Random planar maps & growth-fragmentations

TL;DR

This paper studies the cycle structure of random Boltzmann triangulations sliced at various heights, establishing a limit process described by growth-fragmentation, and offers a new perspective on the Brownian map.

Contribution

It introduces a novel growth-fragmentation process to describe cycle lengths in random triangulations and develops a branching peeling exploration method.

Findings

Established a functional invariance principle for cycle lengths

Identified a self-similar growth-fragmentation as the limit process

Provided a new construction of the Brownian map

Abstract

We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation for this work is to give a new construction of the Brownian map from a growth-fragmentation process.