When: Monday 16 December, 16:00.

Room: Sala di Consiglio, Department of Mathematics, Sapienza University of Rome

Speaker: Ivailo Hartarsky (ENS, Lyon)
Title: Catalan percolation

Abstract: In Catalan percolation, one declares the edges {i,i+1}, for integer i, occupied and each edge {i,j} with j> i+1 open independently with probability p. For k> i+1, we recursively define {i,k} to be occupied, if {i,k} is open and both {i,j} and {j,k} are occupied for some j in {i+1,...,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 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.

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