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A high-genus asymptotic expansion of Weil-Petersson volume polynomials

Nalini Anantharaman, Laura Monk

Abstract

The object under consideration in this article is the total volume $V_{g,n}(x_1, \ldots, x_n)$ of the moduli space of hyperbolic surfaces of genus $g$ with $n$ boundary components of lengths $x_1, \ldots, x_n$, for the Weil-Petersson volume form. We prove the existence of an asymptotic expansion of the quantity $V_{g,n}(x_1, \ldots, x_n)$ in terms of negative powers of the genus $g$, true for fixed $n$ and any $x_1, \ldots, x_n \geq 0$. The first term of this expansion appears in work of Mirzakhani and Petri (2019), and we compute the second term explicitly. The main tool used in the proof is Mirzakhani's topological recursion formula, for which we provide a comprehensive introduction.

A high-genus asymptotic expansion of Weil-Petersson volume polynomials

Abstract

The object under consideration in this article is the total volume of the moduli space of hyperbolic surfaces of genus with boundary components of lengths , for the Weil-Petersson volume form. We prove the existence of an asymptotic expansion of the quantity in terms of negative powers of the genus , true for fixed and any . The first term of this expansion appears in work of Mirzakhani and Petri (2019), and we compute the second term explicitly. The main tool used in the proof is Mirzakhani's topological recursion formula, for which we provide a comprehensive introduction.

Paper Structure

This paper contains 37 sections, 21 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction and statement of the results
  2. First definitions and notations
  3. Asymptotic expansions of Weil--Petersson volumes
  4. Notations
  5. State of the art
  6. Statement of the main result
  7. Coefficient estimate and sketch of the proof
  8. Expansion in negative powers of $g$
  9. Explicit expression for the first orders
  10. Motivation to the study of random compact hyperbolic surfaces
  11. Organisation of the paper
  12. Acknowledgements
  13. Preliminaries: Weil--Petersson volumes and Mirzakhani's topological recursion
  14. Polynomial expression
  15. Mirzakhani's topological recursion formula
  16. ...and 22 more sections

Key Result

Theorem 1.1

For any integers $g \geq 0$, $n \geq 1$ such that $2g-2+n>0$, there exists a family of $n$-variable even polynomial functions $(P_{g,n}^{(N,I_\pm)})_{N, I_\pm}$, for $N \geq 0$ and $I_+ \sqcup I_- \subseteq \{1, \ldots, n\}$, such that for any integer $N \geq 0$ and any length vector $\mathbf{x} \in where Furthermore, there exists constants $D_{n,N}, A_N \geq 0$ such that the polynomial function

Figures (1)

  • Figure 1: Dependency of the coefficients of the volume polynomials $V_{g,n}(\mathbf{x})$ when $|\chi|=2g-2+n \leq 3$. Note that all the coefficients for which $n \neq 0$ can therefore be computed thanks to the coefficients for which $|\chi| = 1$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Example
  • Theorem 2.1: mirzakhani2007
  • Example
  • Lemma 2.2: mirzakhani2013
  • Lemma 2.3
  • ...and 33 more