Maximal Bootstrap Percolation Time on the Hypercube via Generalised Snake-in-the-Box

TL;DR

This paper determines the maximum percolation time for r-neighbour bootstrap percolation on hypercubes for all r ≥ 3, revealing it to be approximately 2^d/d, and uncovers a connection to a generalized Snake-in-the-Box problem.

Contribution

It establishes the asymptotic maximal percolation time for hypercubes in r-neighbour bootstrap percolation for all r ≥ 3 and links this to a generalized Snake-in-the-Box problem.

Findings

Maximal percolation time is approximately 2^d/d for r ≥ 3.

Contrast with r=2 case, which is quadratic in d.

Connection to a generalized Snake-in-the-Box problem.

Abstract

In $r$-neighbour bootstrap percolation, vertices (sites) of a graph $G$ are infected, round-by-round, if they have $r$ neighbours already infected. Once infected, they remain infected. An initial set of infected sites is said to percolate if every site is eventually infected. We determine the maximal percolation time for $r$-neighbour bootstrap percolation on the hypercube for all $r \geq 3$ as the dimension $d$ goes to infinity up to a logarithmic factor. Surprisingly, it turns out to be $\frac{2^d}{d}$, which is in great contrast with the value for $r=2$, which is quadratic in $d$, as established by Przykucki. Furthermore, we discover a link between this problem and a generalisation of the well-known Snake-in-the-Box problem.