Imaginary chaos Janne Junnila (EPFL) Les Diablerets, February 10th - - PowerPoint PPT Presentation

Imaginary chaos

Janne Junnila (EPFL) Les Diablerets, February 10th 2020

joint work with Eero Saksman and Christian Webb; Juhan Aru, Guillaume Bavarez and Antoine Jego

1

Outline

• 1. Introduction
• 2. Moments
• 3. XOR-Ising model
• 4. Regularity, densities and monofractality

2

Introduction

Log-correlated Gaussian fjelds

Let us fjx a bounded simply connected domain 𝐸 ⊂ ℝ𝑒. Heuristic defjnition A Gaussian fjeld 𝑌∶ 𝐸 → ℝ is called log-correlated if

𝔽𝑌(𝑦)𝑌(𝑧) = 𝐷(𝑦, 𝑧) ≔ log 1 |𝑦 − 𝑧| + 𝑕(𝑦, 𝑧)

where 𝑕 is regular. Caveat Such fjelds cannot be defjned pointwise and must instead be understood as distributions (generalized functions). This means that for all 𝜒, 𝜔 ∈ 𝐷∞

𝑑 (ℝ𝑒) we have

𝔽𝑌(𝜒)𝑌(𝜔) = ∫ 𝜒(𝑦)𝜔(𝑧)𝐷(𝑦, 𝑧) 𝑒𝑦 𝑒𝑧.

3

Log-correlated Gaussian fjelds

Let us fjx a bounded simply connected domain 𝐸 ⊂ ℝ𝑒. Heuristic defjnition A Gaussian fjeld 𝑌∶ 𝐸 → ℝ is called log-correlated if

𝔽𝑌(𝑦)𝑌(𝑧) = 𝐷(𝑦, 𝑧) ≔ log 1 |𝑦 − 𝑧| + 𝑕(𝑦, 𝑧)

where 𝑕 is regular. Caveat Such fjelds cannot be defjned pointwise and must instead be understood as distributions (generalized functions). This means that for all 𝜒, 𝜔 ∈ 𝐷∞

𝑑 (ℝ𝑒) we have

𝔽𝑌(𝜒)𝑌(𝜔) = ∫ 𝜒(𝑦)𝜔(𝑧)𝐷(𝑦, 𝑧) 𝑒𝑦 𝑒𝑧.

3

Log-correlated Gaussian fjelds

• We will always assume at least that
• 𝑕 ∈ 𝑀1(𝐸 × 𝐸) ∩ 𝐷(𝐸 × 𝐸)
• 𝑕 is bounded from above
• These properties are enough to ensure that 𝑌 exists.

(Assuming that the kernel 𝐷 is positive defjnite.)

4

Log-correlated Gaussian fjelds

• We will always assume at least that
• 𝑕 ∈ 𝑀1(𝐸 × 𝐸) ∩ 𝐷(𝐸 × 𝐸)
• 𝑕 is bounded from above
• These properties are enough to ensure that 𝑌 exists.

(Assuming that the kernel 𝐷 is positive defjnite.)

4

Example: Tie 2D GFF

Defjnition The 0-boundary GFF 𝛥 in the domain 𝐸 is a Gaussian fjeld with the covariance

𝔽𝛥(𝑦)𝛥(𝑧) = 𝐻𝐸(𝑦, 𝑧)

where 𝐻𝐸 is the Green’s function of the Dirichlet Laplacian in 𝐸.

• universality:
• appears in the scaling limit of various height function models,

random matrices, QFT, …

• a recent characterisation: the only random fjeld with conformally

invariant law and domain Markov property (+some moment condition) [BPR19]

5

Example: Tie 2D GFF

Defjnition The 0-boundary GFF 𝛥 in the domain 𝐸 is a Gaussian fjeld with the covariance

𝔽𝛥(𝑦)𝛥(𝑧) = 𝐻𝐸(𝑦, 𝑧)

where 𝐻𝐸 is the Green’s function of the Dirichlet Laplacian in 𝐸.

• universality:
• appears in the scaling limit of various height function models,

random matrices, QFT, …

• a recent characterisation: the only random fjeld with conformally

invariant law and domain Markov property (+some moment condition) [BPR19]

5

Example: Tie GFF in the unit square

Figure 1: An approximation of the GFF in the unit square.

6

Gaussian Multiplicative Chaos

• In various applications one is interested in measures formally of the

form 𝑓𝛿𝑌(𝑦) 𝑒𝑦 where 𝛿 is a parameter.

• To rigorously defjne them one has to approximate 𝑌 with regular

fjelds 𝑌𝑜 and normalize properly when taking the limit as 𝑜 → ∞. Theorem/Defjnition ([Kah85; RV10; Sha16; Ber17]) For any given 𝛿 ∈ (0, √2𝑒) the functions 𝜈𝑜(𝑦) ≔ 𝑓𝛿𝑌𝑜(𝑦)− 𝛿2

2 𝔽𝑌𝑜(𝑦)2

converge to a random measure 𝜈. We say that 𝜈 = 𝜈𝛿 is a GMC measure associated to 𝑌.

7

Gaussian Multiplicative Chaos

• In various applications one is interested in measures formally of the

form 𝑓𝛿𝑌(𝑦) 𝑒𝑦 where 𝛿 is a parameter.

• To rigorously defjne them one has to approximate 𝑌 with regular

fjelds 𝑌𝑜 and normalize properly when taking the limit as 𝑜 → ∞. Theorem/Defjnition ([Kah85; RV10; Sha16; Ber17]) For any given 𝛿 ∈ (0, √2𝑒) the functions 𝜈𝑜(𝑦) ≔ 𝑓𝛿𝑌𝑜(𝑦)− 𝛿2

2 𝔽𝑌𝑜(𝑦)2

converge to a random measure 𝜈. We say that 𝜈 = 𝜈𝛿 is a GMC measure associated to 𝑌.

7

Gaussian Multiplicative Chaos

• In various applications one is interested in measures formally of the

form 𝑓𝛿𝑌(𝑦) 𝑒𝑦 where 𝛿 is a parameter.

• To rigorously defjne them one has to approximate 𝑌 with regular

fjelds 𝑌𝑜 and normalize properly when taking the limit as 𝑜 → ∞. Theorem/Defjnition ([Kah85; RV10; Sha16; Ber17]) For any given 𝛿 ∈ (0, √2𝑒) the functions 𝜈𝑜(𝑦) ≔ 𝑓𝛿𝑌𝑜(𝑦)− 𝛿2

2 𝔽𝑌𝑜(𝑦)2

converge to a random measure 𝜈. We say that 𝜈 = 𝜈𝛿 is a GMC measure associated to 𝑌.

7

Existence of GMC when 𝛿 ∈ (0, √𝑒)

A simple 𝑀2-computation For any 𝑔 ∈ 𝐷∞

𝑑 (𝐸) we have

𝔽|𝜈𝑜(𝑔)|2 = ∫

𝐸2 𝑔(𝑦)𝑔(𝑧)𝔽𝑓𝛿𝑌𝑜(𝑦)+𝛿𝑌𝑜(𝑧)− 𝛿2

2 𝔽𝑌𝑜(𝑦)2− 𝛿2 2 𝔽𝑌𝑜(𝑧)2 𝑒𝑦 𝑒𝑧

= ∫

𝐸2 𝑔(𝑦)𝑔(𝑧)𝑓𝛿2𝔽𝑌𝑜(𝑦)𝑌𝑜(𝑧) 𝑒𝑦 𝑒𝑧

≲ ‖𝑔‖2

∞ ∫ 𝐸2 𝑓𝛿2 log

1 |𝑦−𝑧| 𝑒𝑦 𝑒𝑧 = ∫

𝐸2

𝑒𝑦 𝑒𝑧 |𝑦 − 𝑧|𝛿2 < ∞.

• If 𝜈𝑜(𝑔) is a martingale this immediately shows convergence.
• Otherwise one can do a similar computation to show that the

sequence is Cauchy in 𝑀2(𝛻).

8

Existence of GMC when 𝛿 ∈ (0, √𝑒)

A simple 𝑀2-computation For any 𝑔 ∈ 𝐷∞

𝑑 (𝐸) we have

𝔽|𝜈𝑜(𝑔)|2 = ∫

𝐸2 𝑔(𝑦)𝑔(𝑧)𝔽𝑓𝛿𝑌𝑜(𝑦)+𝛿𝑌𝑜(𝑧)− 𝛿2

2 𝔽𝑌𝑜(𝑦)2− 𝛿2 2 𝔽𝑌𝑜(𝑧)2 𝑒𝑦 𝑒𝑧

= ∫

𝐸2 𝑔(𝑦)𝑔(𝑧)𝑓𝛿2𝔽𝑌𝑜(𝑦)𝑌𝑜(𝑧) 𝑒𝑦 𝑒𝑧

≲ ‖𝑔‖2

∞ ∫ 𝐸2 𝑓𝛿2 log

1 |𝑦−𝑧| 𝑒𝑦 𝑒𝑧 = ∫

𝐸2

𝑒𝑦 𝑒𝑧 |𝑦 − 𝑧|𝛿2 < ∞.

• If 𝜈𝑜(𝑔) is a martingale this immediately shows convergence.
• Otherwise one can do a similar computation to show that the

sequence is Cauchy in 𝑀2(𝛻).

8

Existence of GMC when 𝛿 ∈ (0, √𝑒)

A simple 𝑀2-computation For any 𝑔 ∈ 𝐷∞

𝑑 (𝐸) we have

𝔽|𝜈𝑜(𝑔)|2 = ∫

𝐸2 𝑔(𝑦)𝑔(𝑧)𝔽𝑓𝛿𝑌𝑜(𝑦)+𝛿𝑌𝑜(𝑧)− 𝛿2

2 𝔽𝑌𝑜(𝑦)2− 𝛿2 2 𝔽𝑌𝑜(𝑧)2 𝑒𝑦 𝑒𝑧

= ∫

𝐸2 𝑔(𝑦)𝑔(𝑧)𝑓𝛿2𝔽𝑌𝑜(𝑦)𝑌𝑜(𝑧) 𝑒𝑦 𝑒𝑧

≲ ‖𝑔‖2

∞ ∫ 𝐸2 𝑓𝛿2 log

1 |𝑦−𝑧| 𝑒𝑦 𝑒𝑧 = ∫

𝐸2

𝑒𝑦 𝑒𝑧 |𝑦 − 𝑧|𝛿2 < ∞.

• If 𝜈𝑜(𝑔) is a martingale this immediately shows convergence.
• Otherwise one can do a similar computation to show that the

sequence is Cauchy in 𝑀2(𝛻).

8

Complex values of 𝛿

ℜ(𝛿) ℑ(𝛿) √𝑒 −√𝑒 −√2𝑒 √2𝑒

Figure 2: The subcritical regime 𝐵 for 𝛿 in the complex plane.

• In fact, 𝛿 ↦ 𝜈𝛿(𝑔) is an analytic function on 𝐵 [JSW19].
• The circle corresponds to the 𝑀2-phase – in particular it contains

the whole subcritical part of the imaginary axis.

9

Complex values of 𝛿

ℜ(𝛿) ℑ(𝛿) √𝑒 −√𝑒 −√2𝑒 √2𝑒

Figure 2: The subcritical regime 𝐵 for 𝛿 in the complex plane.

• In fact, 𝛿 ↦ 𝜈𝛿(𝑔) is an analytic function on 𝐵 [JSW19].
• The circle corresponds to the 𝑀2-phase – in particular it contains

the whole subcritical part of the imaginary axis.

9

Complex values of 𝛿

ℜ(𝛿) ℑ(𝛿) √𝑒 −√𝑒 −√2𝑒 √2𝑒

Figure 2: The subcritical regime 𝐵 for 𝛿 in the complex plane.

• In fact, 𝛿 ↦ 𝜈𝛿(𝑔) is an analytic function on 𝐵 [JSW19].
• The circle corresponds to the 𝑀2-phase – in particular it contains

the whole subcritical part of the imaginary axis.

9

Imaginary multiplicative chaos

Theorem/Defjnition ([JSW18; LRV15]) Let 𝛾 ∈ (0, √𝑒). Then the random functions

𝜈𝑜(𝑦) ≔ 𝑓𝑗𝛾𝑌𝑜(𝑦)+ 𝛾2

2 𝔽𝑌𝑜(𝑦)2

converge in probability in 𝐼−𝑒/2−𝜁(ℝ𝑒) to a random distribution 𝜈.

• Applications: XOR-Ising model [JSW18], two-valued sets of the GFF

[SSV19] and certain random fjelds constructed using the Brownian loop soup [CGPR19].

10

Imaginary multiplicative chaos

Theorem/Defjnition ([JSW18; LRV15]) Let 𝛾 ∈ (0, √𝑒). Then the random functions

𝜈𝑜(𝑦) ≔ 𝑓𝑗𝛾𝑌𝑜(𝑦)+ 𝛾2

2 𝔽𝑌𝑜(𝑦)2

converge in probability in 𝐼−𝑒/2−𝜁(ℝ𝑒) to a random distribution 𝜈.

• Applications: XOR-Ising model [JSW18], two-valued sets of the GFF

[SSV19] and certain random fjelds constructed using the Brownian loop soup [CGPR19].

10

Moments

Moments

Theorem ([JSW18]) There exists 𝐷 > 0 such that for any 𝑔 ∈ 𝐷∞

𝑑 (𝐸) and 𝑂 ≥ 1 we have

𝔽|𝜈(𝑔)|2𝑂 ≤ 𝐷𝑂‖𝑔‖2𝑂

∞ 𝑂

𝛾2 𝑒 𝑂.

Corollary The mixed moments 𝔽𝜈(𝑔)𝑙𝜈(𝑔)

𝑚 determine the distribution of 𝜈(𝑔).

Theorem ([JSW18]) Let 𝑔 ∈ 𝐷∞

𝑑 (𝐸) be non-negative and non-zero, then there exists 𝐷 > 0

such that

𝔽|𝜈(𝑔)|2𝑂 ≥ 𝐷𝑂𝑂

𝛾2 𝑒 𝑂.

11

Moments

Theorem ([JSW18]) There exists 𝐷 > 0 such that for any 𝑔 ∈ 𝐷∞

𝑑 (𝐸) and 𝑂 ≥ 1 we have

𝔽|𝜈(𝑔)|2𝑂 ≤ 𝐷𝑂‖𝑔‖2𝑂

∞ 𝑂

𝛾2 𝑒 𝑂.

Corollary The mixed moments 𝔽𝜈(𝑔)𝑙𝜈(𝑔)

𝑚 determine the distribution of 𝜈(𝑔).

Theorem ([JSW18]) Let 𝑔 ∈ 𝐷∞

𝑑 (𝐸) be non-negative and non-zero, then there exists 𝐷 > 0

such that

𝔽|𝜈(𝑔)|2𝑂 ≥ 𝐷𝑂𝑂

𝛾2 𝑒 𝑂.

11

Moments

Theorem ([JSW18]) There exists 𝐷 > 0 such that for any 𝑔 ∈ 𝐷∞

𝑑 (𝐸) and 𝑂 ≥ 1 we have

𝔽|𝜈(𝑔)|2𝑂 ≤ 𝐷𝑂‖𝑔‖2𝑂

∞ 𝑂

𝛾2 𝑒 𝑂.

Corollary The mixed moments 𝔽𝜈(𝑔)𝑙𝜈(𝑔)

𝑚 determine the distribution of 𝜈(𝑔).

Theorem ([JSW18]) Let 𝑔 ∈ 𝐷∞

𝑑 (𝐸) be non-negative and non-zero, then there exists 𝐷 > 0

such that

𝔽|𝜈(𝑔)|2𝑂 ≥ 𝐷𝑂𝑂

𝛾2 𝑒 𝑂.

11

Bounding moments – the naive way

• 𝔽|𝜈(1)|2𝑂 is (formally) given by

∫

𝐸2𝑂 𝔽 𝑂

∏

𝑘=1

𝑓𝑗𝛾𝑌(𝑦

𝑘)+ 𝛾2 2 𝔽𝑌(𝑦 𝑘)2𝑓−𝑗𝛾𝑌(𝑧 𝑘)+ 𝛾2 2 𝔽𝑌(𝑧 𝑘)2𝑒𝑦𝑘𝑒𝑧𝑘 =

∫

𝐸2𝑂 𝑓𝛾2 ∑1≤𝑘,𝑙≤𝑂 𝐷(𝑦

𝑘,𝑧𝑙)−𝛾2 ∑1≤𝑘<𝑙≤𝑂(𝐷(𝑦 𝑘,𝑦𝑙)+𝐷(𝑧 𝑘,𝑧𝑙))𝑒𝑦1 … 𝑒𝑦𝑂𝑒𝑧1 … 𝑒𝑧𝑂,

where 𝐷(𝑦, 𝑧) is the covariance kernel of 𝑌.

• In the case 𝐷(𝑦, 𝑧) = log

1 |𝑦−𝑧| this is simply the partition function

• f Coulomb gas with 𝑂 positive and 𝑂 negative charges. Estimating

this was done in [GP77] by using an electrostatic inequality due to Onsager [Ons39].

• In the general case 𝐷(𝑦, 𝑧) = log

1 |𝑦−𝑧| + 𝑕(𝑦, 𝑧) with 𝑕 bounded

• ne could simply estimate each 𝐷(𝑦, 𝑧) in the sums by

log

1 |𝑦−𝑧| ± ‖𝑕‖∞, but this would incur an error of order 𝑃(𝑂2).

12

Bounding moments – the naive way

• 𝔽|𝜈(1)|2𝑂 is (formally) given by

∫

𝐸2𝑂 𝔽 𝑂

∏

𝑘=1

𝑓𝑗𝛾𝑌(𝑦

𝑘)+ 𝛾2 2 𝔽𝑌(𝑦 𝑘)2𝑓−𝑗𝛾𝑌(𝑧 𝑘)+ 𝛾2 2 𝔽𝑌(𝑧 𝑘)2𝑒𝑦𝑘𝑒𝑧𝑘 =

∫

𝐸2𝑂 𝑓𝛾2 ∑1≤𝑘,𝑙≤𝑂 𝐷(𝑦

𝑘,𝑧𝑙)−𝛾2 ∑1≤𝑘<𝑙≤𝑂(𝐷(𝑦 𝑘,𝑦𝑙)+𝐷(𝑧 𝑘,𝑧𝑙))𝑒𝑦1 … 𝑒𝑦𝑂𝑒𝑧1 … 𝑒𝑧𝑂,

where 𝐷(𝑦, 𝑧) is the covariance kernel of 𝑌.

• In the case 𝐷(𝑦, 𝑧) = log

1 |𝑦−𝑧| this is simply the partition function

• f Coulomb gas with 𝑂 positive and 𝑂 negative charges. Estimating

this was done in [GP77] by using an electrostatic inequality due to Onsager [Ons39].

• In the general case 𝐷(𝑦, 𝑧) = log

1 |𝑦−𝑧| + 𝑕(𝑦, 𝑧) with 𝑕 bounded

• ne could simply estimate each 𝐷(𝑦, 𝑧) in the sums by

log

1 |𝑦−𝑧| ± ‖𝑕‖∞, but this would incur an error of order 𝑃(𝑂2).

12

Bounding moments – the naive way

• 𝔽|𝜈(1)|2𝑂 is (formally) given by

∫

𝐸2𝑂 𝔽 𝑂

∏

𝑘=1

𝑓𝑗𝛾𝑌(𝑦

𝑘)+ 𝛾2 2 𝔽𝑌(𝑦 𝑘)2𝑓−𝑗𝛾𝑌(𝑧 𝑘)+ 𝛾2 2 𝔽𝑌(𝑧 𝑘)2𝑒𝑦𝑘𝑒𝑧𝑘 =

∫

𝐸2𝑂 𝑓𝛾2 ∑1≤𝑘,𝑙≤𝑂 𝐷(𝑦

𝑘,𝑧𝑙)−𝛾2 ∑1≤𝑘<𝑙≤𝑂(𝐷(𝑦 𝑘,𝑦𝑙)+𝐷(𝑧 𝑘,𝑧𝑙))𝑒𝑦1 … 𝑒𝑦𝑂𝑒𝑧1 … 𝑒𝑧𝑂,

where 𝐷(𝑦, 𝑧) is the covariance kernel of 𝑌.

• In the case 𝐷(𝑦, 𝑧) = log

1 |𝑦−𝑧| this is simply the partition function

• f Coulomb gas with 𝑂 positive and 𝑂 negative charges. Estimating

this was done in [GP77] by using an electrostatic inequality due to Onsager [Ons39].

• In the general case 𝐷(𝑦, 𝑧) = log

1 |𝑦−𝑧| + 𝑕(𝑦, 𝑧) with 𝑕 bounded

• ne could simply estimate each 𝐷(𝑦, 𝑧) in the sums by

log

1 |𝑦−𝑧| ± ‖𝑕‖∞, but this would incur an error of order 𝑃(𝑂2).

12

General Onsager inequalities

Theorem ([JSW18; JSW19]) Assume that either of the following conditions hold:

• 𝑕 ∈ 𝐼𝑒+𝜁

𝑚𝑝𝑑 (𝐸 × 𝐸) for some 𝜁, or

• 𝑒 = 2 and 𝑕 ∈ 𝐷2(𝐸 × 𝐸),

Then around any 𝑨 ∈ 𝐸 there exists a neighbourhood 𝑉 ⊂ 𝐸 and 𝐷 > 0 such that for any 𝑨1, … , 𝑨𝑂 ∈ 𝑉 and 𝑟1, … , 𝑟𝑂 ∈ {−1, 1} we have

− ∑

1≤𝑘<𝑙≤𝑂

𝑟𝑘𝑟𝑙𝐷(𝑨𝑘, 𝑨𝑙) ≤ 1 2

𝑂

∑

𝑘=1

log 1

1 2 min𝑘≠𝑙 |𝑨𝑘 − 𝑨𝑙|

+ 𝐷𝑂.

13

Tie rest of the argument

𝑦1 𝑦2 𝑦3 𝑧1 𝑧2 𝑧3 𝑨1 𝑨2 𝑨3 𝑨4 𝑨5 𝑨6

Onsager

Figure 3: Dependencies between the variables in the integral.

• After applying Onsager the dependencies between the variables can

be reduced to a set of 2-cycles with attached trees.

• The upper bound is now obtained by computing a uniform bound
• ver all the graphs with a given number of components (integrate

variables one by one starting from the leaves) and multiplying by the number of such graphs.

14

Tie rest of the argument

𝑦1 𝑦2 𝑦3 𝑧1 𝑧2 𝑧3 𝑨1 𝑨2 𝑨3 𝑨4 𝑨5 𝑨6

Onsager

Figure 3: Dependencies between the variables in the integral.

• After applying Onsager the dependencies between the variables can

be reduced to a set of 2-cycles with attached trees.

• The upper bound is now obtained by computing a uniform bound
• ver all the graphs with a given number of components (integrate

variables one by one starting from the leaves) and multiplying by the number of such graphs.

14

Tie rest of the argument

𝑦1 𝑦2 𝑦3 𝑧1 𝑧2 𝑧3 𝑨1 𝑨2 𝑨3 𝑨4 𝑨5 𝑨6

Onsager

Figure 3: Dependencies between the variables in the integral.

• After applying Onsager the dependencies between the variables can

be reduced to a set of 2-cycles with attached trees.

• The upper bound is now obtained by computing a uniform bound
• ver all the graphs with a given number of components (integrate

variables one by one starting from the leaves) and multiplying by the number of such graphs.

14

Proof of the Onsager inequality for nice fjelds

• Let 𝐵𝑘 be centered Gaussians. From 𝔽( ∑𝑂

𝑘=1 𝑟𝑘𝐵𝑘) 2 ≥ 0 we get by

expanding and rearranging the inequality

−

𝑂

∑

1≤𝑘<𝑙≤𝑂

𝑟𝑘𝑟𝑙𝔽𝐵𝑘𝐵𝑙 ≤ 1 2

𝑂

∑

𝑘=1

𝔽𝐵2

𝑘 .

• We want to choose 𝐵𝑘 so that 𝔽𝐵𝑘𝐵𝑙 = 𝐷(𝑨𝑘, 𝑨𝑙), but 𝔽𝐵2

𝑘 are

small.

• Assume that 𝑌 has an approximation 𝑌𝑠 with the following

properties:

• 𝑌𝑠(𝑦) is a martingale as 𝑠 → 0
• 𝔽𝑌𝑠(𝑦)2 ≈ log 1

𝑠

• (𝑌𝑣(𝑦) − 𝑌𝑠(𝑦))⊥(𝑌𝑤(𝑧) − 𝑌𝑡(𝑧)) for all 𝑣 < 𝑠 and 𝑤 < 𝑡 if

𝑠 + 𝑡 < |𝑦 − 𝑧|

• By choosing 𝐵𝑘 = 𝑌𝑠

𝑘(𝑨𝑘), where 𝑠

𝑘 = 1 2 min𝑙≠𝑘 |𝑨𝑘 − 𝑨𝑙|, we see

that 𝔽𝐵𝑘𝐵𝑙 = 𝐷(𝑨𝑘, 𝑨𝑙) and the claim follows.

15

Proof of the Onsager inequality for nice fjelds

• Let 𝐵𝑘 be centered Gaussians. From 𝔽( ∑𝑂

𝑘=1 𝑟𝑘𝐵𝑘) 2 ≥ 0 we get by

expanding and rearranging the inequality

−

𝑂

∑

1≤𝑘<𝑙≤𝑂

𝑟𝑘𝑟𝑙𝔽𝐵𝑘𝐵𝑙 ≤ 1 2

𝑂

∑

𝑘=1

𝔽𝐵2

𝑘 .

• We want to choose 𝐵𝑘 so that 𝔽𝐵𝑘𝐵𝑙 = 𝐷(𝑨𝑘, 𝑨𝑙), but 𝔽𝐵2

𝑘 are

small.

• Assume that 𝑌 has an approximation 𝑌𝑠 with the following

properties:

• 𝑌𝑠(𝑦) is a martingale as 𝑠 → 0
• 𝔽𝑌𝑠(𝑦)2 ≈ log 1

𝑠

• (𝑌𝑣(𝑦) − 𝑌𝑠(𝑦))⊥(𝑌𝑤(𝑧) − 𝑌𝑡(𝑧)) for all 𝑣 < 𝑠 and 𝑤 < 𝑡 if

𝑠 + 𝑡 < |𝑦 − 𝑧|

• By choosing 𝐵𝑘 = 𝑌𝑠

𝑘(𝑨𝑘), where 𝑠

𝑘 = 1 2 min𝑙≠𝑘 |𝑨𝑘 − 𝑨𝑙|, we see

that 𝔽𝐵𝑘𝐵𝑙 = 𝐷(𝑨𝑘, 𝑨𝑙) and the claim follows.

15

Proof of the Onsager inequality for nice fjelds

• Let 𝐵𝑘 be centered Gaussians. From 𝔽( ∑𝑂

𝑘=1 𝑟𝑘𝐵𝑘) 2 ≥ 0 we get by

expanding and rearranging the inequality

−

𝑂

∑

1≤𝑘<𝑙≤𝑂

𝑟𝑘𝑟𝑙𝔽𝐵𝑘𝐵𝑙 ≤ 1 2

𝑂

∑

𝑘=1

𝔽𝐵2

𝑘 .

• We want to choose 𝐵𝑘 so that 𝔽𝐵𝑘𝐵𝑙 = 𝐷(𝑨𝑘, 𝑨𝑙), but 𝔽𝐵2

𝑘 are

small.

• Assume that 𝑌 has an approximation 𝑌𝑠 with the following

properties:

• 𝑌𝑠(𝑦) is a martingale as 𝑠 → 0
• 𝔽𝑌𝑠(𝑦)2 ≈ log 1

𝑠

• (𝑌𝑣(𝑦) − 𝑌𝑠(𝑦))⊥(𝑌𝑤(𝑧) − 𝑌𝑡(𝑧)) for all 𝑣 < 𝑠 and 𝑤 < 𝑡 if

𝑠 + 𝑡 < |𝑦 − 𝑧|

• By choosing 𝐵𝑘 = 𝑌𝑠

𝑘(𝑨𝑘), where 𝑠

𝑘 = 1 2 min𝑙≠𝑘 |𝑨𝑘 − 𝑨𝑙|, we see

that 𝔽𝐵𝑘𝐵𝑙 = 𝐷(𝑨𝑘, 𝑨𝑙) and the claim follows.

15

Proof of the Onsager inequality for nice fjelds

• Let 𝐵𝑘 be centered Gaussians. From 𝔽( ∑𝑂

𝑘=1 𝑟𝑘𝐵𝑘) 2 ≥ 0 we get by

expanding and rearranging the inequality

−

𝑂

∑

1≤𝑘<𝑙≤𝑂

𝑟𝑘𝑟𝑙𝔽𝐵𝑘𝐵𝑙 ≤ 1 2

𝑂

∑

𝑘=1

𝔽𝐵2

𝑘 .

• We want to choose 𝐵𝑘 so that 𝔽𝐵𝑘𝐵𝑙 = 𝐷(𝑨𝑘, 𝑨𝑙), but 𝔽𝐵2

𝑘 are

small.

• Assume that 𝑌 has an approximation 𝑌𝑠 with the following

properties:

• 𝑌𝑠(𝑦) is a martingale as 𝑠 → 0
• 𝔽𝑌𝑠(𝑦)2 ≈ log 1

𝑠

• (𝑌𝑣(𝑦) − 𝑌𝑠(𝑦))⊥(𝑌𝑤(𝑧) − 𝑌𝑡(𝑧)) for all 𝑣 < 𝑠 and 𝑤 < 𝑡 if

𝑠 + 𝑡 < |𝑦 − 𝑧|

• By choosing 𝐵𝑘 = 𝑌𝑠

𝑘(𝑨𝑘), where 𝑠

𝑘 = 1 2 min𝑙≠𝑘 |𝑨𝑘 − 𝑨𝑙|, we see

that 𝔽𝐵𝑘𝐵𝑙 = 𝐷(𝑨𝑘, 𝑨𝑙) and the claim follows.

15

Generalizing to other fjelds

Theorem ([JSW19]) Assume that in the covariance 𝐷(𝑦, 𝑧) = log

1 |𝑦−𝑧| + 𝑕(𝑦, 𝑧) the

function 𝑕 lies in 𝐼𝑒+𝜁

𝑚𝑝𝑑 (𝐸 × 𝐸). Then around any point 𝑦0 ∈ 𝐸 there

exists a neighbourhood in which 𝑌 can be decomposed as a sum of independent fjelds, 𝑌 = 𝑀 + 𝑆, where 𝑀 is a nice log-correlated fjeld (in particular it has the properties in the previous slide) and 𝑆 is a regular fjeld with Hölder continuous realisations.

16

XOR-Ising model

Ising model

Figure 4: Critical Ising model

• a model of ferromagnetism

consisting of spins

𝜏(𝑔) ∈ {−1, 1} for all faces 𝑔

• f a square lattice (for us 𝜏 = 1
• n the boundary)
• Gibbs distribution:

ℙ[𝜏] ∝ 𝑓𝛾 ∑𝑔

1∼𝑔 2 𝜏(𝑔 1)𝜏(𝑔 2)

• phase transition at

𝛾 = 𝛾𝑑 = log(1 + √2)/2.

• We denote 𝜏𝜀(𝑦) = 𝜏(𝑔) for

𝑦 ∈ 𝑔 and a given lattice

length 𝜀 > 0.

17

Ising model

Figure 4: Critical Ising model

• a model of ferromagnetism

consisting of spins

𝜏(𝑔) ∈ {−1, 1} for all faces 𝑔

• f a square lattice (for us 𝜏 = 1
• n the boundary)
• Gibbs distribution:

ℙ[𝜏] ∝ 𝑓𝛾 ∑𝑔

1∼𝑔 2 𝜏(𝑔 1)𝜏(𝑔 2)

• phase transition at

𝛾 = 𝛾𝑑 = log(1 + √2)/2.

• We denote 𝜏𝜀(𝑦) = 𝜏(𝑔) for

𝑦 ∈ 𝑔 and a given lattice

length 𝜀 > 0.

17

Ising model

Figure 4: Critical Ising model

• a model of ferromagnetism

consisting of spins

𝜏(𝑔) ∈ {−1, 1} for all faces 𝑔

• f a square lattice (for us 𝜏 = 1
• n the boundary)
• Gibbs distribution:

ℙ[𝜏] ∝ 𝑓𝛾 ∑𝑔

1∼𝑔 2 𝜏(𝑔 1)𝜏(𝑔 2)

• phase transition at

𝛾 = 𝛾𝑑 = log(1 + √2)/2.

• We denote 𝜏𝜀(𝑦) = 𝜏(𝑔) for

𝑦 ∈ 𝑔 and a given lattice

length 𝜀 > 0.

17

Ising model

Figure 4: Critical Ising model

• a model of ferromagnetism

consisting of spins

𝜏(𝑔) ∈ {−1, 1} for all faces 𝑔

• f a square lattice (for us 𝜏 = 1
• n the boundary)
• Gibbs distribution:

ℙ[𝜏] ∝ 𝑓𝛾 ∑𝑔

1∼𝑔 2 𝜏(𝑔 1)𝜏(𝑔 2)

• phase transition at

𝛾 = 𝛾𝑑 = log(1 + √2)/2.

• We denote 𝜏𝜀(𝑦) = 𝜏(𝑔) for

𝑦 ∈ 𝑔 and a given lattice

length 𝜀 > 0.

17

XOR-Ising model

• The XOR-Ising spin fjeld is defjned by 𝑇𝜀(𝑦) ≔ 𝜏𝜀(𝑦)𝜐𝜀(𝑦), where 𝜏

and 𝜐 are two independent Ising spin fjelds.

Figure 5: Ising, Ising, XOR-Ising

18

XOR-Ising and the real part of imaginary chaos

Theorem ([JSW18]) For any 𝑔 ∈ 𝐷∞

𝑑 (𝐸) we have

𝜀−1/4 ∫

𝐸

𝑔(𝑦)𝑇𝜀(𝑦) 𝑒𝑦 → 𝐷2 ∫

𝐸

𝑔(𝑦)(2|𝜒′(𝑦)| ℑ𝜒(𝑦) )

1/4

cos(2−1/2𝛥(𝑦)) 𝑒𝑦

where cos(2−1/2𝛥(𝑦)) denotes the real part of the imaginary chaos distribution 𝜈 with parameter 𝛾 = 1/√2 and 𝜒∶ 𝐸 → ℍ is a conformal bijection.

19

On the proof

• method of moments ⇒ integrals of 𝑜-point correlations

Theorem ([CHI15]) For any distinct 𝑦1, … , 𝑦𝑜 we have

lim

𝜀→0 𝜀−𝑜/8𝔽[𝜏𝜀(𝑦1) … 𝜏𝜀(𝑦𝑜)]

= 𝐷𝑜

𝑜

∏

𝑘=1

( |𝜒′(𝑦𝑘)| 2ℑ𝜒(𝑦𝑘))√2−𝑜/2 ∑

𝜈∈{−1,1}𝑜

∏

1≤𝑙<𝑛≤𝑜

|𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) 𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) |

𝜈𝑙𝜈𝑛 2 .

• A direct computation shows that the moments match formally.
• To justify dominated convergence, we prove an Onsager-type

inequality for the Ising model:

𝜀−𝑜/8𝔽𝜏𝜀(𝑦1) … 𝜏𝜀(𝑦𝑜) ≤ 𝐷𝑜

𝑜

∏

𝑘=1

(min

𝑙≠𝑘 |𝑦𝑘 − 𝑦𝑙|)−1/8

20

On the proof

• method of moments ⇒ integrals of 𝑜-point correlations

Theorem ([CHI15]) For any distinct 𝑦1, … , 𝑦𝑜 we have

lim

𝜀→0 𝜀−𝑜/8𝔽[𝜏𝜀(𝑦1) … 𝜏𝜀(𝑦𝑜)]

= 𝐷𝑜

𝑜

∏

𝑘=1

( |𝜒′(𝑦𝑘)| 2ℑ𝜒(𝑦𝑘))√2−𝑜/2 ∑

𝜈∈{−1,1}𝑜

∏

1≤𝑙<𝑛≤𝑜

|𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) 𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) |

𝜈𝑙𝜈𝑛 2 .

• A direct computation shows that the moments match formally.
• To justify dominated convergence, we prove an Onsager-type

inequality for the Ising model:

𝜀−𝑜/8𝔽𝜏𝜀(𝑦1) … 𝜏𝜀(𝑦𝑜) ≤ 𝐷𝑜

𝑜

∏

𝑘=1

(min

𝑙≠𝑘 |𝑦𝑘 − 𝑦𝑙|)−1/8

20

On the proof

• method of moments ⇒ integrals of 𝑜-point correlations

Theorem ([CHI15]) For any distinct 𝑦1, … , 𝑦𝑜 we have

lim

𝜀→0 𝜀−𝑜/8𝔽[𝜏𝜀(𝑦1) … 𝜏𝜀(𝑦𝑜)]

= 𝐷𝑜

𝑜

∏

𝑘=1

( |𝜒′(𝑦𝑘)| 2ℑ𝜒(𝑦𝑘))√2−𝑜/2 ∑

𝜈∈{−1,1}𝑜

∏

1≤𝑙<𝑛≤𝑜

|𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) 𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) |

𝜈𝑙𝜈𝑛 2 .

• A direct computation shows that the moments match formally.
• To justify dominated convergence, we prove an Onsager-type

inequality for the Ising model:

𝜀−𝑜/8𝔽𝜏𝜀(𝑦1) … 𝜏𝜀(𝑦𝑜) ≤ 𝐷𝑜

𝑜

∏

𝑘=1

(min

𝑙≠𝑘 |𝑦𝑘 − 𝑦𝑙|)−1/8

20

On the proof

• method of moments ⇒ integrals of 𝑜-point correlations

Theorem ([CHI15]) For any distinct 𝑦1, … , 𝑦𝑜 we have

lim

𝜀→0 𝜀−𝑜/8𝔽[𝜏𝜀(𝑦1) … 𝜏𝜀(𝑦𝑜)]

= 𝐷𝑜

𝑜

∏

𝑘=1

( |𝜒′(𝑦𝑘)| 2ℑ𝜒(𝑦𝑘))√2−𝑜/2 ∑

𝜈∈{−1,1}𝑜

∏

1≤𝑙<𝑛≤𝑜

|𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) 𝜒(𝑦𝑙) − 𝜒(𝑦𝑛) |

𝜈𝑙𝜈𝑛 2 .

• A direct computation shows that the moments match formally.
• To justify dominated convergence, we prove an Onsager-type

inequality for the Ising model:

𝜀−𝑜/8𝔽𝜏𝜀(𝑦1) … 𝜏𝜀(𝑦𝑜) ≤ 𝐷𝑜

𝑜

∏

𝑘=1

(min

𝑙≠𝑘 |𝑦𝑘 − 𝑦𝑙|)−1/8

20

Regularity, densities and monofractality

Besov spaces

The spaces 𝐶𝑡

𝑞,𝑟(ℝ𝑒)

• Banach spaces of distributions parametrised by smoothness

parameter 𝑡 ∈ ℝ and two size parameters 𝑞, 𝑟 ∈ [1, ∞].

• Contain both Sobolev and Hölder spaces:
• 𝐶𝑡

2,2(ℝ𝑒) = 𝐼𝑡(ℝ𝑒) (𝑡 ∈ ℝ)

• 𝐶𝑡

∞,∞(ℝ𝑒) = 𝐷𝑡(ℝ𝑒) (𝑡 ∈ (0, ∞) ⧵ ℕ).

• We say that 𝑔 ∈ 𝐶𝑡

𝑞,𝑟,𝑚𝑝𝑑(𝐸) if and only if 𝜔𝑔 ∈ 𝐶𝑡 𝑞,𝑟(ℝ𝑒) for all

𝜔 ∈ 𝐷∞

𝑑 (𝐸).

21

Besov spaces

The spaces 𝐶𝑡

𝑞,𝑟(ℝ𝑒)

• Banach spaces of distributions parametrised by smoothness

parameter 𝑡 ∈ ℝ and two size parameters 𝑞, 𝑟 ∈ [1, ∞].

• Contain both Sobolev and Hölder spaces:
• 𝐶𝑡

2,2(ℝ𝑒) = 𝐼𝑡(ℝ𝑒) (𝑡 ∈ ℝ)

• 𝐶𝑡

∞,∞(ℝ𝑒) = 𝐷𝑡(ℝ𝑒) (𝑡 ∈ (0, ∞) ⧵ ℕ).

• We say that 𝑔 ∈ 𝐶𝑡

𝑞,𝑟,𝑚𝑝𝑑(𝐸) if and only if 𝜔𝑔 ∈ 𝐶𝑡 𝑞,𝑟(ℝ𝑒) for all

𝜔 ∈ 𝐷∞

𝑑 (𝐸).

21

Regularity of 𝜈

Theorem ([JSW18]) We have for all 𝑞, 𝑟 ∈ [1, ∞] that

• 𝑡 < − 𝛾2

2 ⇒ 𝜈 ∈ 𝐶𝑡 𝑞,𝑟,𝑚𝑝𝑑(𝐸)

• 𝑡 > − 𝛾2

2 ⇒ 𝜈 ∉ 𝐶𝑡 𝑞,𝑟,𝑚𝑝𝑑(𝐸)

• 𝜈 is almost surely not a complex measure
• One can get fjniteness of Besov norms by computing moments.
• To show that 𝜈 is not a complex measure it suffjces to show that

𝜈(𝑓−𝑗𝛾𝑌𝜀𝜔) → ∞ as 𝜀 → 0 for some 𝜔 ∈ 𝐷∞

𝑑 (𝐸).

22

Regularity of 𝜈

Theorem ([JSW18]) We have for all 𝑞, 𝑟 ∈ [1, ∞] that

• 𝑡 < − 𝛾2

2 ⇒ 𝜈 ∈ 𝐶𝑡 𝑞,𝑟,𝑚𝑝𝑑(𝐸)

• 𝑡 > − 𝛾2

2 ⇒ 𝜈 ∉ 𝐶𝑡 𝑞,𝑟,𝑚𝑝𝑑(𝐸)

• 𝜈 is almost surely not a complex measure
• One can get fjniteness of Besov norms by computing moments.
• To show that 𝜈 is not a complex measure it suffjces to show that

𝜈(𝑓−𝑗𝛾𝑌𝜀𝜔) → ∞ as 𝜀 → 0 for some 𝜔 ∈ 𝐷∞

𝑑 (𝐸).

22

Regularity of 𝜈

Theorem ([JSW18]) We have for all 𝑞, 𝑟 ∈ [1, ∞] that

• 𝑡 < − 𝛾2

2 ⇒ 𝜈 ∈ 𝐶𝑡 𝑞,𝑟,𝑚𝑝𝑑(𝐸)

• 𝑡 > − 𝛾2

2 ⇒ 𝜈 ∉ 𝐶𝑡 𝑞,𝑟,𝑚𝑝𝑑(𝐸)

• 𝜈 is almost surely not a complex measure
• One can get fjniteness of Besov norms by computing moments.
• To show that 𝜈 is not a complex measure it suffjces to show that

𝜈(𝑓−𝑗𝛾𝑌𝜀𝜔) → ∞ as 𝜀 → 0 for some 𝜔 ∈ 𝐷∞

𝑑 (𝐸).

22

Smooth and bounded densities

Theorem ([ABJJ20]) Assume that 𝑌 is a GFF in some bounded domain 𝐸 and let 𝑔 ∈ 𝑀∞(𝐸) be a non-zero function. Then the random variable 𝜈(𝑔) has a smooth and bounded density in ℂ.

• A rough fjrst idea towards a proof: Look at

∫ 𝑓𝑗𝛾 ∑∞

𝑜=1 𝐵𝑜𝜒𝑜(𝑦)+ 𝛾2 2 ∑∞ 𝑜=1 𝜒𝑜(𝑦)2 𝑒𝑦 and try to show that if one

conditions for instance on 𝐵1, 𝐵2, then with a high probability the continuous map (𝐵1, 𝐵2) ↦ 𝜈(𝑔) sweeps a reasonable area in the complex plane for |𝐵1|, |𝐵2| ≤ 1, say.

• Central diffjculty with this approach: How to rule out the rest of the

chaos 𝑓𝑗𝛾 ∑∞

𝑜=3 𝐵𝑜𝜒𝑜(𝑦)+ 𝛾2 2 ∑∞ 𝑜=3 𝜒𝑜(𝑦)2 being close to 0?

23

Smooth and bounded densities

Theorem ([ABJJ20]) Assume that 𝑌 is a GFF in some bounded domain 𝐸 and let 𝑔 ∈ 𝑀∞(𝐸) be a non-zero function. Then the random variable 𝜈(𝑔) has a smooth and bounded density in ℂ.

• A rough fjrst idea towards a proof: Look at

∫ 𝑓𝑗𝛾 ∑∞

𝑜=1 𝐵𝑜𝜒𝑜(𝑦)+ 𝛾2 2 ∑∞ 𝑜=1 𝜒𝑜(𝑦)2 𝑒𝑦 and try to show that if one

conditions for instance on 𝐵1, 𝐵2, then with a high probability the continuous map (𝐵1, 𝐵2) ↦ 𝜈(𝑔) sweeps a reasonable area in the complex plane for |𝐵1|, |𝐵2| ≤ 1, say.

• Central diffjculty with this approach: How to rule out the rest of the

chaos 𝑓𝑗𝛾 ∑∞

𝑜=3 𝐵𝑜𝜒𝑜(𝑦)+ 𝛾2 2 ∑∞ 𝑜=3 𝜒𝑜(𝑦)2 being close to 0?

23

Smooth and bounded densities

Theorem ([ABJJ20]) Assume that 𝑌 is a GFF in some bounded domain 𝐸 and let 𝑔 ∈ 𝑀∞(𝐸) be a non-zero function. Then the random variable 𝜈(𝑔) has a smooth and bounded density in ℂ.

• A rough fjrst idea towards a proof: Look at

∫ 𝑓𝑗𝛾 ∑∞

𝑜=1 𝐵𝑜𝜒𝑜(𝑦)+ 𝛾2 2 ∑∞ 𝑜=1 𝜒𝑜(𝑦)2 𝑒𝑦 and try to show that if one

conditions for instance on 𝐵1, 𝐵2, then with a high probability the continuous map (𝐵1, 𝐵2) ↦ 𝜈(𝑔) sweeps a reasonable area in the complex plane for |𝐵1|, |𝐵2| ≤ 1, say.

• Central diffjculty with this approach: How to rule out the rest of the

chaos 𝑓𝑗𝛾 ∑∞

𝑜=3 𝐵𝑜𝜒𝑜(𝑦)+ 𝛾2 2 ∑∞ 𝑜=3 𝜒𝑜(𝑦)2 being close to 0?

23

Smooth and bounded densities

• In the case of real chaos on say the unit interval [0, 1] one

heuristically has something like

ℙ[𝜈([0, 1]) ≤ 𝜁] ≤ ℙ[𝜈([0, 1 2]) ≤ 𝜁, 𝜈([1 2, 1]) ≤ 𝜁] ≈ ℙ[𝜈([0, 1]) ≤ 2𝜁]2.

Reasoning along these lines can indeed be made precise and yields the existence of all negative moments for 𝜈([0, 1]).

• The crucial property here was non-negativity, which of course fails

for imaginary chaos.

• In the end our proof goes through Malliavin calculus.

24

Smooth and bounded densities

• In the case of real chaos on say the unit interval [0, 1] one

heuristically has something like

ℙ[𝜈([0, 1]) ≤ 𝜁] ≤ ℙ[𝜈([0, 1 2]) ≤ 𝜁, 𝜈([1 2, 1]) ≤ 𝜁] ≈ ℙ[𝜈([0, 1]) ≤ 2𝜁]2.

Reasoning along these lines can indeed be made precise and yields the existence of all negative moments for 𝜈([0, 1]).

• The crucial property here was non-negativity, which of course fails

for imaginary chaos.

• In the end our proof goes through Malliavin calculus.

24

Smooth and bounded densities

• In the case of real chaos on say the unit interval [0, 1] one

heuristically has something like

ℙ[𝜈([0, 1]) ≤ 𝜁] ≤ ℙ[𝜈([0, 1 2]) ≤ 𝜁, 𝜈([1 2, 1]) ≤ 𝜁] ≈ ℙ[𝜈([0, 1]) ≤ 2𝜁]2.

Reasoning along these lines can indeed be made precise and yields the existence of all negative moments for 𝜈([0, 1]).

• The crucial property here was non-negativity, which of course fails

for imaginary chaos.

• In the end our proof goes through Malliavin calculus.

24

Monofractality

Theorem ([ABJJ20]) Almost surely for all 𝑨 ∈ 𝐸 we have

lim inf

𝑠→0

log |𝜈(𝑅(𝑨, 𝑠))| log 𝑠 = 2 − 𝛾2/2.

• We refjne this in two difgerent ways:
• A law of iterated logarithm -type result: For fjxed 𝑦 we have

lim sup

𝑠→0

|𝜈(𝑅(𝑦, 𝑠))| 𝑠2−𝛾2/2(log | log 𝑠|)𝛾2/4 = 𝑑1(𝛾)

• Existence of exceptional (fast) points:

sup

𝑦∈𝐸

lim sup

𝑠→0

|𝜈(𝑅(𝑦, 𝑠))| 𝑠2−𝛾2/2| log 𝑠|𝛾2/4 = 𝑑2(𝛾)

25

Monofractality

Theorem ([ABJJ20]) Almost surely for all 𝑨 ∈ 𝐸 we have

lim inf

𝑠→0

log |𝜈(𝑅(𝑨, 𝑠))| log 𝑠 = 2 − 𝛾2/2.

• We refjne this in two difgerent ways:
• A law of iterated logarithm -type result: For fjxed 𝑦 we have

lim sup

𝑠→0

|𝜈(𝑅(𝑦, 𝑠))| 𝑠2−𝛾2/2(log | log 𝑠|)𝛾2/4 = 𝑑1(𝛾)

• Existence of exceptional (fast) points:

sup

𝑦∈𝐸

lim sup

𝑠→0

|𝜈(𝑅(𝑦, 𝑠))| 𝑠2−𝛾2/2| log 𝑠|𝛾2/4 = 𝑑2(𝛾)

25

Thanks!

26

References i

[ABJJ20]

• J. Aru, G. Bavarez, A. Jego, and J. Junnila. TBA. Work in progress

(2020). [Ber17]

• N. Berestycki. An elementary approach to Gaussian

multiplicative chaos. Electronic Communications in Probability 22 (2017). [BPR19]

• N. Berestycki, E. Powell, and G. Ray. A characterisation of the

Gaussian free fjeld. Probability Theory and Related Fields (2019). [CGPR19]

• F. Camia, A. Gandolfj, G. Peccati, and Reddy T. R. Brownian

Loops, Layering Fields and Imaginary Gaussian Multiplicative

• Chaos. arXiv:1908.05881 (2019).

27

References ii

[CHI15]

• D. Chelkak, C. Hongler, and K. Izyurov. Conformal invariance of

spin correlations in the planar Ising model. Annals of mathematics (2015), 1087–1138. [GP77]

• J. Gunson and L. S. Panta. Two-dimensional neutral Coulomb
• gas. Communications in Mathematical Physics 52.3 (1977),

295–304. [JSW18]

• J. Junnila, E. Saksman, and C. Webb. Imaginary multiplicative

chaos: Moments, regularity and connections to the Ising

• model. arXiv:1806.02118 (2018).

[JSW19]

• J. Junnila, E. Saksman, and C. Webb. Decompositions of

log-correlated fjelds with applications. Annals of Applied Probability 29.6 (2019), 3786–3820.

28

References iii

[Kah85] J.–P. Kahane. Sur le chaos multiplicatif. Comptes rendus de l’Académie des sciences. Série 1, Mathématique 301.6 (1985), 329–332. [LRV15]

• H. Lacoin, R. Rhodes, and V. Vargas. Complex Gaussian

Multiplicative Chaos. Communications in Mathematical Physics 337.2 (2015), 569–632. [Ons39]

• L. Onsager. Electrostatic Interaction of Molecules. The Journal
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[RV10]

• R. Robert and V. Vargas. Gaussian multiplicative chaos
• revisited. The Annals of Probability 38.2 (2010), 605–631.

[Sha16]

• A. Shamov. On Gaussian multiplicative chaos. Journal of

Functional Analysis 270.9 (2016), 3224–3261.

29

References iv

[SSV19]

• L. Schoug, A. Sepúlveda, and F. Viklund. Dimension of

two-valued sets via imaginary chaos. arXiv:1910.09294 (2019).

30