Cardy random matrix theory

Abstract

We construct and study a ``Cardy random matrix model'': a random matrix theory whose only input, in a maximal-ignorance prescription, is the constraint of $\mathbb{S}$-modular invariance. We regularize this constraint by coarse-graining over energy windows of fixed width, and obtain a potential for a single random matrix whose eigenvalues play the role of the scaling dimensions of primaries in a 2d CFT. The potential drives the coarse-grained density of states to the Cardy profile above a gap at the black hole threshold. We perform Monte Carlo simulations to confirm this behavior and find good agreement for the coarse-grained density of states, while at the same time revealing GUE statistics for nearby eigenvalues. We identify the scaling limit in which these phenomena emerge. In this limit, the model realizes a hierarchy of scales: at the macroscopic scale given by the window size, the occupations are frozen and the bootstrap constraint rigidly fixes the spectrum to the Cardy profile, while at the mesoscopic scale (inside a window) the Vandermonde repulsion rearranges the eigenvalues as in a random matrix theory. It is only at the microscopic scale of the mean level spacing that the individual levels are resolved. Accordingly, our random matrix model displays an effective Thouless time, set by the inverse energy width of the coarse-graining window. Our model provides a first step towards solving the matrix/tensor model dual to 3D quantum gravity on a computer.