Sharp Fuss-Catalan thresholds in graph bootstrap percolation

TL;DR

This paper determines the precise thresholds for graph bootstrap percolation in Erdős-Rényi graphs for all r ≥ 5, revealing complex behavior and connecting thresholds to Fuss-Catalan numbers and tree-like witness graphs.

Contribution

It establishes sharp percolation thresholds for r ≥ 5, extending previous results and introducing new combinatorial structures called K_r-tree witness graphs.

Findings

Sharp thresholds for r ≥ 5 are identified.

Threshold constants are linked to Fuss-Catalan numbers.

In the subcritical regime, the edge density increases by a constant factor.

Abstract

We study graph bootstrap percolation on the Erd\H{o}s-R\'enyi random graph ${\mathcal G}_{n,p}$. For all $r \ge 5$, we locate the sharp $K_r$-percolation threshold $p_c \sim (\gamma n)^{-1/\lambda}$, solving a problem of Balogh, Bollob\'as and Morris. The case $r=3$ is the classical graph connectivity threshold, and the threshold for $r=4$ was found using strong connections with the well-studied $2$-neighbor dynamics from statistical physics. When $r \ge 5$, such connections break down, and the process exhibits much richer behavior. The constants $\lambda=\lambda(r)$ and $\gamma=\gamma(r)$ in $p_c$ are determined by a class of $\left({r\choose2}-1\right)$-ary tree-like graphs, which we call $K_r$-tree witness graphs. These graphs are associated with the most efficient ways of adding a new edge in the $K_r$-dynamics, and they can be counted using the Fuss-Catalan numbers. Also, in the subcritical setting, we determine the asymptotic number of edges added to ${\mathcal G}_{n,p}$, showing that the edge density increases only by a constant factor, whose value we identify.