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arXiv · 2302.07821

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

Konrad Anand·Mark Jerrum
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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.

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Konrad Anand, Mark Jerrum. 2023-02-15. Perfect Sampling of $q$-Spin Systems on $\mathbb Z^2$ via Weak Spatial Mixing. https://arxiv.org/abs/2302.07821

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