On the convergence of monotone Hurwitz generating functions

TL;DR

This paper studies the convergence properties of monotone Hurwitz generating functions, revealing they are absolutely summable in fixed genus, unlike classical counterparts, with bounds on their growth rates and convergence radii.

Contribution

It identifies a key difference in convergence between monotone and classical Hurwitz numbers and provides bounds on the growth rate of monotone Hurwitz numbers in fixed genus.

Findings

Monotone Hurwitz generating functions are absolutely summable in fixed genus.

Classical Hurwitz generating functions are not absolutely summable.

Universal bounds on the radii of convergence for monotone Hurwitz numbers.

Abstract

Monotone Hurwitz numbers were introduced by the authors as a combinatorially natural desymmetrization of the Hurwitz numbers studied in enumerative algebraic geometry. Over the course of several papers, we developed the structural theory of monotone Hurwitz numbers and demonstrated that it is in many ways parallel to that of their classical counterparts. In this note, we identify an important difference between the monotone and classical worlds: fixed-genus generating functions for monotone double Hurwitz numbers are absolutely summable, whereas those for classical double Hurwitz numbers are not. This property is crucial for applications of monotone Hurwitz theory in analysis. We quantify the growth rate of monotone Hurwitz numbers in fixed genus by giving universal upper and lower bounds on the radii of convergence of their generating functions.