Large Unicellular maps in high genus

TL;DR

This paper investigates the geometric properties of large random unicellular maps with high genus, revealing that distances and diameters grow logarithmically with the map size, and the maps are locally planar.

Contribution

It establishes the logarithmic scale of distances and diameters in high-genus unicellular maps and demonstrates their local planarity, using a novel exploration method based on a bijection.

Findings

Distances between vertices are of order log n

Map diameters are of order log n

Maps are locally planar with high probability

Abstract

We study the geometry of a random unicellular map which is uniformly distributed on the set of all unicellular maps whose genus size is proportional to the number of edges of the map. We prove that the distance between two uniformly selected vertices of such a map is of order $\log n$ and the diameter is also of order $\log n$ with high probability. We further prove that the map is locally planar with high probability. The main ingredient of the proofs is an exploration procedure which uses a bijection due to Chapuy, Feray and Fusy.