Critical percolation on slabs with random columnar disorder
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arXiv
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| Format: | Preprint |
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2024
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| author | Castro, Matheus B. Sanchis, Rémy Silva, Roger W. C. |
| author_facet | Castro, Matheus B. Sanchis, Rémy Silva, Roger W. C. |
| contents | We explore a bond percolation model on slabs $\mathbb{S}^+_k=\mathbb{Z}_+\times \mathbb{Z}_+\times\{0,\dots,k\}$ featuring one-dimensional inhomogeneities. In this context, a vertical column on the slab comprises the set of vertical edges projecting to the same vertex on $\mathbb{Z}_+\times\{0,\dots,k\}$. Columns are chosen based on the arrivals of a renewal process, where the tail distributions of inter-arrival times follow a power law with exponent $ϕ>1$. Inhomogeneities are introduced as follows: vertical edges on selected columns are open (closed) with probability $q$ (respectively $1-q$), independently. Conversely, vertical edges within unselected columns and all horizontal edges are open (closed) with probability $p$ (respectively $1-p$). We prove that for all sufficiently large $ϕ$ (depending solely on $k$), the following assertion holds: if $q>p_c(\mathbb{S}^+_k)$, then $p$ can be taken strictly smaller than $p_c(\mathbb{S}^+_k)$ in a manner that percolation still occurs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_10927 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical percolation on slabs with random columnar disorder Castro, Matheus B. Sanchis, Rémy Silva, Roger W. C. Probability We explore a bond percolation model on slabs $\mathbb{S}^+_k=\mathbb{Z}_+\times \mathbb{Z}_+\times\{0,\dots,k\}$ featuring one-dimensional inhomogeneities. In this context, a vertical column on the slab comprises the set of vertical edges projecting to the same vertex on $\mathbb{Z}_+\times\{0,\dots,k\}$. Columns are chosen based on the arrivals of a renewal process, where the tail distributions of inter-arrival times follow a power law with exponent $ϕ>1$. Inhomogeneities are introduced as follows: vertical edges on selected columns are open (closed) with probability $q$ (respectively $1-q$), independently. Conversely, vertical edges within unselected columns and all horizontal edges are open (closed) with probability $p$ (respectively $1-p$). We prove that for all sufficiently large $ϕ$ (depending solely on $k$), the following assertion holds: if $q>p_c(\mathbb{S}^+_k)$, then $p$ can be taken strictly smaller than $p_c(\mathbb{S}^+_k)$ in a manner that percolation still occurs. |
| title | Critical percolation on slabs with random columnar disorder |
| topic | Probability |
| url | https://arxiv.org/abs/2408.10927 |