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Random Matrix Theory for Complexity Growth and Black Hole Interiors

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arxiv 2106.02046 v3 pith:SZMBK5BZ submitted 2021-06-03 hep-th

Random Matrix Theory for Complexity Growth and Black Hole Interiors

classification hep-th
keywords growththeoriesoperatorcomplexitydynamicsholographicmatrixrandom
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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 $\unicode{x2014}$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.

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