Low-temperature Sampling on Sparse Random Graphs

TL;DR

This paper develops a polynomial-time sampling algorithm for low-temperature regimes on sparse Erdős-Rényi graphs, particularly for the Potts and random-cluster models, by extending the polymer method to graphs with weak small-set expansion.

Contribution

It introduces a novel polymer-based sampling approach applicable to sparse random graphs with weak small-set expansion, enabling efficient sampling at all temperatures for certain models.

Findings

Polynomial-time sampling algorithm for Potts and random-cluster models on sparse graphs.

Polymer size bounded by O(log n), facilitating efficient sampling.

Applicable to models with monotone properties across all temperatures.

Abstract

We consider sampling in the so-called low-temperature regime, which is typically characterised by non-local behaviour and strong global correlations. Canonical examples include sampling independent sets on bipartite graphs and sampling from the ferromagnetic $q$-state Potts model. Low-temperature sampling is computationally intractable for general graphs, but recent advances based on the polymer method have made significant progress for graph families that exhibit certain expansion properties that reinforce the correlations, including for example expanders, lattices and dense graphs. One of the most natural graph classes that has so far escaped this algorithmic framework is the class of sparse Erd\H{o}s-R\'enyi random graphs whose expansion only manifests for sufficiently large subsets of vertices; small sets of vertices on the other hand have vanishing expansion which makes them behave independently from the bulk of the graph and therefore weakens the correlations. At a more technical level, the expansion of small sets is crucial for establishing the Kotecky-Priess condition which underpins the applicability of the framework. Our main contribution is to develop the polymer method in the low-temperature regime for sparse random graphs. As our running example, we use the Potts and random-cluster models on $G(n,d/n)$ for $d=\Theta(1)$, where we show a polynomial-time sampling algorithm for all sufficiently large $q$ and $d$, at all temperatures. Our approach applies more generally for models that are monotone. Key to our result is a simple polymer definition that blends easily with the connectivity properties of the graph and allows us to show that polymers have size at most $O(\log n)$.