# Past Probability Seminars Spring 2010

# UW Math Probability Seminar Spring 2010

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.

Organized by Benedek Valk�

## Schedule and Abstracts

|| Thursday, January 21 || || * Daniel Remenik, * Cornell University || || * Brunet-Derrida particle systems, free boundary problems and Wiener-Hopf equations* ||

We consider a branching-selection system in R with N particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased, keeping the number of particles constant. We show that, as N\to\infty, the empirical measure process associated to the system converges in distribution to a deterministic measure-valued process whose densities solve a free boundary integro-differential equation. We also show that this equation has a unique traveling wave solution traveling at speed $c$ or no such solution depending on whether c<=a or c< a, where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process. The traveling wave solutions correspond to solutions of Wiener-Hopf equations. This is joint work with Rick Durrett.

|| Thursday, January 28 || || * Benedek Valk�* UW - Madison || || * Random tridiagonal matrices, beta ensembles and random Schr�dinger operators* ||

For a certain class of random tridiagonal matrices the scaling limit of the eigenvalue equations lead to a system of stochastic differential equations. This can be used to show the existence of the bulk scaling limit of the spectrum and to describe and analyze the limiting point process. I will discuss some of the applications of the method: the scaling limits of beta generalizations of classical random matrix ensembles and also the bulk limit of the spectrum of certain random Schr�dinger operators. (Parts of this work are joint with B. Vir�g, E. Kritchevski and S. Jacquot)

|| Thursday, February 4 || || * Timo Sepp�l�inen, * UW Madison || || * Scaling exponents for a one-dimensional directed polymer* ||

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.

|| Thursday, February 11 || || * Arnab Ganguly, * UW Madison || || * Large deviation principle for stochastic integrals and stochastic differential equations driven by infinite dimensional semimartingales* ||

Let {Y_n} be a sequence of 'infinite dimensional' semimartingales, and {X_n} be a sequence of 'infinite dimensional', adapted, cadlag processes. Assuming that {(X_n,Y_n)} satisfy a large deviation principle (LDP), we describe a uniform exponential tightness condition, under which a LDP holds for the the stochastic integral {X_{n-}.Y_n}. We apply the result to obtain large deviation principles for solutions of stochastic differential equations.

|| Thursday, February 18 || || * Nicos Georgiou, * UW Madison || || * TASEP(s) with discontinuous jump rates* ||

We present an almost sure hydrodynamic limit for the particle density (and current) of a sequence of 1-dimensional TASEPs with discontinuous jump rates that are given by a positive lower semi-continuous speed function. This leads to a connection with scalar conservation laws with discontinuous coefficients. The proof depends on evaluating the last passage time in a wedge of the planar lattice with appropriate exponential weights. As a corollary of this (and as a standing example throughout the talk) we will present the two-phase Corner Growth Model. This is joint work with Timo Seppalainen and Rohini Kumar.

|| Thursday, February 25 || || * Jonathon Peterson, * Cornell University || || * Maximal displacement of bridges for transient random walks in a random environment * ||

A bridge of a random walk is a path of length $2n$ that begins and ends at the origin. We are interested in studying the distribution of the maximal displacement of bridges for random walks. That is, conditioned on the random walk being back at the origin after $2n$ steps, how far did the random walk move away from the origin in the first $2n$ steps? It is easy to see that for any simple one-dimensional random walk the maximal displacement of bridges is of the order $\sqrt{n}$. We are interested in studying the distribution of bridges for transient one-dimensional random walks in a random environment. It turns out that in this case the maximal displacement of bridges is of the order $n^\beta$ for some $\beta$ which depends only on the probability distribution of the environments. Moreover, any $\beta$ between $0$ and $1$ is possible. In this talk I will explain how this result is obtained by using what is called the ``potential* of the environment to study the trapping effects of the environment.*

|| Thursday, March 4 || || * Tiefeng Jiang, * University of Minnesota || || * Moments of Traces for Circular Beta-ensembles * ||

Let x_1, ..., x_n be random variables from Dyson's circular beta-ensemble with probability density function \prod |e^{i x_j} - e^{i x_k}|^{beta}. For each n \geq 2 and beta>0, we obtain inequalities on expectations of p_{mu}(Z_n), where Z_n=(e^{i x_1}, \cdots, e^{i x_n}) and p_{mu} is the power-sum symmetric function for partition mu. When beta=2, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have limit results on the moments. These results apply to the three important ensembles: Circular Orthogonal Ensemble (beta=1), Circular Unitary Ensemble (beta=2) and Circular Symplectic Ensemble (beta=4). The main tool is the Jack function. This is a joint work with Sho Matsumoto.

|| Thursday, March 11 || || * NO SEMINAR* || || * * ||

|| Thursday, March 18 || || * B�lint T�th, * Technical University Budapest || || * Long time asymptotic behaviour of self-repelling random processes * ||

I will present a survey of recent results about the long time asymptotic behaviour of random processes with long memory due to some rather natural local self-intaraction (self-repellence) of the trajectories. Typical examples are the so-called myopic (or "true") self-avoiding random walk and the self-repelling Brownian polymer models. The long time asymptotics of the displacement is expected to be robust (not depending on some microscopic details), but dimension dependent. It is expected that in 1d the motion is strongly superdiffusive, with time-to-the-two-thirds scaling; in 2d the motion is marginally superdiffusive with logarithmic multiplicative correction in the scaling; in three and more dimensions the displacement is diffusive. For some particular models some of these have been recently proved.

|| Thursday, March 25 || || * NO SEMINAR* || || * * ||

|| Thursday, April 1 || || * SPRING BREAK (NO SEMINAR)* || || * * ||

|| Thursday, April 8 || || * L�szl� Erd�s, * Ludwig-Maximilians-Universit�t M�nchen || || *Universality of Wigner random matrices * ||

We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric or hermitian random matrices with independent, identically distributed entries (Wigner matrices) where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. Our main result is that the correlation functions of the local eigenvalue statistics in the bulk of the spectrum coincide with those of the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE), respectively, in the limit $N\to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. As a main input, we establish that the density of eigenvalues converges to the Wigner semicircle law and this holds even down to the smallest possible scale, and, moreover, we show that eigenvectors are fully delocalized. These results hold even without the condition that the matrix elements are identically distributed, only independence is used.

|| Thursday, April 15 || || * Nawaf Bou-Rabee, * Courant Institute, NYU || || *Non-asymptotic Mixing Time of the MALA Algorithm * ||

The Metropolis-Adjusted Langevin Algorithm (MALA), originally introduced to sample exactly the invariant measure of certain stochastic differential equations (SDE) on infinite-time intervals, can also be used to approximate pathwise the solution of these SDEs on finite-time intervals. However, when applied to an SDE with a nonglobally Lipschitz drift coefficient, MALA may not have a spectral gap even when the SDE does. This talk reconciles MALA's lack of a spectral gap with MALA's ergodicity to the invariant measure of the SDE and finite-time accuracy to its solution. In particular, the talk shows that the convergence of MALA to its equilibrium distribution happens at exponential rate up to terms exponentially small in time-stepsize. This result relies on MALA exactly preserving the invariant measure of the SDE and accurately representing its transition probability on finite-time intervals.

|| Friday, April 23, 2:25pm, VV B123 || || * Sunder Sethuraman, * Iowa State University || || *On some limits for a class of preferential attachment graphs * ||

In this talk, we will discuss an embedding into branching processes of some preferential attachment growth schemes from which a LLN, first proved by Bollobas et al 2001, can be recovered, among other limits. This embedding differs from the nice one given by Rudas-Toth-Valko 2007 in that it allows random edge additions, albeit with respect to `linear weights.' We will also discuss large deviations from this LLN with respect to the number of `leaves' (vertices with degree 1).

|| Thursday, April 29 || || * Firas Rassoul-Agha, * Unversity of Utah || || *Almost sure process-level large deviations for random walk in random environment * ||

Random walk in random environment is a generalization of classical random walk. It accounts for the disordered medium in which the particle travels and with which it interacts. The usual limit theorems become more subtle and much harder to prove. We will prove a level-3 large deviation principle, for almost every realization of the environment, with rate function related to an entropy. This is joint work with Timo Seppalainen.