"Negative Dependence and the Symmetric Exclusion Process"
Thomas Liggett, UCLA
Over the past several years, several conjectures related to negative dependence
of Bernoulli random variables have been made. Among them are: (a) the Rayleigh property
(also known as the hereditary negative lattice condition after application of
external fields) implies the ultra logconcavity (ULC) of the rank sequence, (b) the Rayleigh
property implies negative association, and (c) the symmetric exclusion process with
product initial distribution is negatively associated at positive times. We will discuss
these and other conjectures. Among the results: (a) is false and (c) is true, while (b)
is still open. Furthermore, a stronger form of the Rayleigh property does imply both ULC and
negative association. As a consequence of (c), we obtain distributional limit theorems
for certain functionals of the symmetric exclusion process.
Much of this is joint work with J. Borcea and P. Branden.