Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models

TL;DR

This paper demonstrates that Mirzakhani's recursion formulas for moduli space volumes are instances of random matrix loop equations, confirming Kontsevich's integral as a generating function, and proposes a new volume formula.

Contribution

It reveals the connection between Mirzakhani's recursions and matrix models, providing a new perspective and a formula for Weil-Petersson volumes.

Findings

Mirzakhani's recursions are special cases of matrix loop equations

Kontsevich's integral generates moduli space volumes

Proposed a new formula for Weil-Petersson volume Vol(M_{g,0})

Abstract

We show that Mirzakhani's recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix loop equations, and therefore we confirm again that Kontsevitch's integral is a generating function for those volumes. As an application, we propose a formula for the Weil-Petersson volume Vol(M_{g,0}).