Random planar maps and growth-fragmentations
type: Item
instance of: scholarly article
title: Random planar maps and growth-fragmentations
zbMATH Open document ID: 1447.60058
DOI: 10.1214/17-AOP1183
author: Jean Bertoin
author: Nicolas Curien
author: Igor Kortchemski
published in: The Annals of Probability
publication date: 2018-04-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1507.02265
review text: Let \(T^{(p)}\) be a random Boltzmann triangulation of the \(p\)-gon. The main object of interest in the present paper is the sequence of length (or perimeters) of the cycles of \(B_r(T^{(p)})\) ranked in decreasing order, which is denoted by \(L^{(p)}(r)= (L_1^{(p)}, L_2^{(p)},\dots)\). The main result of the paper is a functional invariance principle that describes the scaling limit of the process \(L^{(p)}\). \(X(t)\) is the size of a typical particle or cell at age \(t\). The rest of the present paper is organized as follows. In Section 2, the notion of general branching peeling explorations are introduced. Using results from [the first and third authors, Ann. Appl. Probab. 26, No. 4, 2556--2595 (2016; Zbl 1352.60103)] a scaling limit for the length of the locally largest cycle in large Boltzmann triangulations is obtained. In Section 3, the authors then use the peeling by layers, which is specific peeling algorithm that is the key in this work, to get the scaling limit of the locally largest cycle at given heights. Section 4 introduces the basics on cell systems and their scaling limits in order to prove the following result: Theorem 1. \((\frac{1}{p} L^{(p)}(r \sqrt{p}); r\ge 0) \to(X(\frac{2 t_{\Delta^1}}{a_{\Delta^1}}\times r); r\ge 0)\), as \(p\to \infty\), where convergence holds in distribution in the space of càdlàg process taking values in \(l_3^{\downarrow}\) equipped with the Skorokhod \(J_1\) topology (\(t_{\Delta^1}= \frac{\sqrt{3}}{8\sqrt{\pi}}\), and \(a_{\Delta^1}=\frac{1}{2\sqrt{3}}\)). Finally, Section 5 is devoted to the proof of a result which completes Theorem 1 by a more geometric point of view.
reviewed by: Viktor Ohanyan
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60G18
zbMATH DE Number: 6865122
zbMATH Keywords: planar map
zbMATH Keywords: growth-fragmentation
zbMATH Keywords: self-similar Markov process
zbMATH Keywords: scaling limit
MaRDI profile type: Publication
arXiv ID: 1507.02265
cites work: Q3976721
cites work: Growth and percolation on the uniform infinite planar triangulation
cites work: Percolations on random maps. I: Half-plane models
cites work: Classification of half-planar maps
cites work: Uniform infinite planar triangulations
cites work: Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points
cites work: Self-similar fragmentations
cites work: Compensated fragmentation processes and limits of dilated fragmentations
cites work: Markovian growth-fragmentation processes
cites work: Self-similar scaling limits of Markov chains on the positive integers
cites work: Exponential functionals of Lévy processes
cites work: Scaling limit of random planar quadrangulations with a boundary
cites work: Compact Brownian surfaces. I: Brownian disks
cites work: The peeling process of infinite Boltzmann planar maps
cites work: A glimpse of the conformal structure of random planar maps
cites work: Planar stochastic hyperbolic triangulations
cites work: Percolation on random triangulations and stable looptrees
cites work: The Brownian plane
cites work: Scaling limits for the peeling process on random maps
cites work: The hull process of the Brownian plane
cites work: The Brownian cactus. I: Scaling limits of discrete cactuses
cites work: Q3721531
cites work: Q4778955
cites work: Explicit enumeration of triangulations with multiple boundaries
cites work: Uniform Infinite Planar Triangulation and Related Time-Reversed Critical Branching Process
cites work: Semi-stable Markov processes. I
cites work: Uniqueness and universality of the Brownian map
cites work: The Brownian map: a universal limit for random planar maps
cites work: Analysis.
cites work: Conceptual proofs of \(L\log L\) criteria for mean behavior of branching processes
cites work: Percolation on uniform infinite planar maps
cites work: Self-similar fragmentation derived from the stable tree. I: Splitting at heights
cites work: The Brownian map is the scaling limit of uniform random plane quadrangulations
cites work: An axiomatic characterization of the Brownian map
cites work: Quantum Loewner evolution
cites work: The falling apart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation
cites work: Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation
description: scientific article