Sparse hard-disk packings and local Markov chains

TL;DR

This paper introduces a model based on sparse hard-disk packings to analyze and benchmark Markov-chain Monte Carlo algorithms, revealing different escape time behaviors and discussing the sample space connectivity.

Contribution

It presents a novel application of B"or"oczky packings for analyzing MCMC algorithms and compares the scaling of escape times in different ECMC variants.

Findings

Escape time scales algebraically with relaxation parameter in some ECMC variants.

Escape time scales logarithmically with relaxation parameter in other ECMC variants.

The work provides open-source software for generating packings and running ECMC algorithms.

Abstract

We propose locally stable sparse hard-disk packings, as introduced by B\"or\"oczky, as a model for the analysis and benchmarking of Markov-chain Monte Carlo (MCMC) algorithms. We first generate such packings in a square box with periodic boundary conditions and analyze their properties. We then study how local MCMC algorithms, namely the Metropolis algorithm and several versions of event-chain Monte Carlo (ECMC), escape from configurations that are obtained by slightly reducing all disk radii by a relaxation parameter. A scaling analysis is confirmed by simulation results. We obtain two classes of ECMC, one in which the escape time varies algebraically with the relaxation parameter (as for the local Metropolis algorithm) and another in which the escape time scales as the logarithm of the relaxation parameter. We discuss the connectivity of the hard-disk sample space, the ergodicity of local MCMC algorithms, as well as the meaning of packings in the context of the NPT ensemble. Our work is accompanied by open-source, arbitrary-precision software for B\"or\"oczky packings (in Python) and for straight, reflective, forward, and Newtonian ECMC (in Go).