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, in particular it is constant for each vertex.

Improved Bounds for Sampling Solutions of Random CNF Formulas

“…In addition, both [GGGH22] and [CMM22] use Markov-chain-based algorithms in line with [FGYZ21, FHY21, JPV21a, HSW21, FGW22]. Whereas our algorithm follows the recursive sampler approach developed in [AJ21,HWY22b,HWY22a]. Therefore both the analysis and bounds of the papers are very different.…”

“…We adapt a recent sampling algorithm [AJ21,HWY22b] to random formulas, and present an near-linear time sampler for solutions of random k-CNF formulas as long as α 2 k/3 . Our main approach to analyzing the algorithm is inspired by [GGGY21,HWY22b,HWY22a], and the observation that the structure of random instances is "locally sparse" with high probability.…”

Towards derandomising Markov chain Monte Carlo

“…As mentioned before, the core of our derandomisation method is a logarithmic-cost marginal sampler, which may have independent interest. Our main source of inspiration, and also the first such marginal sampler, is the recent recursive algorithm by Anand and Jerrum [AJ22] for perfect sampling in infinite spin systems. Although the implementation of our marginal sampler also has a recursive structure, there are some quite noticeable distinctions.…”

“…e only thing missing from [AJ22] is tail bounds for their algorithm's running time. We provide such analysis and collate the implications in Appendix B.…”

“…ere is an FPTAS for the number of proper -vertex-colourings for any finite subgraph of the following la ice graphs: For simplicity, as the input spin system S and the parameter ℓ are never changed throughout the algorithm, we drop it from the list of parameters. We remark that this is slightly different from the "bounded" variant appeared in [AJ22] as that truncates the algorithm by a certain depth instead of the number of calls. e correctness of the untruncated algorithm is summarised as follows.…”