# UW Math Probability Seminar Spring 2011

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

Organized by Benedek Valkó

## Schedule and Abstracts

|| Monday, January 24, 2:25PM, (B129 Van Vleck) || || Note the unusual time and place * Sunil Chhita, * Brown University || || * Particle Systems arising from an Anti-ferromagnetic Ising Model* ||

We present a low temperature anisotropic anti-ferromagnetic 2D Ising model through the guise of a certain dimer model. This model also has a bijection with a one-dimensional particle system equipped with creation and annihilation. We can find the exact phase diagram, which determines two significant values (the independent and critical value). We also highlight some of the behavior of the model in the scaling window at criticality and at independence.

|| Thursday, February 10, || || * Alex Bloemendal, * University of Toronto || || * Finite rank perturbations of large random matrices * ||

Finite (or fixed) rank perturbations of large random matrices arise in a number of applications. The main phenomenon is a phase transition in the largest eigenvalues as a function of the strength of the perturbation. I will describe joint work with Balint Virag in which we introduce a new way to study these models. The starting point is a reduction to a natural band form; under the soft edge scaling, it converges to a souped-up version of the known continuum random Schr�dinger operator on the half-line. We describe the near-critical fluctuations in several ways, solving a known open problem in the real case. One characterization also yields a new route to the Painlev� structure in the celebrated Tracy-Widom laws.

|| Thursday, February 24, || || * Marton Balazs, * Technical University Budapest || || * Modelling flocks and prices: jumping particles with an attractive interaction * ||

I will introduce a model of a finite number of competing particles on R. Real-life phenomena that could be modeled this way includes the evolution of stocks in a market, or herding behavior of animals. Given a particle configuration, the center of mass of the particles is computed by simply averaging the particle locations. The evolution is a continuous time Markov jump process: given a configuration and thus the center of mass, each particle jumps with a rate that depends on the particle's relative position compared to the center of mass. Those left behind have a higher jump rate than those in front of the center of mass. When a jump of a particle occurs, the jump length is chosen independently of everything from a positive distribution. Hence we see that the dynamics tries to keep the particles together.

The main point of interest is the behavior of the model as the number of particles goes to infinity. We first heuristically wrote up a differential equation on the evolution of particle density. I will explain the heuristics, and show traveling wave solutions in a few cases. I will also present a surprising connection to extreme value statistics. Then I will briefly sketch a hydrodynamic argument which proves that the evolution of the system indeed converges to that governed by the differential equation.
(Joint work with Miklos Racz and Balint Toth)


|| Wednesday, March 2, 3:30PM, (B115 Van Vleck) || || Note the unusual time and place * Alan Hammond, * Oxford || || * The sharpness of the phase transition for speed for biased walk in supercritical percolation* ||

I will discuss a joint work with Alex Fribergh in which we study the biased random walk on the infinite cluster of supercritical percolation. Fixing any $d \geq 2$ and supercritical parameter $p >p_c$, the model has a parameter $\lambda > 0$ for the degree of bias of the walker in a certain preferred direction (which is another parameter, in $S^{d-1}$). We prove that the model has a sharp phase transition, that is, that there exists a critical value $\lambda_c > 0$ of the bias such that the walk moves at positive speed if $lambda < \lambda_c$ and at zero speed if $\lambda > \lambda_c$. This means that a stronger preference for the walker to move in a given direction actually causes the walk to slow down. The reason for this effect is a trapping phenomenon, and, as I will explain, our result is intimately tied to understanding the random geometry of the local environment that is trapping the particle at late time in the case when motion is sub-ballistic.

|| March 31, Thursday || || * Stefan Grosskinsky, * Warwick || || * Zero-range condensation at criticality* ||

Zero-range processes with decreasing jump rates exhibit a condensation transition, where a positive fraction of all particles condenses on a single lattice site when the total density exceeds a critical value. We study the onset of condensation at criticality, and establish a law of large numbers for the excess mass fraction in the maximum, which turns out to jump from zero to a positive value. Our results also include distributional limits for the fluctuations of the maximum, changing from standard extreme value statistics to Gaussian when the density crosses the critical point, as well as for the fluctuations of the bulk, showing that the mass outside the maximum is distributed homogeneously. The rigorous limit theorems can be extended heuristically to understand finite size effects and metastable dynamics in applications, which is confirmed by simulation results.

This is joint work with Ines Armendariz, Michalis Loulakis and Paul Chleboun.


|| Wednesday, April 6, 3:30PM, (B115 Van Vleck) || || Note the unusual time and place || || * Richard Sowers, * University of Illinois at Urbana-Champaign || || * Very Hard to Borrow Stocks* ||

We take a look at a possible dynamics for very hard-to-borrow (i.e., bankrupt) stocks. The dynamics are 2-dimensional, and stems from a model proposed by Avellaneda and Lipkin for hard-to-borrow stocks. We motivate and understand the model. We understand how to calibrate it and how to price puts on the stocks. This is work in progress with Xiao Li and Mike Lipkin.

|| Thursday, April 14, || || * Janos Englander, * University of Colorado - Boulder || || * Some particle models with self interaction and in random environment * ||

Recently a number of particle models have been studied where individuals move in space and also interact via the center of the system (given by the center of mass). I will review some of my results as well as those of Gill, Balazs and Racz. Time permitting, I will also report on some (simulation) results concerning a branching random walk in a random "cookie" environment. This latter work is joint with N. Sieben.

|| Thursday, April 21, || || * Nicos Georgiou, * UW - Madison || || * Large deviations for directed polymers* ||

We present recent results about large deviations for directed polymers in random environment. In a completely general setting, we prove quenched large deviations (and compute the rate function explicitly) for the exit point of the polymer chain and the polymer chain itself. We also prove existence of the upper tail large deviation rate function for the logarithm of the partition function. In the case where the environment weights have certain log-gamma distributions the computations are tractable and allow us to compute the rate function explicitly. This is joint work with Timo Sepp�l�inen.

|| Thursday, April 28, || || * John Fricks, * Penn State || || * Multiple Scales in Molecular Motor Models * ||

Molecular motors, such as kinesin and dynein, carry cargos through a cell along a microtubule network. The heads of these motors step along a microtubule and are on the order of nanometers, while the cargo can be on the order of hundreds of nanometers. In addition, multiple motors may be connected to a single cargo. In this talk, we will discuss the application of semi-Markov models to link the actions of the heads to the stepping of the motor. In addition, we analyze multiple motors combined with the cargo across relevant temporal and spatial scales by performing a dimensional analysis and applying relevant averaging theorems.

|| Thursday, May 5, || || * Soumik Pal , * University of Washington || || * The Aldous diffusion on continuum trees* ||

Consider a Markov chain on the space of rooted binary trees that randomly removes leaves and reinserts them on a random edge. This chain was introduced by David Aldous in '99 who conjectured a diffusion limit of this chain, as the size of the tree grows, on the space of continuum trees. We talk about how to prove this conjecture. Our approach involves taking an explicit scaled limit which is novel in the area of Markov processes on real trees.

|| Thursday, May 12, || || * Jordan Ellenberg, * UW - Madison || || * Random l-adic matrices * ||

The notion of "random matrix in G(K)" makes sense whenever K is a ring or field nice measure, so that GL_n(K) has Haar measure. The usual theory of random matrices concerns itself with the case K = R or K = C. I will explain how problems in number theory lead one naturally to the case where K is a finite field like Z/lZ or the ring of l-adic integers Z_l. (I will explain what this means.) In many respects the situation ("what are the interesting questions, and what are the answers to those questions?") turns out to be largely parallel. In particular, one is always interested in the distribution of the eigenvalues of a large random matrix drawn from some algebraic group (symplectic, orthogonal, general linear...)