Cap amplitudes in random matrix models

TL;DR

The paper introduces the cap amplitude $\psi(b)$ as the expansion coefficient of the 1-form $y\,dx$ on the spectral curve of general one-matrix models and shows that the dilaton equation for the discrete volumes $N_{g,n}$ is realized by gluing the cap along a boundary, reducing the boundary count. It then demonstrates that the genus-$g$ free energies $F_g$ are obtained by gluing the cap to $N_{g,1}$, and that the entire hierarchy of $N_{g,n}$ and $F_g$ is determined by the cap data through the moments $M_k,J_k$, which themselves are encoded by $\psi(b)$. The framework is validated in explicit Gaussian and ETH matrix-model examples, with genus expansions matching known results (e.g., Ambjørn–1992gw) and with precise small-$q$ expansions in the ETH/DSSYK case. These results provide a geometric and combinatorial bridge between spectral-curve data, topological recursion, and discrete moduli-space volumes, offering a compact, cap-centered way to compute the full genus expansion from boundary-cap data, with potential links to CFT and holography.

Abstract

For general one-matrix models in the large $N$ limit, we introduce the cap amplitude $ψ(b)$ as the expansion coefficient of the 1-form $ydx$ on the spectral curve. We find that the dilaton equation for the discrete volume $N_{g,n}$ of the moduli space of genus-$g$ Riemann surfaces with $n$ boundaries is interpreted as gluing the cap amplitude along one of the boundaries. In this process, one of the boundaries is capped and the number of boundaries decreases by one. In a similar manner, the genus-$g$ free energy $F_g$ is obtained by gluing the cap amplitude to $N_{g,1}$.