# Probability Seminar

# Spring 2014

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

**
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## Thursday, January 23, CANCELED--NO SEMINAR

## Thursday, February 6, Jay Newby, Mathematical Biosciences Institute

Title: Applications of large deviation theory in neuroscience

Abstract: The membrane voltage of a neuron is modeled with a piecewise deterministic stochastic process. The membrane voltage changes deterministically while the population of open ion channels, which allow current to flow across the membrane, is constant. Ion channels open and close randomly, and the transition rates depend on voltage, making the process nonlinear. In the limit of infinite transition rates, the process becomes deterministic. The deterministic process is the well known Morris-Lecar model. Under certain conditions, the deterministic process has one stable fixed point and is excitable. An excitable event, called an action potential, is a single large transient spike in voltage that eventually returns to the stable steady state. I will discuss recent development of large deviation theory to study noise induced action potentials.

## Thursday, February 13, Diane Holcomb, UW-Madison

Title: Large deviations for point process limits of random matrices.

Abstract: The Gaussian Unitary ensemble (GUE) is one of the most studied Hermitian random matrix model. When appropriately rescaled the eigenvalues in the bulk of the spectrum converge to a translation invariant limiting point process called the Sine process. On large intervals one expects the Sine process to have a number of points that is roughly the length of the interval times a fixed constant (the density of the process). We solve the large deviation problem which asks about the asymptotic probability of seeing a different density in a large interval as the size of the interval tends to infinity. Our proof works for a one-parameter family of models called beta-ensembles which contain the Gaussian orthogonal, unitary and symplectic ensembles as special cases.

## Thursday, February 20, Philip Matchett Wood, UW-Madison

Title: The empirical spectral distribution (ESD) of a fixed matrix plus small random noise.

Abstract: A fixed matrix has a distribution of eigenvalues in the complex
plane. Small random noise can be formed by a random matrix with iid mean 0
variance 1 entries scaled by [math]n^{-\gamma -1/2}[/math] for [math]\gamma \gt 0[/math], which
by itself has eigenvalues collapsing to the origin. What happens to the
eigenvalues when you add a small random noise matrix to the fixed matrix?
There are interesting cases where the eigenvalue distribution is known to
change dramatically when small Gaussian random noise is added, and this talk
will focus on what happens when the noise is *not* Gaussian..

## Thursday, February 27, Jack Hanson, Indiana University Bloomington

Title: **Subdiffusive Fluctuations in First-Passage Percolation**

Abstract: First-passage percolation is a model consisting of a random metric t(x,y) generated by random variables associated to edges of a graph. Many questions and conjectures in this model revolve around the fluctuating properties of this metric on the graph Z^d. In the early 1990s, Kesten showed an upper bound of Cn for the variance of t(0,nx); this was improved to Cn/log(n) by Benjamini-Kalai-Schramm and Benaim-Rossignol for particular choices of distribution. I will discuss recent work (with M. Damron and P. Sosoe) extending this upper bound to general classes of distributions.

## Thursday, March 20, No Seminar due to Spring Break

## Thursday, March 27, Cécile Ané, UW-Madison Department of Statistics

Title: ** Application of a birth-death process to model gene gains and losses on a phylogenetic tree **

Abstract: Over time, genes can duplicate or be lost. The history of a gene family is a tree whose nodes represent duplications, speciations, or losses. A birth-and-death process is used to model this gene family tree, embedded within a species tree. I will present this phylogenetic version of the birth and death tree process, along with a probability model for whole-genome duplications. If there is interest and time, I will talk about learning birth and death rates and detecting ancient whole-genome duplications from genomic data.

## Thursday, April 10, Dan Romik UC-Davis

Title: **Connectivity patterns in loop percolation and pipe percolation**

Abstract:

Loop percolation is a random collection of closed cycles in the square lattice Z^2, that is closely related to critical bond percolation. Its "connectivity pattern" is a random noncrossing matching associated with a loop percolation configuration that encodes information about connectivity of endpoints. The same probability measure on noncrossing matchings arises in several different and seemingly unrelated settings, for example in connection with alternating sign matrices, the quantum XXZ spin chain, and another type of percolation model called pipe percolation. In the talk I will describe some of these connections and discuss some results about the study of pipe percolation from the point of view of the theory of interacting particle systems. I will also mention the "rationality phenomenon" which causes the probabilities of certain natural connectivity events to be dyadic rational numbers such as 3/8, 97/512 and 59/1024. The reasons for this are not completely understood and are related to certain algebraic conjectures that I will discuss separately in Friday's talk in the Applied Algebra seminar.

## Thursday, May 1, Antonio Auffinger U Chicago

Title: *Strict Convexity of the Parisi Functional*

Spin glasses are magnetic systems exhibiting both quenched disorder and frustration, and have often been cited as examples of "complex systems." As mathematical objects, they provide several fascinating structures and conjectures. This talk will cover recent progress that shed more light in the mysterious and beautiful solution proposed 30 years ago by G. Parisi. We will focus on properties of the free energy of the famous famous Sherrington-Kirkpatrick model and we will explain a recent proof of the strict convexity of the Parisi functional. Based on a joint work with Wei-Kuo Chen.

## Thursday, May 8, Steve Goldstein, WID