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

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