The peeling process of infinite Boltzmann planar maps

TL;DR

This paper analyzes the peeling process of infinite Boltzmann planar maps by connecting it to finite maps, identifying the perimeter process as a biased random walk, and deriving its scaling limits.

Contribution

It introduces a novel approach to study the peeling process of IBPMs by conditioning finite map processes, providing new insights into their perimeter and volume scaling limits.

Findings

Perimeter process is a biased random walk.

Peeling process on IBPM is obtained by conditioning finite map processes.

Scaling limits of perimeter and volume processes are derived.

Abstract

We start by studying a peeling process on finite random planar maps with faces of arbitrary degrees determined by a general weight sequence, which satisfies an admissibility criterion. The corresponding perimeter process is identified as a biased random walk, in terms of which the admissibility criterion has a very simple interpretation. The finite random planar maps under consideration were recently proved to possess a well-defined local limit known as the infinite Boltzmann planar map (IBPM). Inspired by recent work of Curien and Le Gall, we show that the peeling process on the IBPM can be obtained from the peeling process of finite random maps by conditioning the perimeter process to stay positive. The simplicity of the resulting description of the peeling process allows us to obtain the scaling limit of the associated perimeter and volume process for arbitrary regular critical weight sequences.