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arXiv · 2106.02046

Random Matrix Theory for Complexity Growth and Black Hole Interiors

Arjun Kar·Lampros Lamprou·Moshe Rozali·James Sully
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Abstract

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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Arjun Kar, Lampros Lamprou, Moshe Rozali, James Sully. 2021-06-03. Random Matrix Theory for Complexity Growth and Black Hole Interiors. https://doi.org/10.1007/jhep01(2022)016

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