### "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 *c*_{e} to *e* \in *E*. For each edge *e*,
there is a Poisson process \Pi_{e} of
rate *c*_{e}. 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 \Pi_{e}. 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*=*Z*^{1} and *c*_{e}=1 for
each edge. Consider the symmetric exclusion process starting with the
configuration ... 1 1 1 0 0 0 .... , and let *N*_{t} 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 *N*_{t} 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 *l*_{1} and
*l*_{2} be these values for the two processes. It is elementary
that *l*_{2} =< *l*_{1}. Based on explicit
computations in some special cases, D. Aldous conjectured in 1992 that
*l*_{1}=*l*_{2}. 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.

Enigmas of Chance;
Mathematics

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