An OSSS-type inequality for uniformly drawn subsets of fixed size

TL;DR

This paper extends the OSSS inequality to k-out-of-n measures, which are non-monotonic, and applies it to analyze crossing events in a specific percolation model.

Contribution

It introduces a new version of the OSSS inequality applicable to non-monotonic measures like k-out-of-n, expanding its scope beyond previous monotonic measure cases.

Findings

Derived OSSS inequality for k-out-of-n measures

Applied the inequality to percolation crossing events

Provided new tools for analyzing non-monotonic measures

Abstract

The OSSS inequality [O'Donnell, Saks, Schramm and Servedio, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), Pittsburgh (2005)] gives an upper bound for the variance of a function f of independent 0-1 valued random variables, in terms of the influences of these random variables and the computational complexity of a (randomised) algorithm for determining the value of f. Duminil-Copin, Raoufi and Tassion [Annals of Mathematics 189, 75-99 (2019)] obtained a generalization to monotonic measures and used it to prove new results for Potts models and random-cluster models. Their generalization of the OSSS inequality raises the question if there are still other measures for which a version of that inequality holds. We derive a version of the OSSS inequality for a family of measures that are far from monotonic, namely the k-out-of-n measures (these measures correspond with drawing k elements from a set of size n uniformly). We illustrate the inequality by studying the event that there is an occupied horizontal crossing of an R times R box on the triangular lattice in the site percolation model where exactly half of the vertices in the box are occupied.