Concentration of Lipschitz Functionals of Determinantal and Other Strong Rayleigh Measures

Abstract

Let {X1 , . . , Xn} be a collection of binary-valued random variables and let f : {0, 1}n → $\mathbb{R}$ be a Lipschitz function. Under a negative dependence hypothesis known as the strong Rayleigh condition, we show that f − ${\mathbb E}$f satisfies a concentration inequality. 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, any Lipschitz-1 function of the edges of a uniform spanning tree on vertex set V (e.g., the number of leaves) satisfies the Gaussian concentration inequality \begin{linenomath}$${{\mathbb P} (f - {\mathbb E} f \geq a) \leq \exp \biggl( - \frac{a^2}{8 \, |V|} \biggr) }.$$\end{linenomath} We also prove a continuous version for concentration of Lipschitz functionals of a determinantal point process.