A spectral independence view on hardspheres via block dynamics

TL;DR

This paper introduces a polynomial-time Markov chain Monte Carlo algorithm for approximating the partition function of the hard-sphere model in multiple dimensions, leveraging spectral independence and block dynamics techniques.

Contribution

It develops a novel discretization approach and applies clique dynamics with spectral analysis to efficiently approximate the partition function up to the known uniqueness regime.

Findings

Algorithm runs in polynomial time for certain fugacity values.

Proves rapid mixing of clique dynamics up to the tree threshold.

Relates clique dynamics to block dynamics for spectral analysis.

Abstract

The hard-sphere model is one of the most extensively studied models in statistical physics. It describes the continuous distribution of spherical particles, governed by hard-core interactions. An important quantity of this model is the normalizing factor of this distribution, called the partition function. We propose a Markov chain Monte Carlo algorithm for approximating the grand-canonical partition function of the hard-sphere model in $d$ dimensions. Up to a fugacity of $\lambda < \text{e}/2^d$, the runtime of our algorithm is polynomial in the volume of the system. This covers the entire known real-valued regime for the uniqueness of the Gibbs measure. Key to our approach is to define a discretization that closely approximates the partition function of the continuous model. This results in a discrete hard-core instance that is exponential in the size of the initial hard-sphere model. Our approximation bound follows directly from the correlation decay threshold of an infinite regular tree with degree equal to the maximum degree of our discretization. To cope with the exponential blow-up of the discrete instance we use clique dynamics, a Markov chain that was recently introduced in the setting of abstract polymer models. We prove rapid mixing of clique dynamics up to the tree threshold of the univariate hard-core model. This is achieved by relating clique dynamics to block dynamics and adapting the spectral expansion method, which was recently used to bound the mixing time of Glauber dynamics within the same parameter regime.