On the large genus of Hurwitz numbers

TL;DR

This paper develops a unified elementary framework for analyzing the large genus asymptotics of various Hurwitz numbers, covering many known types and applications in geometry and physics.

Contribution

It introduces a general elementary approach to large genus asymptotics that applies to a wide range of Hurwitz numbers, including new and classical cases.

Findings

Unified method recovers classical Hurwitz asymptotics

Applies to multiple types of Hurwitz numbers including monotone and weighted variants

Includes applications to matrix models, dessins d'enfant, and Gromov-Witten invariants

Abstract

Hurwitz theory provides a large variety of enumerative problems related to algebraic geometry, mathematical physics, and combinatorics. We give a general framework to approach the large genus asymptotics of Hurwitz theory using only elementary methods and apply it to several types of Hurwitz numbers: single, double, or with an arbitrary numbers of fixed ramifications; simple and / or including completed cycles type of ramification and / or finitely many blocks of weakly monotone and / or strictly monotone types of ramifications. These, to the best of our knowledge, cover most of the Hurwitz numbers studied, and include for instance correlators of the HCIZ matrix model, Grothendieck dessins d'enfant, weighted Hurwitz numbers, and Gromov-Witten invariants of the Riemann sphere. We also apply our method to b-content Hurwitz numbers. As a specialisation, we recover some previously known about the large genus asymptotics of Hurwitz theory, namely classical results by Hurwitz and recent results of Do-He-Robertson, C. Yang, and results connected to recent work of X. Li.