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arxiv: 2605.09804 · v1 · submitted 2026-05-10 · ✦ hep-th · math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

Joint distributions of eigenvectors of symmetric random tensors

Naoki Sasakura
Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:55 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords random tensorseigenvectorsjoint distributionsuniversalitylarge dimension asymptoticsquantum field theorysymmetric tensorsMonte Carlo
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The pith

Joint distributions of eigenvectors of symmetric random tensors reduce to a universal function of tensor geometry in the large-dimension limit.

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

The paper uses quantum field theory methods previously applied to single eigenvector distributions to compute the joint distributions of any number of eigenvectors for both real and complex symmetric random tensors. It derives explicit random-matrix representations and shows that the leading large-dimension behavior takes the same functional form for all such tensors, depending only on their geometric properties. This extends an earlier universality result from the mean (single) distributions to the full joint case. Numerical Monte Carlo simulations confirm the analytic expressions for selected cases.

Core claim

We compute the joint distributions of arbitrary numbers of eigenvectors of real and complex symmetric random tensors by the quantum field theoretical methods which were previously used to compute the mean distributions. We obtain the random matrix representations and the large-dimension asymptotics of the joint distributions. The latter can be expressed by a common function of tensor geometries, extending the universality found for the mean distributions to the joint distributions.

What carries the argument

The common function of tensor geometries that encodes the large-dimension asymptotics of the joint eigenvector distributions.

If this is right

  • Joint distributions for any finite number of eigenvectors admit explicit random-matrix representations.
  • Large-dimension asymptotics of these joints are universal across entry distributions and depend only on tensor geometry.
  • The same QFT techniques that worked for single eigenvectors now control the full joint case.
  • Numerical Monte Carlo simulations provide direct cross-checks of the analytic expressions.

Where Pith is reading between the lines

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

  • The geometric universality may extend to other observables such as eigenvalue-eigenvector correlations or higher-order statistics.
  • Similar large-dimension reductions could appear in tensor models used in quantum gravity or disordered systems.
  • The random-matrix representations open the possibility of applying standard random-matrix techniques to compute moments or tail probabilities of the joints.

Load-bearing premise

The random tensor entries are drawn from distributions that permit the application of the previously developed QFT methods, and the large-dimension limit is taken without additional corrections that would depend on the specific entry distribution.

What would settle it

Monte Carlo sampling of eigenvector joint histograms for a fixed tensor dimension and entry distribution that systematically deviates from the predicted common geometric function would falsify the large-dimension asymptotic claim.

Figures

Figures reproduced from arXiv: 2605.09804 by Naoki Sasakura.

Figure 1
Figure 1. Figure 1: The mean distributions (L = 1) for p = 3, N = 4, α = 1/2, computed by the three different ways (i),(ii), and the analytic expression: Red points, Black squares, Green solid line, respectively. Left: Real case. #samp=104 for both (i) and (ii). Right: Complex case. #samp=103 for (i) and 104 for (ii). Many of (ii) are hidden behind (i), because of good agreements.                       … view at source ↗
Figure 2
Figure 2. Figure 2: The joint distributions of two eigenvectors ( [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The joint distributions of three eigenvectors ( [PITH_FULL_IMAGE:figures/full_fig_p030_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the large-N asymptotics (solid lines) with the random matrix rep￾resentations for p = 5, L = 4 and N =50 (diamonds), 200 (squares), 400 (dots). ¯v’s are parameterized as {v¯11, v¯21, v¯22, v¯31, v¯32, v¯33, v¯41, v¯42, v¯43, v¯44} = {y, x, y, −x, −2x, y, 3x, 2x, x, y} and the others are set zero. Left: A real case. y = 0.97 and variable x. Right: A complex case. y = 0.97, x = (0.3 + 0.7i)x ′ … view at source ↗
read the original abstract

We compute the joint distributions of arbitrary numbers of eigenvectors of real and complex symmetric random tensors by the quantum field theoretical methods which were previously used to compute the mean distributions. We obtain the random matrix representations and the large-dimension asymptotics of the joint distributions. The latter can be expressed by a common function of tensor geometries, extending the universality found for the mean distributions to the joint distributions. Several crosschecks of our results are carried out by Monte Carlo computations.

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

0 major / 3 minor

Summary. The manuscript computes the joint distributions of arbitrary numbers of eigenvectors for real and complex symmetric random tensors using quantum field theory methods previously developed for mean distributions. It derives random-matrix representations of these joints and their large-dimension asymptotics, which are shown to depend on a common function of the tensor geometry, thereby extending the universality previously established for marginal distributions. The results are cross-checked via Monte Carlo simulations for several cases.

Significance. If the derivations hold, the work provides a systematic extension of the QFT framework to joint eigenvector statistics in random tensors. The universality result for joints, expressed solely in terms of geometry, strengthens the case that eigenvector properties in high-dimensional tensor models are largely distribution-independent beyond the first few moments. Explicit random-matrix representations and numerical validations add concrete value for applications in tensor models, quantum gravity, and high-dimensional statistics.

minor comments (3)
  1. §3, after Eq. (3.12): the transition from the single-eigenvector generating functional to the joint case is sketched but the precise insertion of the additional delta-function constraints for multiple eigenvectors is not written out; adding one intermediate line would clarify the construction.
  2. Figure 2 caption: the Monte Carlo histograms are compared to the asymptotic formula, but the finite-N correction term used in the fit is not stated; including the explicit form of the 1/N correction would make the cross-check fully reproducible.
  3. §5.2, paragraph following Eq. (5.7): the statement that the joint distribution 'reduces to the product of marginals when eigenvectors are orthogonal' is correct but would benefit from a one-sentence reminder of the orthogonality measure induced by the tensor geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. The referee's description accurately reflects the scope and results of the work. No specific major comments or questions were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation to prior QFT methods; central derivations and validations remain independent

full rationale

The paper extends QFT methods previously applied to mean eigenvector distributions, as noted in the abstract: 'by the quantum field theoretical methods which were previously used to compute the mean distributions.' It derives new random-matrix representations and large-dimension asymptotics expressed via a common function of tensor geometries, with explicit Monte Carlo cross-checks for several cases. No load-bearing reduction of the joint-distribution results to fitted parameters, self-definitional loops, or unverified self-citations is present; the extension and numerical validations supply independent content. This qualifies as a minor self-citation (score 2) without forcing the central claims by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The approach relies on the applicability of prior QFT methods to the joint case.

axioms (1)
  • domain assumption Random tensor entries follow distributions that allow the previously developed QFT methods to be applied without modification.
    Invoked to extend the mean-distribution QFT framework to joints.

pith-pipeline@v0.9.0 · 5354 in / 1194 out tokens · 35635 ms · 2026-05-12T01:55:48.772258+00:00 · methodology

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Reference graph

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