Martingales in self-similar growth-fragmentations and their connections with random planar maps

TL;DR

This paper develops a theory of self-similar growth-fragmentation processes using martingales, connects them to stable Lévy processes, and shows their relevance as scaling limits in random planar maps with large degrees.

Contribution

It introduces a comprehensive framework for growth-fragmentations, establishes many-to-one formulas, and links these processes to the geometry of random planar maps.

Findings

Martingales are fundamental in self-similar growth-fragmentations.

Stable Lévy processes are closely related to certain growth-fragmentations.

Growth-fragmentations serve as scaling limits for perimeter processes in random planar maps.

Abstract

The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable L\'evy processes, which are then shown to arise as the scaling limit of the perimeter process in Markovian explorations of certain random planar maps with large degrees (which are, roughly speaking, the dual maps of the stable maps of Le Gall & Miermont. As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growth-fragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in arXiv:1507.02265 .