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Concentration of Lipschitz Functionals of Determinantal and Other Strong Rayleigh Measures

Published online by Cambridge University Press:  19 September 2013

ROBIN PEMANTLE  and
YUVAL PERES
ROBIN PEMANTLE
Affiliation:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA (e-mail: pemantle@math.upenn.edu)
YUVAL PERES
Affiliation:
Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, USA (e-mail: peres@microsoft.com)
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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.

Keywords

60G55
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 1 , January 2014 , pp. 140 - 160
Copyright
Copyright © Cambridge University Press 2013 

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