Abstract: Over the past several years, a number of conjectures related to negative dependence of Bernoulli random variables X_i have been made by R. Pemantle, D. Wagner and others. Among them are:

(a) The Rayleigh property implies the ultra logconcavity (ULC) of the rank sequence a_k=P (sum X_i=k).

(b) The Rayleigh property implies negative association.

(c) The symmetric exclusion process with product initial distribution is negatively associated at positive times.

These are motivated by problems in probability, mathematical physics, and combinatorics. In the latter context, the connection is with Mason's conjecture for certain classes of matroids.

We will discuss these and other conjectures. Among the results: (a) is false and (c) is true, while (b) remains 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. A key tool in the positive results is a property of polynomials known as stability. Much of this is joint work with J. Borcea and P. Branden.