Convergence and coupling for spin glasses and hard spheres

TL;DR

This paper explores convergence and coupling in Markov chains, introduces a local-patch algorithm for exact sampling in complex models like spin glasses and hard spheres, and demonstrates its improved performance at lower temperatures and higher densities.

Contribution

It presents a novel local-patch algorithm that extends exact sampling capabilities for spin glasses and hard-sphere models, surpassing previous methods in efficiency and range.

Findings

Works at lower temperatures for spin glasses than previous methods

Enables exact sampling of hard disks at higher densities

Potential to reach the spin-glass transition temperature in 3D

Abstract

We discuss convergence and coupling of Markov chains, and present general relations between the transfer matrices describing these two processes. We then analyze a recently developed local-patch algorithm, which computes rigorous upper bound for the coupling time of a Markov chain for non-trivial statistical-mechanics models. Using the coupling from the past protocol, this allows one to exactly sample the underlying equilibrium distribution. For spin glasses in two and three spatial dimensions, the local-patch algorithm works at lower temperatures than previous exact-sampling methods. We discuss variants of the algorithm which might allow one to reach, in three dimensions, the spin-glass transition temperature. The algorithm can be adapted to hard-sphere models. For two-dimensional hard disks, the algorithm allows us to draw exact samples at higher densities than previously possible.