Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing

Abstract

We present a simple algorithm that perfectly samples configurations from the unique Gibbs measure of a spin system on a potentially infinite graph G. The sampling algorithm assumes strong spatial mixing together with subexponential growth of G. It produces a finite window onto a perfect sample from the Gibbs distribution. The runtime is linear in the size of the window.

Fast and Perfect Sampling of Subgraphs and Polymer Systems

“…We note that only when λ < λ * (∆) does there exist a suitable percolation probability p that results in a subcritical percolation process, so that the size of the active component has finite expectation and exponential tails. Sampling a graphlet by exploring a random component and performing a rejection step has been used in the past (most notably in the recent work of Bressan [9] to sample uniformly random graphlets of size k; see also [2]). The weighted model we sample from is particularly well suited to this type of exploration algorithm because of the direct connection to a subcritical percolation process.…”