Concentration of Lipschitz functionals of determinantal and other strong Rayleigh measures

TL;DR

This paper establishes Gaussian concentration inequalities for Lipschitz functions of negatively dependent random variables, including determinantal measures and strong Rayleigh measures, extending classical results to dependent settings.

Contribution

It introduces a concentration inequality for Lipschitz functionals under the strong Rayleigh condition, encompassing determinantal and other negatively dependent measures.

Findings

Proves concentration inequalities for strong Rayleigh measures.

Extends Gaussian concentration to dependent random variables.

Provides applications to spanning trees and determinantal point processes.

Abstract

Let X_1 ,..., X_n be a collection of binary valued random variables and let f : {0,1}^n -> R be a Lipschitz function. Under a negative dependence hypothesis known as the {\em strong Rayleigh} condition, we show that f - E f satisfies a concentration inequality generalizing the classical Gaussian concentration inequality for sums of independent Bernoullis: P (S_n - E S_n > a) < exp (-2 a^2 / n). The class of strong Rayleigh measures includes determinantal measures, weighted uniform matroids and exclusion measures; some familiar examples from these classes are generalized negative binomials and spanning tree measures. For instance, the number of vertices of odd degree in a uniform random spanning tree of a graph satisfies a Gaussian concentration inequality with n replaced by |V|, the number of vertices. We also prove a continuous version for concentration of Lipschitz functionals of a determinantal point process.