Short cycles in high genus unicellular maps

TL;DR

This paper investigates the distribution of short cycles and vertex degrees in large high-genus unicellular maps, revealing Poisson and asymptotic laws that enhance understanding of discrete hyperbolic geometry.

Contribution

It provides the first analysis of short cycle counts and vertex degree distributions in high genus unicellular maps, connecting geometric features with probabilistic laws.

Findings

Number of short cycles converges to a Poisson distribution.

Law of the systole in high genus maps is established.

Vertex degrees follow a specific asymptotic distribution.

Abstract

We study large uniform random maps with one face whose genus grows linearly with the number of edges, which are a model of discrete hyperbolic geometry. In previous works, several hyperbolic geometric features have been investigated. In the present work, we study the number of short cycles in a uniform unicellular map of high genus, and we show that it converges to a Poisson distribution. As a corollary, we obtain the law of the systole of uniform unicellular maps in high genus. We also obtain the asymptotic distribution of the vertex degrees in such a map.