Bootstrap percolation on a generalized Hamming cube

TL;DR

This paper analyzes the bootstrap percolation process on a generalized Hamming cube, deriving critical probabilities for percolation and examining minimal percolating sets for various parameters.

Contribution

It provides new asymptotic results for critical probabilities and minimal percolating sets in bootstrap percolation on a generalized Hamming cube, extending previous models.

Findings

Asymptotic formulas for critical percolation probabilities for r=2,3

Results on the size of smallest percolating sets for k≥2

Extension of bootstrap percolation analysis to generalized Hamming graphs

Abstract

We consider the $r$-neighbor bootstrap percolation process on the graph with vertex set $V=\{0,1\}^n$ and edges connecting the pairs at Hamming distance $1,2,\dots,k$, where $k\ge 2$. We find asymptotics of the critical probability of percolation for $r=2,3$. In the deterministic setting, we obtain several results for the size of the smallest percolating set for $k\ge 2$.