Matrix Correlators as Discrete Volumes of Moduli Space I: Recursion Relations, the BMN-limit and DSSYK

TL;DR

This paper establishes a connection between matrix model correlators and discrete volumes of moduli space, revealing recursion relations and limits that relate to known geometric volumes and confirming a conjecture in the context of DSSYK.

Contribution

It introduces a framework linking matrix correlators to discrete moduli space volumes, extending the understanding of matrix models beyond the double-scaling limit and proving a conjecture about DSSYK.

Findings

Correlators define discrete moduli space volumes with Mirzakhani-like recursion.

In a BMN-like limit, recursion transitions to a continuous form matching Kontsevich volumes.

DSSYK matrix integral provides a q-analog of Weil--Petersson volumes, confirming Okuyama's conjecture.

Abstract

We show certain correlators in generic one-matrix models define a notion of ``discrete'' volumes of the moduli space of Riemann surfaces, generalizing the connection between random matrices and JT gravity. We prove they obey a discrete, Mirzakhani-like recursion relation. Their fundamental discreteness crucially relies upon studying these matrix integrals away from the usual double-scaling limit. In a BMN-like limit of large traces, this recursion universally goes over to a continuous one, and the correlators asymptote to the volumes of Kontsevich. Finally, we demonstrate that the ETH matrix integral for DSSYK furnishes a discrete, $q$-analog of the Weil--Petersson volumes, thereby proving a conjecture due to K. Okuyama.