Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs

TL;DR

This paper develops algorithms for approximately sampling from the Potts model on random regular graphs within the uniqueness regime, covering both ferromagnetic and antiferromagnetic cases, using novel resampling and percolation techniques.

Contribution

It introduces new algorithms for sampling from the Potts model on random regular graphs in the uniqueness regime, utilizing correlation decay and random-cluster representations.

Findings

Algorithms work throughout the uniqueness regime for all q ≥ 3 and Δ ≥ 3.

Antiferromagnetic algorithm efficiently resamples bichromatic classes using correlation decay.

Ferromagnetic algorithm simplifies sampling via the random-cluster model and percolation methods.

Abstract

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers $q\geq 3$ and $\Delta\geq 3$, we develop algorithms that produce samples within error $o(1)$ from the $q$-state Potts model on random $\Delta$-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases. The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes may induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of $q$ and $\Delta$ in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value. In the case of the ferromagnetic Potts model, we simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we show that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters $p,q$ on random $\Delta$-regular graphs for all values of $q\geq 1$ and $p<p_c(q,\Delta)$, where $p_c(q,\Delta)$ corresponds to a uniqueness threshold for the model on the $\Delta$-regular tree. When restricted to integer values of $q$, this yields a simplified algorithm for the ferromagnetic Potts model on random $\Delta$-regular graphs.