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

Jean Bertoin, Timothy Budd, Nicolas Curien, Igor Kortchemski

Published 2016-05-02Version 1

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. As an application, 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). This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in arXiv:1507.02265 .