No abstract

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 $\{2,3\}$.

We introduce and study a new random surface, which we call the hyperbolic Brownian plane and which is the near-critical scaling limit of the hyperbolic triangulations constructed in [16]. The law of the hyperbolic Brownian plane is obtained after biasing the law of the Brownian plane [17] by an explicit martingale depending on its perimeter and volume processes studied in [19]. Although the hyperbolic Brownian plane has the same local properties as those of the Brownian plane, its large scale structure is much different since we prove e.g. that is has exponential volume growth. * ENS Paris and Université Paris-Saclay, thomas.budzinski@ens.fr Properties of P h . We also establish some properties of the hyperbolic Brownian plane. Since the density (3) goes to 1 as r goes to 0, the hyperbolic Brownian plane is "locally isometric" to the Brownian plane (and hence also to the Brownian map). More precisely, for all ε > 0, there is a δ > 0 and a coupling between P and P h such that, with probability at least 1 − ε, they have the same ball of radius δ around the origin. We also prove that P h almost surely has Hausdorff dimension 4 and is homeomorphic to the plane.