A Matrix Model for Black Hole Thermalization

TL;DR

The paper presents a tractable matrix-model toy for AdS black hole thermodynamics, coupling a $U(N)$ adjoint harmonic oscillator to a fundamental oscillator to study thermalization. In the planar limit, the fundamental correlator decays to zero at long times, captured by a closed Schwinger-Dyson recursion that is exactly solvable at zero temperature and extensible to finite temperature, where the decay behavior transitions from power-law to exponential with increasing temperature or coupling. A Hawking-Page-like transition is engineered by enforcing a singlet constraint, illustrating how phase structure can alter quasinormal behavior in this setting. The discussion connects large-$N$ dissipativity to the black hole information problem, exploring how nonperturbative effects or saddle-point contributions might restore information and highlighting the model’s limitations and interpretations within holography and the information paradox.

Abstract

We present a matrix model which is intended as a toy model of the gauge dual of an AdS black hole. In particular, it captures the key property that at infinite $N$ correlators decay to zero on long time scales, while at finite $N$ this cannot happen. The model consists of a harmonic oscillator in the adjoint which acts as a heat bath for a particle in the fundamental representation. The Schwinger-Dyson equation reduces to a closed recursion relation, which we study by various analytical and numerical methods. We discuss some implications for the information problem.

Figures (5)

• Figure 1: The basic graphical unit studied in Ref. Festuccia:2006sa. Iteration of this leads to breakdown of perturbation theory at long times. Our model iterates a basic unit which is just one side of this, above the dashed line.
• Figure 2: Schwinger-Dyson equation for planar contributions to $\tilde{G}(\omega)$ (propagator with shaded rectangle) in terms of $\tilde{G}_0(\omega)$ and $\tilde{K}_0(\omega)$.
• Figure 3: a) The real part of $\tilde{G}(\omega)$ for $\nu = 1$, $m=0.05$, evaluated 0.01 units above the real axis to give the delta functions finite width. b) The same function evaluated 0.1 units above the real axis: the poles merge into an approximate semicircle distribution.
• Figure 4: The real part of $\tilde{G}(\omega)$ for $\nu_T = 1$, $m=0.80$, and various values of $y = e^{- m/T}$. The vertical axis is rescaled at each temperature for best visibility (the actual area under the curve is $\pi$ at all temperatures), and at zero temperature $\omega$ is taken slightly above the real axis for the same reason.
• Figure 5: The logarithm of the real time infinite temperature correlator, $\ln |G(t)|$, for $\nu_T = 1$, $m =0.8$, and $T= \infty$. For clarity, time differences of $2\pi/m$ are displayed: on shorter intervals the correlator shows strong oscillations due to interference of singularities spaced by multiples of $m$.