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Recent advances in AdS/CFT holography have suggested that the near-horizondynamics of black holes can be described by random matrix systems. We study howthe energy spectrum of a system with a generic random Hamiltonian matrixaffects its early and late time thermalization behaviour using the spectralform factor (which captures the time-dependence of two-point correlationfunctions). We introduce a simple statistical framework for generating randomspectra in terms of the nearest neighbor spacing statistics of energyeigenvalues, enabling us to compute the averaged spectral form factor in aclosed form. This helps to easily illustrate how the spectral form factorchanges with different choices of nearest neighbor statistics ranging from thePoisson to Wigner surmise statistics. We suggest that it is possible to havelate time oscillations in random matrix models involving $\beta$-ensembles(generalizing classical Gaussian ensembles). We also study the form factor ofrandomly coupled oscillator systems and show that at weak coupling, suchsystems exhibit regular decaying oscillations in the spectral form factormaking them interesting toy models for gravitational wave echoes. We speculateon the holographic interpretation of a system of coupled oscillators, andsuggest that they describe the thermalization behaviour of a black holegeometry with a membrane that cuts off the geometry at the stretched horizon.

Spacing Statistics of Energy Spectra: Random Matrices, Black Hole Thermalization, and Echoes