Combinatorics and asymptotic behavior for double Hurwitz numbers

TL;DR

This paper derives Pandharipande-type equations for double Hurwitz numbers from the 2-Toda hierarchy and studies their asymptotic behavior in large genus and degree regimes.

Contribution

It extends the Pandharipande equation framework to double Hurwitz numbers and analyzes their asymptotics using integrable hierarchy methods.

Findings

Derived Pandharipande-type equations for double Hurwitz numbers.

Established polynomial-time algorithms for computing these numbers.

Analyzed large genus and degree asymptotic behaviors.

Abstract

Polynomial-in-time algorithms for computing classical Hurwitz numbers were given in [4] based on the Pandharipande equation. The paritition function of double Hurwitz numbers was proved [21] to satisfy the 2-Toda hierarchy. In this paper, similar to [21] we derive Pandharipande-type equations for double Hurwitz numbers from 2-Toda hierarchy. Based on these equations and a method from [4], we study large genus as well as large degree asymptotics of double Hurwitz numbers.