## September 09, 2009

### "Two Problems on Stirring Processes --- and their Solutions" (This Week at the Statistics Seminar)

Attention conservation notice: Irrelevant unless you are (a) interested in interacting particle systems and/or stochastic processes on graphs, and (b) in Pittsburgh.

This week (in fact, tomorrow!) at the statistics seminar:

Thomas M. Liggett, "Two Problems on Stirring Processes — and their Solutions"
Abstract: Consider a finite or infinite graph G=(V,E), together with an assignment of nonnegative rates ce to e \in E. For each edge e, there is a Poisson process \Pie of rate ce. A Markov process is defined on this structure by placing labels on V, and then interchanging the labels at the two vertices joined by e at the event times of \Pie. If one considers the motion of just one label, this is a random walk on V. If all labels are 0 or 1, it is the symmetric exclusion process on V. If all labels are distinct, it is a random walk on the set of permutations of V. In this talk, I will describe recent solutions to two problems about these processes:
1. Suppose G=Z1 and ce=1 for each edge. Consider the symmetric exclusion process starting with the configuration ... 1 1 1 0 0 0 .... , and let Nt be the number of 1's to the right of the origin at time t. This is sum of negatively correlated Bernoulli random variables. In 2000, R. Pemantle asked whether Nt satisfies a central limit theorem. I will explain how a new negative dependence concept leads to a a positive answer to this question. This is partly based on joint work with J. Borcea and P. Brändén.
2. Suppose G is the complete graph on n vertices, and consider both the random walk on V and the random walk on the set of permutations of V. Each is a reversible, finite state, Markov chain, with n and n! states respectively. The exponential rate of convergence to equilibrium (which is the uniform distribution) for such a chain is determined by the smallest non-zero eigenvalue of -Q, where Q is the transition rate matrix of the chain. Let l1 and l2 be these values for the two processes. It is elementary that l2 =< l1. Based on explicit computations in some special cases, D. Aldous conjectured in 1992 that l1=l2. I will describe some elements of the approach that leads to a proof of this conjecture. This is joint work with P. Caputo and T. Richthammer.
Time and Place: Thursday, 10 September 2009, 4:30--5:30 pm, Giant Eagle Auditorium (Baker Hall A53)

The seminar is free and open to the public.

Posted at September 09, 2009 14:24 | permanent link