The sharp threshold for bootstrap percolation in all dimensions

TL;DR

This paper establishes the precise threshold for bootstrap percolation in all dimensions, showing how the critical probability depends on the dimension, infection threshold, and iterated logarithm of the grid size.

Contribution

It proves the existence of a sharp percolation threshold in all dimensions and explicitly determines the constant L(d,r) for each pair (d,r).

Findings

Sharp threshold for bootstrap percolation in all dimensions.

Explicit formula for the critical probability involving iterated logarithms.

Determination of the constant L(d,r) for all dimension-threshold pairs.

Abstract

In r-neighbour bootstrap percolation on a graph G, a (typically random) set A of initially 'infected' vertices spreads by infecting (at each time step) vertices with at least r already-infected neighbours. This process may be viewed as a monotone version of the Glauber dynamics of the Ising model, and has been extensively studied on the d-dimensional grid $[n]^d$. The elements of the set A are usually chosen independently, with some density p, and the main question is to determine $p_c([n]^d,r)$, the density at which percolation (infection of the entire vertex set) becomes likely. In this paper we prove, for every pair $d \ge r \ge 2$, that there is a constant L(d,r) such that $p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r-1) (n)]^{d-r+1}$ as $n \to \infty$, where $log_r$ denotes an r-times iterated logarithm. We thus prove the existence of a sharp threshold for percolation in any (fixed) number of dimensions. Moreover, we determine L(d,r) for every pair (d,r).