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

Abstract

In this paper we investigate the critical probability $p_c(Q_n,r)$ for bootstrap percolation with the infection threshold $r$ on the $n$-dimensional hypercube $Q_n$ with vertex set $V(Q_n)=\{0,1\}^n$ and edges connecting the pairs at Hamming distance $1$. More precisely, by utilizing the techniques developed by Balogh, Bollob{\'a}s, and Morris (2009), we determine the first-order term of $p_c(Q_n,n^a)$ where $\frac{2}{3}<a< 1$. Additionally, 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 $N=\sum_{i=1}^k\binom{n}{i}$. More precisely, we obtain the first-order term of $p_c(Q_{k,n},\frac{N}{2})$ and some bounds on the second-order term by extending the main theorem from Balogh, Bollob{\'a}s, and Morris (2009).