Large Unicellular maps in high genus

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.