Upper bound of some character ratios and large genus asymptotic behavior of Hurwitz numbers

TL;DR

This paper extends the understanding of Hurwitz numbers by deriving asymptotic bounds for arbitrary Riemann surfaces with specific profile types, generalizing previous results on the Riemann sphere.

Contribution

It generalizes previous asymptotic results of Hurwitz numbers from the sphere to arbitrary compact Riemann surfaces with specific profile configurations.

Findings

Derived upper bounds for character ratios.

Established large genus asymptotics for Hurwitz numbers.

Extended previous results to more general surfaces and profiles.

Abstract

In [14] we found the large genus asymptotics of Hurwitz numbers for the Riemann sphere with a fixed number of general profiles and some (2,1^{d-2}) profiles. In this paper, motivated from [3], we generalize these results to Hurwitz numbers of an arbitrary compact Riemann surface with a fixed number of general profiles and some (r,1^{d-r}) profiles.