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

TL;DR

This paper explores how the energy spectrum's statistical properties influence thermalization and echoes in black hole models, using random matrix theory and spectral form factors to connect quantum chaos with gravitational phenomena.

Contribution

It introduces a statistical framework for generating and analyzing random spectra, linking spectral statistics to black hole thermalization and echoes in holographic models.

Findings

Spectral form factor varies with different nearest neighbor statistics.

Late time oscillations can occur in $eta$-ensemble random matrix models.

Weakly coupled oscillators exhibit decaying spectral form factor oscillations.

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.