Negative dependence and the geometry of polynomials

Abstract

We introduce the class of 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 and 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.

A Randomized Rounding Approach to the Traveling Salesman Problem

“…Borcea et al in [8] proved that the strongly Rayleigh measure satisfy the strongest form of negative association known as CNA+ which we do not describe here.…”

“…Part (ii) on the structure of near minimum cuts builds on the cactus structure [17] and polygon representation [5] of minimum and near-minimum cuts, respectively. Finally, the last part uses techniques from a recent and very interesting study of strongly Rayleigh measures [8] and their properties to prove results on the joint distribution of the parity of the number of edges across multiple cuts.…”

“…The next set of results are the main tools in proving the goodness of the trivial edges and are mostly restated from a recent work of Borcea, Branden and Ligget [8] on strongly Rayleigh probability measures. Strongly Rayleigh measures include determinantal measures (in a form similar to the above formula) and in particular uniform and λ-uniform random spanning tree measures.…”

Stochastic models for large interacting systems and related correlation inequalities

“…Whereas the intuition is fairly clear, it took 35 years to find the correct version of the connection between the symmetric exclusion process and negative association (45)(46)(47). Here is one consequence of the general statement for the symmetric exclusion process that is proved in ref.…”

“…Here is an application of the more elaborate version proved in ref. 47 By Theorem 5, for each t > 0, the summands above are negatively correlated, and there are independent Bernoulli random variables ζ t ðxÞ so that W ðtÞ has the same distribution as ∑ x>0 ζ t ðxÞ:…”