Percolations on random maps I: half-plane models

TL;DR

This paper investigates Bernoulli percolation on random half-plane maps derived from uniform planar triangulations and quadrangulations, providing universal formulas for critical thresholds and exponents using peeling processes.

Contribution

It introduces a universal formula for percolation thresholds on random half-plane maps and computes related critical exponents using the peeling process technique.

Findings

Universal formula for critical percolation thresholds

Explicit computation of off-critical and critical exponents

Application of peeling process to analyze percolation on random maps

Abstract

We study Bernoulli percolations on random lattices of the half-plane obtained as local limit of uniform planar triangulations or quadrangulations. Using the characteristic spatial Markov property or peeling process of these random lattices we prove a surprisingly simple universal formula for the critical threshold for bond and face percolations on these graphs. Our techniques also permit us to compute off-critical and critical exponents related to percolation clusters such as the volume and the perimeter.