Late-Time Saturation of Black Hole Complexity

TL;DR

This paper investigates the late-time behavior of black hole complexity, showing that it saturates exponentially in entropy due to non-perturbative effects, and introduces a toy model to illustrate this saturation.

Contribution

It connects the saturation of black hole complexity to non-perturbative corrections in JT gravity and presents an analytically solvable toy model demonstrating late-time saturation.

Findings

Complexity grows linearly at late times in classical theory.

Non-perturbative effects cause complexity to saturate exponentially in entropy.

A simple discretized model confirms late-time saturation behavior.

Abstract

The holographic complexity of a static spherically symmetric black hole, defined as the volume of an extremal surface, grows linearly with time at late times in general relativity. The growth comes from a region at a constant transverse area inside the black hole and continues forever in the classical theory. In this region the volume complexity of any spherically symmetric black hole in $d+1$ spacetime dimensions reduces to a geodesic length in an effective two-dimensional JT-gravity theory. The length in JT-gravity has been argued to saturate at very late times via non-perturbative corrections obtained from a random matrix description of the gravity theory. The same argument, applied to our effective JT-gravity description of the volume complexity, leads to complexity saturation at times of exponential order in the Bekenstein-Hawking entropy of a $d+1$-dimensional black hole. Along the way, we explore a simple toy model for complexity growth, based on a discretisation of Nielsen complexity geometry, that can be analytically shown to exhibit the expected late-time complexity saturation.