Ivailo Hartarsky and Bruno Schapira: Catalan percolation

Abstract:

In Catalan percolation, one declares the edges $\{i,i+1\}$ for $i\in\mathbb Z$ \emph{occupied} and each edge $\{i,j\}\subset\mathbb Z$ with $j\ge i+2$ \emph{open} independently with probability $p$ . For $k\ge i+2$ , we recursively define $\{i,k\}$ to be \emph{occupied}, if $\{i,k\}$ is open and both $\{i,j\}$ and $\{j,k\}$ are occupied for some $j\in\{i+1,\dots,k-1\}$ . The model was introduced by Gravner and Kolesnik in the context of polluted bootstrap percolation, but is tightly linked with Catalan structures and oriented percolation. We establish that the critical parameter of the model is strictly between the natural lower and upper bounds given by $1/4$ and the critical probability of oriented site percolation on $\mathbb Z^2$ respectively. The most challenging part of the proof is a strict inequality for the critical parameter of an oriented percolation model with non-decaying infinite range dependencies, not relying on the Aizenman–Grimmett argument for essential enhancements. It can be viewed as an oriented version of the Brochette percolation model.

The talk is based on joint work with Eleanor Archer, Brett Kolesnik, Sam Olesker-Taylor and Daniel Valesin available at https://arxiv.org/abs/2404.19583.

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Ivailo Hartarsky and Bruno Schapira: Catalan percolation

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