Catalan percolation

TL;DR

This paper investigates Catalan percolation, establishing bounds for its critical probability and demonstrating that an enhanced oriented percolation model can have a lower critical point, using novel proof techniques without traditional inequalities.

Contribution

The paper proves that the critical probability of Catalan percolation lies strictly between that of oriented site percolation and the Catalan growth rate, introducing a new approach to analyze models with infinite-range dependencies.

Findings

Critical probability of Catalan percolation is strictly between two known bounds.

Enhanced oriented percolation models can have lower critical thresholds.

New proof techniques avoid traditional inequalities like Aizenman--Grimmett enhancements.

Abstract

In Catalan percolation, all nearest-neighbor edges $\{i,i+1\}$ along $\mathbb Z$ are initially occupied, and all other edges are open independently with probability $p$. Open edges $\{i,j\}$ are occupied if some pair of edges $\{i,k\}$ and $\{k,j\}$, with $i<k<j$, become occupied. This model was introduced by Gravner and the third author, in the context of polluted graph bootstrap percolation. We prove that the critical $p_{\mathrm c}$ is strictly between that of oriented site percolation on $\mathbb Z^2$ and the Catalan growth rate $1/4$. Our main result shows that an enhanced oriented percolation model, with non-decaying infinite-range dependency, has a strictly smaller critical parameter than the classical model. This is reminiscent of the work of Duminil-Copin, Hil\'ario, Kozma and Sidoravicius on brochette percolation. Our proof differs, however, in that we do not use Aizenman--Grimmett enhancements or differential inequalities. Two key ingredients are the work of Hil\'ario, S\'a, Sanchis and Teixeira on stretched lattices, and the Russo--Seymour--Welsh result for oriented percolation by Duminil-Copin, Tassion and Teixeira.