The local limit of unicellular maps in high genus

Omer Angel,Nicolas Curien,Gourab Ray,Guillaume Chapuy

doi:10.1214/ecp.v18-3037

Abstract

We show that the local limit of unicellular maps whose genus is proportional to the number of edges is a supercritical geometric Galton-Watson tree conditioned to survive. The proof relies on enumeration results obtained via the recent bijection given by the second author together with Feray and Fusy.

Highlights

The last author of this note studied the large scale structure of random unicellular maps whose genus grows linearly with their size [12]
Motivated by the theory of two-dimensional quantum gravity, the study of local limits of random planar maps and graphs has been rapidly developing over the last years, since the introduction of the Uniform Infinite Planar Triangulation (UIPT) by Angel & Schramm [2]
The UIPT is defined as the local limit in distribution of uniform random triangulations of the sphere, when their size tends to infinity

Summary

Introduction

The last author of this note studied the large scale structure of random unicellular maps whose genus grows linearly with their size [12]. For θ = 0, the genus is much smaller than the size of the map, so it is not surprising that the local limit is the same as that of a critical tree conditioned to survive. Note that the mean of the geometric offspring distribution in Theorem 1 is given by (1 + βθ)/(1 − βθ) > 1 and in particular the Galton-Watson tree is supercritical. In order to prove Theorem 1 we first determine the root degree distribution of unicellular maps using the bijection of [6]

Enumeration and root degree distribution

The low genus case

The local limit

Identifying the limit

Questions and remarks