Profile of a self-similar growth-fragmentation

TL;DR

This paper investigates the properties of self-similar growth-fragmentations, establishing conditions for their profiles to be absolutely continuous, approximating these profiles, and applying findings to random planar maps.

Contribution

It extends existing results on pure fragmentations to growth-fragmentations, linking profile regularity to self-similarity index and cumulant, and explores applications in random map models.

Findings

Conditions for absolutely continuous profiles are derived.

Profiles can be approximated by functions of small fragments.

Hausdorff dimension is computed in the singular case.

Abstract

A self-similar growth-fragmentation describes the evolution of particles that grow and split as time passes. Its genealogy yields a self-similar continuum tree endowed with an intrinsic measure. Extending results of Haas for pure fragmentations, we relate the existence of an absolutely continuous profile to a simple condition in terms of the index of self-similarity and the so-called cumulant of the growth-fragmentation. When absolutely continuous, we approximate the profile by a function of the small fragments, and compute the Hausdorff dimension in the singular case. Applications to Boltzmann random planar maps are emphasized, exploiting recently established connections between growth-fragmentations and random maps by Bertoin et al.