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High genus one part monotone Hurwitz numbers

Simon Barazer, Baptiste Louf

Abstract

We obtain bivariate asymptotics for one part monotone Hurwitz numbers in high genus (i.e. as both the size and the genus go to infinity). To do so, we start with a linear recurrence for these numbers obtained by Do and Chaudhuri. Then, we apply a recent method developped by Elvey-Price, Fang, Wallner and the second author to extract asymptotics from such recurrences.

High genus one part monotone Hurwitz numbers

Abstract

We obtain bivariate asymptotics for one part monotone Hurwitz numbers in high genus (i.e. as both the size and the genus go to infinity). To do so, we start with a linear recurrence for these numbers obtained by Do and Chaudhuri. Then, we apply a recent method developped by Elvey-Price, Fang, Wallner and the second author to extract asymptotics from such recurrences.

Paper Structure

This paper contains 16 sections, 15 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. High genus: geometry and asymptotics
  3. Monotone Hurwitz numbers
  4. Result
  5. Definitions and heuristic guessing
  6. Heuristic guessing
  7. Rigorous definition
  8. Asymptotic properties in $\theta=0$
  9. Asymptotic properties in $\theta=\infty$
  10. Proof ideas
  11. Proof of the assumptions
  12. $\alpha+\beta=1$
  13. Low genus
  14. Intermediate and high genus
  15. Proof of proposition \ref{['prop_alpha_beta']}
  16. ...and 1 more sections

Key Result

Theorem 1

As $n\rightarrow\infty$, for any sequence $g=g_n$, we have where $f$ and $j$ are explicit functions defined in section sec_def. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (31)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3: Coefficient $n^{2g-2}$
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • ...and 21 more