The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue
statistics of large random matrices in the bulk exhibit
universal behavior depending only on the symmetry
class of the matrix ensemble. For invariant matrix models, the eigenvalue distributions are given
by a log gas with a potential $V$ and inverse temperature $\beta = 1, 2, 4$, corresponding to
the orthogonal, unitary and symplectic ensembles.
The universality conjecture for invariant ensembles asserts that
the local eigenvalue statistics are independent of $V$ for all positive real $\beta$.
In this talk, we review our recent solution to the universality conjecture for both
invariant and non-invariant ensembles. The special role played by the logarithmic Sobolev inequality
and Dyson Brownian motion will be discussed.
The Gaussian free field is a random field that is associated with many random surface models. The goal of this talk is to explain how they arise from Pfaffian point processes.
I will introduce a measure on plane partitions that is a Pfaffian point process. I will show that the height fluctuations around the limit shape converge to a pullback of the Gaussian process whose covariance is given by the Green's function for the Laplacian with Dirichlet boundary conditions on the first quadrant.
The result was obtained by computing higher moments using the steepest descent analysis, based on an idea of Kenyon. The argument can be generalized to a class of Pfaffian processes whose kernels possess certain properties.
It has been observed by several authors that a large family of stochastic partial differential equations have
solutions that are highly intermittent; this means that the solution tends to develop tall peaks that are distributed over
small sets ["islands"]. In this talk we will argue that the solution of such stochastic PDEs can be "chaotic," and that this
property leads to the onset of intermittence.
This talk is based on joint works with Daniel Conus, Mathew Joseph, and Shang-Yuan Shiu.
For several classical matrix models the joint density of the eigenvalues can be written as an expression involving a Vandermonde determinant raised to the power of 1, 2 or 4. Most of these examples have beta-generalizations where this exponent is replaced by a parameter beta>0.
In recent years the point process limits of various beta ensembles have been derived. The limiting processes are usually described as the spectrum of certain stochastic operators or with the help of a coupled system of SDEs.
In the bulk beta Hermite case (which is the generalization of GUE) there is a nice geometric construction of the point process involving a Brownian motion in the hyperbolic plane, this is the Brownian carousel. Surprisingly, there are a number of other limit processes that have carousel like representation. We will discuss a couple of examples and some applications of these new representations.
Joint with Balint Virag.
We consider the model of cylinders' percolation in three dimensions. This model was introduced by I. Benjamini and consists of removing from R^d a set of infinite cylinders having radius one and axis given by the lines in the support of a Poisson point process in the space of lines of R^d.
An nonnegative intensity parameter u controls the amount of cylinders to be removed. In 2010, J. Tykesson and D. Windisch showed that for d>3 the connectivity of the complementary set undergoes a phase transition as u varies. They also showed that for d=3 and u large enough there is no percolation.
In this work we prove the existence of a percolative phase for d=3. More than that, we prove that the complementary set percolates in a sufficiently thick slab if u is small enough, in spite of the fact that its restriction to any 2-dimensional subspace does not percolate regardless of the value of u.
Joint work with A. Teixeira (ENS and IMPA) and V. Sidoravicius (IMPA).
We study some problems relating to polynomials evaluated either at random Gaussian or random Bernoulli inputs. We present some new work on a structure theorem for degree-d polynomials with Gaussian inputs. In particular, if p is a given degree-d polynomial, then p can be written in terms of some bounded number of other polynomials
q_1,...,q_m so that the joint probability density function of q_1(G),...,q_m(G) is close to being bounded. This says essentially that any abnormalities in the distribution of p(G) can be explained by the way in which p decomposes into the q_i. We then present some applications of this result.
The goal of the talk is to survey recent progress in understanding statistics of certain exactly solvable growth models, particle systems, directed polymers in one space dimension, and stochastic PDEs. A remarkable connection to representation theory and integrable systems is at the heart of Macdonald processes, which provide an overarching theory for this solvability. This is based off of joint work with Alexei Borodin.
We compute the almost-sure Hausdorff dimension of the set of points surrounded by more or fewer than the typical number of loops in a realization of the conformal loop ensemble. More precisely, this is the set of points z for which the number of loops surrounding the disk of radius r centered at z is (theta + o(1)) log(1/r), where theta is a nonnegative parameter. We will also discuss a generalization of this result in which the loops are assigned iid weights. Using a coupling of CLE with the Gaussian free field, this leads to a new way of understanding GFF extremes.
Joint work with Jason Miller and David Wilson.
The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that disperses like a linear solution, plus a localized component that behaves like a soliton or multi-soliton solution. Considered to be one of the fundamental problems in the area of nonlinear dispersive equations, this conjecture has eluded a proof or even a precise formulation till date. I will present a theorem that proves a "statistical version" of this conjecture at mass-subcritical nonlinearity. The proof involves a combination of techniques from large deviations, PDE, harmonic analysis and bare hands probability theory.
We are interested in various models of random plane non-crossing
configurations consisting of diagonals of convex polygons, such as uniform
triangulations, dissections, non-crossing partitions or non-crossing
trees. For all these models, we prove convergence in distribution towards
Aldous' Brownian triangulation of the disk. We will discuss interesting
combinatorial applications concerning the length of the longest diagonal
or the maximal vertex degree. This is joint work with Nicolas Curien.
We discuss the asymptotic analysis of vertex observables of dimer
height functions and their relations to correlators in the Ising
I will explain the construction of an interacting particle system which has a unique stationary state but which is not ergodic. In other words, the dynamics started from different initial states never converge to each other, while the stationary state of the dynamics is unique. The result can be interpreted as the case of the non-uniqueness of the steady state. Joint work with Christian Maes.
It is a know theorem that in low dimensional models of statistical mechanics, such as the Random Field Ising Model (RFIM) first order phase transitions are unstable with respect to the introduction of arbitrarily weak disorder in the field conjugate to the order parameter. The result (derived jointly with J. Wehr) will be explained in the lecture, along with its more recent generalization to quantum systems (derived jointly with R. Greenblatt and J. Lebowitz). The talk will however focus on the question whether there is, nevertheless, a phase transition in the the behavior of RFIM as function of the disorder strength. Its manifestation could be power law decay of correlations at weak disorder, which at high disorder changes to exponential decay. It is proposed that a relevant point of reference for the scaling limit of the ground state's ``sensitivity percolation'' could be Mandelbrot's percolation model. (Work in progress with Jack Hanson)
In a pioneering work, J.P. Kahane introduced the theory of
Gaussian Multiplicative Chaos. Such an object is roughly speaking a random measure with density w.r.t. the Lebesgue measure given by the exponential of a Gaussian log-correlated field up to a multiplicative factor, called the intermittency parameter. When the intermittency parameter is small enough, these measures are non trivial and are characterized by a functional equation, which we call star-equation. Beyond some critical value, these measures vanish. However the star equation admits non trivial solutions, which we call generalized Gaussian multiplicative chaos. It turns out that Gaussian multiplicative chaos and generalized Gaussian multiplicative chaos satisfy a duality relation together with the KPZ formula, giving precise mathematical meaning to the KPZ duality introduced by theoretical physicists.
In statistical physics, systems
like percolation and Ising models are of particular interest
at their critical points. Critical systems have long-range
correlations that typically decay like inverse powers. Their
continuum scaling limits are expected to have universal dimension-dependent properties.
Critical two-dimensional scaling limits
have been studied by Schramm, Lawler, Werner, Smirnov, Sheffield
and others with a focus on the boundaries of large clusters. In the
scaling limit these can be described by Schramm-Loewner Evolution
In this talk, I'll discuss a different but related approach,
which focuses on cluster area measures. In the case of the
two-dimensional Ising model, this leads to a representation of
the continuum Ising magnetization field in terms of sums of
certain measure ensembles with random signs. This is based
on joint work with F. Camia and on work in progress
with F. Camia and C. Garban.
We prove that the linearly reinforced random walk on a bounded degree graph with sufficiently small initial weights is recurrent. On non-amenable graphs we also establish transience for sufficiently large initial weights. The vertex reinforced jump process is also treated. Join with Nicholas Crawford and Gady Kozma.