# Universality for random surfaces in unconstrained genus

## TL;DR

This paper demonstrates the universality of geometric properties in random surfaces formed from polygons, showing convergence towards a Poisson--Dirichlet distribution and analyzing the diameter of random maps.

## Contribution

It provides an alternative probabilistic proof of known results and extends the understanding of universal properties in random surfaces and maps.

## Key findings

- Degree sequence converges to Poisson--Dirichlet distribution.
- Diameter of a random map with n edges converges to a variable in {2,3}.
- Several geometric properties are universal across configurations.

## 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 and Pittel (arXiv:1503.01816) 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 and 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 $\{2,3\}$.