Abstract: ‘Recent works have proposed that pure gravity in $AdS_3$ may be understood as an ensemble average over approximate $\mathrm{CFT}_2$ data. We explore the spectral content of this idea by constructing a “Cardy random matrix theory”: an RMT whose only bootstrap input is coarse-grained $\mathbb{S}$-modular invariance.
When we prescribe the vacuum to be the only state below the black-hole threshold, the matrix potential fixes, at leading order, the number of eigenvalues in each window, and its minimum reproduces in the high-energy regime the coarse-grained Cardy density. Monte Carlo simulations confirm this behavior and exhibit, within a high-energy window, GUE statistics: Wigner–Dyson level-spacings and a linear ramp in the connected spectral form factor. In a complementary dynamical analysis, we identify a scaling regime in which vacuum occupation is favored and Cardy growth emerges in the heavy spectrum. The model therefore provides a simple picture in which modular invariance determines the coarse-grained spectral density, while RMT governs local spectral correlations. In a tractable high-energy limit, we relate the crossover between these regimes to a Thouless time parametrically set by the inverse local energy width of a coarse-graining window.’