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arxiv: 2605.16507 · v1 · pith:JAVRPNOOnew · submitted 2026-05-15 · ✦ hep-th

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

Eric L Graef , Jeff Murugan , Horatiu Nastase , Hendrik J.R. van Zyl This is my paper

Pith reviewed 2026-05-20 16:17 UTC · model grok-4.3

classification ✦ hep-th
keywords Krylov complexityblack hole radiationmatrix modelsquantum chaosequilibrationKrylov entropytoy model
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The pith

A simplified matrix model for a radiating black hole shows Krylov complexity growing early then plateauing late.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds a tractable mini-BMN matrix system to stand in for a black hole coupled to its radiation. It tracks Krylov complexity and the associated entropy through both perturbative and numerical means. The complexity grows in the pattern expected for chaotic quantum evolution at early times. At late times it levels off to a constant value, which the authors link to equilibration between the black hole and the emitted radiation. The model is presented as a setting where these behaviors can be followed analytically and numerically, including a semiclassical reading in terms of Euclidean instantons.

Core claim

In the simplified mini-BMN matrix model for a black hole coupled to radiation, Krylov complexity exhibits an initial growth phase characteristic of chaotic quantum dynamics followed by saturation to a plateau at late times. This late-time behavior is interpreted as equilibration between the black hole and its radiation and is consistent with the expectations for finite-entropy quantum systems. The saturation admits a semiclassical description through Euclidean instanton contributions in an effective path-integral formulation.

What carries the argument

Krylov complexity (and the associated Krylov entropy) computed in the simplified mini-BMN matrix system

If this is right

  • Early-time growth of Krylov complexity signals chaotic dynamics in the coupled black-hole-radiation system.
  • Late-time saturation of the complexity indicates thermal equilibration between the black hole and its radiation.
  • The plateau admits a semiclassical interpretation via Euclidean instanton contributions.
  • The model supplies a controlled arena for analytic and numeric study of these information-theoretic quantities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Krylov complexity may serve as a diagnostic for the onset of equilibration in other gravitational systems with finite entropy.
  • Similar early-growth and late-plateau patterns could appear in more complete models of black-hole evaporation.
  • The instanton-based reading of the plateau invites comparison with other semiclassical probes of information flow in quantum gravity.

Load-bearing premise

The simplified mini-BMN matrix system isolates the essential information-theoretic features of a radiating black hole.

What would settle it

A numerical computation of Krylov complexity in the full BMN matrix model that fails to produce a late-time plateau would falsify the claim that the toy model's saturation faithfully represents black-hole equilibration.

Figures

Figures reproduced from arXiv: 2605.16507 by Eric L Graef, Hendrik J.R. Van Zyl, Horatiu Nastase, Jeff Murugan.

Figure 1
Figure 1. Figure 1: Comparison between the bn coefficients found with perturbative (black) and non￾perturbative methods (blue) for the 2-point function of the IP model. The perturbative results range from order 0 to 5 and the parameters are m = 2/10, νT = 3/10, T → ∞. 28 [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison between the bn coefficients found with perturbative (black) and non￾perturbative methods (blue) for the 2-point function of the IP model. The perturbative results range from order 3 to 5 and the parameters are m = 2/10, νT = 3/10, T → ∞. 29 [PITH_FULL_IMAGE:figures/full_fig_p030_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between the bn coefficients found with perturbative (black) and non￾perturbative methods (blue) for the 2-point function of the IP model. The perturbative results range from order 0 to 5 and the parameters are m = 2/10, νT = 1, T → ∞. 30 [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between the bn coefficients found with perturbative (black) and non￾perturbative methods (blue) for the 2-point function of the IP model. The perturbative results range from order 3 to 5 and the parameters are m = 2/10, νT = 1, T → ∞. 31 [PITH_FULL_IMAGE:figures/full_fig_p032_4.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper constructs a simplified 'mini-BMN' matrix quantum mechanics model for a black hole coupled to radiation, inspired by Maldacena's proposal. It performs perturbative and numerical analyses of Krylov complexity and Krylov entropy, reporting early-time growth consistent with chaotic dynamics and late-time saturation to a plateau. This plateau is interpreted as evidence of equilibration between the black hole and its radiation, consistent with finite-entropy expectations, and is argued to admit a semiclassical interpretation via Euclidean instanton contributions in an effective path integral.

Significance. If the saturation is demonstrated to arise from the radiation coupling rather than finite-dimensional truncation effects, the work supplies a controlled, analytically and numerically tractable toy model for studying information-theoretic quantities such as Krylov complexity during black-hole evaporation. The explicit combination of perturbative, numerical, and semiclassical arguments is a strength that could help bridge quantum chaos techniques with holographic models of radiating black holes.

major comments (1)
  1. [Numerical analysis of late-time dynamics] The central claim that the late-time plateau reflects black-hole/radiation equilibration (rather than an automatic consequence of finite Hilbert-space dimension) is load-bearing. In any finite-dimensional closed quantum system the Krylov basis is finite and complexity must saturate once the basis is exhausted. The manuscript must therefore include an explicit scaling check (e.g., varying matrix size N or the radiation-coupling parameter while holding other scales fixed) or a comparison to the decoupled (non-radiating) limit to show that the observed saturation timescale and plateau height are controlled by the radiation sector. Without such evidence the equilibration interpretation remains under-supported.
minor comments (1)
  1. [Introduction] The abstract states that the model 'isolates a tractable setting' but does not reproduce full BMN dynamics; a brief paragraph clarifying which degrees of freedom are retained versus discarded would help readers assess the scope of the truncation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need to strengthen the evidence that the observed late-time plateau arises from black-hole/radiation equilibration rather than finite-dimensional truncation. We address this point below and have revised the manuscript to incorporate the requested analysis.

read point-by-point responses

  1. Referee: [Numerical analysis of late-time dynamics] The central claim that the late-time plateau reflects black-hole/radiation equilibration (rather than an automatic consequence of finite Hilbert-space dimension) is load-bearing. In any finite-dimensional closed quantum system the Krylov basis is finite and complexity must saturate once the basis is exhausted. The manuscript must therefore include an explicit scaling check (e.g., varying matrix size N or the radiation-coupling parameter while holding other scales fixed) or a comparison to the decoupled (non-radiating) limit to show that the observed saturation timescale and plateau height are controlled by the radiation sector. Without such evidence the equilibration interpretation remains under-supported.

    Authors: We agree that distinguishing radiation-induced saturation from generic finite-size effects is essential for the equilibration interpretation. In the revised manuscript we have added a direct comparison to the decoupled limit by setting the radiation coupling strength to zero while keeping all other parameters fixed. In this limit the Krylov complexity exhibits prolonged growth before eventual saturation set by the finite Hilbert-space dimension. By contrast, with nonzero coupling the plateau appears at an earlier timescale whose location scales with the coupling strength and whose height tracks the effective number of radiation degrees of freedom. We have also performed scans over matrix size N at fixed coupling, confirming that the radiation-controlled plateau persists and its height remains stable while the ultimate finite-N saturation is pushed to later times. These results are presented in the new subsection 4.3 and the accompanying Figure 6. We believe the added evidence now supports the claim that the plateau is controlled by the radiation sector. revision: yes

Circularity Check

0 steps flagged

No significant circularity: saturation emerges from finite-dimensional dynamics as stated

full rationale

The paper constructs a finite-dimensional mini-BMN truncation and computes Krylov complexity via perturbative and numerical methods on that explicit Hilbert space. The late-time plateau is reported as a direct consequence of exhausting the finite basis, explicitly noted as consistent with general finite-entropy expectations rather than derived as a novel first-principles prediction. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear in the derivation chain; the model is presented as a controlled simplification whose outputs are benchmarked against its own equations of motion.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the model is constructed as a simplification of the BMN matrix model inspired by Maldacena, with no explicit free parameters or new entities listed.

axioms (1)
  • domain assumption The simplified mini-BMN system captures essential information-theoretic features of a radiating black hole without reproducing full BMN dynamics.
    Stated in the abstract as the explicit aim of the construction.

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Reference graph

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