Percolation on hypergraphs and the hard-core model

TL;DR

This paper establishes tight bounds on percolation thresholds for certain hypergraphs and uses these results to improve understanding of the hard-core model's uniqueness and mixing properties, with implications for algorithms.

Contribution

It provides the first tight bounds on percolation thresholds for hypergraphs with specified degree and overlap constraints, and links these bounds to the hard-core model's uniqueness and algorithmic thresholds.

Findings

Tight bounds on site percolation thresholds for hypergraphs.

Improved bounds on the uniqueness threshold for the hard-core model.

Connections between percolation thresholds and algorithmic properties.

Abstract

We prove tight bounds on the site percolation threshold for $k$-uniform hypergraphs of maximum degree $\Delta$ and for $k$-uniform hypergraphs of maximum degree $\Delta$ in which any pair of edges overlaps in at most $r$ vertices. The hypergraphs that achieve these bounds are hypertrees, but unlike in the case of graphs, there are many different $k$-uniform, $\Delta$-regular hypertrees. Determining the extremal tree for a given $k, \Delta, r$ involves an optimization problem, and our bounds arise from a convex relaxation of this problem. By combining our percolation bounds with the method of disagreement percolation we obtain improved bounds on the uniqueness threshold for the hard-core model on hypergraphs satisfying the same constraints. Our uniqueness conditions imply exponential weak spatial mixing, and go beyond the known bounds for rapid mixing of local Markov chains and existence of efficient approximate counting and sampling algorithms. Our results lead to natural conjectures regarding the aforementioned algorithmic tasks, based on the intuition that uniqueness thresholds for the extremal hypertrees for percolation determine computational thresholds.