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". Roughly speaking, the paving conjecture states that every positive semi-definite contraction with small diagonal entries can be "paved" by a small number of principal submatrices with small operator norms. We extend this result to real stable polynomials. We will prove that assuming mild conditions on the leading coefficients of a multi-affine real stable polynomial, it is possible to partition the set of variables to a small number of subsets such that the roots of the "restrictions" of the polynomial to each set of variables are small. We will use this generalized paving theorem to show that for every strongly Rayleigh point process, it is possible to partition the underlying space into a small number of subsets such that the points of the restrictions of the point process to each subset are "weakly correlated". This result is intuitively appealing since it implies that the repulsive force among the points of a negatively dependent point process cannot be strong everywhere. To prove this result, we will introduce the notion of the kernel polynomial for strongly Rayleigh processes. This notion is a generalization of the notion of the kernel of determinantal processes and provides a unified framework for studying these two families of point processes. We will also prove an entropy lower for strongly Rayleigh processes in terms of the roots of the kernel polynomial.