On site percolation in random quadrangulations of the half-plane

TL;DR

This paper develops a new method using Angel's peeling process to analyze site percolation thresholds on random quadrangulations of the half-plane, providing explicit bounds and extending to other maps.

Contribution

A novel approach for applying Angel's peeling process to obtain bounds on percolation thresholds in half-planar maps.

Findings

Percolation threshold bounds: 0.5511 ≤ p_c ≤ 0.5581

Method applicable to other half-planar maps with domain Markov property

Rigorous bounds derived for site percolation on quadrangulations

Abstract

We study site percolation on uniform quadrangulations of the upper half plane. The main contribution is a method for applying Angel's peeling process, in particular for analyzing an evolving boundary condition during the peeling. Our method lets us obtain rigorous and explicit upper and lower bounds on the percolation threshold $p_\mathrm{c}$, and thus show in particular that $0.5511\leq p_\mathrm{c}\leq 0.5581$. The method can be extended to site percolation on other half-planar maps with the domain Markov property.