Local Gibbs sampling beyond local uniformity

TL;DR

This paper introduces efficient local Gibbs samplers for complex spin systems, including the first near-critical Ising model sampler and a sampler for proper colorings, surpassing previous limitations of local uniformity assumptions.

Contribution

It develops the first local samplers for permissive and truly repulsive spin systems that operate beyond the local uniformity threshold, advancing local sampling capabilities.

Findings

Linear-time local sampler for the Ising model in near-critical regimes

First local sampler for uniform proper q-colorings with q=O(Δ)

Local sampling complexity comparable to global sampling for spin systems

Abstract

Local samplers are algorithms that generate random samples based on local queries to high-dimensional distributions, ensuring the samples follow the correct induced distributions while maintaining time complexity that scales locally with the query size. These samplers have broad applications, including deterministic approximate counting [He, Wang, Yin, SODA '23; Feng et.al., FOCS '23], sampling from infinite or high-dimensional Gibbs distributions [Anand, Jerrum, SICOMP '22; He, Wang, Yin, FOCS '22], and providing local access to large random objects [Biswas, Rubinfield, Yodpinyanee, ITCS '20]. In this work, we present local samplers for Gibbs distributions of spin systems. Specifically, we design linear-time local samplers for: - permissive spin systems, including the first local sampler for the Ising model in near-critical regimes; - truly repulsive spin systems, represented by the first local sampler for uniform proper $q$-colorings, with $q=O(\Delta)$ colors on graphs with maximum degree $\Delta$. These local samplers are efficient beyond the "local uniformity" threshold, which imposes unconditional marginal lower bounds -- a key assumption required by all prior local samplers. Our results show that, in general, local sampling is not significantly harder than global sampling for spin systems. As an application, our results also imply local algorithms for probabilistic inference in the same near-critical regimes.