Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Large unicellular maps in high genus

Gourab Ray

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We study the geometry of a random unicellular map which is uniformly distributed on the set of all unicellular maps whose genus size is proportional to the number of edges. We prove that the distance between two uniformly selected vertices of such a map is of order $\log n$ and the diameter is also of order $\log n$ with high probability. We further prove a quantitative version of the result that the map is locally planar with high probability. The main ingredient of the proofs is an exploration procedure which uses a bijection due to Chapuy, Feray and Fusy (J. Combin. Theory Ser. A 120 (2013) 2064–2092).


Nous étudions la géometrie d’une carte aléatoire unicellulaire qui est distribuée uniformement sur l’ensemble de toutes les cartes unicellulaires dont le genre est proportionnel au nombre des arrêtes. Nous prouvons que la distance entre deux sommets choisis uniformement d’une telle carte est de l’ordre $\log n$ et le diamètre est aussi de l’ordre $\log n$ avec une forte probabilité. Nous prouvons aussi une version quantitative du résultat que la carte est localement planaire avec une forte probabilité. L’ingrédient principal de la preuve est une procédure d’exploration qui utilise une bijection due au Chapuy, Féray et Fusy (J. Combin. Theory Ser. A 120 (2013) 2064–2092).

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 4 (2015), 1432-1456.

Received: 10 July 2013
Revised: 14 January 2014
Accepted: 25 March 2014
First available in Project Euclid: 21 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B05: Probability measures on topological spaces 60B10: Convergence of probability measures 97K50: Probability theory

Unicellular maps High genus maps Hyperbolic Diameter Typical distance $C$-permutations


Ray, Gourab. Large unicellular maps in high genus. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 4, 1432--1456. doi:10.1214/14-AIHP618.

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