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

TL;DR

This work introduces a discrete Mirzakhani-type recursion for pruned matrix-model traces in generic one-cut ensembles, revealing a lattice-point view of moduli-space volumes that persists beyond the usual double-scaling limit. In a BMN-like large-trace regime, the discrete recursion flows to the continuous Kontsevich volumes, yielding an Airy-universality description independent of the potential. The authors then show that the DSSYK ETH matrix integral furnishes a discrete $q$-analog of Weil--Petersson volumes, proving Okuyama's conjecture via a $q$-deformed Mirzakhani recursion and a precise limiting procedure. Together, these results connect random matrix theory, moduli space geometry, and low-dimensional gravity through discrete versus continuous volumes and their universal limits. The framework provides new computational tools and a unifying perspective across hyperbolic, Strebel, and lattice-based moduli-space descriptions, with potential extensions to CohFTs and holographic duals.

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.

Figures (7)

• Figure 1: A Pictorial Representation of the Discrete Recursion: The pruned correlators $\langle \prod_{i=1}^n \frac{1}{b_i} \! :\mathrel{\mathop{\mathrm{Tr}}\nolimits{M^{b_i}}}: \rangle_{g,\mathrm{c}}$ define a discrete notion of volume of the moduli space of Riemann surfaces, denoted $N_{g,n}(b_1,\ldots,b_n)$. They satisfy a discrete recursion relation that parallels Mirzakhani's formula for the Weil--Petersson volumes. The recursion kernels $B$ and $C$ can be computed directly from the matrix model potential.
• Figure 2: A Moduli Space Perspective on the BMN-like Limit: In the limit of large traces, many more Wick contractions are possible. Since, each such Feynman diagram maps onto one point on $\mathcal{M}_{g,n}$, more and more points populate the moduli space. Our recursion relation proves that, generically, the $N_{g,n}$ converge to the well-known continuum Kontsevich volumes, $V^{\textup{Kon}}_{g,n}$.
• Figure 3: Riemann Surfaces as Metrized Ribbon Graphs: The Strebel differential foliates any Riemann surface by a a unique set of curves, called horizontal trajectories (in red). A measure zero subset, the critical trajectories, assign a unique Strebel graph to the surface (left panel). The moduli are encoded as edge lengths, $\ell_{e}$, providing the basis for the combinatorial description of the moduli space, $\mathcal{M}_{g,n}^{\textup{comb}}$. The sum of these lengths around a face of the Strebel graph must equal the length of the boundary. Geometrically, this decomposes the surface as a collection of semi-infinite flat cylinders, glued to the Strebel graph.
• Figure 4: Geometric Origin of the Kernels: By shooting an orthogeodesic (in red) from the first boundary component of the surface $\Sigma$, one determines one or two simple closed curves (in green). Different behaviors can arise: on the left, the orthogeodesic intersects the boundary component $\partial_m \Sigma$ ($B_m$-type), determining a single internal geodesic $\gamma$. In the two other cases, the orthogeodesic intersects $\partial_1 \Sigma$ or itself ($C$-type), determining two internal geodesics $\gamma$ and $\gamma'$. The kernels compute the probability of these different behaviors occurring.
• Figure 5: Matrix correlators as lattice point counts on $\mathcal{M}_{g,n}$. Expanding matrix model correlators in terms of Feynman diagrams allows one to reinterpret them as a count of discrete lattice points on the moduli space. Collapsing homotopic edges of a Feynman diagram yields a skeleton graph with integer edge lengths. By taking its graph dual, one obtains am integer Strebel graph parameterizing the corresponding point on moduli space.

Theorems & Definitions (14)

• Theorem A
• Theorem B
• Theorem C: Okuyama's conjecture
• Theorem 2.1: Mirzakhani
• Theorem 2.2: Kontsevich et al.
• Theorem 2.3: Norbury
• Proposition 6.1
• Lemma B.1
• proof
• Lemma B.2