# Past Probability Seminars Spring 2008

## Contents

## UW Math Probability Seminar Spring 2008

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

Organized by Tom Kurtz

### Schedule and Abstracts

|| Thursday, January 24, 2008 || || No Seminar || || ||

|| Thursday, January 31, 2008 || || *Fedor Nazarov*, University of Wisconsin-Madison || || *Zeroes of Gaussian analytic functions* ||

|| Thursday, February 7, 2008 || || *Firas Rassoul-Agha, *University of Utah*� * || || *On the almost-sure invariance principle for random walk in random environment* ||

Consider a crystal formed of two types of atoms placed at the nodes of the integer lattice. The type of each atom is chosen at random, but the crystal is statistically shift-invariant. Consider next an electron hopping from atom to atom. This electron performs a random walk on the integer lattice with randomly chosen transition probabilities (since the configuration seen by the electron is different at each lattice site).� This process is highly non-Markovian, due to the interaction between the walk and the environment.

We will present a martingale approach to proving the invariance principle (i.e. Gaussian fluctuations from the mean) for such a process (called random walk in random environment).

This is joint work with Timo Seppalainen.

|| Thursday, February 14, 2008 || || *Ankit Gupta*, University of Wisconsin-Madison || || *Stochastic model for concentration in Yeast cell* ||

We study the model proposed by Altschuler, Angenent and Wu for the diffusion of particles in a yeast cell. For any finite population size N we model the phenomenon as a discrete Markov chain. We show that under suitable scaling these discrete particle processes converge to a measure valued diffusion process as N goes to infinity.� We characterize its stationary distribution and draw inferences about localization of particles on the membrane in the infinite population limit.

|| Thursday, February 21, 2008 || || *Tom Kurtz*, University of Wisconsin-Madison || || *Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type* ||

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes.

|| Thursday, February 28, 2008 || || No Seminar || || ||

|| Thursday, March 6, 2008 || || *Jonathan Mattingly*, Duke University || || *Troubles with a chain of stochastic oscillators* ||

I will describe a simple chain of coupled, stochastically forced oscillators used to model heat conduction. Eventually one would like to understand the structure of the energy flux at equilibrium.� I will focus on the less lofty goal of showing that there is an equilibrium. This turns out to be surprisingly difficult. The results I will present make use of a detailed analysis of the multiscale in time structure of the problem.

|| Thursday, March 13, 2008 || || *David Anderson,* University of Wisconsin-Madison || || *Product form stationary distributions for deficiency zero chemical **reaction networks* ||

The dynamics of chemical reaction networks can be modeled either deterministically or stochastically.� The deficiency zero theorem for deterministically modeled systems gives easily checked conditions under which a unique equilibrium value with strictly positive components exists within each stoichiometric compatibility class (subset of Euclidean space in which trajectories are bounded).� The conditions of the theorem actually imply the stronger result that there exist concentrations for which the network is ``complex balanced.*� That observation in turn implies that the standard stochastic model for the reaction network has a product form stationary distribution.*

|| Thursday, March 20, 2008 || || No Seminar || || Spring Break ||

|| Thursday, March 27, 2008 || || No Seminar || || ||

|| Thursday, April 3, 2008 || || *Joel Zinn*, Texas A&M University || || *Probability Inequalities: Some old and new conjectures* ||

I will discuss the Gaussian Correlation Conjecture. Starting with a discussion of known cases of this conjecture and ending with a newer approach. If there is time, I would also like to discuss a recent, strange inequality of Mark Brown.

|| Thursday, April 10, 2008 || || || || ||

|| Thursday, April 17, 2008 || || *Brian Rider*, University of Colorado || || *Diffusion at the random matrix hard edge* ||

The "hard edge" refers to the minimal eigenvalues of a (natural) one-parameter generalization of Gaussian sample covariance matrices.� We show that, in the large dimensional limit, the law of these points is shared by that of the spectrum of a certain random second-order differential operator.� The latter may be viewed as the generator of a Brownian motion with white noise drift.� By a Riccati transformation we obtain a second diffusion description of the hard edge in terms of hitting times.� (Joint with J. Ramirez)

|| Thursday, April 24, 2008 || || *Jim Kuelbs*, University of Wisconsin-Madison || || *Asymptotic Inference for High Dimensional Data* ||

In this talk we present limit theorems for high dimensional data that is characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve situations in which (i) the number of parameters increase with the sample size (that is allowed to be random) and (ii) there is a possibility of missing data. Under a variety of tail conditions on the components of the data, we provide conditions for the law of large numbers, as well as various results concerning the rate of convergence in these models.� We also present central limit theorems in this setting, some which involve data driven coordinate-wise normalizations.� (Joint with Anand Vidyashankar)

|| Thursday, May 1, 20084:00 PM (NOTE change of time)B139 Van Vleck (NOTE change of room) || || *Amir Dembo*, Stanford University || || *Finite size scaling for the core of large random hyper-graphs*** ||

The (two) core of an hyper-graph is the maximal collection of �hyper-edges such that no vertex appears in only one of them. �It is of importance in tasks such as efficiently solving a large linear system� over GF(2), or iterative decoding of low-density parity-check codes used over the binary erasure channel. Similar structures emerge in a variety of NP-hard combinatorial optimization and decision problems, from vertex cover to satisfiability.

For a uniformly chosen random hyper-graph of M=r N vertices and N hyper-edges, each consisting of the same fixed number L > 2 of vertices, the size of the core exhibits for large N a first order phase transition, changing from o(N) for r>b to a positive fraction of N for r<b, with a transition window size of order 1/sqrt(N) around b.

Analyzing the corresponding `leaf removal' algorithm, we determine the associated finite size scaling behavior.� In particular, if r is inside the scaling window, the probability of a core whose size is a positive fraction of N has a limit strictly between 0 and 1, and a leading correction of order N^{-1/6}, explicitly characterized in terms of the distribution of a Brownian motion with quadratic drift, from which it inherits the scaling with N.

This talk is based on a joint work with Andrea Montanari.

- May 2-4, 2008***

- Keynote Speakers *

### ��������� Amir Dembo, Stanford University

### ��������� Davar Khoshnevisan, University of Utah.

|| Thursday, May 8, 2008 || || *Ruth Williams*, University of California, San Diego || || *Stochastic delay differential equations with state constraints* ||

Deterministic dynamic models with delayed feedback and state constraints arise in modeling Internet rate control and biochemical reactions involving transcription and translation.� Much of the analysis of such models has focused on local stability analysis of equilibrium points or on proving global asymptotic stability of such points.� There is interest in understanding when such systems can have sustained oscillations, and what effect noise has on system behavior.� Here we consider a one dimensional stochastic delay differential equation as a simple prototype model for a noisy deterministic system with delayed (negative) feedback. We obtain sufficient conditions for the noiseless model to have slowly oscillating periodic solutions and for the noisy model to have stationary solutions. (This talk is based on joint work with Michael Kinnally)