Low-temperature Sampling on Sparse Random Graphs

TL;DR

The paper tackles low-temperature sampling on sparse graphs by extending the polymer method to Erdos-Renyi graphs $G(n,d/n)$ through a new ordered-polymer framework that capably handles small non-expanding components. By isolating the giant component and proving weak spatial mixing within the ordered and disordered phases, the authors obtain a polynomial-time sampler for the random-cluster and Potts models when $q$ is large and $d= heta(1)$, for all temperatures, complemented by a deterministic $Z_G$-estimator with sub-polynomial overhead. The approach hinges on a multiplicative polymer decomposition with polymers of size $O( obreak amark{log} n$ and a careful analysis of the giant core and kernel expansions. The results establish tractability for low-temperature sampling on sparse random graphs and provide tools that may extend to other monotone models and tree-like graph neighborhoods, with potential implications for related #BIS-hardness questions. Overall, the work advances a practical, theory-supported framework for efficient sampling and counting in challenging low-temperature regimes on sparse random graphs.

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 Erdős-Rényi 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 $d=Θ(1)$, 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 $O(\log n)$.

Theorems & Definitions (103)

• Theorem 1
• Theorem 2
• Definition 3: Phases
• Theorem 4
• Definition 5
• Lemma 6
• Lemma 6
• Lemma 6
• Lemma 6
• proof : Proof of Theorem \ref{['thm:mainthm2']}