Negative dependence and the geometry of polynomials

TL;DR

This paper introduces strongly Rayleigh measures characterized by stable generating polynomials, demonstrating their negative dependence properties, proving several conjectures, and providing counterexamples to others, thereby advancing understanding of polynomial geometry in probability.

Contribution

It defines strongly Rayleigh measures via polynomial stability, proves their negative dependence, and resolves multiple conjectures while constructing counterexamples.

Findings

Strongly Rayleigh measures exhibit negative dependence.

The paper proves several conjectures by Liggett, Pemantle, and Wagner.

Counterexamples are provided to certain conjectures on ultra log-concavity.

Abstract

We introduce the class of {\em strongly Rayleigh} probability measures by means of geometric properties of their generating polynomials that amount to the stability of the latter. This class covers important models such as determinantal measures (e.g. product measures, uniform random spanning tree measures) and distributions for symmetric exclusion processes. We show that strongly Rayleigh measures enjoy all virtues of negative dependence and we also prove a series of conjectures due to Liggett, Pemantle, and Wagner, respectively. Moreover, we extend Lyons' recent results on determinantal measures and we construct counterexamples to several conjectures of Pemantle and Wagner on negative dependence and ultra log-concave rank sequences.