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

Abstract

Recent advances in AdS/CFT holography have suggested that the near-horizon dynamics of black holes can be described by random matrix systems. We study how the energy spectrum of a system with a generic random Hamiltonian matrix affects its early and late time thermalization behaviour using the spectral form factor (which captures the time-dependence of two-point correlation functions). We introduce a simple statistical framework for generating random spectra in terms of the nearest neighbor spacing statistics of energy eigenvalues, enabling us to compute the averaged spectral form factor in a closed form. This helps to easily illustrate how the spectral form factor changes with different choices of nearest neighbor statistics ranging from the Poisson to Wigner surmise statistics. We suggest that it is possible to have late time oscillations in random matrix models involving $\beta$-ensembles (generalizing classical Gaussian ensembles). We also study the form factor of randomly coupled oscillator systems and show that at weak coupling, such systems exhibit regular decaying oscillations in the spectral form factor making them interesting toy models for gravitational wave echoes. We speculate on the holographic interpretation of a system of coupled oscillators, and suggest that they describe the thermalization behaviour of a black hole geometry with a membrane that cuts off the geometry at the stretched horizon.

60 pages, 22 figures, consistent with published JHEP version

References in corpus (13)

• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Echoes of ECOs: gravitational-wave signatures of exotic compact objects and of quantum corrections at the horizon scale
• Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
• Chaos, Complexity, and Random Matrices
• A recipe for echoes from exotic compact objects
• Introduction to Random Matrices - Theory and Practice
• Lectures on Gravity and Entanglement
• The Black Hole Information Problem
• Wormholes and Spectral Statistics in the Narain Ensemble
• From operator statistics to wormholes
• Homology Conditions for RT Surfaces in Double Holography
• Nearest-neigbor spacing distributions of the beta-Hermite ensemble of random matrices
• Ringdown of Charged Compact Objects using Membrane Paradigm