Abstract:

G.-C. Rota once said: "The one contribution of mine that I hope will be remembered has consisted in just pointing out that all sorts of problems of combinatorics can be viewed as problems of location of the zeros of certain polynomials and in giving these zeros a combinatorial interpretation."

Many examples of and conjectures on log-concavity, unimodality and negative association in Probability, Combinatorics and Statistical Mechanics are considered instances of "Negative Dependence". Pemantle recently stressed the need for a "Theory of Negative Dependence" as there are several disparate important instances but no unifying theory.

Following Rota's philosophy we point out that often negative dependence is a question about zeros of polynomials. We provide a large and natural class of measures defined in terms of the zero-set of the generating polynomial and prove that these measures enjoy all virtues of negative dependence. The results are used to prove conjectures of Liggett and Wagner. We also disprove some of Pemantle's conjectures, most notably the "Big Conjecture" as it was coined by Wagner.

This is joint work with J. Borcea (SU) and T. M. Liggett (UCLA).