Universal aspects of critical percolation on random half-planar maps

TL;DR

This paper investigates critical percolation on random half-planar maps, deriving exact thresholds and demonstrating the universality of crossing probabilities, independent of specific model details.

Contribution

It computes the site percolation threshold for quadrangulations and proves the universality of crossing probabilities in half-planar maps.

Findings

Exact site percolation threshold for quadrangulations

Universality of crossing probabilities across models

Extension of Angel's scaling limit result

Abstract

We study a large class of Bernoulli percolation models on random lattices of the half- plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact value of the site percolation threshold in the quadrangular case using the so-called peeling techniques. Then, we generalize a result of Angel about the scaling limit of crossing probabilities, that are a natural analogue to Cardy's formula in (non-random) plane lattices. Our main result is that those probabilities are universal, in the sense that they do not depend on the percolation model neither on the degree of the faces of the map.