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

02/15/2023

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by   Konrad Anand, et al.

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We present a perfect marginal sampler of the unique Gibbs measure of a spin system on 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 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.