# Beyond the Gaussian Universality Class

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The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers, particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.

# Distributional limits for the symmetric exclusion process.

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Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and Gaussian distributions for various functionals of the process.

# The one-dimensional Kadar-Parisi-Zhang equation and universal height statistics.

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The KPZ equation is a stochastic PDE describing the motion of an interface

between a stable and an unstable phase. We will discuss solutions of the one-dimensional

equation with sharp wedge initial conditions. For long times the Tracy-Widom distribution

of GUE random matrices is recovered.

The talk is based on recent joint work with Tomohiro Sasamoto.

# Hermitian random matrix model with spiked external source

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If a random Hermitian Gaussian matrix (GUE matrix) is perturbed additively by a matrix of small rank, the largest eigenvalue undergoes a transition depending on the spectrum of the added matrix. We consider a generalization of this case with general potential. When the potential is convex, the transition phenomenon is universal. However, for non-convex potentials, new types of transition may occur. This is a joint work with Dong Wang.

# Gaussian fluctuations for Plancherel partitions

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The limit shape of Young diagrams under the Plancherel

measure was found by Vershik \& Kerov (1977) and Logan \& Shepp

(1977). We obtain a central limit theorem for fluctuations of Young

diagrams in the bulk of the partition '`spectrum''. More

specifically, under a suitable (logarithmic) normalization, the

corresponding random process converges (in the FDD sense) to a

Gaussian process with independent values. We also discuss a link

with an earlier result by Kerov (1993) on the convergence to a

generalized Gaussian process. The proof is based on poissonization

of the Plancherel measure and an application of a general central

limit theorem for determinantal point processes. (Joint work with

Zhonggen Su.)

# scaling exponents for a one-dimensional directed polymer

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We study a 1+1-dimensional directed polymer in a random

environment on the integer lattice with log-gamma distributed

weights and both endpoints of the polymer path fixed.

We show that under appropriate boundary conditions

the fluctuation exponents for the free energy and

the polymer path take the values conjectured in the

theoretical physics literature. Without the boundary

we get the conjectured upped bounds on the exponents.

# A propagation-of-chaos type result in stochastic averaging

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Stochastic averaging goes back to Khasminskii in the 1960's. The

standard result is that, given a separation of scales, one can find effective dynamics

for slow components. We investigate the motion of two particles in such a system, in

particular in a randomly-perturbed twist map. The nub of the issue

is how two points escape from a 1-1 resonance zone. Results of Pinsky

and Wihstutz indicate that there is a third scale at work, which we can use to study

the escape from resonance.

# The Ghirlanda-Guerra identities and ultrametricity in the Sherrington-Kirkpatrick model.

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The Parisi theory of the Sherrington-Kirkpatrick model completely describes the geometry of the Gibbs sample in a sense that it predicts the limiting joint distribution of all scalar products, or overlaps, between i.i.d. replicas. One of the main predictions is that asymptotically the Gibbs measure concentrates on an ultrametric subset of all spin configurations. Another part of the theory are the Ghirlanda-Guerra identities which in various formulations have been proved rigorously. It is well known that together these two properties completely determine the joint distribution of the overlaps and for this reason they were always considered complementary. We show that in the case when overlaps take finitely many values the Ghirlanda-Guerra identities actually imply ultrametricity.

# On the existence and position of the farthest peaks of a family of stochastic heat and wave equations.

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We study the stochastic heat equation &#8706;tu = u+&#963;(u)w in (1+1) dimensions, where w is space-time white noise, &#963;:R&#8594;R is Lipschitz continuous, and is the generator of a Lvy process. We assume that the underlying Lvy process has finite exponential moments in a neighborhood of the origin and u0 has exponential decay at &#8734;. Then we prove that under natural conditions on &#963;: (i) The &#957;th absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.

This is joint work with Daniel Conus (University of Utah)