Martingales in self-similar growth-fragmentations and their connections with 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 Lévy 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 [36]). As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growthfragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in [8].

Brownian geometry

“…Finally, in Section 9, we present the very recent results of [49] studying the sequence of boundary sizes of the connected components of {x ∈ D : H(x) > r} as a process parameterized by r. We show that this process is a growthfragmentation process whose distribution is completely determined. The latter result is very closely related to the recent papers [13,12] investigating scaling limits for a similar process associated with triangulations with a boundary.…”

“…As a consequence of Theorem 19 and known asymptotics [12,Corollary 4.5] for the distribution of the extinction time of a growth-fragmentation process, we derive the following corollary about the tail of the distribution of the maximal height in a Brownian disk.…”

“…Theorem 18 then suggests that this process enjoys properties similar to those of the growth-fragmentation processes that have been studied recently by several authors. In fact, Bertoin, Curien and Kortchemski [13] (see also [12] for extensions) have considered a process analogous to X for triangulations with a boundary and showed that the scaling limit of this process (when the boundary size tends to infinity) is a well-identified growth-fragmentation process. Still it does not seem easy to apply the results of [13] in order to identify the distribution of the process X, but the excursion theory of Section 7 can be used instead to compute this distribution.…”

Self-similar Growth Fragmentations as Scaling Limits of Markov Branching Processes

“…The purpose of the present work is to study more general dynamics which incorporate growth, that is the addition of new balls in the system (see Figure 1). One example of recent interest lies in the exploration of random planar maps [7,6], which exhibits "holes" (the yet unexplored areas) that split or grow depending on whether the new edges being discovered belong to an already known face or not. We thus consider a Markov branching system in discrete time and space where at each step, every particle is replaced by either one particle with a bigger size (growth) or by two smaller particles in a conservative way (fragmentation).…”

“…Indeed, the scaling limits of integer-valued Markov chains investigated in [8], which we build our work upon, belong to the class of so called positive self-similar Markov processes (pssMp), and these processes constitute the cornerstone of Bertoin's self-similar growth-fragmentations [5,6]. Besides, in the context of random planar maps [7,6], they have already been identified as scaling limits for the sequences of perimeters of the separating cycles that arise in the exploration of large Boltzmann triangulations. Informally, a self-similar growth-fragmentation Y depicts a system of particles which all evolve according to a given pssMp and whose each negative jump −y < 0 begets a new independent particle with initial size y.…”