The Time of Bootstrap Percolation in High Dimensions

TL;DR

This paper analyzes the percolation time in high-dimensional bootstrap percolation on a torus, extending previous results from 2D to higher dimensions with infection threshold equal to the dimension.

Contribution

It extends the understanding of bootstrap percolation time from 2D to high-dimensional tori with infection threshold equal to the dimension.

Findings

Percolation time determined up to a constant factor with high probability

Results apply to initial infection probabilities in a specific range

Generalizes previous 2D results to higher dimensions

Abstract

We consider the $d$-neighbor bootstrap percolation process on the $d$-dimensional torus, with vertex set $V=\{1,\cdots,n\}^d$ and edge set $\{xy:\sum_{i=1}^d|x_i-y_i (\text{mod} \; n)|=1\}$. We determine the percolation time up to a constant factor with high probability when the initial infection probability is in a certain range and the infection threshold is $d$, extending one of the two main theorems from Balister,Bollob{\'a}s, and Smith (2016) about the percolation time with the infection threshold equal to $2$ on the two-dimensional torus.