Sampling from a polytope and hard-disk Monte Carlo

TL;DR

This paper reformulates hard-disk Monte Carlo algorithms as sampling from a polytope, analyzing their convergence and exploring parallelization strategies for improved computational efficiency.

Contribution

It introduces a polytope-based perspective on hard-disk Monte Carlo algorithms and analyzes their convergence properties and parallelization potential.

Findings

Local Markov-chain Monte Carlo as high-dimensional polytope walks

Event-chain algorithm corresponds to molecular dynamics evolution

Parallelization strategies for event-chain Monte Carlo

Abstract

The hard-disk problem, the statics and the dynamics of equal two-dimensional hard spheres in a periodic box, has had a profound influence on statistical and computational physics. Markov-chain Monte Carlo and molecular dynamics were first discussed for this model. Here we reformulate hard-disk Monte Carlo algorithms in terms of another classic problem, namely the sampling from a polytope. Local Markov-chain Monte Carlo, as proposed by Metropolis et al. in 1953, appears as a sequence of random walks in high-dimensional polytopes, while the moves of the more powerful event-chain algorithm correspond to molecular dynamics evolution. We determine the convergence properties of Monte Carlo methods in a special invariant polytope associated with hard-disk configurations, and the implications for convergence of hard-disk sampling. Finally, we discuss parallelization strategies for event-chain Monte Carlo and present results for a multicore implementation.