Paving property for real stable polynomials and strongly Rayleigh processes

Abstract

One of the equivalent formulations of the Kadison–Singer problem which was resolved in 2013 by Marcus, Spielman and Srivastava, is the “paving conjecture”. In this paper, we first extend this result to real stable polynomials. We prove that for every multi-affine real stable polynomial satisfying a simple condition, it is possible to partition its set of variables to a small number of subsets such that the “restriction” of the polynomial to each subset has small roots. Then, we derive a probabilistic interpretation of this result. We show that there exists a partition of the underlying space of every strongly Rayleigh process into a small number of sets such that the restriction of the point process to each set has “almost independent” points. This result implies that the dependence structure of strongly Rayleigh processes is constrained—a phenomenon that is to be expected in negatively dependent measures. To prove this result, we introduce and study the notion of kernel polynomial for strongly Rayleigh processes. This notion is a natural generalization of the kernel of determinantal processes. We also derive an entropy bound for strongly Rayleigh processes in terms of the roots of the kernel polynomial which is interesting on its own.