Bootstrap percolation on the Hamming torus with threshold 2

Abstract

This paper analyzes various questions pertaining to bootstrap percolation on the $d$-dimensional Hamming torus where each node is open with probability $p$ and the percolation threshold is 2. For each $d^{\prime }<d$ we find the critical exponent for the event that a $d^{\prime }$-dimensional subtorus becomes open and compute the limiting value of its probability under the critical scaling. For even $d^{\prime }$, we use the Chen-Stein method to show that the number of $d^{\prime }$-dimensional subtori that become open can be approximated by a Poisson random variable.

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