Black Holes and Random Matrices

TL;DR

The paper investigates how the late-time dynamics of horizon fluctuations in large AdS black holes reflect random-matrix dynamics typical of quantum chaotic systems, using the SYK model as a controllable black-hole toy.By analyzing the analytically continued partition function $|Z(β+it)|^2$ and correlation functions, it uncovers a robust dip–ramp–plateau structure in the spectral form factor that matches RMT predictions and depends on symmetry class (N mod 8).A double-scaling analysis yields exact early-time behavior and provides a plausible estimate for the crossover time to RMT behavior; extrapolations suggest these features generalize to large AdS black holes, including those dual to ${\cal N}=4$ SYM, with the ramp controlled by spectral rigidity and the plateau fixed by degeneracies.The work discusses the bulk interpretation, challenges of nonperturbative effects, and outlines a path toward understanding how holographic black holes realize the random-matrix universality seen in SYK and related theories.

Abstract

We argue that the late time behavior of horizon fluctuations in large anti-de Sitter (AdS) black holes is governed by the random matrix dynamics characteristic of quantum chaotic systems. Our main tool is the Sachdev-Ye-Kitaev (SYK) model, which we use as a simple model of a black hole. We use an analytically continued partition function $|Z(β+it)|^2$ as well as correlation functions as diagnostics. Using numerical techniques we establish random matrix behavior at late times. We determine the early time behavior exactly in a double scaling limit, giving us a plausible estimate for the crossover time to random matrix behavior. We use these ideas to formulate a conjecture about general large AdS black holes, like those dual to 4D super-Yang-Mills theory, giving a provisional estimate of the crossover time. We make some preliminary comments about challenges to understanding the late time dynamics from a bulk point of view.

Figures (15)

• Figure 1: A log-log plot of SYK $g(t; \beta=5)$, plotted against time for $N=34$. Here we use the dimensionless combination $tJ$ for time. Initially the value drops quickly, through a region we call the slope, to a minimum, which we call the dip. After that the value increases roughly linearly, $\sim t$, until it smoothly connects to a plateau around $tJ = 3\times 10^4$. We call this increase the ramp, and the time at which the extrapolated linear fit of the ramp in the log-log plot crosses the fitted plateau level the plateau time. The data was taken using $90$ independent samples, and the disorder average was taken for the numerator and denominator separately.
• Figure 2: A log-log plot of $g(t; \beta=5)$ against time for GUE random matrices, dimension $L=2^{12}$. A dip, ramp and plateau structure similar to Fig. \ref{['fig:g-SYK']} is apparent.
• Figure 3: Unfolded nearest-neighbor level spacing distribution for SYK vs. RMT. Here $s$ is measured in units of the mean spacing. Semi-analytical exact large $L$ results (correcting the Wigner surmise) for the RMT $P(s)$ are available mehta1960statisticalgaudin1961loi, but we computed the RMT curves from $L = 12870$ exact diagonalization data.
• Figure 4: SYK $g(t, \beta)$ with $\beta=0,1,5$ and various $N$ values. The value at late times, which is equal to plateau height $g_p$, matches with $N_E Z(2\beta)/Z(\beta)^2$ as discussed in Appendix \ref{['App:dtime']}. Here $N_E$ is the eigenvalue degeneracy, $2$ for $(N~\mathrm{mod}~8)\neq 0$ and $1$ for $(N~\mathrm{mod}~8)= 0$. As explained in the main text, the shape of the ramp and the plateau depends on the symmetry class, and the agreements with the counterparts in the RMT with GUE, GOE, and GSE are good. The numbers of samples are 1 200 000 ($N=16$), 600 000 ($N=18$), 240 000 ($N=20$), 120 000 ($N=22$), 48 000 ($N=24$), 10 000 ($N=26$), 3 000 ($N=28$), 914 ($N=30$), 516 ($N=32$), 90 ($N=34$).
• Figure 5: Shown are SYK thermodynamic $\langle E(T) \rangle /N$ for different values of $N$, computed by exact diagonalization. We also plot the point-wise extrapolation obtained by fitting the eight values of $N$ to a three-tem expansion in $1/N$ and taking the leading term. This is almost indistinguishable from the exact large $N$ result obtained by solving the Schwinger-Dyson equations numerically.