Session S04 - Random Walks and Related Topics

Monday, July 19, 19:20 UTC-3

Phase transition for percolation on randomly stretched lattices

Augusto Teixeira

IMPA - Rio de Janeiro, Brazil   -   augusto@impa.br

In this talk we study the existence/absence of phase transitions for Bernoulli percolation on a class of random planar graphs. More precisely, the graphs we consider have vertex sets given by $\mathbb{Z}^2$ and we start by adding all horizontal edges connecting nearest neighbor vertices. This gives us a disconnected graph, composed of infinitely many copies of $\mathbb{Z}$, with the trivial behavior $p_c(\mathbb{Z}) = 1$. We now add to $G$ vertical lines of edges at $\{X_i\} \times \mathbb{Z}$, where the points $X_i$ are given by an i.i.d. integer-valued renewal process with inter arrivals distributed as $T$. This graph $G$ now looks like a randomly stretched version of the nearest neighbor graph on $\mathbb{Z}^2$. In this talk we show an interesting phenomenon relating the existence of phase transition for percolation on $G$ with the moments of the variable $T$. Namely, if $E(T^{1+\epsilon})$ is finite, then $G$ almost surely features a non-trivial phase transition. While if $E(T^{1-\epsilon})$ is infinite, then $p_c(G) = 1$.

Joint work with Marcelo Hilário (Universidade Federal de Minas Gerais, Brazil), Remy Sanchis (Universidade Federal de Minas Gerais, Brazil) and Marcos Sá (IMPA, Brazil).

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