Conformal Loop Ensembles and the Gaussian Free Fieldmath.mit.edu/~sswatson/pdfs/CLEtalk.pdf ·...

107
Conformal Loop Ensembles and the Gaussian Free Field Sam Watson PuMaGraSS 14 October, 2011

Transcript of Conformal Loop Ensembles and the Gaussian Free Fieldmath.mit.edu/~sswatson/pdfs/CLEtalk.pdf ·...

Conformal Loop Ensembles andthe Gaussian Free Field

Sam WatsonPuMaGraSS

14 October, 2011

SLE

Introduction

What is the Schramm-Loewner evolution?

SLE(κ) is a non-self-crossing random fractal curve in a planardomain, from one boundary point to another (chordal) orfrom a boundary point to an interior point (radial)

Sample SLE traces from 0 to ∞ in the upper half-plane:

0∑= 8/3

0∑= 4

0∑= 6

SLE

Introduction

What is the Schramm-Loewner evolution?

SLE(κ) is a non-self-crossing random fractal curve in a planardomain, from one boundary point to another (chordal) orfrom a boundary point to an interior point (radial)

Sample SLE traces from 0 to ∞ in the upper half-plane:

0∑= 8/3

0∑= 4

0∑= 6

SLE

Introduction

A scaling limit of discrete paths

Brownian motion arises as a natural limit of random walks

Simple Random Walk Brownian Motion

Analogously, SLE arises as a natural limit for a variety ofdiscrete non-self-intersecting paths

Self-avoiding Walk SLE

SLE

Introduction

A scaling limit of discrete paths

Brownian motion arises as a natural limit of random walks

Simple Random Walk Brownian Motion

Analogously, SLE arises as a natural limit for a variety ofdiscrete non-self-intersecting paths

Self-avoiding Walk SLE

SLE

Introduction

A scaling limit of discrete paths

SLE arises in many two-dimensional probabilistic models

Loop-erased random walk, κ = 2 [Lawler, Schramm, Werner]Boundary of Brownian motion, κ = 8/3 [Lawler, Schramm,Werner]Ising model interface , κ = 3 [Smirnov, Chelkak and Hongler,Kytola]Level lines of DGFF, κ = 4, [Schramm and Sheffield]Percolation interface, κ = 6, [Smirnov and Camia, Newman]Uniform spanning tree, κ = 8, [Lawler, Schramm, Werner]

We will discuss percolation and the DGFF

SLE

Introduction

A scaling limit of discrete paths

SLE arises in many two-dimensional probabilistic models

Loop-erased random walk, κ = 2 [Lawler, Schramm, Werner]

Boundary of Brownian motion, κ = 8/3 [Lawler, Schramm,Werner]Ising model interface , κ = 3 [Smirnov, Chelkak and Hongler,Kytola]Level lines of DGFF, κ = 4, [Schramm and Sheffield]Percolation interface, κ = 6, [Smirnov and Camia, Newman]Uniform spanning tree, κ = 8, [Lawler, Schramm, Werner]

We will discuss percolation and the DGFF

SLE

Introduction

A scaling limit of discrete paths

SLE arises in many two-dimensional probabilistic models

Loop-erased random walk, κ = 2 [Lawler, Schramm, Werner]Boundary of Brownian motion, κ = 8/3 [Lawler, Schramm,Werner]

Ising model interface , κ = 3 [Smirnov, Chelkak and Hongler,Kytola]Level lines of DGFF, κ = 4, [Schramm and Sheffield]Percolation interface, κ = 6, [Smirnov and Camia, Newman]Uniform spanning tree, κ = 8, [Lawler, Schramm, Werner]

We will discuss percolation and the DGFF

SLE

Introduction

A scaling limit of discrete paths

SLE arises in many two-dimensional probabilistic models

Loop-erased random walk, κ = 2 [Lawler, Schramm, Werner]Boundary of Brownian motion, κ = 8/3 [Lawler, Schramm,Werner]Ising model interface , κ = 3 [Smirnov, Chelkak and Hongler,Kytola]

Level lines of DGFF, κ = 4, [Schramm and Sheffield]Percolation interface, κ = 6, [Smirnov and Camia, Newman]Uniform spanning tree, κ = 8, [Lawler, Schramm, Werner]

We will discuss percolation and the DGFF

SLE

Introduction

A scaling limit of discrete paths

SLE arises in many two-dimensional probabilistic models

Loop-erased random walk, κ = 2 [Lawler, Schramm, Werner]Boundary of Brownian motion, κ = 8/3 [Lawler, Schramm,Werner]Ising model interface , κ = 3 [Smirnov, Chelkak and Hongler,Kytola]Level lines of DGFF, κ = 4, [Schramm and Sheffield]

Percolation interface, κ = 6, [Smirnov and Camia, Newman]Uniform spanning tree, κ = 8, [Lawler, Schramm, Werner]

We will discuss percolation and the DGFF

SLE

Introduction

A scaling limit of discrete paths

SLE arises in many two-dimensional probabilistic models

Loop-erased random walk, κ = 2 [Lawler, Schramm, Werner]Boundary of Brownian motion, κ = 8/3 [Lawler, Schramm,Werner]Ising model interface , κ = 3 [Smirnov, Chelkak and Hongler,Kytola]Level lines of DGFF, κ = 4, [Schramm and Sheffield]Percolation interface, κ = 6, [Smirnov and Camia, Newman]

Uniform spanning tree, κ = 8, [Lawler, Schramm, Werner]

We will discuss percolation and the DGFF

SLE

Introduction

A scaling limit of discrete paths

SLE arises in many two-dimensional probabilistic models

Loop-erased random walk, κ = 2 [Lawler, Schramm, Werner]Boundary of Brownian motion, κ = 8/3 [Lawler, Schramm,Werner]Ising model interface , κ = 3 [Smirnov, Chelkak and Hongler,Kytola]Level lines of DGFF, κ = 4, [Schramm and Sheffield]Percolation interface, κ = 6, [Smirnov and Camia, Newman]Uniform spanning tree, κ = 8, [Lawler, Schramm, Werner]

We will discuss percolation and the DGFF

SLE

Introduction

A scaling limit of discrete paths

SLE arises in many two-dimensional probabilistic models

Loop-erased random walk, κ = 2 [Lawler, Schramm, Werner]Boundary of Brownian motion, κ = 8/3 [Lawler, Schramm,Werner]Ising model interface , κ = 3 [Smirnov, Chelkak and Hongler,Kytola]Level lines of DGFF, κ = 4, [Schramm and Sheffield]Percolation interface, κ = 6, [Smirnov and Camia, Newman]Uniform spanning tree, κ = 8, [Lawler, Schramm, Werner]

We will discuss percolation and the DGFF

SLE

Discrete models

Percolation

SLE

Discrete models

Percolation

SLE

Discrete models

Percolation

SLE

Discrete models

Percolation

SLE

Discrete models

Percolation

SLE

Discrete models

Percolation

Theorem (Smirnov, 2001)

As the mesh size goes to 0, the law of the percolation interfacepath converges to SLE(6)

We’re thinking of the sample paths of the interfaces andSLE(6) as elements of the metric space of equivalence classesof curves from a to b modulo reparametrization, with metric

d([γ], [γ ′]) = infϕ

supu∈[0,1]

∣∣γ(u)− γ′(ϕ(u))

∣∣

Hence the laws of the interfaces and SLE(6) are measures onthis metric space with its Borel σ -algebra

SLE

Discrete models

Percolation

Theorem (Smirnov, 2001)

As the mesh size goes to 0, the law of the percolation interfacepath converges to SLE(6)

We’re thinking of the sample paths of the interfaces andSLE(6) as elements of the metric space of equivalence classesof curves from a to b modulo reparametrization, with metric

d([γ], [γ ′]) = infϕ

supu∈[0,1]

∣∣γ(u)− γ′(ϕ(u))

∣∣Hence the laws of the interfaces and SLE(6) are measures onthis metric space with its Borel σ -algebra

SLE

Discrete models

The Discrete Gaussian Free Field

The discrete Gaussian free field h is a random real-valuedfunction on a finite vertex set D⊂ Z2.

h is required to agree with some specified functionψ : ∂D→ R, and its density is given by

Z−1 exp

(−1

2 ∑x∼y

(h(x)−h(y))2

),

where the sum is over all pairs of neighbors in D.So the DGFF is a multivariate Gaussian in R|D|−|∂D|.

SLE

Discrete models

The Discrete Gaussian Free Field

The discrete Gaussian free field h is a random real-valuedfunction on a finite vertex set D⊂ Z2.h is required to agree with some specified functionψ : ∂D→ R, and its density is given by

Z−1 exp

(−1

2 ∑x∼y

(h(x)−h(y))2

),

where the sum is over all pairs of neighbors in D.

So the DGFF is a multivariate Gaussian in R|D|−|∂D|.

SLE

Discrete models

The Discrete Gaussian Free Field

The discrete Gaussian free field h is a random real-valuedfunction on a finite vertex set D⊂ Z2.h is required to agree with some specified functionψ : ∂D→ R, and its density is given by

Z−1 exp

(−1

2 ∑x∼y

(h(x)−h(y))2

),

where the sum is over all pairs of neighbors in D.So the DGFF is a multivariate Gaussian in R|D|−|∂D|.

SLE

Discrete models

The Discrete Gaussian Free Field

If we use appropriate boundary conditions (±√

π/8), thenthere is a unique curve connecting a to b on which h is 0

SLE

Discrete models

The Discrete Gaussian Free Field

Theorem (Schramm & Sheffield, 2008)

As the mesh size goes to 0, the zero contour line of the discreteGaussian free field converges in law to SLE(4)

SLE

Definition and Properties of SLE

Definition of (radial) SLE

Let Bt be a standard 1D Brownian motion, and letWt = exp(i

√κBt) wander on the unit circle

For each point z, define (gt(z))t∈[0,∞) by the ODE g0(z) = z and

∂tgt(z) =−gt(z)gt(z)+Wt

gt(z)−Wt

So z is subjected to a force with the following vector field:

Wt

SLE

Definition and Properties of SLE

Definition of (radial) SLE

Let Bt be a standard 1D Brownian motion, and letWt = exp(i

√κBt) wander on the unit circle

For each point z, define (gt(z))t∈[0,∞) by the ODE g0(z) = z and

∂tgt(z) =−gt(z)gt(z)+Wt

gt(z)−Wt

So z is subjected to a force with the following vector field:

Wt

SLE

Definition and Properties of SLE

Definition of (radial) SLE

Let Bt be a standard 1D Brownian motion, and letWt = exp(i

√κBt) wander on the unit circle

For each point z, define (gt(z))t∈[0,∞) by the ODE g0(z) = z and

∂tgt(z) =−gt(z)gt(z)+Wt

gt(z)−Wt

So z is subjected to a force with the following vector field:

Wt

SLE

Definition and Properties of SLE

Definition of (radial) SLE

Since Wt is an attractor in the radial direction, some values ofz eventually get eaten

In other words, the ODE reaches a singularity at a randomtime T(z) and is killed

We define the hull Kt = {z : T(z)≤ t} defined to be the set ofpoints eaten by time t

SLE

Definition and Properties of SLE

Definition of (radial) SLE

Since Wt is an attractor in the radial direction, some values ofz eventually get eaten

In other words, the ODE reaches a singularity at a randomtime T(z) and is killed

We define the hull Kt = {z : T(z)≤ t} defined to be the set ofpoints eaten by time t

SLE

Definition and Properties of SLE

Definition of (radial) SLE

Since Wt is an attractor in the radial direction, some values ofz eventually get eaten

In other words, the ODE reaches a singularity at a randomtime T(z) and is killed

We define the hull Kt = {z : T(z)≤ t} defined to be the set ofpoints eaten by time t

SLE

Definition and Properties of SLE

Definition of (radial) SLE

Since Wt is an attractor in the radial direction, some values ofz eventually get eaten

In other words, the ODE reaches a singularity at a randomtime T(z) and is killed

We define the hull Kt = {z : T(z)≤ t} defined to be the set ofpoints eaten by time t

Theorem (Rohde & Schramm, 2001)

Kt is almost surely generated by a continuouscurve γ(t), meaning that Kt = the set of pointscut off from 0 by γ[0, t].

SLE

Definition and Properties of SLE

Definition of (radial) SLE

Since Wt is an attractor in the radial direction, some values ofz eventually get eaten

In other words, the ODE reaches a singularity at a randomtime T(z) and is killed

We define the hull Kt = {z : T(z)≤ t} defined to be the set ofpoints eaten by time t

Theorem (Rohde & Schramm, 2001)

Kt is almost surely generated by a continuouscurve γ(t), meaning that Kt = the set of pointscut off from 0 by γ[0, t].

Kt

γt

SLE

Definition and Properties of SLE

Properties of SLE

SLE(κ) is almost surely simple when 0≤ κ ≤ 4, neither simplenor space-filling when 4 < κ < 8, and space-filling when κ ≥ 8

SLE(κ) is conformally invariant: if γ is an SLE(κ) in from ato b in D and ϕ : D→ D′ is a conformal map, then ϕ ◦ γ is anSLE(κ) from ϕ(a) to ϕ(b) in D′

ϕ

γ ϕ(γ)a

b

ϕ(a)

ϕ(b)

The Hausdorff dimension of SLE(κ) interpolates linearlybetween 1 and 2: dimH(SLE(κ)) = 1+ κ

8 (Beffara, 2007)

SLE

Definition and Properties of SLE

Properties of SLE

SLE(κ) is almost surely simple when 0≤ κ ≤ 4, neither simplenor space-filling when 4 < κ < 8, and space-filling when κ ≥ 8

SLE(κ) is conformally invariant: if γ is an SLE(κ) in from ato b in D and ϕ : D→ D′ is a conformal map, then ϕ ◦ γ is anSLE(κ) from ϕ(a) to ϕ(b) in D′

ϕ

γ ϕ(γ)a

b

ϕ(a)

ϕ(b)

The Hausdorff dimension of SLE(κ) interpolates linearlybetween 1 and 2: dimH(SLE(κ)) = 1+ κ

8 (Beffara, 2007)

SLE

Definition and Properties of SLE

Properties of SLE

SLE(κ) is almost surely simple when 0≤ κ ≤ 4, neither simplenor space-filling when 4 < κ < 8, and space-filling when κ ≥ 8

SLE(κ) is conformally invariant: if γ is an SLE(κ) in from ato b in D and ϕ : D→ D′ is a conformal map, then ϕ ◦ γ is anSLE(κ) from ϕ(a) to ϕ(b) in D′

ϕ

γ ϕ(γ)a

b

ϕ(a)

ϕ(b)

The Hausdorff dimension of SLE(κ) interpolates linearlybetween 1 and 2: dimH(SLE(κ)) = 1+ κ

8 (Beffara, 2007)

SLE

Computing Percolation Critical Exponents

What is SLE good for?

Consider percolation on the honeycomb lattice, with unitmesh size

What is the probability of a path from the origin to radius n?

Physicists (Den Nijs, B. Nienhuis, etc.) conjectured a powerlaw in the late 1970s and early 1980s

Theorem (Lawler, Schramm, Werner (2001))

P(0↔ ∂Λ(n)) = n−5/48+o(1)

Proof method uses convergence of interface to to SLE(6), andgeneralizes to other “critical arm exponents”

SLE

Computing Percolation Critical Exponents

What is SLE good for?

Consider percolation on the honeycomb lattice, with unitmesh size

What is the probability of a path from the origin to radius n?

Physicists (Den Nijs, B. Nienhuis, etc.) conjectured a powerlaw in the late 1970s and early 1980s

Theorem (Lawler, Schramm, Werner (2001))

P(0↔ ∂Λ(n)) = n−5/48+o(1)

Proof method uses convergence of interface to to SLE(6), andgeneralizes to other “critical arm exponents”

SLE

Computing Percolation Critical Exponents

What is SLE good for?

Consider percolation on the honeycomb lattice, with unitmesh size

What is the probability of a path from the origin to radius n?

Physicists (Den Nijs, B. Nienhuis, etc.) conjectured a powerlaw in the late 1970s and early 1980s

Theorem (Lawler, Schramm, Werner (2001))

P(0↔ ∂Λ(n)) = n−5/48+o(1)

Proof method uses convergence of interface to to SLE(6), andgeneralizes to other “critical arm exponents”

SLE

Computing Percolation Critical Exponents

What is SLE good for?

Consider percolation on the honeycomb lattice, with unitmesh size

What is the probability of a path from the origin to radius n?

Physicists (Den Nijs, B. Nienhuis, etc.) conjectured a powerlaw in the late 1970s and early 1980s

Theorem (Lawler, Schramm, Werner (2001))

P(0↔ ∂Λ(n)) = n−5/48+o(1)

Proof method uses convergence of interface to to SLE(6), andgeneralizes to other “critical arm exponents”

SLE

Computing Percolation Critical Exponents

What is SLE good for?

Consider percolation on the honeycomb lattice, with unitmesh size

What is the probability of a path from the origin to radius n?

Physicists (Den Nijs, B. Nienhuis, etc.) conjectured a powerlaw in the late 1970s and early 1980s

Theorem (Lawler, Schramm, Werner (2001))

P(0↔ ∂Λ(n)) = n−5/48+o(1)

Proof method uses convergence of interface to to SLE(6), andgeneralizes to other “critical arm exponents”

CLE

Introduction

What is CLE?

For both percolation and GFF, we had to work to get a singlepath out of the model

More natural questions involve the full model:

Does the collection of all percolation cluster interfaces have alimiting law?Does the zero-set for a zero-boundary-conditions DGFF have alimit?

What about all of the contours?

Zero-height contours of the DGFF on a 300 ⇥ 300 square in Z2.

Jason Miller and Scott She�eld (MSR and MIT) The Gaussian Free Field and CLE(4) February 18, 2011 12 / 34

CLE

Introduction

What is CLE?

For both percolation and GFF, we had to work to get a singlepath out of the model

More natural questions involve the full model:

Does the collection of all percolation cluster interfaces have alimiting law?Does the zero-set for a zero-boundary-conditions DGFF have alimit? What about all of the contours?

Zero-height contours of the DGFF on a 300 ⇥ 300 square in Z2.

Jason Miller and Scott She�eld (MSR and MIT) The Gaussian Free Field and CLE(4) February 18, 2011 12 / 34

CLE

Introduction

What is CLE?

For both percolation and GFF, we had to work to get a singlepath out of the model

More natural questions involve the full model:

Does the collection of all percolation cluster interfaces have alimiting law?

Does the zero-set for a zero-boundary-conditions DGFF have alimit? What about all of the contours?

Zero-height contours of the DGFF on a 300 ⇥ 300 square in Z2.

Jason Miller and Scott She�eld (MSR and MIT) The Gaussian Free Field and CLE(4) February 18, 2011 12 / 34

CLE

Introduction

What is CLE?

For both percolation and GFF, we had to work to get a singlepath out of the model

More natural questions involve the full model:

Does the collection of all percolation cluster interfaces have alimiting law?Does the zero-set for a zero-boundary-conditions DGFF have alimit?

What about all of the contours?

Zero-height contours of the DGFF on a 300 ⇥ 300 square in Z2.

Jason Miller and Scott She�eld (MSR and MIT) The Gaussian Free Field and CLE(4) February 18, 2011 12 / 34

CLE

Introduction

What is CLE?

For both percolation and GFF, we had to work to get a singlepath out of the model

More natural questions involve the full model:

Does the collection of all percolation cluster interfaces have alimiting law?Does the zero-set for a zero-boundary-conditions DGFF have alimit? What about all of the contours?

Zero-height contours of the DGFF on a 300 ⇥ 300 square in Z2.

Jason Miller and Scott She�eld (MSR and MIT) The Gaussian Free Field and CLE(4) February 18, 2011 12 / 34

CLE

Introduction

What is CLE?

CLE(κ) is a random countably infinite collection of loops,each of which “looks like” SLE(κ) locally

No loop of a CLE crosses itself or another loop

Each point in D is almost surely surrounded by infinitely manyloops

For each z ∈ D, let Lk(z) be the kthlargest loop around z.

The sequence Tk(z) =− logCR(Lk(z),z)+ logCR(Lk−1(z),z) isiid with well-understood distribution

CLE

Introduction

What is CLE?

CLE(κ) is a random countably infinite collection of loops,each of which “looks like” SLE(κ) locally

No loop of a CLE crosses itself or another loop

Each point in D is almost surely surrounded by infinitely manyloops

For each z ∈ D, let Lk(z) be the kthlargest loop around z.

The sequence Tk(z) =− logCR(Lk(z),z)+ logCR(Lk−1(z),z) isiid with well-understood distribution

CLE

Introduction

What is CLE?

CLE(κ) is a random countably infinite collection of loops,each of which “looks like” SLE(κ) locally

No loop of a CLE crosses itself or another loop

Each point in D is almost surely surrounded by infinitely manyloops

For each z ∈ D, let Lk(z) be the kthlargest loop around z.

The sequence Tk(z) =− logCR(Lk(z),z)+ logCR(Lk−1(z),z) isiid with well-understood distribution

CLE

Introduction

What is CLE?

CLE(κ) is a random countably infinite collection of loops,each of which “looks like” SLE(κ) locally

No loop of a CLE crosses itself or another loop

Each point in D is almost surely surrounded by infinitely manyloops

For each z ∈ D, let Lk(z) be the kthlargest loop around z.

The sequence Tk(z) =− logCR(Lk(z),z)+ logCR(Lk−1(z),z) isiid with well-understood distribution

∂D

L1(z)

L2(z)

CLE

Introduction

What is CLE?

CLE(κ) is a random countably infinite collection of loops,each of which “looks like” SLE(κ) locally

No loop of a CLE crosses itself or another loop

Each point in D is almost surely surrounded by infinitely manyloops

For each z ∈ D, let Lk(z) be the kthlargest loop around z.

The sequence Tk(z) =− logCR(Lk(z),z)+ logCR(Lk−1(z),z) isiid with well-understood distribution

∂D

L1(z)

L2(z)

CLE

Introduction

What is CLE?

CLE(κ) is a random countably infinite collection of loops,each of which “looks like” SLE(κ) locally

No loop of a CLE crosses itself or another loop

Each point in D is almost surely surrounded by infinitely manyloops

For each z ∈ D, let Lk(z) be the kthlargest loop around z.

The sequence Tk(z) =− logCR(Lk(z),z)+ logCR(Lk−1(z),z) isiid with well-understood distribution

∂D

L1(z)

L2(z)

x

density

CLE

Introduction

Exceptional Loop Counts

The conformal radius informationallows us to study the set Φ(θ) ofpoints z for which the numberN(z,ε) of loops surrounding B(z,ε)grows like θ log(1/ε).

∑Nk=1 Tk concentrates around

NET1, so setting NET1 = log(1/ε)tells us that typical behavior isN(z,ε) = 1

ET1log(1/ε)

We can compute the (almost sure)Hausdorff dimension of Φ(θ)

The y-intercept is the dimension ofthe CLE gasket, the set of pointsnot surrounded by a loop

CLE

Introduction

Exceptional Loop Counts

The conformal radius informationallows us to study the set Φ(θ) ofpoints z for which the numberN(z,ε) of loops surrounding B(z,ε)grows like θ log(1/ε).

∑Nk=1 Tk concentrates around

NET1, so setting NET1 = log(1/ε)tells us that typical behavior isN(z,ε) = 1

ET1log(1/ε)

We can compute the (almost sure)Hausdorff dimension of Φ(θ)

The y-intercept is the dimension ofthe CLE gasket, the set of pointsnot surrounded by a loop

CLE

Introduction

Exceptional Loop Counts

The conformal radius informationallows us to study the set Φ(θ) ofpoints z for which the numberN(z,ε) of loops surrounding B(z,ε)grows like θ log(1/ε).

∑Nk=1 Tk concentrates around

NET1, so setting NET1 = log(1/ε)tells us that typical behavior isN(z,ε) = 1

ET1log(1/ε)

We can compute the (almost sure)Hausdorff dimension of Φ(θ)

The y-intercept is the dimension ofthe CLE gasket, the set of pointsnot surrounded by a loop

θθ0 ≈ 0.7107

1ET1

dimH(points with θ log(1/ε) loops)

2

CLE

Introduction

Exceptional Loop Counts

The conformal radius informationallows us to study the set Φ(θ) ofpoints z for which the numberN(z,ε) of loops surrounding B(z,ε)grows like θ log(1/ε).

∑Nk=1 Tk concentrates around

NET1, so setting NET1 = log(1/ε)tells us that typical behavior isN(z,ε) = 1

ET1log(1/ε)

We can compute the (almost sure)Hausdorff dimension of Φ(θ)

The y-intercept is the dimension ofthe CLE gasket, the set of pointsnot surrounded by a loop

θθ0 ≈ 0.7107

1ET1

dimH(points with θ log(1/ε) loops)

2

CLE

Introduction

Exceptional Loop Counts

Idea for computing dimH(Φ(θ)): arrange disjoint annuli ofradii r1� r2 in the domain

Watch the sequence of loops surrounding the annulus centersuntil a loop hits the inner boundary

Keep only the annuli with loop count approximatelyθ log(r2/r1)

Repeat inside the inner boundary

CLE

Introduction

Exceptional Loop Counts

Idea for computing dimH(Φ(θ)): arrange disjoint annuli ofradii r1� r2 in the domain

Watch the sequence of loops surrounding the annulus centersuntil a loop hits the inner boundary

Keep only the annuli with loop count approximatelyθ log(r2/r1)

Repeat inside the inner boundary

CLE

Introduction

Exceptional Loop Counts

Idea for computing dimH(Φ(θ)): arrange disjoint annuli ofradii r1� r2 in the domain

Watch the sequence of loops surrounding the annulus centersuntil a loop hits the inner boundary

Keep only the annuli with loop count approximatelyθ log(r2/r1)

Repeat inside the inner boundary

CLE

Introduction

Exceptional Loop Counts

Idea for computing dimH(Φ(θ)): arrange disjoint annuli ofradii r1� r2 in the domain

Watch the sequence of loops surrounding the annulus centersuntil a loop hits the inner boundary

Keep only the annuli with loop count approximatelyθ log(r2/r1)

Repeat inside the inner boundary

CLE

Introduction

Exceptional Loop Counts

The probability of having θ log(r2/r1) loops in a given annuluscan be computed using large deviations

Theorem (Cramer)

Suppose (Xi)i∈I are iid and let Λ∗ be the Legendre transform of thelogarithmic moment generating function of the distribution of X1.Then for a < EX1,

P

(1n

n

∑i=1

Xi ≤ a

)= e−nΛ∗(a)+o(1)

Having at least θ log(r2/r1) loops is the same as

∑θ log(r2/r1)k=1 Tk ≤ log(r2/r1)

Substituting into Cramer’s theorem tells us that theprobability of this is (r2/r1)

−θΛ∗(1/θ)+o(1). SodimH(Φ(θ)) = 2−θΛ∗(1/θ)

CLE

Introduction

Exceptional Loop Counts

The probability of having θ log(r2/r1) loops in a given annuluscan be computed using large deviations

Theorem (Cramer)

Suppose (Xi)i∈I are iid and let Λ∗ be the Legendre transform of thelogarithmic moment generating function of the distribution of X1.Then for a < EX1,

P

(1n

n

∑i=1

Xi ≤ a

)= e−nΛ∗(a)+o(1)

Having at least θ log(r2/r1) loops is the same as

∑θ log(r2/r1)k=1 Tk ≤ log(r2/r1)

Substituting into Cramer’s theorem tells us that theprobability of this is (r2/r1)

−θΛ∗(1/θ)+o(1). SodimH(Φ(θ)) = 2−θΛ∗(1/θ)

CLE

Introduction

Exceptional Loop Counts

The probability of having θ log(r2/r1) loops in a given annuluscan be computed using large deviations

Theorem (Cramer)

Suppose (Xi)i∈I are iid and let Λ∗ be the Legendre transform of thelogarithmic moment generating function of the distribution of X1.Then for a < EX1,

P

(1n

n

∑i=1

Xi ≤ a

)= e−nΛ∗(a)+o(1)

Having at least θ log(r2/r1) loops is the same as

∑θ log(r2/r1)k=1 Tk ≤ log(r2/r1)

Substituting into Cramer’s theorem tells us that theprobability of this is (r2/r1)

−θΛ∗(1/θ)+o(1). SodimH(Φ(θ)) = 2−θΛ∗(1/θ)

CLE

Introduction

Exceptional Loop Counts

The probability of having θ log(r2/r1) loops in a given annuluscan be computed using large deviations

Theorem (Cramer)

Suppose (Xi)i∈I are iid and let Λ∗ be the Legendre transform of thelogarithmic moment generating function of the distribution of X1.Then for a < EX1,

P

(1n

n

∑i=1

Xi ≤ a

)= e−nΛ∗(a)+o(1)

Having at least θ log(r2/r1) loops is the same as

∑θ log(r2/r1)k=1 Tk ≤ log(r2/r1)

Substituting into Cramer’s theorem tells us that theprobability of this is (r2/r1)

−θΛ∗(1/θ)+o(1). SodimH(Φ(θ)) = 2−θΛ∗(1/θ)

CLE

Introduction

What is CLE good for?

Can CLE be used to enhance understanding or docomputations?

It is a classical result that for critical percolation, infinitelymany clusters surround the origin

The convergence to CLE gives us some sense of the rate ofgrowth (i.e., logarithmic), via the conformal radius distribution

We can also use CLE to study the continuum Gaussian freefield

, , . . .→ ?

CLE

Introduction

What is CLE good for?

Can CLE be used to enhance understanding or docomputations?

It is a classical result that for critical percolation, infinitelymany clusters surround the origin

The convergence to CLE gives us some sense of the rate ofgrowth (i.e., logarithmic), via the conformal radius distribution

We can also use CLE to study the continuum Gaussian freefield

, , . . .→ ?

CLE

Introduction

What is CLE good for?

Can CLE be used to enhance understanding or docomputations?

It is a classical result that for critical percolation, infinitelymany clusters surround the origin

The convergence to CLE gives us some sense of the rate ofgrowth (i.e., logarithmic), via the conformal radius distribution

We can also use CLE to study the continuum Gaussian freefield

, , . . .→ ?

CLE

Introduction

What is CLE good for?

Can CLE be used to enhance understanding or docomputations?

It is a classical result that for critical percolation, infinitelymany clusters surround the origin

The convergence to CLE gives us some sense of the rate ofgrowth (i.e., logarithmic), via the conformal radius distribution

We can also use CLE to study the continuum Gaussian freefield

, , . . .→ ?

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

The discrete Gaussian free field on {0,ε,2ε,3ε, . . .} (suitablyrescaled) converges to one-dimensional Brownian motion asε → 0

The DGFF on a two-dimensional lattice should converge to ananalogue of Brownian motion with two time dimensions

It turns out that in 2D, it is not possible to define such a limitpointwise

Nevertheless, we can define a limit in the distributional sense

The DGFF h : D→ R can be written as a finite sum ∑i αifi,where αi are iid standard normals and {fi} form anorthonormal basis of the Hilbert space of real-valued functionson D with Dirichlet inner product

(f ,g) = ∑x∼y

(f (x)− f (y))(g(x)−g(y)).

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

The discrete Gaussian free field on {0,ε,2ε,3ε, . . .} (suitablyrescaled) converges to one-dimensional Brownian motion asε → 0The DGFF on a two-dimensional lattice should converge to ananalogue of Brownian motion with two time dimensions

It turns out that in 2D, it is not possible to define such a limitpointwise

Nevertheless, we can define a limit in the distributional sense

The DGFF h : D→ R can be written as a finite sum ∑i αifi,where αi are iid standard normals and {fi} form anorthonormal basis of the Hilbert space of real-valued functionson D with Dirichlet inner product

(f ,g) = ∑x∼y

(f (x)− f (y))(g(x)−g(y)).

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

The discrete Gaussian free field on {0,ε,2ε,3ε, . . .} (suitablyrescaled) converges to one-dimensional Brownian motion asε → 0The DGFF on a two-dimensional lattice should converge to ananalogue of Brownian motion with two time dimensions

It turns out that in 2D, it is not possible to define such a limitpointwise

Nevertheless, we can define a limit in the distributional sense

The DGFF h : D→ R can be written as a finite sum ∑i αifi,where αi are iid standard normals and {fi} form anorthonormal basis of the Hilbert space of real-valued functionson D with Dirichlet inner product

(f ,g) = ∑x∼y

(f (x)− f (y))(g(x)−g(y)).

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

The discrete Gaussian free field on {0,ε,2ε,3ε, . . .} (suitablyrescaled) converges to one-dimensional Brownian motion asε → 0The DGFF on a two-dimensional lattice should converge to ananalogue of Brownian motion with two time dimensions

It turns out that in 2D, it is not possible to define such a limitpointwise

Nevertheless, we can define a limit in the distributional sense

The DGFF h : D→ R can be written as a finite sum ∑i αifi,where αi are iid standard normals and {fi} form anorthonormal basis of the Hilbert space of real-valued functionson D with Dirichlet inner product

(f ,g) = ∑x∼y

(f (x)− f (y))(g(x)−g(y)).

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

The discrete Gaussian free field on {0,ε,2ε,3ε, . . .} (suitablyrescaled) converges to one-dimensional Brownian motion asε → 0The DGFF on a two-dimensional lattice should converge to ananalogue of Brownian motion with two time dimensions

It turns out that in 2D, it is not possible to define such a limitpointwise

Nevertheless, we can define a limit in the distributional sense

The DGFF h : D→ R can be written as a finite sum ∑i αifi,where αi are iid standard normals and {fi} form anorthonormal basis of the Hilbert space of real-valued functionson D with Dirichlet inner product

(f ,g) = ∑x∼y

(f (x)− f (y))(g(x)−g(y)).

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

Similarly, the Gaussian free field is defined via the formal sumh = ∑i αifi, where fi is a fixed orthonormal basis for the Hilbertspace completion of the set of smooth, compactly supportedfunctions on D with inner product

(f ,g) =∫

D(∇f ·∇g) dA

h is not in the Hilbert space, because its L2 norm is infinite

But h is defined as a distribution, since (h,∑i cifi) := ∑i αici

converges almost surely

In particular, h has well-defined circle averages and diskaverages

In fact, (he−t(z))t∈[0,∞) is a Brownian motion, where hr(z) isthe average of h on a circle of radius r centered at z

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

Similarly, the Gaussian free field is defined via the formal sumh = ∑i αifi, where fi is a fixed orthonormal basis for the Hilbertspace completion of the set of smooth, compactly supportedfunctions on D with inner product

(f ,g) =∫

D(∇f ·∇g) dA

h is not in the Hilbert space, because its L2 norm is infinite

But h is defined as a distribution, since (h,∑i cifi) := ∑i αici

converges almost surely

In particular, h has well-defined circle averages and diskaverages

In fact, (he−t(z))t∈[0,∞) is a Brownian motion, where hr(z) isthe average of h on a circle of radius r centered at z

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

Similarly, the Gaussian free field is defined via the formal sumh = ∑i αifi, where fi is a fixed orthonormal basis for the Hilbertspace completion of the set of smooth, compactly supportedfunctions on D with inner product

(f ,g) =∫

D(∇f ·∇g) dA

h is not in the Hilbert space, because its L2 norm is infinite

But h is defined as a distribution, since (h,∑i cifi) := ∑i αici

converges almost surely

In particular, h has well-defined circle averages and diskaverages

In fact, (he−t(z))t∈[0,∞) is a Brownian motion, where hr(z) isthe average of h on a circle of radius r centered at z

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

Similarly, the Gaussian free field is defined via the formal sumh = ∑i αifi, where fi is a fixed orthonormal basis for the Hilbertspace completion of the set of smooth, compactly supportedfunctions on D with inner product

(f ,g) =∫

D(∇f ·∇g) dA

h is not in the Hilbert space, because its L2 norm is infinite

But h is defined as a distribution, since (h,∑i cifi) := ∑i αici

converges almost surely

In particular, h has well-defined circle averages and diskaverages

In fact, (he−t(z))t∈[0,∞) is a Brownian motion, where hr(z) isthe average of h on a circle of radius r centered at z

CLE

CLE and the Gaussian free field

The continuum Gaussian free field

Similarly, the Gaussian free field is defined via the formal sumh = ∑i αifi, where fi is a fixed orthonormal basis for the Hilbertspace completion of the set of smooth, compactly supportedfunctions on D with inner product

(f ,g) =∫

D(∇f ·∇g) dA

h is not in the Hilbert space, because its L2 norm is infinite

But h is defined as a distribution, since (h,∑i cifi) := ∑i αici

converges almost surely

In particular, h has well-defined circle averages and diskaverages

In fact, (he−t(z))t∈[0,∞) is a Brownian motion, where hr(z) isthe average of h on a circle of radius r centered at z

CLE

CLE and the Gaussian free field

Maxima of the Gaussian free field

What is the extreme behavior of the GFF?

Maxima aren’t defined as such, but we can ask: what is theHausdorff dimension of the set Θ(α) of points z for which

limε→0

avg(h,B(z,ε))log(1/ε)

= α?

Typical behavior corresponds to α = 0, since Bt/t→ 0 almostsurely

CLE

CLE and the Gaussian free field

Maxima of the Gaussian free field

What is the extreme behavior of the GFF?

Maxima aren’t defined as such, but we can ask: what is theHausdorff dimension of the set Θ(α) of points z for which

limε→0

avg(h,B(z,ε))log(1/ε)

= α?

Typical behavior corresponds to α = 0, since Bt/t→ 0 almostsurely

CLE

CLE and the Gaussian free field

Maxima of the Gaussian free field

What is the extreme behavior of the GFF?

Maxima aren’t defined as such, but we can ask: what is theHausdorff dimension of the set Θ(α) of points z for which

limε→0

avg(h,B(z,ε))log(1/ε)

= α?

Typical behavior corresponds to α = 0, since Bt/t→ 0 almostsurely

CLE

CLE and the Gaussian free field

Maxima of the Gaussian free field

Theorem (Hu, Miller, Peres, 2009)

The Hausdorff dimension of Θ(α) isalmost surely 2−πα2.

Idea of proof: assemble disjointannuli and watch the circle-averageBrownian motions evolve in eachannulus

Some of these will stay pretty closethe line αt

For those that do, repeat in theinner disk

CLE

CLE and the Gaussian free field

Maxima of the Gaussian free field

Theorem (Hu, Miller, Peres, 2009)

The Hausdorff dimension of Θ(α) isalmost surely 2−πα2.

Idea of proof: assemble disjointannuli and watch the circle-averageBrownian motions evolve in eachannulus

Some of these will stay pretty closethe line αt

For those that do, repeat in theinner disk

CLE

CLE and the Gaussian free field

Maxima of the Gaussian free field

Theorem (Hu, Miller, Peres, 2009)

The Hausdorff dimension of Θ(α) isalmost surely 2−πα2.

Idea of proof: assemble disjointannuli and watch the circle-averageBrownian motions evolve in eachannulus

Some of these will stay pretty closethe line αt

For those that do, repeat in theinner disk

αt

CLE

CLE and the Gaussian free field

Maxima of the Gaussian free field

Theorem (Hu, Miller, Peres, 2009)

The Hausdorff dimension of Θ(α) isalmost surely 2−πα2.

Idea of proof: assemble disjointannuli and watch the circle-averageBrownian motions evolve in eachannulus

Some of these will stay pretty closethe line αt

For those that do, repeat in theinner disk

αt

CLE

CLE and the Gaussian free field

GFF via CLE

One may construct the GFF using CLE (Miller & Sheffield)

Sample CLE(4) and flip coins to assign to each loop ±√

π/2

The nth such iterate converges to the GFF in the space ofdistributions as n→ ∞

CLE

CLE and the Gaussian free field

GFF via CLE

One may construct the GFF using CLE (Miller & Sheffield)

Sample CLE(4) and flip coins to assign to each loop ±√

π/2

The nth such iterate converges to the GFF in the space ofdistributions as n→ ∞

CLE

CLE and the Gaussian free field

GFF via CLE

One may construct the GFF using CLE (Miller & Sheffield)

Sample CLE(4) and flip coins to assign to each loop ±√

π/2

−λ +λ

−λ

−λ

−λ −λ

−λ

The nth such iterate converges to the GFF in the space ofdistributions as n→ ∞

CLE

CLE and the Gaussian free field

GFF via CLE

One may construct the GFF using CLE (Miller & Sheffield)

Sample CLE(4) and flip coins to assign to each loop ±√

π/2

−λ +λ

−λ

−λ

−λ −λ

−λ

+2λ

+2λ0

0+2λ

0

0

0

The nth such iterate converges to the GFF in the space ofdistributions as n→ ∞

CLE

CLE and the Gaussian free field

GFF via CLE

One may construct the GFF using CLE (Miller & Sheffield)

Sample CLE(4) and flip coins to assign to each loop ±√

π/2

−λ

−λ

−λ −λ

−λ

+2λ

+2λ0

0+2λ

0

0

0

00

0

−2λ

−2λ−2λ

−2λ

The nth such iterate converges to the GFF in the space ofdistributions as n→ ∞

CLE

CLE and the Gaussian free field

GFF via CLE

One may construct the GFF using CLE (Miller & Sheffield)

Sample CLE(4) and flip coins to assign to each loop ±√

π/2

−λ

−λ

−λ −λ

−λ

+2λ

+2λ0

0+2λ

0

0

0

00

0

−2λ

−2λ−2λ

−2λ

The nth such iterate converges to the GFF in the space ofdistributions as n→ ∞

CLE

CLE and the Gaussian free field

GFF via CLE

One may construct the GFF using CLE (Miller & Sheffield)

Sample CLE(4) and flip coins to assign to each loop ±√

π/2

−λ

−λ

−λ −λ

−λ

+2λ

+2λ0

0+2λ

0

0

0

00

0

−2λ

−2λ−2λ

−2λ

The nth such iterate converges to the GFF in the space ofdistributions as n→ ∞

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

Constructing GFF using CLE paves the way for a CLE-basedtreatment of the GFF maxima

Key idea: finding the points with large ε-disk averagesamounts to finding the points with large sums ∑

Ki=1 ξi, where

ξi =±λ and K is the first index k for which Lk hits the ball ofradius ε.

There is a tradeoff: large K makes it more likely for the sumto be large, but if we ask for K to be too large then we don’tget enough points

(I.e., the sums ∑Ki=1 ξi will have larger K, but there will be

fewer of them and thus there are less likely to be some thatare large)

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

Constructing GFF using CLE paves the way for a CLE-basedtreatment of the GFF maxima

Key idea: finding the points with large ε-disk averagesamounts to finding the points with large sums ∑

Ki=1 ξi, where

ξi =±λ and K is the first index k for which Lk hits the ball ofradius ε.

There is a tradeoff: large K makes it more likely for the sumto be large, but if we ask for K to be too large then we don’tget enough points

(I.e., the sums ∑Ki=1 ξi will have larger K, but there will be

fewer of them and thus there are less likely to be some thatare large)

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

Constructing GFF using CLE paves the way for a CLE-basedtreatment of the GFF maxima

Key idea: finding the points with large ε-disk averagesamounts to finding the points with large sums ∑

Ki=1 ξi, where

ξi =±λ and K is the first index k for which Lk hits the ball ofradius ε.

There is a tradeoff: large K makes it more likely for the sumto be large, but if we ask for K to be too large then we don’tget enough points

(I.e., the sums ∑Ki=1 ξi will have larger K, but there will be

fewer of them and thus there are less likely to be some thatare large)

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

Constructing GFF using CLE paves the way for a CLE-basedtreatment of the GFF maxima

Key idea: finding the points with large ε-disk averagesamounts to finding the points with large sums ∑

Ki=1 ξi, where

ξi =±λ and K is the first index k for which Lk hits the ball ofradius ε.

There is a tradeoff: large K makes it more likely for the sumto be large, but if we ask for K to be too large then we don’tget enough points

(I.e., the sums ∑Ki=1 ξi will have larger K, but there will be

fewer of them and thus there are less likely to be some thatare large)

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

The optimisation problem is just Cramer’s theorem again

Given α,θ , we can determine the Hausdorff dimension of theset of points surrounded by θ log(1/ε) loops and having GFFdisk average growth α log(1/ε)

We get

P(N ≈ θ log(1/ε),avg≈ α log(1/ε))

≈ P(N ≈ θ log(1/ε))P

(θ log(1/ε)

∑i=1

ξi ≥ α log(1/ε)

)≈ ε

θΛ∗(1/θ)+θΛ∗ξ(α/θ)

This can easily be made precise since Cramer’s theoremguarantees that such exceptional events are highlyconcentrated at the extremes

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

The optimisation problem is just Cramer’s theorem again

Given α,θ , we can determine the Hausdorff dimension of theset of points surrounded by θ log(1/ε) loops and having GFFdisk average growth α log(1/ε)

We get

P(N ≈ θ log(1/ε),avg≈ α log(1/ε))

≈ P(N ≈ θ log(1/ε))P

(θ log(1/ε)

∑i=1

ξi ≥ α log(1/ε)

)≈ ε

θΛ∗(1/θ)+θΛ∗ξ(α/θ)

This can easily be made precise since Cramer’s theoremguarantees that such exceptional events are highlyconcentrated at the extremes

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

The optimisation problem is just Cramer’s theorem again

Given α,θ , we can determine the Hausdorff dimension of theset of points surrounded by θ log(1/ε) loops and having GFFdisk average growth α log(1/ε)

We get

P(N ≈ θ log(1/ε),avg≈ α log(1/ε))

≈ P(N ≈ θ log(1/ε))P

(θ log(1/ε)

∑i=1

ξi ≥ α log(1/ε)

)≈ ε

θΛ∗(1/θ)+θΛ∗ξ(α/θ)

This can easily be made precise since Cramer’s theoremguarantees that such exceptional events are highlyconcentrated at the extremes

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

The optimisation problem is just Cramer’s theorem again

Given α,θ , we can determine the Hausdorff dimension of theset of points surrounded by θ log(1/ε) loops and having GFFdisk average growth α log(1/ε)

We get

P(N ≈ θ log(1/ε),avg≈ α log(1/ε))

≈ P(N ≈ θ log(1/ε))P

(θ log(1/ε)

∑i=1

ξi ≥ α log(1/ε)

)≈ ε

θΛ∗(1/θ)+θΛ∗ξ(α/θ)

This can easily be made precise since Cramer’s theoremguarantees that such exceptional events are highlyconcentrated at the extremes

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

So for each value of α, there is a unique value of θ thatminimizes θΛ∗(1/θ)+θΛ∗

ξ(α/θ)

This value of θ is the “main contributor” to the set of thepoints having GFF growth rate α log(1/ε)

Recall that the supremum of the growth rates which givenonzero Hausdorff dimension was

√2/π (since the dimension

was 2−πα2.

What loop growth rate corresponds to α =√

2/π?

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

So for each value of α, there is a unique value of θ thatminimizes θΛ∗(1/θ)+θΛ∗

ξ(α/θ)

This value of θ is the “main contributor” to the set of thepoints having GFF growth rate α log(1/ε)

Recall that the supremum of the growth rates which givenonzero Hausdorff dimension was

√2/π (since the dimension

was 2−πα2.

What loop growth rate corresponds to α =√

2/π?

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

So for each value of α, there is a unique value of θ thatminimizes θΛ∗(1/θ)+θΛ∗

ξ(α/θ)

This value of θ is the “main contributor” to the set of thepoints having GFF growth rate α log(1/ε)

Recall that the supremum of the growth rates which givenonzero Hausdorff dimension was

√2/π (since the dimension

was 2−πα2.

What loop growth rate corresponds to α =√

2/π?

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

So for each value of α, there is a unique value of θ thatminimizes θΛ∗(1/θ)+θΛ∗

ξ(α/θ)

This value of θ is the “main contributor” to the set of thepoints having GFF growth rate α log(1/ε)

Recall that the supremum of the growth rates which givenonzero Hausdorff dimension was

√2/π (since the dimension

was 2−πα2.

What loop growth rate corresponds to α =√

2/π?

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

It turns out that optimum number of loops to get themaximum growth rate of

√2/π log(1/ε) is about 89.5% of

the maximum number of loops.

There is a one-to-one correspondence between these twoportions of the dimension graphs, given byθ = λα/ tanh(π2λα)

θθ0

dimH(points with θ log(1/ε) loops)

2

dimH(points with α log(1/ε) disk averages)

α

CLE

CLE and the Gaussian free field

Maxima of GFF via CLE

It turns out that optimum number of loops to get themaximum growth rate of

√2/π log(1/ε) is about 89.5% of

the maximum number of loops.There is a one-to-one correspondence between these twoportions of the dimension graphs, given byθ = λα/ tanh(π2λα)

θθ0

dimH(points with θ log(1/ε) loops)

2

dimH(points with α log(1/ε) disk averages)

α

CLE

CLE and the Gaussian free field

Thanks!