High genus one part monotone Hurwitz numbers

TL;DR

This paper derives asymptotic formulas for high genus one-part monotone Hurwitz numbers by analyzing a linear recurrence and applying a recent asymptotic extraction method.

Contribution

It introduces a new asymptotic analysis for monotone Hurwitz numbers in high genus using a novel method for extracting asymptotics from recurrences.

Findings

Derived bivariate asymptotics for high genus monotone Hurwitz numbers

Applied a recent method to extract asymptotics from linear recurrences

Extended understanding of the growth behavior of these combinatorial numbers

Abstract

We obtain bivariate asymptotics for one part monotone Hurwitz numbers in high genus (i.e. as both the size and the genus go to infinity). To do so, we start with a linear recurrence for these numbers obtained by Do and Chaudhuri. Then, we apply a recent method developped by Elvey-Price, Fang, Wallner and the second author to extract asymptotics from such recurrences.