# Probability Seminar

## Spring 2013

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. If you would like to receive announcements about upcoming seminars, please visit this page to sign up for the email list.

## Thursday, January 31, Bret Larget, UW-Madison

Title: Approximate conditional independence of separated subtrees and phylogenetic inference

Abstract: Bayesian methods to reconstruct evolutionary trees from aligned DNA sequence data from different species depend on Markov chain Monte Carlo sampling of phylogenetic trees from a posterior distribution. The probabilities of tree topologies are typically estimated with the simple relative frequencies of the trees in the sample. When the posterior distribution is spread thinly over a very large number of trees, the simple relative frequencies from finite samples are often inaccurate estimates of the posterior probabilities for many trees. We present a new method for estimating the posterior distribution on the space of trees from samples based on the approximation of conditional independence between subtrees given their separation by an edge in the tree. This approximation procedure effectively spreads the estimated posterior distribution from the sampled trees to the larger set of trees that contain clades (sets of species in subtrees) that have been sampled, even if the full tree is not part of the sample. The approximation is shown to be accurate for many data sets and is theoretically justified. We also explore a consequence of this result that may lead to substantial increases in computational efficiency for sampling trees from posterior distributions. Finally, we present an open problem to compare rates of convergence between the simple relative frequency approach and the approximation approach.

## Thursday, February 14, Jean-Luc Thiffeault, UW-Madison

Title: Biomixing and large deviations

Abstract: As fish, micro-organisms, or other bodies move through a fluid, they stir their surroundings. This can be beneficial to some fish, since the plankton they eat depends on a well-stirred medium to feed on nutrients. Bacterial colonies also stir their environment, and this is even more crucial for them since at small scales there is no turbulence to help mixing. I will discuss a simple model of the stirring action of moving bodies through a fluid. An attempt will be made to explain existing data on the displacements of small particles, which exhibits probability densities with exponential tails. A large-deviation approach helps to explain some of the data, but mysteries remain.

## Tuesday, March 5, 2:30pm VV B341, Janosch Ortmann, University of Toronto

Title: Product-form Invariant Measures for Brownian Motion with Drift Satisfying a Skew-symmetry Type Condition

Abstract: Motivated by recent developments on positive-temperature polymer models we propose a generalisation of reflected Brownian motion (RBM) in a polyhedral domain. Our process is obtained by replacing the singular drift on the boundary by a continuous one which depends, via a potential U, on the position of the process relative to the domain. We show that our generalised process has an invariant measure in product form, under a certain skew-symmetry condition that is independent of the choice of potential. Applications include TASEP-like particle systems, generalisations of Brownian motion with rank-dependent drift and diffusions connected to the generalised Pitman transform.

## Thursday, March 14, Brian Rider, Temple University

Title: Universality for the stochastic Airy operator

Abstract: The stochastic Airy operator (SAO) has the form second derivative plus shifted white noise potential. Its reason for being is that it describes the Tracy-Widom laws extended to "general beta" (from the original beta=1,2,4 laws tied to real, complex, and quaternion symmetries). More to the point, SAO is known to be the operator limit for certain random tridiagonal matrices which realize, for example, log-gas distributions on the line with quadratic potential (the "beta Hermite ensembles"), scaled to the edge of their spectrum. Here we show that SAO characterizes edge universality for a more general class of log-gases, defined by more general polynomial potentials beyond the quadratic case. Joint work with M. Krishnapur and B. Virag.

## Thursday, March 21, Timo Seppalainen (UW Madison)

Title: Limits of ratios of partition functions for the log-gamma polymer

Abstract: For the model known as the directed polymer in a random medium, the definition of weak disorder is that normalized partition functions converge to a positive limit. In strong disorder this limit vanishes. In the log-gamma polymer we can show that ratios of point-to-point and point-to-line partition functions converge to gamma-distributed limits. One consequence of this is that the quenched polymer measure converges to a random walk in a correlated random environment. This RWRE can be regarded as a positive temperature analogue of the competition interface of last-passage percolation, or the second class particle.

## Thursday, April 11, Kevin Lin, University of Arizona

Title: TBA

Abstract: TBA

## Thursday, April 18, Lee DeVille, University of Illinois

Title: TBA

Abstract: TBA

## Thursday, April 25, Fraydoun Rezakhanlou, UC - Berkeley

Title: TBA

Abstract: TBA

## Wednesday, May 1, Bálint Vető, University of Bonn

Title: TBA

Abstract: TBA