Matrix Models for the Black Hole Information Paradox

Abstract

We study various matrix models with a charge-charge interaction as toy models of the gauge dual of the AdS black hole. These models show a continuous spectrum and power-law decay of correlators at late time and infinite N, implying information loss in this limit. At finite N, the spectrum is discrete and correlators have recurrences, so there is no information loss. We study these models by a variety of techniques, such as Feynman graph expansion, loop equations, and sum over Young tableaux, and we obtain explicitly the leading 1/N^2 corrections for the spectrum and correlators. These techniques are suggestive of possible dual bulk descriptions. At fixed order in 1/N^2 the spectrum remains continuous and no recurrence occurs, so information loss persists. However, the interchange of the long-time and large-N limits is subtle and requires further study.

Figures (9)

• Figure 1: a) Typical Feynman graph for the fundamental correlator (\ref{['correlator']}) in the trilinear model (\ref{['ham']}). The only poles in the lower half $\omega_{1,2,3}$-planes are from the adjoint propagators. After evaluating residues this leaves b), where $k_{r+1} = k_r \pm 1$. The most singular graphs alternate between $k = 0$ and $k = \pm 1$.
• Figure 2: Planar Feynman graphs for the correlator (\ref{['correlator']}) in the model (\ref{['tilde']}). The shaded rectangles mark the full planar propagators. Arrows point from creation operators toward annihilation operators. The graphs for $n = 0,1,2$ are shown.
• Figure 3: Schwinger-Dyson equation for the full propagator. The sum runs over all I$^*$ graphs. Lines with shaded semicircles denote full propagators. Vertices with shaded semicircles contain the summed trivial forward propagators. When we refer to the genus of this graph (as in Sec. 3.2, where we sum all genus one graphs) we mean the explicit genus, not taking into account the non-planar corrections implicit in the shaded circles.
• Figure 4: Disk with one handle. The $\times$ indicates the point where the ends of the fundamental line are joined. The dashed lines mark the $A$ and $B$ cycles.
• Figure 5: a) The I$^*$ graph $n_1 =2$, $n_2 = n_3 = n_4 = n_5 = 1$ drawn on the disk with handle. b) The same graph drawn in the usual double line notation. c) The same graph drawn as a torus with a hole.