Krylov complexity from a simple quantum mechanical model for a radiating black hole

TL;DR

This paper studies Krylov complexity in a simplified quantum model of a black hole and its radiation, revealing growth and saturation behaviors consistent with chaos and equilibration.

Contribution

It introduces a tractable quantum mechanical model inspired by the BMN matrix model to analyze Krylov complexity in black hole radiation dynamics.

Findings

Krylov complexity exhibits initial growth characteristic of chaos.

Complexity saturates to a plateau at late times, indicating equilibration.

Plateau behavior can be interpreted semiclassically via Euclidean instantons.

Abstract

We investigate Krylov complexity in a simple quantum mechanical model describing a black hole coupled to its radiation. The model is constructed as a simplified ``mini-BMN" matrix system inspired by a recent proposal of Maldacena. Our aim is not to reproduce the full dynamics of the BMN matrix model, but rather to isolate a tractable setting in which the information-theoretic behaviour of a radiating black hole can be studied explicitly. We analyze both the early- and late-time behaviour of Krylov complexity and the associated Krylov entropy. At early times, perturbative and numerical analyses reveal the expected growth characteristic of chaotic quantum dynamics. At late times, however, the dynamics saturates to a plateau, consistent with equilibration between the black hole and its radiation and with general expectations from finite-entropy quantum systems. We argue that this plateau behaviour admits a semiclassical interpretation in terms of Euclidean instanton contributions in an effective path-integral. The toy model studied here offers a controlled framework in which these features can be investigated analytically and numerically.