Random Matrix Theory for Complexity Growth and Black Hole Interiors

TL;DR

This paper introduces a refined notion of operator complexity in holographic quantum theories, revealing universal chaotic behavior characterized by exponential and linear growth phases, and connects it to random matrix theory and black hole interior geometry.

Contribution

It develops a microcanonical K-complexity framework applicable to quantum field theories and establishes a universal random matrix description of operator dynamics after scrambling.

Findings

Operator complexity exhibits exponential then linear growth in holographic theories.

Linear growth regime is described by universal random matrix dynamics.

Empirical match between K-complexity and maximal volume in gravity duals.

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.