Beyond the Gaussian Universality Class

Speaker: 

Professor Ivan Corwin

Institution: 

NYU

Time: 

Thursday, December 2, 2010 - 11:00am

Location: 

340P

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.

Speaker: 

Professor Thomas Liggett

Institution: 

UCLA

Time: 

Tuesday, November 9, 2010 - 11:00am

Location: 

RH 306

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.

Speaker: 

Professor Herbert Spohn

Institution: 

TU Muenchen

Time: 

Tuesday, November 30, 2010 - 11:00am

Location: 

RH 306

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

Speaker: 

Professor Jinho Baik

Institution: 

University of Michigan

Time: 

Tuesday, October 26, 2010 - 11:00am

Location: 

RH 306

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

Speaker: 

Professor Leonid Bogachev

Institution: 

University of Leeds, UK

Time: 

Tuesday, October 12, 2010 - 11:00am

Location: 

RH 306

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

Speaker: 

Professor Timo Seppalainen

Institution: 

University of Wisconsin

Time: 

Wednesday, June 2, 2010 - 2:00pm

Location: 

MSTB 114

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

Speaker: 

Professor Richard Sowers

Institution: 

University of Illinois

Time: 

Tuesday, May 25, 2010 - 11:00am

Location: 

RH 306

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.

Speaker: 

Professor Dmitry Panchenko

Institution: 

Texas A&M

Time: 

Monday, May 10, 2010 - 11:00am

Location: 

RH 306

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.

Speaker: 

Professor Davar Khoshnevisan

Institution: 

University of Utah

Time: 

Tuesday, April 20, 2010 - 11:00am

Location: 

RH 306

We study the stochastic heat equation ∂tu = u+σ(u)w in (1+1) dimensions, where w is space-time white noise, σ:R→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 ∞. Then we prove that under natural conditions on σ: (i) The νth 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)

Pages

Subscribe to RSS - Combinatorics and Probability