Black Holes and Random Matrices (1611.04650v3)

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 $|Z(\beta +it)|^2$ 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.

• The paper demonstrates that late-time AdS black hole dynamics exhibit universal spectral properties akin to quantum chaotic systems.
• The study employs the SYK model to compute the spectral form factor, uncovering a ramp-plateau structure consistent with random matrix predictions.
• The research implies significant contributions to the AdS/CFT correspondence and advances our understanding of quantum gravity and black hole microstates.

Overview of "Black Holes and Random Matrices"

This paper investigates the late-time behavior of anti-de Sitter (AdS) black holes using tools from both quantum chaos and random matrix theory (RMT). Specifically, the authors argue that the fluctuations of horizons within large AdS black holes at late times can be effectively described by the dynamics typical of quantum chaotic systems, governed by random matrices. The research utilizes the Sachdev-Ye-Kitaev (SYK) model, a quantum mechanical model of $N$ fermions with random interactions, as a proxy for studying black holes, focusing on the parallels between quantum chaotic behavior in the model and the theoretical expectations for black holes based on the holographic principle.

Key Points and Methods

1. Use of SYK Model: The SYK model, known for showcasing maximal quantum chaos, serves as the foundation for analyzing black hole dynamics. Due to its solvability in large $N$ limits, the SYK model allows for the precise handling of complex interactions typical of black holes, particularly through the tools of disorder-averaged observables and replica symmetry methods.
2. Spectral Form Factor and Random Matrix Theory: The authors compute the spectral form factor, $g(t)$, for the SYK model, which indicates fluctuations in quantum systems' energy levels. Utilizing numerical methods, the behavior of $g(t)$ resembles that observed in RMT: starting with an initial decay (slope), followed by a linearly increasing region (ramp), and finally stabilizing at a plateau, suggesting a close correspondence between quantum chaotic systems and the expected behavior within AdS black holes.
3. Numerical Evidence and Analytic Approaches: By numerically analyzing both the SYK model and corresponding random matrix ensembles (GUE, GOE, GSE), the authors observe a consistent late-time ramp and plateau at exponential scales in $N$. These suggest the SYK model's microstate dynamics can serve as indicators for black hole behavior. They further conclude that at large $N$, the late-time spectral form has universal characteristics of spectral rigidity typical in chaotic systems.
4. Implications for AdS/CFT Correspondence: Extending these findings to a conjecture within the AdS/CFT framework, the paper posits that such spectral rigidity and the ramp-plateau dynamics should emerge naturally in large AdS black holes described by conformal field theories (CFTs) on the boundary, such as in $N=4$ Super-Yang-Mills.
5. Limitations and Further Inquiry: The paper acknowledges the challenge of matching theoretical predictions to experimental observations when dealing with the non-self-averaging behavior post-dip time and points to the open question of understanding the bulk interpretation of RMT effects. The authors hint that more refined techniques, potentially involving an appeal to sparse random matrix theory, may be required.

Implications and Future Research

This investigation pushes forward our understanding of the intersection between quantum chaos, gravity, and high-energy physics. The insights gained from this paper, particularly regarding the dynamics of black holes and their holographic duals, open avenues to explore fundamental aspects of quantum gravity. Future research could focus on refining these analytic and numerical techniques, better understanding the origins of RMT in holographic systems, and potentially uncovering new physical phenomena at play in quantum chaotic systems.

Moreover, further elucidation of the bulk interpretation of these dynamics could offer a more nuanced perspective on the nature of black hole microstates, the nature of information scrambling, and how quantum information is preserved across horizons. This research may hold implications for resolving practical and theoretical questions, such as the black hole information paradox.

In summary, "Black Holes and Random Matrices" provides a comprehensive examination of late-time dynamics in quantum chaotic systems, leveraging the SYK model as a vital tool for approximating behavior expected in AdS black holes, thus enriching our overall comprehension of quantum gravity theories.