A Sharp Threshold for Bootstrap Percolation in a Random Hypergraph

TL;DR

This paper establishes a precise threshold for the bootstrap percolation process in random hypergraphs, confirming a conjecture and extending results to graph bootstrap processes using differential equations.

Contribution

It proves a sharp threshold for bootstrap percolation in hypergraphs generated from regular hypergraphs, confirming Morris's conjecture and generalizing prior graph results.

Findings

Sharp threshold identified for hypergraph bootstrap percolation

Confirms Morris's conjecture on hypergraph thresholds

Extends results to graph bootstrap processes for 2-balanced graphs

Abstract

Given a hypergraph $\mathcal{H}$, the $\mathcal{H}$-bootstrap process starts with an initial set of infected vertices of $\mathcal{H}$ and, at each step, a healthy vertex $v$ becomes infected if there exists a hyperedge of $\mathcal{H}$ in which $v$ is the unique healthy vertex. We say that the set of initially infected vertices percolates if every vertex of $\mathcal{H}$ is eventually infected. We show that this process exhibits a sharp threshold when $\mathcal{H}$ is a hypergraph obtained by randomly sampling hyperedges from an approximately $d$-regular $r$-uniform hypergraph satisfying some mild degree and codegree conditions; this confirms a conjecture of Morris. As a corollary, we obtain a sharp threshold for a variant of the graph bootstrap process for strictly $2$-balanced graphs which generalises a result of Kor\'{a}ndi, Peled and Sudakov. Our approach involves an application of the differential equations method.