Random matrix theory for complexity growth and black hole interiors. (English) Zbl 1521.83125
Summary: We study a precise and computationally tractable notion of operator complexity in holographic quantum theories, including the ensemble dual of Jackiw-Teitelboim gravity and two-dimensional holographic conformal field theories. This is a refined, “microcanonical” version of K-complexity that applies to theories with infinite or continuous spectra (including quantum field theories), and in the holographic theories we study exhibits exponential growth for a scrambling time, followed by linear growth until saturation at a time exponential in the entropy – a behavior that is characteristic of chaos. We show that the linear growth regime implies a universal random matrix description of the operator dynamics after scrambling. Our main tool for establishing this connection is a “complexity renormalization group” framework we develop that allows us to study the effective operator dynamics for different timescales by “integrating out” large K-complexities. In the dual gravity setting, we comment on the empirical match between our version of K-complexity and the maximal volume proposal, and speculate on a connection between the universal random matrix theory dynamics of operator growth after scrambling and the spatial translation symmetry of smooth black hole interiors.
MSC:
| 83C57 | Black holes |
| 15B52 | Random matrices (algebraic aspects) |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 83C80 | Analogues of general relativity in lower dimensions |
| 83C45 | Quantization of the gravitational field |
| 81R15 | Operator algebra methods applied to problems in quantum theory |
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