Universality for random surfaces in unconstrained genus

Thomas Budzinski¹ ; Nicolas Curien² ; Bram Petri³

¹ENS Paris
²Université Paris-Saclay and Institut Universitaire de France
³Universität Bonn

The electronic journal of combinatorics, Tome 26 (2019) no. 4

Résumé

Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov & Pittel have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson–Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov & Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.

Détail

DOI : 10.37236/8623

Classification : 05C80, 60C05, 05C07, 05C12, 05C10, 05C78
Mots-clés : configuration model, surfaces, polygonal discs, random permutations, irreducible characters, Euler characteristic, limit distributions