Bivariate asymptotics via random walks: application to large genus maps

TL;DR

This paper develops a method to derive bivariate asymptotics for the enumeration of combinatorial maps as size and genus increase, connecting multivariate asymptotics and large genus geometry.

Contribution

It introduces a general theorem for asymptotics of linear recurrences, applicable to combinatorial maps and potentially other models.

Findings

Derived bivariate asymptotics for map enumeration with growing size and genus

Established a general theorem for asymptotics of linear recurrences

Linked multivariate asymptotics with large genus geometry

Abstract

We obtain bivariate asymptotics for the number of (unicellular) combinatorial maps (a model of discrete surfaces) as both the size and the genus grow. This work is related to two research topics that have been very active recently: multivariate asymptotics and large genus geometry. Our method consists of studying a linear recurrence for these numbers, and can be applied to many other linear recurrences. In particular, we include a general theorem that yields asymptotics for such recurrences, provided that some assumptions are satisfied.