MiniWorkshop on LogCorrelated Random Fields
Date: December 1214, 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
LouisPierre 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 
9:4510:45

Sheffield

Rhodes 
Bourgade

10:4511

coffee

coffee 
coffee 
1112

Lupu 
Subag 
Remy

122

lunch

lunch 
lunch 
23

Arguin

Ding

Louidor

33:15

coffee

coffee

coffee

3:154:15

Hartung 
Holden

Aru

Abstracts
Scott Sheffield
Random walks in "scalefree" 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
SLEdecorated 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 nonGaussian.
LouisPierre 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
logcorrelated 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 orderone 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
DuplantierMillerSheffield. 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 highdimensional 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 matingoftrees 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 nontrivial 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 nonuniform random planar maps. Both parts of the talk are
based on a matingoftrees 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 HorngTzer Yau.
Guilaume Rémy
The FyodorovBouchaud 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 onepoint
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 spinglass 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 spinglass system with logarithmic correlations.
Taking temperature to be below the freezing point, we then study
this process both at preequilibrium and inequilibrium 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 Kprocess, driven by the
limiting extremal process of the field, or alternatively, by a
supercritical 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 SenetaHeyde rescaling 3) by taking a certain
limit of the subcritical measures.
This is joint work with E. Powell and A. Sepúlveda.