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 $N=\sum_{i=1}^k\binom{n}{i}$. 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).
From: Fengxing Zhu [view email]
[v1]
Sun, 16 Mar 2025 18:43:05 UTC (18 KB)
[v2]
Sat, 22 Mar 2025 17:00:56 UTC (19 KB)
[v3]
Tue, 17 Jun 2025 14:46:41 UTC (20 KB)
[v4]
Mon, 31 Aug 2026 17:42:31 UTC (23 KB)
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