Perfect Sampling of $q$-Spin Systems on $\mathbb Z^2$ via Weak Spatial Mixing

TL;DR

This paper introduces a perfect sampling algorithm for $q$-spin systems on $Z^2$ that operates efficiently under weak spatial mixing conditions, broadening applicability to models like the ferromagnetic Potts and Ising models.

Contribution

It adapts a previous lazy depth-first sampling method to work with weak spatial mixing, enabling perfect sampling in broader settings.

Findings

Linear run-time for systems with weak spatial mixing

Effective sampling for ferromagnetic Potts model at supercritical temperatures

Applicable to ferromagnetic Ising model with external field at any non-zero temperature

Abstract

We present a perfect marginal sampler of the unique Gibbs measure of a spin system on $\mathbb Z^2$. The algorithm is an adaptation of a previous `lazy depth-first' approach by the authors, but relaxes the requirement of strong spatial mixing to weak. It exploits a classical result in statistical physics relating weak spatial mixing on $\mathbb Z^2$ to strong spatial mixing on squares. When the spin system exhibits weak spatial mixing, the run-time of our sampler is linear in the size of sample. Applications of note are the ferromagnetic Potts model at supercritical temperatures, and the ferromagnetic Ising model with consistent non-zero external field at any non-zero temperature.