Universality for Random Surfaces in Unconstrained Genus

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 $\{2,3\}$.

A model for random three--manifolds

“…The analogous result is known to hold for surfaces obtained by randomly gluing polygons together [23,15,12].…”

A product of invariant random permutations has the same small cycle structure as uniform

“…Similar results have been obtained in [9] for permutations that are equicontinuous in both coordinates and converging as a permuton (see definitions there). With the motivation of random gluing of polygons, a stronger convergence (in total variation distance) was established in [4] when one of the permutations has all its cycles of length at least 3 (see also [5]) and in [3] when one of the permutations is an involution without fixed point. None of the previous results covers for example the product of two Ewens distributions.…”

Large genus bounds for the distribution of triangulated surfaces in moduli space

“…In [20], Guth, Parlier and Young showed that both random triangulated surfaces and random hyperbolic surfaces have large total pants decomposition length. In [7], Budzinski, Curien and Petri computed the asymptotic diameter of a random triangulated surface with respect to the hyperbolic metric, while in [10], Budzinski, Curien and Petri computed the asymptotic minimal diamater of a hyperbolic surface, and these quantities simply differ by a factor of 2. These results suggest a close relationship between random triangulated surfaces and random hyperbolic surfaces.…”