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
π (β π ) we have π½π(π)π(π) = β« π(π¦)π(π§)π·(π¦, π§) ππ¦ ππ§. Log-correlated Gaussian fjelds Let us fjx a bounded simply connected domain πΈ β β π . Heuristic defjnition A Gaussian fjeld πβΆ πΈ β β is called log-correlated if π½π(π¦)π(π§) = π·(π¦, π§) β log 1 where π is regular. Caveat Such fjelds cannot be defjned pointwise and must instead be understood as distributions (generalized functions). This means that for all π, π β π· β 3 |π¦ β π§| + π(π¦, π§)
Log-correlated Gaussian fjelds Let us fjx a bounded simply connected domain πΈ β β π . Heuristic defjnition A Gaussian fjeld πβΆ πΈ β β is called log-correlated if π½π(π¦)π(π§) = π·(π¦, π§) β log 1 where π is regular. Caveat Such fjelds cannot be defjned pointwise and must instead be understood as distributions (generalized functions). This means that for all π, π β π· β 3 |π¦ β π§| + π(π¦, π§) π (β π ) we have π½π(π)π(π) = β« π(π¦)π(π§)π·(π¦, π§) ππ¦ ππ§.
Log-correlated Gaussian fjelds β’ We will always assume at least that β’ π β π 1 (πΈ Γ πΈ) β© π·(πΈ Γ πΈ) β’ π is bounded from above β’ These properties are enough to ensure that π exists. (Assuming that the kernel π· is positive defjnite.) 4
Log-correlated Gaussian fjelds β’ We will always assume at least that β’ π β π 1 (πΈ Γ πΈ) β© π·(πΈ Γ πΈ) β’ π is bounded from above β’ These properties are enough to ensure that π exists. (Assuming that the kernel π· is positive defjnite.) 4
Example: Tie 2D GFF Defjnition The 0 -boundary GFF π₯ in the domain πΈ is a Gaussian fjeld with the covariance π½π₯(π¦)π₯(π§) = π» πΈ (π¦, π§) β’ universality: β’ appears in the scaling limit of various height function models, random matrices, QFT, β¦ β’ a recent characterisation: the only random fjeld with conformally invariant law and domain Markov property (+some moment condition) [BPR19] 5 where π» πΈ is the Greenβs function of the Dirichlet Laplacian in πΈ .
Example: Tie 2D GFF Defjnition The 0 -boundary GFF π₯ in the domain πΈ is a Gaussian fjeld with the covariance π½π₯(π¦)π₯(π§) = π» πΈ (π¦, π§) β’ universality: β’ appears in the scaling limit of various height function models, random matrices, QFT, β¦ β’ a recent characterisation: the only random fjeld with conformally invariant law and domain Markov property (+some moment condition) [BPR19] 5 where π» πΈ is the Greenβs function of the Dirichlet Laplacian in πΈ .
Example: Tie GFF in the unit square Figure 1: An approximation of the GFF in the unit square. 6
fjelds π π and normalize properly when taking the limit as π β β . For any given πΏ β (0, β2π) the functions π π (π¦) β π πΏπ π (π¦)β πΏ2 2 π½π π (π¦) 2 converge to a random measure π . We say that π = π πΏ is a GMC measure Gaussian Multiplicative Chaos β’ In various applications one is interested in measures formally of the β’ To rigorously defjne them one has to approximate π with regular Theorem/Defjnition ([Kah85; RV10; Sha16; Ber17]) associated to π . 7 form π πΏπ(π¦) ππ¦ where πΏ is a parameter.
For any given πΏ β (0, β2π) the functions π π (π¦) β π πΏπ π (π¦)β πΏ2 2 π½π π (π¦) 2 converge to a random measure π . We say that π = π πΏ is a GMC measure Gaussian Multiplicative Chaos β’ In various applications one is interested in measures formally of the β’ To rigorously defjne them one has to approximate π with regular Theorem/Defjnition ([Kah85; RV10; Sha16; Ber17]) associated to π . 7 form π πΏπ(π¦) ππ¦ where πΏ is a parameter. fjelds π π and normalize properly when taking the limit as π β β .
Gaussian Multiplicative Chaos β’ In various applications one is interested in measures formally of the β’ To rigorously defjne them one has to approximate π with regular Theorem/Defjnition ([Kah85; RV10; Sha16; Ber17]) associated to π . 7 form π πΏπ(π¦) ππ¦ where πΏ is a parameter. fjelds π π and normalize properly when taking the limit as π β β . For any given πΏ β (0, β2π) the functions π π (π¦) β π πΏπ π (π¦)β πΏ2 2 π½π π (π¦) 2 converge to a random measure π . We say that π = π πΏ is a GMC measure
8 β² βπβ 2 For any π β π· β sequence is Cauchy in π 2 (π») . β’ Otherwise one can do a similar computation to show that the β’ If π π (π) is a martingale this immediately shows convergence. ππ¦ ππ§ πΈ 2 1 A simple π 2 -computation Existence of GMC when πΏ β (0, βπ) π (πΈ) we have πΈ 2 π(π¦)π(π§)π½π πΏπ π (π¦)+πΏπ π (π§)β πΏ2 2 π½π π (π¦) 2 β πΏ2 2 π½π π (π§) 2 ππ¦ ππ§ π½|π π (π)| 2 = β« πΈ 2 π(π¦)π(π§)π πΏ 2 π½π π (π¦)π π (π§) ππ¦ ππ§ = β« πΈ 2 π πΏ 2 log |π¦βπ§| ππ¦ ππ§ = β« β β« |π¦ β π§| πΏ 2 < β.
8 β² βπβ 2 For any π β π· β sequence is Cauchy in π 2 (π») . β’ Otherwise one can do a similar computation to show that the β’ If π π (π) is a martingale this immediately shows convergence. ππ¦ ππ§ πΈ 2 1 A simple π 2 -computation Existence of GMC when πΏ β (0, βπ) π (πΈ) we have πΈ 2 π(π¦)π(π§)π½π πΏπ π (π¦)+πΏπ π (π§)β πΏ2 2 π½π π (π¦) 2 β πΏ2 2 π½π π (π§) 2 ππ¦ ππ§ π½|π π (π)| 2 = β« πΈ 2 π(π¦)π(π§)π πΏ 2 π½π π (π¦)π π (π§) ππ¦ ππ§ = β« πΈ 2 π πΏ 2 log |π¦βπ§| ππ¦ ππ§ = β« β β« |π¦ β π§| πΏ 2 < β.
8 β² βπβ 2 For any π β π· β sequence is Cauchy in π 2 (π») . β’ Otherwise one can do a similar computation to show that the β’ If π π (π) is a martingale this immediately shows convergence. ππ¦ ππ§ πΈ 2 1 A simple π 2 -computation Existence of GMC when πΏ β (0, βπ) π (πΈ) we have πΈ 2 π(π¦)π(π§)π½π πΏπ π (π¦)+πΏπ π (π§)β πΏ2 2 π½π π (π¦) 2 β πΏ2 2 π½π π (π§) 2 ππ¦ ππ§ π½|π π (π)| 2 = β« πΈ 2 π(π¦)π(π§)π πΏ 2 π½π π (π¦)π π (π§) ππ¦ ππ§ = β« πΈ 2 π πΏ 2 log |π¦βπ§| ππ¦ ππ§ = β« β β« |π¦ β π§| πΏ 2 < β.
Complex values of πΏ β(πΏ) β(πΏ) βπ ββπ ββ2π β2π Figure 2: The subcritical regime π΅ for πΏ in the complex plane. β’ In fact, πΏ β¦ π πΏ (π) is an analytic function on π΅ [JSW19]. β’ The circle corresponds to the π 2 -phase β in particular it contains the whole subcritical part of the imaginary axis. 9
Complex values of πΏ β(πΏ) β(πΏ) βπ ββπ ββ2π β2π Figure 2: The subcritical regime π΅ for πΏ in the complex plane. β’ In fact, πΏ β¦ π πΏ (π) is an analytic function on π΅ [JSW19]. β’ The circle corresponds to the π 2 -phase β in particular it contains the whole subcritical part of the imaginary axis. 9
Complex values of πΏ β(πΏ) β(πΏ) βπ ββπ ββ2π β2π Figure 2: The subcritical regime π΅ for πΏ in the complex plane. β’ In fact, πΏ β¦ π πΏ (π) is an analytic function on π΅ [JSW19]. β’ The circle corresponds to the π 2 -phase β in particular it contains the whole subcritical part of the imaginary axis. 9
Imaginary multiplicative chaos Theorem/Defjnition ([JSW18; LRV15]) converge in probability in πΌ βπ/2βπ (β π ) to a random distribution π . β’ Applications: XOR-Ising model [JSW18], two-valued sets of the GFF [SSV19] and certain random fjelds constructed using the Brownian loop soup [CGPR19]. 10 Let πΎ β (0, βπ) . Then the random functions π π (π¦) β π ππΎπ π (π¦)+ πΎ2 2 π½π π (π¦) 2
Imaginary multiplicative chaos Theorem/Defjnition ([JSW18; LRV15]) converge in probability in πΌ βπ/2βπ (β π ) to a random distribution π . β’ Applications: XOR-Ising model [JSW18], two-valued sets of the GFF [SSV19] and certain random fjelds constructed using the Brownian loop soup [CGPR19]. 10 Let πΎ β (0, βπ) . Then the random functions π π (π¦) β π ππΎπ π (π¦)+ πΎ2 2 π½π π (π¦) 2
Moments
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