Correlation decay for hard spheres via Markov chains

TL;DR

This paper uses Markov chain techniques from computer science to improve bounds on the critical parameters of the hard sphere model in multiple dimensions, showing rapid mixing at low fugacities.

Contribution

It introduces a novel application of Markov chain analysis to establish tighter bounds on the critical fugacity and density for hard spheres in higher dimensions.

Findings

Improved lower bounds on critical fugacity and density in 2+ dimensions.

Proved rapid mixing of a sampling Markov chain at low fugacities.

Established an equivalence between spatial and temporal mixing for hard spheres.

Abstract

We improve upon all known lower bounds on the critical fugacity and critical density of the hard sphere model in dimensions two and higher. As the dimension tends to infinity our improvements are by factors of $2$ and $1.7$, respectively. We make these improvements by utilizing techniques from theoretical computer science to show that a certain Markov chain for sampling from the hard sphere model mixes rapidly at low enough fugacities. We then prove an equivalence between optimal spatial and temporal mixing for hard spheres to deduce our results.