Random planar maps and growth-fragmentations

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.

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

“…Thus, without any "local" assumption on the reproduction law at small sizes, the number of small particles may grow exponentially and we cannot in general expect X (n) (⌊a n ·⌋)/n to be tight in ℓ q↓ , for some q > 0. However, this happens to be the case for the perimeters of the cycles in the branching peeling process of random Boltzmann triangulations [7], where versions of Theorems 1 and 2 hold for γ = 1/2, q * = 3, and M = 0, although κ n (3) ≤ 0 seems fulfilled only for M ≥ 3 (which should mean that the holes with perimeter 1 or 2 do not contribute to a substantial part of the triangulation).…”

“…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.…”

Brownian geometry

“…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. Before stating our last result, we need to recall a few basic facts about growth-fragmentation processes (see [11] for more details).…”

A short remark on a growth–fragmentation equation

“…Motivated by the study of growth-fragmentation stochastic processes [3], this type of equation was considered recently by J. Bertoin and A. R. Watson in [5], with the initial data u(0, x) = δ(x − 1),…”