Simple peeling of planar maps with application to site percolation

TL;DR

This paper explores a simple peeling process for planar maps, establishing its relation to lazy-peeling, and applies it to determine site percolation thresholds on random maps with complex boundary structures.

Contribution

It introduces a simple peeling exploration method for planar maps, linking it to lazy-peeling, and analyzes critical Boltzmann maps for site percolation studies.

Findings

Established deep relations between simple and lazy-peeling processes.

Defined and studied the half-plane Boltzmann map with a simple boundary.

Analyzed percolation thresholds on complex random planar maps.

Abstract

The peeling process, which describes a step-by-step exploration of a planar map, has been instrumental in addressing percolation problems on random infinite planar maps. Bond and face percolation on maps with faces of arbitrary degree are conveniently studied via so-called lazy-peeling explorations. During such explorations distinct vertices on the exploration contour may at later stage be identified, making the process less suited to the study of site percolation. To tackle this situation and to explicitly identify site-percolation thresholds, we come back to the alternative "simple" peeling exploration of Angel and uncover deep relations with the lazy-peeling process. Along the way we define and study the random Boltzmann map of the half-plane with a simple boundary for an arbitrary critical weight sequence. Its construction is nontrivial especially in the "dense regime" where the half-planar random Boltzmann map does not possess an infinite simple core.