The Time of Bootstrap Percolation in High 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{á}s, and Smith (2016) about the percolation time with the infection threshold equal to $2$ on the two-dimensional torus.