

ℓ=1

ΣKσ(2ℓ−1)σ(2ℓ) =:∑

N1, ..., Nd

∑D

d∏ν=1

(±)ωNνν .

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Proof. (ii) follows from (i) by Aut(D).

To see (i): detΣK = det(−(ΣK)T) = det(−ΣK) = (−1)n detΣK implies

detΣK:= strictly positive-definite square of a rational function of array ΣKij ,∀ i, j=1, . . . , n, n := even, on the Leibniz’s second-index permutations:

∑σ∈Sn

n∏i=1

(−1)t(σ)ΣKiσ(i)=n!/2∑σ=1

n∏

i=1

σ(i) := given byeven permutations

σ∈Sn

(−1)t(σ)ΣKiσ(i) +n∏

i=1

σ(i) := given byodd permutations

σ∈Sn

(−1)t(σ)ΣKiσ(i)

where

σ : {(i∈N | 1≤ i≤n)} −→ {(σ(i∈N) | 1≤ i≤n)}

and,t(σ) := (σ(i∈N) | 1≤ i≤n) −→ (i∈N | 1≤ i≤n).

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In particular, skew-symmetry ΣKiσ(i)=−ΣKσ(i) i | i≥σ(i) implies non-vanishingmonomials given by:

n!/(n

2

)! 2

n2∑

σ=1

(−1)t(σ)n2∏

i=1

ΣKτ1(2i−1)τ1(2i)

n2∏

i=1

ΣKτ2(2i)τ2(2i−1)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∃ΣKτ1(2i−1)τ1(2i)=−ΣKτ2(2i)τ2(2i−1),∀ΣKτ2(2i)τ2(2i−1);

t(σ)=even for n2 :=even,t(σ)=odd for n2 :=odd

+

2

(n!/(n2

)! 2

n2

2

)∑σ=1

(−1)t(σ)n∏

i=1

ΣKiσ(i)

∣∣∣∣∣∣t(σ)=odd forn2 :=even,

t(σ)=even for n2 :=odd.

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That is,

n!/(n

2

)! 2

n2∑

σ=1

(−1)t(σ)+n2

( n2∏i=1

ΣKτ1(2i−1)τ1(2i)

)2 ∣∣∣∣∣∣t(σ)=even forn2 :=even,

t(σ)=odd for n2 :=odd

+

2

(n!/(n2

)! 2

n2

2

)∑σ=1

(−1)t(σ)n∏

i=1

ΣKiσ(i)

∣∣∣∣∣∣t(σ)=odd forn2 :=even,

t(σ)=even for n2 :=odd.

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Furthermore, the non-vanishing monomials are given by

n!/(n

2

)! 2

n2∑

σ̃=[σ̃]=1

(−1)t(σ)+n2+2×t(σ̃)

( n2∏ℓ=1

ΣKσ̃(2ℓ−1)σ̃(2ℓ)

)2 ∣∣∣∣∣∣t(σ)=even forn2 :=even,

t(σ)=odd for n2 :=odd

+

2 ×

(n!/(n2

)! 2

n2

2

)∑

{σ̃= [σ̃] ̸= τ̃= [̃τ]}=1

(−1)t(σ̃)n2∏

ℓ=1

ΣKσ̃(2ℓ−1)σ̃(2ℓ) × (−1)t(̃τ)

n2∏

ℓ=1

ΣKτ̃(2ℓ−1)̃τ(2ℓ)

∀ [σ] :=Sn/(Sn

2×S

n22

).

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That is,

n!/(n

2

)! 2

n2∑

σ̃=[σ̃]=1

(−1)2×t(σ̃)( n2∏

ℓ=1

ΣKσ̃(2ℓ−1)σ̃(2ℓ)

)2 ∣∣∣∣∣∣t(σ̃) :=(σ̃(i∈N) |1≤ i≤n)−→ (i∈N |1≤ i≤n)+

2 ×

(n!/(n2

)! 2

n2

2

)∑

{σ̃= [σ̃] ̸= τ̃= [̃τ]}=1

(−1)t(σ̃)+t(̃τ)n2∏

ℓ=1

ΣKσ̃(2ℓ−1)σ̃(2ℓ)

n2∏

ℓ=1

ΣKτ̃(2ℓ−1)̃τ(2ℓ)

∀ [σ] :=Sn/(Sn

2×S

n22

)=

( ∑σ= [σ]

(−1)t(σ)n2∏

ℓ=1

ΣKσ(2ℓ−1)σ(2ℓ)

)2 ∣∣∣∣∣∣t(σ) :=(σ(i∈N) |1≤ i≤n)−→ (i∈N |1≤ i≤n)∀ [σ] :=Sn

/(Sn

2×S

n22

). □

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0 1 1 1 1 11 0 1 1 1 11 1 0 1 1 11 1 1 0 1 11 1 1 1 0 11 1 1 1 1 0

0 := non-adjoined i, j1,1,1 := adjoined i, j.

To see |Pf(ΣK)|=Z, write on the non-vanishings, by partitions σ=[σ], ∀D :

Z =∑

σ= [σ]

∑D(σ)

εKσ(1)σ(2) ωσ(1)σ(2) · · · εKσ(n−1)σ(n) ωσ(n−1)σ(n) =

=∑

σ= [σ]

∑D(σ)

n2∏

ℓ=1

εKσ(2ℓ−1)σ(2ℓ) ωσ(2ℓ−1)σ(2ℓ).

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1

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10

0 := non-adjoined i, j1,1,1 := adjoined i, j

0 1 1 0 0 1 1 1 0 01 0 1 0 0 1 1 1 0 01 1 0 1 1 1 1 1 1 10 0 1 0 1 0 0 1 1 10 0 1 1 0 0 0 1 1 11 1 1 0 0 0 1 1 0 01 1 1 0 0 1 0 1 0 01 1 1 1 1 1 1 0 1 10 0 1 1 1 0 0 1 0 10 0 1 1 1 0 0 1 1 0

.

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But the derived Pfaffian is given by

Pf(ΣK)=

∑

σ ∈ Sn/(Sn

2×S

n22

)

=

[σ]

(−1)t(σ)n2∏

ℓ=1

ΣKσ(2ℓ−1)σ(2ℓ)︸ ︷︷ ︸depends only on σ

=∑

σ= [σ]

depends only on σ︷ ︸︸ ︷∑D(σ)

(−1)t(σ)n2∏

ℓ=1

εKσ(2ℓ−1)σ(2ℓ)

n2∏

ℓ=1

ωσ(2ℓ−1)σ(2ℓ).

Therefore, write

Pf(ΣK) =∑D(σ)

(−1)t(σ)n2∏

ℓ=1

εKσ(2ℓ−1)σ(2ℓ)

n2∏

ℓ=1

ωσ(2ℓ−1)σ(2ℓ) = (sgn) · Z=

(−1)t(σ) .

where the first equality is well-defined by all partitions [σ], ∀ ℓ∈D. □

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Corollary. ⟨ k∏i, j=1

σ(i)σ(j)⟩

= Pf((ΣK)−1ξη

) ∣∣∣ ξ=1, . . . , kη=1, . . . , k.

Proof. ♡.

Remark. Pf(ΣK) := partition function, (ΣK)−1 := correlation functions.

Remark. Kasteleyn theorem allows polynomial-time; skew-symmetric Gausselimination algorithm, by Pf(BABT )=det(B)Pf(A), does O(m3) time forPfaffian of size 2m matrix.

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1.5 Grassmann integral

Definition. Let (v1, . . . , vn) ∈ V := vector space, where∧nV := exterior

algebra of V is defined by:(i) Elements:

n∧i=1

vi := v1∧· · ·∧vn =

=1

n!

∑σ∈Sn

(−1)t(σ)n⊗

i=1

vσ(i) :=1

n!

∑σ∈Sn

(−1)t(σ)vσ(1)⊗ · · · ⊗ vσ(n);

(ii) Multiplication:( n∧i=1

vi

)∧( n∧i=1

wi

):= (v1∧· · ·∧vn) ∧ (w1∧· · ·∧wn) == v1∧· · ·∧vn ∧ w1∧· · ·∧wn.

Then, with chosen basis a=(a1, . . . , an)∈V, the exterior algebra∧nV := {ai | aiaj+ajai = 0}is a Grassmann algebra.Proposition. The Grassmann algebra

∧nV is generated by V.Proof. ♡.

32

Definition. The Grassmann integral on∧nV with respect to x is given by∫

∧nV f = fx , f = fx x + · · ·︸︷︷︸lowerorder termswhere x∈

∧nV ⊂R is an orientation.Lemma. x=

∏ni=1 ai if (a1, . . . , an) is a basis in V.

Proposition. The formal Grassmann constraints are given by the integral:∫n∏

i=1

ai

n∏i=1

dai = (−1)n(n−1)

2

∫n∏

i=1

ai dai = (−1)n(n−1)

2 .

where:(i).

∫n∏

ν=1

aiν da ={0 , k

Theorem. Let a=(ai)∈V be basis, where A∗=Aa=exp(12

∑ij aiAijaj

)satisfies the Grassmann constraints, uniquely generalizing A∗ for allKasteleyn matrices A satisfying the Grassmann constraints. Then:

(i) Pf(A) =∫∧nV

exp(1

2

∑ij

aiAijaj

)da.

(ii) Pf(

0 A−At 0

)= det(A).

(iii) (Pf(A))2 = det(A).

(iv) ∂∂Ai1j1

· · · ∂∂Aikjk

Pf(A) =

= Pf(A) · Pf((ΣK)−1ab )∣∣∣ a = i1, . . . , ikb = j1, . . . , jk .

34

Proof.(i). Write:∫∧nV exp

(1

2⟨a,Aa⟩

)da =

1(n2

)!

1

2n2

∫∧nV ⟨a,Aa⟩

n2 da

such that∫⟨a,Aa⟩

n2 da =

∫ai1aj1 · · · ain

2ajn

2Ai1 j1 · · ·Ain

2jn2

= (−1)σ Ai1 j1 · · ·Ain2jn2

σ :((i1, j1) , . . . ,

(in2, jn

2

))→ (1, . . . , n) .

This implies ∫∧nV exp

(1

2⟨a,Aa⟩

)da =

1(n2

)!

1

2n2

Pf(A) .

Use the integral formula to prove II, III, IV. ♡.

35

Proof (hints).

(ii). Choosing splitting V = W ⊕W∗ by matrix block structure, suchthat the Grassmann algebra of V ∼= algebra (tensor product) generated byci, bi | i=1, . . . , n2 , with relations cicj=−cjci, cibj=−bjci, and b1bj=−bjbi:

(a1, . . . , an) =

=(c1, . . . , cn2︸ ︷︷ ︸

basis inW

, b1, . . . , bn2︸ ︷︷ ︸basis inW ∗

)then ⟨

a ,

(0 A−At 0

)a

⟩= 2 ⟨c, Ab⟩ .

Hence, prove ∫∧n(W⊕W∗)

exp (⟨c, Ab⟩) dc db = det (A).

(iii). Similar.

36

(iv).∫

exp(1

2⟨a,Aa⟩ + ⟨a,η⟩

)da =

=

∫exp(1

2

⟨a + A−1η, A

(a + A−1η

)⟩− 1

2

⟨η, A−1η

⟩)da

= Pf(A) exp(−12

⟨η, A−1η

⟩).

∂

∂Ai1j1· · · ∂

∂AikjkPf(A) =

=

∫ai1aj1 · · · aikajk exp

(1

2⟨a,Aa⟩

)da

=

(∂

∂η

)2k ∫exp(1

2⟨a,Aa⟩ + ⟨η, a⟩

)da.

Prove the Pfaffian formula for the correlation functions. ♡.

37

1.6 Kasteleyn solution for bipartite graph

Given Γ⊂R2 endowed with configuration and Kasteleyn orientation

ZΓ= εKΓ

∫exp(12

∑ij

ai (ΣKΓ )ij aj

)da

∣∣∣∣ εKΓ :=(−1)σ εKσ1σ2· · · εKσn−1σn∈{±1}n= |V (Γ)|.In addition, Γ:= bipartite implies

ΣKΓ =

(0 BKΓ

−(BK)tΓ

0

) ∣∣∣∣∣∣∣∣∣∣BK : RV◦(Γ)→RV•(Γ)

RV (Γ)=RV•(Γ)⊕RV◦(Γ)

dim(RV•(Γ))=dim(RV◦(Γ))= n2V (Γ)=V•(Γ) ⊔ V◦(Γ), |V (Γ)|=n.

Identifying V•(Γ), V◦(Γ) via a diagram {b}∼{ω} with ‘‘hole”

ΣKΓ =

(0 CKΓ

−(CKΓ )t 0

)for RV•(Γ)⊕ RV◦(Γ)←↩

∣∣∣CKΓ :=RV◦(Γ)←↩where ←↩ denotes recursive invocations.

38

Then:

(i). ZΓ = |det(CKΓ )|

(ii).⟨

σb1w1 · · ·σbkwk⟩=

∂

∂ ω(b1w1)· · · ∂

∂ ω(bkwk)ln ZΓ =

= det(((CKΓ )

−1)b̂ w

) ∣∣∣∣ b̂ = b̂1, . . . , b̂kw = w1, . . . , wkwhere the inverse-matrix element b̂ is the white vertex identified with b.

Remark. The ‘‘Physical” meaning of (ii) is given by⟨σb1w1 · · ·σbkwk

⟩=

=

∫ψ∗b1w1· · ·ψ

∗bkwk× exp

(ψ∗CKΓ ψ

)×(∫

exp(ψ∗CKΓ ψ

)dψ∗ dψ

).

That is, it corresponds to correlation functions in free fermionic model.

39

1.7 Dimer monomer relation

Γ

removevertices andadjacent edges

2

4

5

29 32

36

37

229

36

3765

66

145

3

7 8

12

21 22

33

39

41

42

17

22

2342

46

51

52

62

68

126

23

46

51

68

70

74

76 81

87127

Monomers Dimers.

40

For monomer coverΓw1w2b1b2

b1

b2

w1

w2

the monomer-monomer correlation functions are given by

Mb1···bnw1···wn =ZΓw1···wnb1···bn

ZΓ.

41

Remark. If w1 and b1 are adjacent, then we have a dimerΓ

.

Remark. The monomer-monomer correlation functions are a special case ofdimer models on surfaces with nontrivial fundamental group.

.In this case, #([K])=22g+n−1.

42

1.8 Partition function as sum of Pfaffians

Theorem.Z

dimers

Γ⊂Mg =1

2g

∑[K]

Arf(qKD0)εK(D0) · Pf(ΣK)

∣∣Arf(·) := sgn.where:

[K] := equivalence classes of Kasteleyn orientations, 22g in total

[qKD0] := quadratic form on H1(Σ,Z2) associated with Kasteleyn

orientation on reference configuration D0εK(D0) := (−1)σεKσ1σ2· · · ε

Kσn−1σn

σ∈Sn, [σ] := Sn/(S

n22 ×Sn2

)n := number of vertices.

Proof. ♡.

43

Remark. In particular:

I. For bipartite graphs on Mg :

height function :== section of the non-trivialZ-bundle.

II. For fundamental cycles (a1, . . . , ag, b1, . . . , bg) :

Z(Ha1, . . . ,Hag,Hb1, . . . ,Hbg

)=

=∑D

∏ℓ∈D

ω(ℓ)g∏

i=1

exp(∑

i

Hai∆aih +

+∑i

Hbi∆bih)

where ∆Ch := change in height function along every noncontractible cycleC onMg.

44

1.9 Thermodynamic limit of bulk interactions

N

N

N

uniform measureProb(h)= 1|HΓ|N→∞ .

45

Theorem (Schur process; Okounkov & R). Let φε :Z2 ↪→R2 |D⊂R2.

ε

ε

D such that :{ ε→0, |Dε|→∞Dε := φε

(Z2)∩D

.

Then, for stacks of cubes with measure

Prob(π) =

∏tqπ(t)t∑

π

∏tqπ(t)t

∣∣∣∣ π∈HΓ , π∼=D,there is an existence of

Thermodynamic limit (|Dε|→∞) ++ Scaling limit (q=e−ε, ε→+0).

Proof. ♡.

46

a1N b1N a2N b2N

| uN | vN |where u+v = a1+a2+b1+b2, N =ε−1, q=e−ε.

47

1.10 Enumeration of perfect matchings in graphs

Involves computation of matching polynomials using the derived Kasteleyndeterminantal method:

• Γ⊂Mg :=m×n planar square lattice without closed boundary:µ(Γ;m,n) :=

2mn2

m∏i=1

n2∏

j=1

√cos2

(πi

m + 1

)+ cos2

(πj

n + 1

)n := even.

Or,= 0, for n := odd.

Proof. ♡.

48

• Γ⊂Mg :=m×n cylindrical square lattice:µ(Γ;m,n) :=

2mn2

m∏i=1

n2∏

j=1

√sin2

(π(2i−1)

m

)+ cos2

(πj

n + 1

)n := even.

Or,µ(Γ;m,n) :=

2mn2 −

m2 +1

m∏i=1

n2∏

j=1

√sin2

(π(2i−1)

m

)+ cos2

(πj

n + 1

)n := odd.

Proof. ♡.

49

• Γ⊂Mg :=m×n toroidal square lattice:µ(Γ;m,n) :=

2mn2 −1

m∏i=1

n2∏

j=1

√sin2

(π(2i−1)

m

)+ sin2

(2πj

n

)+

m∏i=1

n2∏

j=1

√sin2

(2πi

m

)+ sin2

(π(2j−1)

n

)+

m∏i=1

n2∏

j=1

√sin2

(π(2i−1)

m

)+ sin2

(π(2j−1)

n

)

n := even.

Or,= 0, for n := odd.

Proof. ♡.

50

• Γ⊂Mg := 6×8 planar square lattice without closed boundary:µ(Γ;m,n) :=

16777216

(1

4+ sin2

( π14

))(sin2

( π18

)+ sin2

( π14

))(14+ sin2

(3π

14

))×(

sin2( π18

)+ sin2

(3π

14

))(1

4+ cos2

(π7

))(cos2

(π9

)+ cos2

(π7

))×(

cos2(π7

)+ cos2

(2π

9

))(sin2

( π18

)+ cos2

(π7

))×(

sin2( π14

)+ cos2

(π9

))(sin2

( π14

)+ cos2

(2π

9

))×(

sin2(3π

14

)+ cos2

(π9

))(sin2

(3π

14

)+ cos2

(2π

9

))

51

• Γ⊂Mg := 6×8 cylindrical square lattice:µ(Γ;m,n) :=

5242880

(1

4+ sin2

( π18

))2 (1 + sin2

( π18

))(14+ cos2

(π9

))2(1 + cos2

(π9

))(14+ cos2

(2π

9

))2(1 + cos2

(2π

9

))• Γ⊂Mg := 6×8 toroidal square lattice:µ(Γ;m,n) :=

8388608

(18225

131072+

(1

4+ sin2

(π8

))4(1 + sin2

(π8

))2(14+ cos2

(π8

))4(1 + cos2

(π8

))2+ sin4

(π8

)(34+ sin2

(π8

))4cos4

(π8

)(34+ cos2

(π8

))4)

52

2 Special cases

Points:(i). Reformulate Kasteleyn Grassmann integral by transfer-matrices (for

special domains)

(ii). Compute inverse of Kasteleyn operator

(iii). Find scaling limit

(iv). Derive variational principle for limit topologies.

53

2.1 Grassmann integral kernel

Let∧nV :={ai | aiaj+ajai = 0} be Grassmann algebra generated by V, for

a chosen basis a=(a1, . . . , an)∈V.For vectors ψ∈

∧nV , writeψ(a)=

n∑k=0

n∑i1 .

Pair∧nV ∗⊗∧nV →R by⟨

φ(a∗), ψ(a)

⟩ def==

n∑{i}<

φik···i1 ψi1···ik .

54

Choosinga1, . . . , an ∈

∧nV , a∗n , . . . , a∗1 ∈ ∧nV ∗and

a∗n , . . . , a∗1 , a1, . . . , an ∈∧nV ∗⊗∧nV

then ∫n∏

ν=1

a∗iνn∏

ν=1

ajν da∗ da =

{0 , k ̸= n

(−1)(

σ + τ + n(n−1)2)

, k = n

σ : (i1, . . . , in)→ (1, . . . , n)τ : (j1, . . . , jn)→ (1, . . . , n).

Proposition.⟨φ(a∗), ψ(a)

⟩=

∫exp(∑

i

a∗i ai)

φ(a∗) ψ(a) da∗ da .

Proof. Exercise.

55

Proposition. Let A :V →V such thatψA(a) =

∑{i}< , {j}<

a{i}< A{i}

2.2 Vertex operators

(i). Fermionic Fock space: For ⟨Vm⟩∈CZ+12

F :=

{Vm1 ∧ Vm2 ∧ · · ·

∣∣ mi ∈ Z+ 12mi+1 = mi−1, i≫ 1

}.

(ii). Clifford algebra:

ClZ :=

⟨ψm, ψ∗m

∣∣∣ m ∈ Z+12

⟩ψm ψm′ + ψm′ ψm = ψ∗m ψ∗m′ + ψ∗m′ ψ∗m = 0ψm ψ∗m′ + ψ∗m′ ψm = δmm′ .

(iii). Clifford algebra acts on Fock space F :ψm vm1 ∧ vm2 ∧ · · · = vm ∧ vm1 ∧ vm2 · · ·

ψ∗m vm1 ∧ vm2 ∧ · · · =∞∑i=1

(−1)i δmim ×

× vm1 ∧ · · · ∧ v̂m1 ∧ · · ·

57

(iv). Heisenberg algebra:⟨αn∣∣ n∈Z\{0}⟩ : [αn,αn′] = −n δn,−n′ .

(v). Heisenberg algebra acts on F :• As part of Bose-Fermi correspondence in 1D:

αn 7−→∑

m ∈ Z+12

ψm+n ψ∗m .

• As operator in F :[αn,ψk] = ψk+n[αn,ψ∗k

]= −ψ∗k−n .

(vi). Vertex operators (in F ) :

Γ±(x) = exp( ∞∑

n=1

xn

nα±n

)(Γ−(x) v, w) = (v, Γ+(x)w) = (Γ+(x)w, v).

58

(vii). Commutation relations:

Γ+(x) Γ−(y) = (1−x) · Γ−(y) Γ+(x)

Γ+(x)ψ(z) = (1−z−1 x)−1 · ψ(z) Γ+(x)

Γ−(x)ψ(z) = (1−x z)−1 · ψ(z) Γ−(x)

Γ+(x)ψ∗(z) = (1−z−1 x) · ψ∗(z) Γ+(x)

Γ−(x)ψ∗(z) = (1−z x) · ψ∗(z) Γ−(x).

(viii). Eigenvectors:

Γ−(x)∏i

ψ∗(wi)∏j

ψ∗(zj)V (n)0 =

=∏i

(1−x zi)−1∏j

(1−xwj)∏i

ψ∗(wi)∏j

ψ∗(zj)V (n)0

where V (n)0 = vn−12 ∧ vn−32 ∧ · · ·

59

2.3 Fermionic Kasteleyn operator

[Diagram] [Diagram]

where b(h, t)=(h, t− 12

), and w(h, t)=

(h, t+ 12

).

The K-matrix over (b, w), using above-chosen diagram b ∼ w is:

K(h, t) = (h, t)−(h+

1

2, t+1

)+ xh,t

(h−1

2, t+1

).

[Diagram]

60

Placing fermions a∗h, t and ah, t , respectively, at b(h, t) and w(h, t), then

a∗Ka =∑h,t

a∗h,t ah,t −∑h,t

a∗h+12 , t+1

ah,t +∑h,t

a∗h−12 , t+1

ah,t xh,t =

=∑t

(a∗t at + atV a∗t+1 + atV −1xt a∗t+1

)in addition to considering boundary conditions

[Diagram]

Prob (π) ∝∝∏tq|π(t)|t in previousnotations

qh,t=qt

where the assumption is that xh,t=xt .

61

Theorem. WriteZ =

∫exp(a∗Aa

)da∗ da =

=⟨Γ−(x−12

)· · ·Γ−

(xu0+

12

)Γ+(x1

2

)· · ·Γ+

(xu1+

12

)V

(0)0 , V

(0)0

⟩.

Proof. Outline:∫· · · exp

(a∗t−1 at−1

)· exp

(at−1

(V −V −1Xt

)a∗t)·

· exp(a∗t at

)· exp

(at(V −V −1Xt

)a∗t+1

)· · · =

= · · ·(V −V −1Xt−1

)∼−1︸ ︷︷ ︸

νt−1

·(V −V −1Xt

)∼−1︸ ︷︷ ︸

νt

· · ·

where Γ+(xt)=νt−1 or Γ−(xt)=νt depending on t, Ã :=A such that V ←↩is lifted to

∧∞2 V , given by V :=

⊕m∈Z+12

C vh , with boundary conditions, etc.

Remark. Direct proof exists combinatorially, without K-orientation formula.

62

Remark. Write

Z =

u1−12∏m=12

−12∏m′=u0+

12

(1−x−m′ x

+m

)−1.

Theorem. (Okounkov & R., 2005).⟨σ(h1t1) · · ·σ(hktk)

⟩= det(K((ti, hi), (tj, hj)))1⩽i, j⩽k

K((ti, hi), (tj, hj)) =

=1

(2πi)2

∫|z|

2.4 Thermodynamic limit with scaling

[Diagram]

x+m = aqm

x−m = a−1qm

}assumed

corresponds to Prob(π) ∝ q |π| .

64

2.5 Asymptotics of partition functions

Consider limit ε→0, q=e−ε, u1=ε−1v1, u0=ε−1v0, where v1, v0 are fixed:

Z =∏

u0

Consider limit ε→0 for ti=ε−1τi, h1=ε−1χi, where τi, χi are fixed:

[Diagram](τi, χi)

in the bulkK((t1, h1), (t2, h2)) =

=1

(2πi)2

∫Cz

∫Cw

exp(ε−1 (S(z, t1, χ1)− S(z, t2, χ2))

)·

· (zw)1/2 (z−w)−1 dz dw .S(z, t, χ) =

= −(

χ+τ2−u0

)lnZ +

+ Li2(ze−v0

)+ Li2

(ze−v1

)− Li2(z)− Li2

(ze−τ

)where

Li2(z) =z∫0

t−1 ln(1−t) dt .

66

2.6 Critical points of S (z)

exp(

χ+τ2

)=

(1−ze−v0) (1−ze−v1)(1−z)

(1−ze−τ

)gives quadratic equation. That is, 2 real solutions, or 2 complex conjugate,or zero discriminant.

[Diagram]

∂χh0 (τ, χ) =1

πarg(z0)

⟨σ(h, t)

⟩= K((t, h), (t, h))→ ε ∂χh0(τ, χ)

67

2.7 Results of steepest descent

K((t1, h1), (t2, h2)) = −ε2π·( exp(ε−1(S1(z1)−S2(w2)))(z1−w2)

√−w2S ′′2 (w2)

√z1S ′′1 (z1)

−

−exp(ε−1(S1(z1)−S2(w2))

)(z1−w2)

√−w2S ′′2 (w2)

√z1S ′′1 (z1)

+ c. c.

)· (1 +O(1))

where z0(χ, τ) := inner of limit shape H+ :={z∈C, Im z>0} such thatz1 :=z0(χ1, τ1), w2 :=z0(χ, τ).

Therefore,

K((t1, h1), (t2, h2)) =ε2π

exp(ε−1(Re(S(z0(χ1, τ1)))−Re(S(z0(χ2, τ2))))

)·

·(exp(iε−1(Im(S ′(z1))−Im(S(w2))))

(z1−w2)+

exp(iε−1(Im(S ′(z1))−Im(S(w2)))

)(z1−w2)

+ c. c.

)· (1 +O(1)) (∗) .

68

This suggests convergence of K-orientation fermions to free Dirac fermions:1√ε

ψx⃗ = exp(ε−1Re(S(z0))

)·(

ψ+(z0) exp(iε−1Im(S(z0))

)+

+ ψ−(z0) exp(iε−1Im(S(z0))

))· (1+O(1))

1√ε

ψ∗x⃗ = exp(ε−1Re(S(z0))

)·(

ψ∗+(z0) exp(iε−1Im(S(z0))

)+

+ ψ∗−(z0) exp(iε−1Im(S(z0))

))· (1+O(1))

whereE(ψ∗±(z) ψ±(w)

)=

1

z − wE(ψ∗±(z) ψ∓ (w)

)= E

(ψ∗ψ∗

)= E(ψ ψ) = 0

ψ∗±(z), ψ±(w) are spinors:

ψ∗±(z) = ψ∗±(w)

öw

∂z

ψ±(z) = ψ±(w)√

∂w

∂z.

69

The correlation functions are derived as follows:

9 5

13 10

16 14x⃗

∣∣∣∣∣∣∣∣∣∣∣∣∣∣

⟨(σx⃗1−⟨σx⃗1⟩) (σx⃗2−⟨σx⃗2⟩)

⟩= K12K21 =

= ε2

(2π)2

( ∂z1∂x1

∂w2∂x2

(z1−w2)2−

∂z1∂x1

∂ w2∂x2

(z1−w2)2+ c. c.

)×

× (1+O(1)).

That is, correlation functions are given byσx⃗1−⟨σx⃗1⟩ = ε ∂x φ(z0(τ, x)) + · · ·

where φ(z) := Gaussian free field on H+.

The Green’s function of the Dirichlet problem on H+ is given by

⟨φ(z)φ(w)⟩ = 12π

ln∣∣∣z−wz−w

∣∣∣.And, the Bose-Fermi correspondence is given by

∂xφ = : ψ̃(z, z) ψ̃(z, z) : · · · .

70

2.8 Scaling limit in the Kasteleyn operator

For lattice L such that Γ=Dε := φε(L)∩D

Let ΣKΓ := difference operator. Then(ΣKΓ)x·Gx,y = δx,y

where ε→0 in asymptotic for Gx,y.Particular cases.(i). Hexagonal lattices, with weights as above, such that

qt = e−ε f (t), t = τ

ε, ε→0.

Theorem. Gx,y := same as (⋆) , with different z0(τ, x).

(ii). Periodic lattices.

71

2.9 Limit shapes and variational principle

(i). For the N×M torus[Diagram]

Z(H, V ) =∑D

∏ℓ

ω(ℓ) exp(H∆ahD + V∆bhD)

=1

2

(Pf(AK1

)+ Pf

(AK2

)+ Pf

(AK3

)− Pf

(AK4

))where N,M →∞, NM :=fixed.And, ω (ℓ)=1 gives Kasteleyn matrices’ eigenvalues by Fourier transform.

Theorem. (McCoy & Wu, 1969; Kenyon & Okounkov, 2005).

limN,M→∞

1

NMlnZNM =

∮ ∮ln |1+zw| dz

z

dw

w

= f (H, V ) :=

{|z| = eH|w| = eV .

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(ii). Taking Legendre transformσ(s, t) = max

H,V(Hs + Vt − f (H, V ))

then ∑D

1 =∑D

∏D

w (e) = exp(NM σ(s, t) · (1 +O(1)))

where∆ahDN

= s,∆bhDM

= t, N,M →∞, NM

fixed.

(iii). For the domain[Diagram]

∆ah = sN, ∆bh = tM .

Theorem. (Cohn, Kenyon, & Propp, 2000). As a result,∑D

1 = exp(NM σ(s, t) · (1 +O(1)))

with those boundary conditions of height function hD.

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(iv). For the domain[Diagram] Ni ×Mj

ZDε =∑

values of

height functionson

boundariesbetween rectangles

Z

Mj

Ni(h bound)

=∑

{∆xh, ∆yh}ij

exp( ∑

Mj

Ni

NiMj σ(∆xhMj

,∆yhNi

))

= exp(

ε−2∫D

σ (∂xh0, ∂yh0) dx dy (1 +O (1)))

where h0 :=minimizer forS [h] =

∫D

σ (∂xh0, ∂yh0) dx dy .

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Theorem. (Cohn, Kenyon, & Propp, 2000). In addition,

limε→0

ε2 lnZDε =∫D

σ(−→∇h0

)dx dy

such that• h0 is a minimizer• 0 < ∂xh, ∂yh < 1• h0

∣∣∂D = b, the boundary condition appearing in the limit ε→0.

[Diagram]

for height functionh = ε−1h0 + φ = ε−1 (h0 + εφ)

where h0 := the limit shape, and φ := fluctuations.

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2.10 Physics way of describing fluctuations

S[h0 + εφ] = S[h0] +ε2

2

∫∫D

aij(x)∂iφ ∂jφ d2x

aij(x) = ∂i∂j φ(s, t)s = ∂1h0t = ∂2h0Partition function is given by

Z = exp(ε−2S(h0))∫

exp(12

∫∫D

aij(x)∂iφ ∂jφ d2x)Dφ

where D := scalar field with Riemannian metric induced by h0.Correlation functions are given by

⟨φ(x)φ(y)⟩ = G(x, y)where G :=Green’s function for ∆=∂i(aij∂j).

Conjecture. G := same as from asymptotics of Kasteleyn operators.

Remark. The conjecture := theorem always in certain cases (R., et. al.).

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2.11 Conclusion - limit topology phenomena, ongoing

1. How to make such pictures of (i.e. simulate) random configuration:(i). Monte Carlo for exp

(∝10002

)(ii). Sampling around most probable region(iii). Markov Chain Monte Carlo method.

2. How to describe limit topologies and fluctuations analytically:(i). Kasteleyn matrix solution and correlation function(ii). Variational principle: Minimizing large deviation functionals(iii). Boundary conditions.

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Thank You!

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