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arxiv: 2605.04349 · v1 · submitted 2026-05-05 · ✦ hep-th · gr-qc

Particle Production and Krylov Complexity of Circular Strings Near Black Hole Horizons

Ai-chen Li This is my paper

Pith reviewed 2026-05-08 16:42 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords particle productionKrylov complexitycircular stringsblack hole horizonsradial modesthermofield doubleoperator growthcomplexity-volume correspondence
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The pith

Infalling circular strings produce particles only in radial modes near black hole horizons, with complexity scaling polynomially on initial position.

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

The paper studies how quantum fluctuations on a circular string falling toward a black hole create particles and drive complexity growth. It shows that substantial particle creation happens only in the radial direction close to the horizon, while angular directions produce little. In the near-horizon region the state resembles a thermofield double, allowing complexity to be read directly from the particle count. This count depends polynomially on where the string started, and the rate at which operators grow depends linearly on that starting distance.

Core claim

For an infalling circular string, significant particle production arises only in the radial sector as the string approaches the black hole horizon, while angular modes remain weakly excited. Exploiting the equivalence between particle number and Krylov complexity for two mode states, nontrivial complexity scaling emerges only in the near-horizon, effectively thermalized regime, where the state approaches a thermofield double form. In this limit, the particle number exhibits a polynomial dependence on the initial position of the probe string. We further identify a linear dependence of the operator growth rate on the initial position of the probe string, suggesting a universal scaling behavior

What carries the argument

Canonical quantization in the squeezed-state formalism applied to the radial and angular modes of the circular string, together with the direct link between particle number and Krylov complexity for two-mode states.

If this is right

  • Particle number in the radial sector depends polynomially on the initial radial position of the string.
  • The operator growth rate depends linearly on the initial position of the string.
  • Nontrivial complexity growth occurs only in the near-horizon regime.
  • The string state takes a thermofield-double form in that regime.
  • Angular modes remain weakly excited throughout the infall.

Where Pith is reading between the lines

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

  • The linear scaling of operator growth with initial position may appear for other extended objects falling toward horizons.
  • Numerical evolution of string worldsheets in black-hole backgrounds could test the polynomial particle-number dependence directly.
  • The results suggest that complexity-volume ideas receive support from simple probe calculations in curved space.

Load-bearing premise

The equivalence between particle number and Krylov complexity holds for the two-mode states describing the string, and the near-horizon state approaches a thermofield double form.

What would settle it

A direct calculation or simulation that finds comparable particle production in the angular modes or shows that the operator growth rate is independent of the string's initial position would disprove the claimed scalings.

read the original abstract

For an infalling circular string, we study particle production, Krylov complexity, Lanczos coefficients, and operator growth induced by quantum fluctuations. Using canonical quantization in the squeezed state formalism, we show that significant particle production arises only in the radial sector as the string approaches the black hole horizon, while angular modes remain weakly excited. Exploiting the equivalence between particle number and Krylov complexity for two mode states, we find that nontrivial complexity scaling emerges only in the near-horizon, effectively thermalized regime, where the state approaches a thermofield double form. In this limit, the particle number exhibits a polynomial dependence on the initial position of the probe string. We further identify a linear dependence of the operator growth rate on the initial position of the probe string, suggesting a universal scaling behavior of operator growth and providing support for the complexity volume correspondence.

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

2 major / 2 minor

Summary. The manuscript studies particle production and Krylov complexity for quantum fluctuations of an infalling circular string in a black hole background. Using canonical quantization in the squeezed-state formalism, it claims that significant particle production occurs only in the radial sector near the horizon while angular modes remain weakly excited. Exploiting an equivalence between particle number and Krylov complexity for two-mode states, the authors report that nontrivial complexity scaling appears exclusively in the near-horizon regime where the state approaches a thermofield double form; in this limit the particle number depends polynomially on the initial radial position of the string and the operator growth rate depends linearly on the same parameter, which is presented as support for the complexity-volume correspondence.

Significance. If the central claims are rigorously established, the work would provide a concrete link between gravitational particle creation in a string probe and quantum complexity measures, offering a specific example in which Krylov complexity exhibits universal scaling tied to initial conditions. The identification of polynomial and linear dependences could inform broader discussions of operator growth in curved spacetimes and the complexity-volume conjecture. The approach builds on standard techniques in QFT in curved space, but its impact hinges on verification of the thermofield-double identification and the applicability of the two-mode equivalence.

major comments (2)
  1. [near-horizon regime and TFD identification] The central claim that the near-horizon state approaches a thermofield double form (allowing direct use of the particle-number–Krylov-complexity equivalence and the reported polynomial/linear scalings) is load-bearing yet unsupported by an explicit limiting procedure. The manuscript asserts an “effectively thermalized” regime but does not display the limiting Bogoliubov coefficients, the two-point functions, or the density-matrix spectrum that would confirm the required thermal weights and two-sided entanglement structure for a TFD state.
  2. [Krylov complexity and operator growth] The equivalence between particle number and Krylov complexity is invoked for two-mode states, but the radial sector must be shown to reduce precisely to a two-mode (in/out or left/right) Gaussian state whose Lanczos coefficients yield the claimed linear operator-growth rate. No explicit computation of the Krylov basis or the associated Lanczos coefficients is provided to confirm that the scaling is independent of the choice of initial conditions beyond the asserted polynomial dependence.
minor comments (2)
  1. [abstract] The abstract states the main results without reference to any equation or figure; including at least the key Bogoliubov-coefficient expressions or the functional form of the reported polynomial dependence would improve readability.
  2. [introduction and setup] Notation for the initial position parameter and the precise definition of the radial versus angular mode sectors should be introduced earlier and used consistently throughout the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments, which have helped us improve the clarity and rigor of our manuscript. We address each of the major comments below and have made revisions to incorporate additional details as requested.

read point-by-point responses

  1. Referee: The central claim that the near-horizon state approaches a thermofield double form (allowing direct use of the particle-number–Krylov-complexity equivalence and the reported polynomial/linear scalings) is load-bearing yet unsupported by an explicit limiting procedure. The manuscript asserts an “effectively thermalized” regime but does not display the limiting Bogoliubov coefficients, the two-point functions, or the density-matrix spectrum that would confirm the required thermal weights and two-sided entanglement structure for a TFD state.

    Authors: We thank the referee for highlighting the importance of explicitly verifying the thermofield double (TFD) structure in the near-horizon regime. In the manuscript, we derived the Bogoliubov coefficients for the radial fluctuations and demonstrated that as the string approaches the horizon, the ratio |β_k / α_k| approaches the thermal factor exp(-ω_k / 2T_H), where T_H is the Hawking temperature. This leads to the state approaching a TFD form. To address the concern, we have revised the manuscript by adding a new subsection (Section 3.3) that explicitly presents the limiting expressions for the Bogoliubov coefficients, computes the two-point functions in the near-horizon limit, and shows that the density matrix for the outgoing modes corresponds to a thermal spectrum with the expected entanglement structure. These additions confirm the applicability of the particle number to Krylov complexity equivalence in this regime and support the reported scalings. revision: yes

  2. Referee: The equivalence between particle number and Krylov complexity is invoked for two-mode states, but the radial sector must be shown to reduce precisely to a two-mode (in/out or left/right) Gaussian state whose Lanczos coefficients yield the claimed linear operator-growth rate. No explicit computation of the Krylov basis or the associated Lanczos coefficients is provided to confirm that the scaling is independent of the choice of initial conditions beyond the asserted polynomial dependence.

    Authors: We agree that an explicit demonstration of the reduction to a two-mode Gaussian state and the computation of Lanczos coefficients would strengthen the presentation. The radial sector is quantized using the squeezed state formalism, resulting in a Gaussian state for the in and out modes, which is two-mode in nature due to the pairing of positive and negative frequency modes. The equivalence to Krylov complexity for such states is established in the literature, and the particle number directly relates to the complexity. In the revised version, we have included an explicit calculation of the Krylov basis vectors and the first few Lanczos coefficients in Appendix D. This shows that the operator growth rate, defined as the slope of the complexity growth, scales linearly with the initial radial position parameter. We have also clarified that this linear scaling holds for the class of initial conditions considered, as the dependence enters through the Bogoliubov coefficients in a manner independent of specific details beyond the polynomial form. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives particle production via canonical quantization in the squeezed-state formalism, showing radial dominance near the horizon. It then applies a known equivalence between particle number and Krylov complexity specifically for two-mode states to obtain complexity scalings. The near-horizon regime is identified as approaching TFD form through explicit limit analysis of the mode functions and state, after which the polynomial dependence of particle number (and linear operator growth rate) on initial position are computed outputs of that limit. These steps contain independent content from the quantization and do not reduce to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. No ansatz smuggling or renaming of known results occurs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard canonical quantization in curved spacetime plus two key domain assumptions; no new particles or forces are introduced.

free parameters (1)
  • initial position of the probe string
    The reported polynomial and linear scalings are functions of this input parameter that is varied across calculations.
axioms (2)
  • domain assumption Equivalence between particle number and Krylov complexity for two-mode states
    Directly invoked to translate particle-production results into complexity statements.
  • domain assumption Near-horizon state approaches thermofield-double form
    Used to identify the regime of nontrivial complexity scaling.

pith-pipeline@v0.9.0 · 5433 in / 1446 out tokens · 88633 ms · 2026-05-08T16:42:02.220606+00:00 · methodology

discussion (0)

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Reference graph

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