Mini-Workshop on Log-Correlated Random Fields

Date: December 12-14, 2017
Location: 407 Mathematics building, Columbia University

We expect partial support to be available for travel and accommodation for accepted participants.

Organizers: Julien Dubedat, Fredrik Viklund

The workshop is supported by the National Science Foundation (DMS 1308476)

Confirmed participants

Louis-Pierre Arguin
Juhan Aru
Guillaume Baverez
Stephane Benoist
Paul Bourgade
Jian Ding
Julien Dubedat
Bertrand Duplantier
Julian Gold
Ewain Gwynne
Jack Hanson
Lisa Hartung
Nina Holden
Oren Louidor
Titus Lupu
Joshua Pfeffer
Guillaume Remy
Remi Rhodes
Jay Rosen
Hao Shen
Eliran Subag
Xin Sun
Scott Sheffield
Vincent Vargas
Fredrik Viklund

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Schedule

	Tue Dec 12	Wed Dec 13	Thu Dec 14
9:9:45	breakfast (508 Math)	breakfast	breakfast
9:45-10:45	Sheffield	Rhodes	Bourgade
10:45-11	coffee	coffee	coffee
11-12	Lupu	Subag	Remy
12-2	lunch	lunch	lunch
2-3	Arguin	Ding	Louidor
3-3:15	coffee	coffee	coffee
3:15-4:15	Hartung	Holden	Aru

Abstracts

Scott Sheffield
Random walks in "scale-free" random environments

What is the right notion of "ergodic theory" for random planar environments that do not have a single globally defined length scale? This seems like a fundamental question for anyone interested in random walks on conformally embedded random planar maps or on discretized Liouville quantum gravity surfaces. I will discuss one way to formally pose and answer this question.
For our purposes, an "environment'' consists of an infinite random planar map embedded in the plane, each of whose edges comes with a positive real conductance. Our biggest result is that under modest constraints (a "modulo scaling'' analog of ergodicity and certain bounds on the expected local Dirichlet energy of the map and its dual) a random walk in this kind of environment has Brownian motion as a scaling limit.
This theory has applications to random planar maps, as well as to random walks on discretizations of Liouville quantum gravity measures. In fact, this theory provides one way to prove that certain random planar maps (harmonically embedded in the plane via the so called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity.  This is joint work with Gwynne and Miller.

Titus Lupu
Excursion decomposition of the GFF in 2D and Minkowski content

We consider the continuum Gaussian free field in dimension 2 and construct a "decomposition in excursions" for it. There is a countable family of disjoint random compact subsets, such that on each of it the GFF is a measure, either positive or negative, with equal probability, and such that the GFF is zero outside this family. We call these compact subsets "excursion sets", by analogy with Brownian excursions. We show that the measure induced by the GFF on each of these excursion sets is a Minkowski content in the gauge | log r |^(1/2) r^2. Moreover, using the same excursion decomposition, one can construct a whole family of other random fields, which are all conformal invariant in law, have the same covariance as the GFF, but in general are non-Gaussian.


Louis-Pierre Arguin
The maxima of the Riemann zeta function in a short interval of the critical line

A conjecture of Fyodorov, Hiary & Keating states that the maxima of the modulus of the Riemann zeta function on an interval of the critical line behave similarly to the maxima of a log-correlated process. In this talk, we will discuss a proof of this conjecture to leading order, unconditionally on the Riemann Hypothesis. We will highlight the connections between the number theory problem and the probabilistic models including the branching random walk. We will also discuss the relations with the freezing transition for this problem. This is joint work with D. Belius (Zurich), P. Bourgade (NYU), M. Radizwill (McGill), and K. Soundararajan (Stanford).


Lisa Hartung
Extreme Level Sets of Branching Brownian Motion

We study the structure of extreme level sets of a standard one dimensional branching Brownian motion, namely the sets of particles whose height is within a fixed distance from the order of the global maximum. It is well known that such particles congregate at large times in clusters of order-one genealogical diameter around local maxima which form a Cox process in the limit. We add to these results by finding the asymptotic size of extreme level sets and the typical height and shape of those clusters which carry such level sets. We also find the right tail decay of the distribution of the distance between the two highest particles. These results confirm two conjectures of Brunet and Derrida.(joint work with A. Cortines, O Louidor)


Rémi Rhodes
Reflection coefficient in Liouville theory and tails of Gaussian multiplicative chaos

In  this talk, I will discuss the construction of correlation functions in Liouville conformal field theory (LCFT) with  a special emphasis on
the two point correlation function, also known as reflection coefficient. The reflection coefficient also appears as the partition function of the quantum sphere introduced by Duplantier-Miller-Sheffield. Based on the recent proof of the DOZZ formula (joint work with A.Kupiainen and V. Vargas) I will present an integrability formula for the reflection coefficient. As an application, we are able to derive exact asymptotic expansions for the tails of Gaussian multiplicative chaos. These results seem to be new, even from the physics perspective.


Eliran Subag
The geometry of the Gibbs measure and temperature chaos in some spherical spin glasses

Spherical spin glasses are general models of random Gaussian functions on the high-dimensional sphere. I will begin by describing a geometric picture for the associated Gibbs measure of the pure spherical models at low enough temperature: it concentrates on spherical `bands' centered at the deepest critical points. In contrast to the mixed models, this implies the absence of temperature chaos. I will also describe a similar picture for mixed models that are close enough to pure and explain how it is compatible with the occurrence of temperature chaos in those models.
Based on joint work with Gerard Ben Arous and Ofer Zeitouni.


Jian Ding
Liouville heat kernel and Liouville graph distance

This talk concerns two aspects of a planar GFF: the heat kernel for the Liouville Brownian motion (where we focus on the regime on how (un)likely the LBM will travel far in very short amount of time) and the Liouville graph distance (which roughly speaking is the minimal number of balls with comparable LQG measure whose union contains a path between two points). I will present a joint work with O. Zeitouni and F. Zhang, where we relate the the exponent in the Liouville heat kernel to that of the Liouville graph distance.


Nina Holden
A mating-of-trees approach to distances in random planar maps and Liouville quantum gravity

In the first part of the talk we introduce a graph representing a natural discretization of a Liouville quantum gravity (LQG) surface. We conjecture that this gives a metric on the LQG surface in the scaling limit. We prove non-trivial upper and lower bounds for the cardinality of a metric ball in the graph, and the existence of an exponent describing expected distances. In the second part of the talk we transfer these estimates to certain natural non-uniform random planar maps. Both parts of the talk are based on a mating-of-trees encoding of the surfaces. Based on joint works with Ewain Gwynne and Xin Sun.


Paul Bourgade
The 2D Coulomb gas and the Gaussian free field

We prove a quantitative central limit theorem for linear statistics of particles in the complex plane, with Coulomb interaction at any temperature. This generalizes works by Rider, Virag, Ameur, Hendenmalm and Makarov obtained for the inverse temperature beta=2. The main tools are a multi scale analysis and the Ward identity (or loop equation). This is joint work with Roland Bauerschmidt, Miika Nikula and Horng-Tzer Yau.


Guilaume Rémy
The Fyodorov-Bouchaud formula and Liouville conformal field theory

Starting from the restriction of a 2d Gaussian free field (GFF) to the unit circle one can define a Gaussian multiplicative chaos (GMC) measure whose density is formally given by the exponential of the GFF. In 2008 Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of this GMC. In this talk we will give a rigorous proof of this formula. Our method is inspired by the technology developed by Kupiainen, Rhodes and Vargas to derive the DOZZ formula in the context of Liouville conformal field theory on the Riemann sphere. In our case the key observation is that the negative moments of the total mass of GMC on the circle determine its law and are equal to one-point correlation functions of Liouville theory in the unit disk. Finally we will discuss applications in random matrix theory, asymptotics of the maximum of the GFF, and tail expansions of GMC.


Oren Louidor
Dynamical freezing in a spin-glass with logarithmic correlations.

We consider a continuous time random walk on the 2D torus, governed by the exponential of the discrete Gaussian free field acting as potential. This process can be viewed as Glauber Dynamics for a spin-glass system with logarithmic correlations. Taking temperature to be below the freezing point, we then study this process both at pre-equilibrium and in-equilibrium time scales. In the former case, we show that the system exhibits aging and recover the arcsine law as asymptotics for a natural two point temporal correlation function. In the latter case, we show that the dynamics admits a functional scaling limit, with the limit given by a variant of Kolmogorov's K-process, driven by the limiting extremal process of the field, or alternatively, by a super-critical Liouville Brownian motion.
Joint work with A. Cortines, J. Gold and A. Svejda.


Juhan Aru
3 ways to construct the critical GMC measure for the 2D GFF

I will try to explain how a recent way of seeing the GMC measure for the 2D continuum GFF as a multiplicative cascade allows us to give a relatively simple treatment of the critical regime. We discuss three constructions: 1) via the derivative martingale 2) using Seneta-Heyde rescaling 3) by taking a certain limit of the subcritical measures.
This is joint work with E. Powell and A. Sepúlveda.