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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.12607 |
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| _version_ | 1866916796541435904 |
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| author | Zhu, Fengxing |
| author_facet | Zhu, Fengxing |
| contents | 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{á}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{á}s, and Morris (2009). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bootstrap percolation on a generalized Hamming cube \MakeUppercase{\romannumeral 2} Zhu, Fengxing Combinatorics Probability 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{á}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{á}s, and Morris (2009). |
| title | Bootstrap percolation on a generalized Hamming cube \MakeUppercase{\romannumeral 2} |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2503.12607 |