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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.11763 |
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| _version_ | 1866909738428530688 |
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| author | Guedes, Isadora Lima, Paulo C. Sá, Marcos Sanchis, Remy |
| author_facet | Guedes, Isadora Lima, Paulo C. Sá, Marcos Sanchis, Remy |
| contents | In this article, we study a bond percolation model on a horizontally stretched square lattice, constructed by stretching the distances between the columns of $\mathbb{Z}_+^2$ according to a collection of independent and identically distributed (i.i.d.) copies of a non-negative random variable $ξ$. We assume that $ξ$ satisfies the integrability condition \[ \mathbb{E}\big[ξ\, e^{c(\log ξ)^{1/2}} \,\mathbb{1}_{\{ξ\geq 1\}}\big] < \infty, \] for some constant $c > 8\sqrt{\log 96}$. In this random environment, each vertical edge is independently declared open with probability $p$, while each horizontal edge is open with probability $p^{|e|}$, where $|e|$ denotes the Euclidean length of the edge. We develop a multiscale renormalization scheme adapted to this geometry and use it to prove that percolation occurs for all sufficiently large values of $p < 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11763 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase transition on randomly horizontally stretched square lattice Guedes, Isadora Lima, Paulo C. Sá, Marcos Sanchis, Remy Probability In this article, we study a bond percolation model on a horizontally stretched square lattice, constructed by stretching the distances between the columns of $\mathbb{Z}_+^2$ according to a collection of independent and identically distributed (i.i.d.) copies of a non-negative random variable $ξ$. We assume that $ξ$ satisfies the integrability condition \[ \mathbb{E}\big[ξ\, e^{c(\log ξ)^{1/2}} \,\mathbb{1}_{\{ξ\geq 1\}}\big] < \infty, \] for some constant $c > 8\sqrt{\log 96}$. In this random environment, each vertical edge is independently declared open with probability $p$, while each horizontal edge is open with probability $p^{|e|}$, where $|e|$ denotes the Euclidean length of the edge. We develop a multiscale renormalization scheme adapted to this geometry and use it to prove that percolation occurs for all sufficiently large values of $p < 1$. |
| title | Phase transition on randomly horizontally stretched square lattice |
| topic | Probability |
| url | https://arxiv.org/abs/2508.11763 |