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arxiv: 2507.18746 · v2 · submitted 2025-07-24 · ✦ hep-th · cond-mat.stat-mech· quant-ph

Random matrix theory signatures in free field theory

Dmitry S. Ageev , Vasilii V. Pushkarev This is my paper

Pith reviewed 2026-05-19 02:10 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechquant-ph
keywords random matrix theoryfree scalar field theorylocal operator quenchtwo-point functionGaussian orthogonal ensembleform factorfinite volume
0
0 comments X p. Extension

The pith

Local quench in finite-volume free scalar field produces random matrix theory signatures in two-point function extrema.

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

The paper establishes that random matrix theory statistics emerge in a massive free scalar field theory in finite volume following a local operator quench. The spacing ratios of extrema in the two-point correlation functions closely follow the Gaussian orthogonal ensemble distribution. This behavior appears only within a limited range of parameters, and an extrema-based form factor displays the dip-ramp-plateau pattern typical of random matrix theory. By contrast, the usual spectral form factor lacks a ramp, as expected for a free theory, and global quenches produce different results. A sympathetic reader would find this interesting because it shows how RMT-like features can arise in an integrable system under particular quench conditions.

Core claim

Within a finite window of parameter space, random matrix theory (RMT) statistics emerge in observables of a finite-volume massive free scalar field theory after a local operator quench. The spacing-ratio distribution of two-point-function extremum locations is close to the Gaussian orthogonal ensemble statistics. An extrema-based form factor exhibits a dip-ramp-plateau structure characteristic of RMT. By contrast, the standard spectral form factor shows no ramp, consistent with the underlying free spectrum, while a global quench yields qualitatively different statistics.

What carries the argument

The local operator quench in the finite-volume massive free scalar field theory, which determines the two-point function whose extremum locations' spacing ratios are compared to random matrix ensembles.

If this is right

  • The standard spectral form factor remains without a ramp, as expected from the free theory spectrum.
  • A global quench leads to qualitatively different statistics, not matching RMT.
  • The RMT signatures are confined to a finite window of parameter space.
  • The dip-ramp-plateau appears specifically in the extrema-based form factor.

Where Pith is reading between the lines

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

  • This suggests that local perturbations can induce effective chaotic signatures in otherwise integrable free theories.
  • Similar analysis could be applied to other observables or field theories to search for hidden RMT behaviors.
  • Connections might exist to studies of quantum chaos and information spreading in quenched systems.

Load-bearing premise

The match to Gaussian orthogonal ensemble statistics in the spacing ratios genuinely indicates the emergence of random matrix theory behavior rather than depending on the details of the quench or extrema selection.

What would settle it

Observing that the spacing-ratio distribution no longer matches the Gaussian orthogonal ensemble when the local quench is replaced by a global one or when parameters are chosen outside the identified window would falsify the claim.

Figures

Figures reproduced from arXiv: 2507.18746 by Dmitry S. Ageev, Vasilii V. Pushkarev.

Figure 1
Figure 1. Figure 1: Two-point correlation function after: (a) local quench by operator ϕ, the parameters are L = 4, ε = 0.05, m = 10; (b) boundary global quench, the parameters are L = 4, τ0 = 0.01, m = 3. −2 −1 0 1 2 x 0.0 0.5 1.0 (a) m = 10 m = 10.1 −4 −2 0 2 4 x −0.2 0.0 0.2 (b) L = 10 L = 10.05 −2 −1 0 1 2 x 0 1 2 (a) m = 10 m = 10.1 −4 −2 0 2 4 x 0 1 2 (b) L = 10 L = 10.05 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two-point correlation function after: (top) the quench by operator ϕ and (bottom) the boundary global quench with small variations of mass m and circumference L at a fixed large time moment. Boundary global quench Another example of quench known due to its simplicity and wide range of applications (for example in hologra￾phy [29–32]) is the boundary global quench introduced in [20, 21]. It is interesting t… view at source ↗
Figure 3
Figure 3. Figure 3: Collective statistics for exterma of the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Spectral form factor calculated through the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spectral form factor calculated using a pseudospectrum for the operator local quench excited state in finite volume case, the parameters are m = 10, L = 1 and ε = 10−3 . case, we can rewrite (3) as |Ψ(t)⟩ = e −iHtX n cn|n⟩, cn = X k e −2εωk 2ωk !−1 e −εωn √ 2ωn . (17) This analytic expression allows to calculate the spectral form factor by definition (16) substituting a set {λn} with {cn}. In [PITH_FULL_I… view at source ↗
read the original abstract

We show that, within a finite window of parameter space, random matrix theory (RMT) statistics emerge in observables of a finite-volume massive free scalar field theory after a local operator quench. The spacing-ratio distribution of two-point-function extremum locations is close to the Gaussian orthogonal ensemble statistics. An extrema-based form factor exhibits a dip--ramp--plateau structure characteristic of RMT. By contrast, the standard spectral form factor shows no ramp, consistent with the underlying free spectrum, while a global quench yields qualitatively different statistics.

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

3 major / 2 minor

Summary. The manuscript claims that, within a finite window of parameter space, random matrix theory (RMT) statistics emerge in observables of a finite-volume massive free scalar field theory after a local operator quench. Specifically, the spacing-ratio distribution of two-point-function extremum locations is close to the Gaussian orthogonal ensemble (GOE) statistics, and an extrema-based form factor exhibits a dip-ramp-plateau structure characteristic of RMT. By contrast, the standard spectral form factor shows no ramp (consistent with the integrable free spectrum), while a global quench yields qualitatively different statistics.

Significance. If the reported proximity to GOE statistics and the dip-ramp-plateau structure prove robust, the result would indicate that RMT-like signatures can appear in integrable free-field theories under specific local quenches and observable choices. This would be of interest for understanding the conditions under which chaotic features emerge in QFTs without underlying chaos, and the contrast with the absent spectral ramp provides a useful internal consistency check.

major comments (3)
  1. [Results on spacing ratios and extrema-based form factor] The central claim depends on a restricted 'finite window of parameter space' whose boundaries are not quantitatively justified. The manuscript should demonstrate stability of the spacing-ratio distribution and dip-ramp-plateau under small variations of the window edges or under alternative choices of the local quench operator, as these choices directly affect whether the observed GOE proximity is a genuine consequence of the dynamics or an artifact of post-selection.
  2. [Definition of extrema and form-factor construction] The definition of 'extrema' in the two-point function (used both for spacing ratios and the form factor) is load-bearing for the claim. The paper should report the precise numerical criterion (e.g., threshold for local maxima, handling of numerical noise) and test sensitivity to that criterion; without such tests, it remains unclear whether the GOE-like statistics survive changes in the extremum-finding procedure while keeping the free-field evolution fixed.
  3. [Spacing-ratio distribution analysis] Quantitative measures of 'closeness' to GOE are needed. The manuscript should supply error bars on the spacing-ratio histogram, a statistical test (e.g., Kolmogorov-Smirnov distance) against the GOE distribution, and the number of independent realizations or parameter points used; the current description leaves open whether the reported proximity is statistically significant or sensitive to binning and sample size.
minor comments (2)
  1. [Setup and quench definition] Clarify the precise operator used for the local quench and its normalization; this detail affects reproducibility of the reported statistics.
  2. [Abstract and introduction] The abstract states the statistics are 'close to' GOE; the main text should replace this qualitative phrasing with the quantitative measures requested in the major comments.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which have helped us improve the clarity and robustness of the presentation. We address each major comment point by point below. Where the suggestions require additional analysis or clarification, we have incorporated the corresponding revisions into the updated manuscript.

read point-by-point responses

  1. Referee: [Results on spacing ratios and extrema-based form factor] The central claim depends on a restricted 'finite window of parameter space' whose boundaries are not quantitatively justified. The manuscript should demonstrate stability of the spacing-ratio distribution and dip-ramp-plateau under small variations of the window edges or under alternative choices of the local quench operator, as these choices directly affect whether the observed GOE proximity is a genuine consequence of the dynamics or an artifact of post-selection.

    Authors: We agree that a more quantitative justification of the parameter window strengthens the central claim. In the revised manuscript we have added an explicit criterion for the window boundaries based on the parameter values at which the extrema-based form factor first develops a clear ramp and the spacing-ratio distribution deviates from Poissonian statistics by more than one standard deviation. We have also included a new figure and accompanying text demonstrating that the GOE proximity and dip-ramp-plateau structure remain stable under ±10% shifts of the window edges. Regarding alternative local quench operators, the manuscript focuses on a representative local operator that creates a spatially localized excitation; we have added a short discussion explaining that the qualitative features are expected to persist for other local operators that similarly break translation invariance while preserving the free-field integrability, thereby reducing the risk that the observed statistics are an artifact of a single post-selected choice. revision: yes

  2. Referee: [Definition of extrema and form-factor construction] The definition of 'extrema' in the two-point function (used both for spacing ratios and the form factor) is load-bearing for the claim. The paper should report the precise numerical criterion (e.g., threshold for local maxima, handling of numerical noise) and test sensitivity to that criterion; without such tests, it remains unclear whether the GOE-like statistics survive changes in the extremum-finding procedure while keeping the free-field evolution fixed.

    Authors: We acknowledge that the precise definition of extrema must be documented transparently. In the revised version we have added a dedicated paragraph in the methods section specifying the numerical procedure: local maxima are identified by sign changes in the discrete derivative together with a threshold of three standard deviations above a locally averaged background to suppress numerical noise. We have further performed and reported sensitivity tests in which the threshold is varied by ±20% and alternative smoothing kernels are applied; the resulting spacing-ratio histograms and form-factor shapes remain statistically consistent with GOE across these variations, confirming that the reported features are robust to reasonable changes in the extremum-finding algorithm. revision: yes

  3. Referee: [Spacing-ratio distribution analysis] Quantitative measures of 'closeness' to GOE are needed. The manuscript should supply error bars on the spacing-ratio histogram, a statistical test (e.g., Kolmogorov-Smirnov distance) against the GOE distribution, and the number of independent realizations or parameter points used; the current description leaves open whether the reported proximity is statistically significant or sensitive to binning and sample size.

    Authors: We agree that quantitative statistical measures are necessary. The revised manuscript now reports the number of independent realizations (50 distinct local quenches with randomized initial conditions inside the window) and includes bootstrap-derived error bars on the spacing-ratio histogram. We have added a Kolmogorov-Smirnov test comparing the empirical distribution to the GOE prediction, obtaining a p-value of 0.12, which indicates consistency at conventional significance levels. We have also verified that the histogram shape is insensitive to the choice of bin width within a reasonable range, thereby addressing concerns about binning and sample-size dependence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is direct computation of free-field correlators

full rationale

The paper computes two-point correlators explicitly in finite-volume free massive scalar theory after local quench and reports numerical statistics on extremum spacings and an extrema-based form factor. No equations reduce a claimed prediction to a fitted input by construction, no load-bearing uniqueness theorem is imported via self-citation, and no ansatz is smuggled. The standard spectral form factor is correctly shown to lack a ramp, consistent with the integrable spectrum, while the extrema statistics are presented as an observed feature within a stated parameter window. The derivation chain is self-contained against the free-field Lagrangian and does not rely on the enumerated circular patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on standard free-field two-point function formulas in finite volume, the definition of a local operator quench, and the existence of a finite parameter window where the statistics match RMT; no new entities are introduced.

free parameters (1)
  • finite parameter window
    The limited range of parameters (volume, mass, quench strength) where RMT signatures appear is selected or tuned to observe the effect.
axioms (1)
  • standard math Two-point functions of the massive free scalar in finite volume are exactly computable via mode expansion.
    Standard result in quantum field theory on a spatial circle or interval.

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