Sampling from the Hardcore Model on Random Regular Bipartite Graphs above the Uniqueness Threshold

TL;DR

This paper introduces an efficient sampling algorithm for the hardcore model on random regular bipartite graphs, extending the range of parameters for which sampling and partition function approximation are feasible.

Contribution

The paper develops two new Markov chains and proves their rapid mixing using topological spectral expansion, enabling sampling above the uniqueness threshold.

Findings

Efficient sampling algorithm for $\lambda \\lesssim 1/\sqrt{\Delta}$

FPRAS for the partition function at any fugacity

Analysis of simplicial complexes as spectral expanders

Abstract

We design an efficient sampling algorithm to generate samples from the hardcore model on random regular bipartite graphs as long as $\lambda \lesssim \frac{1}{\sqrt{\Delta}}$, where $\Delta$ is the degree. Combined with recent work of Jenssen, Keevash and Perkins this implies an FPRAS for the partition function of the hardcore model on random regular bipartite graphs at any fugacity. Our algorithm is shown by analyzing two new Markov chains that work in complementary regimes. Our proof then proceeds by showing the corresponding simplicial complexes are top-link spectral expanders and appealing to the trickle-down theorem to prove fast mixing.