Bootstrap percolation on a generalized Hamming cube \MakeUppercase{\romannumeral 2}

Abstract

In this paper we obtain the critical probability $p_c(Q_{k,n},r)$ for bootstrap percolation with the infection threshold $r=\frac{N}{2}$ on the generalized $n$-dimensional hypercube $Q_{k,n}$ with vertex set $V(Q_{k,n})=\{0,1{\}}^n$ and edges connecting the pairs at Hamming distance $1,2,\dots ,k$, where $k\ge 2$ and $N={\sum }_{i=1}^k\left(\genfrac{}{}{0ex}{}{n}{i}\right)$. More precisely, we obtain the exact first-order and second-order terms of $p_c(Q_{k,n},\frac{N}{2})$ by extending the main theorem of Balogh, Bollob{á}s, and Morris (2009).

This paper and a forthcoming paper will supersede V1-V4 of arXiv:2503.12607 [math.CO] as the results have been significantly improved and a main mistake has been fixed