  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.
