The structures of simple Hurwitz numbers and monotone Hurwitz numbers with varying genus

TL;DR

The paper analyzes the structures of ordinary simple Hurwitz numbers $H_{g;g}$ and monotone Hurwitz numbers $_{g;g}$ as genus $g$ varies. Using KP hierarchy and tau-function formalisms, it proves that $_{g;g}$ is a genus-independent linear combination of products of exponentials and polynomials, while $H_{g;g}$ is a linear combination of exponentials, and it derives their large-genus asymptotics. It further confirms Do–He–Robertson's conjecture for simple Hurwitz numbers and refutes their monotone conjecture by providing counterexamples such as $g=(3,3)$. These results illuminate the interplay between integrable systems, representation theory, and branched-cover enumerations, and they sharpen our understanding of the asymptotic behavior of Hurwitz-type counts. The methods provide a unified approach to structural decompositions and asymptotics across both simple and monotone Hurwitz numbers, with potential applications to related enumerative problems.

Abstract

We study the structures of ordinary simple Hurwitz numbers and monotone Hurwitz numbers with varying genus. More precisely, we prove that when the ramification type is fixed and the genus is treated as a variable, the connected monotone Hurwitz number is a linear combination of products of exponentials and polynomials, and the ordinary simple Hurwitz number is a linear combination of exponentials. Using these structural properties, we also derive the large genus asymptotics of these two kinds of Hurwitz numbers. As a result, we prove one conjecture, and disprove another, both proposed by Do, He and Robertson.

Theorems & Definitions (23)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3: Do--He--Robertson's Theorem 1.5 and Conjecture 4.1 DHR
• Remark 2.1
• Remark 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Theorem 3.4: = Theorem \ref{['thm:main thm for mono']}
• proof