Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Geodesics in Brownian surfaces (Brownian maps)

Jérémie Bettinelli

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Abstract

We define a class a metric spaces we call Brownian surfaces, arising as the scaling limits of random maps on general orientable surfaces with a boundary and we study the geodesics from a uniformly chosen random point. These metric spaces generalize the well-known Brownian map and our results generalize the properties shown by Le Gall on geodesics in the latter space. We use a different approach based on two ingredients: we first study typical geodesics and then all geodesics by an “entrapment” strategy. In particular, we give geometrical characterizations of some subsets of interest, in terms of geodesics, boundary points and concatenations of geodesics forming a loop that is not homotopic to $0$.

Résumé

On définit une classe d’espaces métriques aléatoires que nous appelons surfaces browniennes : ces objets apparaissent comme limites d’échelle de cartes aléatoires sur des surfaces orientables à bord générales. Dans un second temps, on étudie les géodésiques émanant d’un point choisi uniformément au hasard. Les surfaces browniennes généralisent la fameuse carte brownienne et nos résultats généralisent les propriétés obtenues par Le Gall sur les géodésiques dans cet espace. On utilise une approche différente reposant sur deux ingrédients : on étudie d’abord les géodésiques aux points typiques et on attrape ensuite les autres points en les « encerclant » par de telles géodésiques. En particulier, on obtient des caractérisations géométriques de certains sous-ensembles d’intérêt en termes de géodésiques, points du bord et concaténations des géodésiques formant une boucle non homotope à $0$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 52, Number 2 (2016), 612-646.

Dates
Received: 8 October 2014
Accepted: 19 December 2014
First available in Project Euclid: 4 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1462367887

Digital Object Identifier
doi:10.1214/14-AIHP666

Mathematical Reviews number (MathSciNet)
MR3498003

Zentralblatt MATH identifier
1342.60043

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 57N05: Topology of $E^2$ , 2-manifolds 05C80: Random graphs [See also 60B20] 05C12: Distance in graphs

Keywords
Brownian surfaces Brownian map Geodesics Random maps Scaling limits Gromov–Hausdorff topology Random metric spaces Bijections

Citation

Bettinelli, Jérémie. Geodesics in Brownian surfaces (Brownian maps). Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 612--646. doi:10.1214/14-AIHP666. https://projecteuclid.org/euclid.aihp/1462367887


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