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arXiv · 1912.03320

Phase transition for percolation on a randomly stretched lattice

Marcelo R. Hilario·Marcos Sá·Remy Sanchis·Augusto Teixeira
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Abstract

Let $\{ξ_i\}_{i \geq 1}$ be a sequence of i.i.d.\ positive random variables. Starting from the usual square lattice replace each horizontal edge that links a site in $i$-th vertical column to another in the $(i+1)$-th vertical column by an edge having length $ξ_i$. Then declare independently each edge $e$ in the resulting lattice open with probability $p_e=p^{|e|}$ where $p\in[0,1]$ and $|e|$ is the length of $e$. We relate the occurrence of nontrivial phase transition for this model to moment properties of $ξ_1$. More precisely, we prove that the model undergoes a nontrivial phase transition when $\mathbb{E}(ξ_1^η)<\infty$, for some $η>1$ whereas, when $\mathbb{E}(ξ_1^η)=\infty$ for some $η<1$, no phase transition occurs.

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Marcelo R. Hilario, Marcos Sá, Remy Sanchis, Augusto Teixeira. 2020-10-19. Phase transition for percolation on a randomly stretched lattice. https://arxiv.org/abs/1912.03320

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