Talagrand's inequality for Interacting Particle Systems satisfying a log-Sobolev inequality

TL;DR

This paper extends Talagrand's inequality to interacting particle systems satisfying a log-Sobolev inequality, establishing equivalences and exploring implications for influence and probability of increasing events.

Contribution

It generalizes Talagrand's inequality to IPS under log-Sobolev conditions and shows their equivalence with a stronger inequality, advancing understanding of dependencies in such systems.

Findings

Talagrand's inequality is extended to IPS satisfying a log-Sobolev inequality.

A stronger version of Talagrand's inequality is shown to be equivalent to a log-Sobolev inequality.

The relation between event probability and influences in IPS is analyzed.

Abstract

Talagrand's inequality for independent Bernoulli random variables is extended to many interacting particle systems (IPS). The main assumption is that the IPS satisfies a log-Sobolev inequality. In this context it is also shown that a slightly stronger version of Talagrand's inequality is equivalent to a log-Sobolev inequality. Additionally we also look at a common application, the relation between the probability of increasing events and the influences on that event by changing a single spin.