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

Xiang Li

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.

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

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.
Paper Structure (3 sections, 10 theorems, 43 equations)

This paper contains 3 sections, 10 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['cH2']} and Theorem \ref{['cH1']}
  3. Further remarks

Key Result

Theorem 1.1

For any fixed $d\geq7,2\leq r\leq d-2,s\geq0$ and $\mu^{(1)},\dots,\mu^{(s)}\vdash d$, we have where $d!^{2g(X)}\prod_{i=1}^s\frac{d!}{z_{\mu^{(i)}}}b_r^{X}(\mu^{(1)},\dots,\mu^{(s)},m)$ are integers with 1. $b^X_r(\mu^{(1)},\dots,\mu^{(s)},\frac{d!}{r(d-r)!})=1$; 2. $b^X_r(\mu^{(1)},\,\cdots,\,\mu^{(s)},m)=0$ for $\frac{(d-1)!}{r\cdot(d-r-1)!}<m<\frac{d!}{r(d-r)!}$; 3. $b^X_r(\mu^{(1)},\,\cd

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 11 more