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REVIEW 3 major objections 4 minor 60 references

Computing marginal eigenvalue distributions for the Gaussian and Laguerre orthogonal ensembles

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper computes every marginal eigenvalue distribution for the Gaussian and Laguerre orthogonal ensembles to high precision at large matrix size.

desk verdict The new Pfaffian/Fourier method for all GOE/LOE marginals is a real step forward, but the printed Gaussian recurrence in Prop. 2.2 is false, so the paper is not reproducible as written. read the letter →

arxiv 2411.15635 v1 pith:MDVEFTNG submitted 2024-11-23 math-ph math.MPmath.STstat.TH

Peter J. Forrester , Santosh Kumar , Bo-Jian Shen This is my paper · ORCID
classification math-phmath.MPmath.STstat.TH MSC 60B2015B52
keywords GaussianorthogonalensembleLaguerremarginaleigenvaluedistributionPfaffianfiniteFourierseriesgapprobabilitylocalcentrallimittheoremcountingstatistics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to compute every marginal eigenvalue distribution—the probability density of the k-th largest eigenvalue—for the Gaussian orthogonal ensemble (GOE) and the Laguerre orthogonal ensemble (LOE), to high numerical precision and for matrices large enough to reach asymptotic regimes. Earlier Pfaffian-based machinery handled only the largest eigenvalue; the paper adds a generating function for the conditioned gap probabilities and a finite Fourier decomposition that recovers all k simultaneously. If the scheme is correct, it turns a previously special-case computation into a general-purpose numerical tool, with applications to central limit theorems, local central limit theorems, and finite-size corrections. The paper also uses the method to test asymptotic formulas for the number of positive GOE eigenvalues and the number of LOE eigenvalues above a threshold, and to produce exact cumulant tables for GOE marginals up to N=12.

What carries the argument

The load-bearing object is the polynomial generating function $\Xi_N((s,\infty);\zeta)$ for the gap probabilities; its Pfaffian expression turns each Fourier coefficient evaluation into a numerical linear algebra problem. The supporting mechanism is the recurrence ladder that generates the Pfaffian entries $H_0, H_1, H_2$ and the boundary vector $\nu$ without computing double integrals, together with the finite Fourier formula that extracts $E_N(k)$ from $\Xi_N$ at $\zeta=e^{2\pi i l/(N+1)}$. Because each Pfaffian has numerical entries, a known direct skew-symmetric Pfaffian algorithm can evaluate them without the square-root branch issue of $\mathrm{Pf}(A)^2=\det A$.

What would settle it

Compute the Gaussian Pfaffian entries $H_0(j,k;s)$, $H_1(j,k;s)$, and $H_2(j,k;s)$ for a fixed case such as $N=20$, $s=1$ directly by high-precision numerical quadrature of the defining double integrals, assemble the same Pfaffians and Fourier sum, and compare the resulting $E_N(k;(1,\infty))$ and $f_N(k;1)$ with the paper's recurrence-based output; disagreement beyond the claimed precision would refute the recurrences, while agreement would confirm the numerical pipeline.

Watch

Extended reading notes

Core claim

The central claim is that the combination of (i) the generating function $\Xi_N((s,\infty);\zeta)=\sum_{k=0}^N \zeta^k E_N(k;(s,\infty))$, (ii) Pfaffian formulas for $\Xi_N$ at any $\zeta$, (iii) second-order recurrences in the Gaussian case and first-order recurrences in the Laguerre case that build every matrix element from error-function and incomplete-gamma seeds, and (iv) a finite Fourier sum over $\zeta$ at roots of unity, yields all conditioned gap probabilities $E_N(k;(s,\infty))$ and, via the derivative identity $f_N(k;s)=\frac{d}{ds}\sum_{l=0}^{k-1}E_N(l;(s,\infty))$, all marginal densities. Numerical evaluations then use a direct skew-symmetric Pfaffian routine, avoiding square-root branch choices, with working precision increased as $N$ grows to control ill-conditioning. The authors report high-precision results for GOE up to $N=100$ and LOE up to $N=90$, and use them to confirm and refine known asymptotics.

Load-bearing premise

The load-bearing premise is that the recurrence formulas for the Pfaffian matrix elements are correct for all indices and that raising working precision in the computer algebra system fully controls the growing ill-conditioning as $N$ increases.

Editorial extensions

If this is right

  • All marginal densities, not just the largest eigenvalue, become numerically accessible in the regime where $N$ is large enough to compare with asymptotic formulas.
  • The number of positive GOE eigenvalues and the number of LOE eigenvalues above a threshold can be checked against local central limit theorems, with computed correction terms.
  • The exact cumulants of GOE marginals, previously tabulated to $N=7$, are extended to $N=12$ through the exact functional forms produced by the Pfaffian computation.
  • The data support an interlacing property: the means of the marginal densities interlace with the zeros of the Hermite polynomial in the GOE case and with the zeros of a Laguerre polynomial in the LOE case.
  • The variance of the positive-eigenvalue count is obtained to ten decimal places for $N$ up to 100, enabling fits of the leading constant that match the known value $3\log 2 + \gamma + 1 - \pi^2/8$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The Fourier-extraction step is ensemble-agnostic: any Pfaffian point process whose generating function is a degree-$N$ polynomial in $\zeta$ and whose matrix elements admit recurrences could be treated by the same pipeline.
  • The $N+1$ Pfaffian evaluations at roots of unity are the computational bottleneck, so replacing the Fourier transform by a low-rank or sparse evaluation scheme could push the method to larger $N$, although the paper does not explore that direction.
  • The observed interlacing of means with zeros of Hermite and Laguerre polynomials suggests a general ordering statement for beta ensembles, but the paper reports it only as numerical evidence.
  • The exact GOE functional forms generated by the Pfaffian expansion remain valid for larger $N$ even when symbolic simplification becomes impractical, so high-precision cumulant tables could likely be extended well beyond $N=12$ with numerical coefficient extraction alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a computational scheme for the marginal eigenvalue densities f_N(k;s) and the conditioned gap probabilities E_N(k;(s,\infty)) for the Gaussian and Laguerre orthogonal ensembles. Starting from a generating function for conditioned gap probabilities, a Pfaffian representation, and recurrences for the matrix elements, the authors use a finite Fourier sum in the parameter \zeta together with Wimmer's numerical Pfaffian algorithm to obtain high-precision values for all k. The method is applied to large-N questions: the variance of the number of positive GOE eigenvalues, local central limit theorems, bulk central-limit rates, Laguerre counting statistics, and tables of cumulants and interlacing properties of means.

Significance. If correct, the method is a useful extension of Chiani's largest-eigenvalue formalism to every marginal distribution, reaching matrix sizes around N=100 and providing numerical checks of asymptotic formulas. The reproducible Mathematica notebooks, the internal consistency checks, and the agreement with known asymptotic constants such as 3\log 2+\gamma+1-\pi^2/8 in Section 3.1 are strengths. However, the central Gaussian recurrence is printed incorrectly, and the claimed numerical accuracies are not supported by an error analysis; these issues must be resolved before the results can be relied upon.

major comments (3)
  1. [§2.1, Eq. (2.10)] The recurrence for I^G in Proposition 2.2 is incorrect as printed. Integration by parts gives the last coefficient as 2^{-(j+k)/2} \Gamma((j+k)/2)/(\Gamma(j/2+1)\Gamma(k/2)) times \Psi^G(j+k;\sqrt{2}x), not with \Gamma(k/2+1) in the denominator; the same correction applies to the tilde recurrence. The printed version fails already for j=0, k=1, x=0: it yields I^G(2,1;0)=0.2929, whereas direct evaluation of the definition (2.7) gives I^G(2,1;0)=1/2-1/\sqrt{2}=-0.2071. Since the Gaussian matrix elements H^G_0 and H^G_2 in (2.8) are built from I^G, a reader implementing the printed Proposition 2.2 will not reproduce the GOE tables or any Gaussian result. Please correct the formula and supply the derivation or an exact reference.
  2. [§2, Props. 2.2–2.3] Propositions 2.2 and 2.3 are asserted to follow by âstraightforward integration by partsâ but no derivation or exact reference is given. In the case of Proposition 2.2 this is not a harmless omission, since the printed formula is wrong. For load-bearing recurrences of this type, the authors should provide the derivation, including boundary terms at x\to\pm\infty and the case j=0, or point to the precise equations in Chiani [18] where each formula appears.
  3. [§3, Tables 1–6] The numerical accuracy claims in Section 3 are not backed by an error analysis. After the remark in §3 that Pfaffian computations become ill-conditioned as N grows, the paper states only that Mathematica's working precision can be increased; no precision-doubling comparison, residual check, or other convergence test is reported for the N=60–100 entries. Consequently, statements such as â10 decimal place accurate variancesâ (Tables 1 and 5) and â8 digit accurate meansâ (Table 4) are not verifiable from the manuscript. Please add explicit accuracy checks or soften the claimed precision.
minor comments (4)
  1. [§2.1, Eq. (2.11)] The initial condition for I^G(1,1;x) is written with a lowercase \psi, while the definitions in (2.7) use a capital \Psi; the notation should be made uniform.
  2. [§3.2, Table 5 caption] The caption of Table 5 refers to âvariances (3.5)â but the displayed formula is (3.19); the cross-reference should be corrected.
  3. [Appendix A, Table A.1] In Table A.1, the N=10, k=4 row reports \gamma_1=0.2109507 and \gamma_2=0.0314529, which are inconsistent with the neighboring rows (e.g., N=11,k=4: 0.0371329 and -0.0073705; N=12,k=4: 0.0416807 and -0.0067366). This appears to be a shifted decimal point or an erroneous entry; the whole table should be rechecked.
  4. [§3.1, Eq. (3.6)] The constants c1, c2, c3 in (3.6) are obtained by fitting âvarious combinations of 3 rowsâ without stating the fitting rule, residuals, or sensitivity; the tentative conclusion about the smallness of c2 should be presented with that caveat explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained and does not assume the target marginals.

full rationale

The paper's computation chain starts from the known joint eigenvalue densities (1.2)/(1.3), the exact generating function (2.1), and the Pfaffian expressions in Proposition 2.1, which are standard consequences of the underlying orthogonal-symmetry structure. The matrix elements are defined independently by the integrals in (2.2)-(2.5) and (2.7)/(2.21); the recurrences in Propositions 2.2 and 2.3 are stated as integration-by-parts consequences and, regardless of their correctness, they are not definitions of the target marginal distributions. The finite Fourier series (3.1) is an exact algebraic extraction of the coefficients of a polynomial in zeta from its values at roots of unity, so it does not smuggle in the quantities being computed. The Pfaffian evaluations use Wimmer's independent numerical algorithm, and the asymptotic checks in Section 3 are compared with external results such as (3.10) and (3.20). Although the paper cites several works by the same authors, those citations are contextual or provide comparisons; the central computation does not reduce to them. The skeptic's specific concern about the gamma-function denominator in (2.10) is a correctness issue, not circularity: the objects I^G and tilde-I^G are defined separately in (2.7), so even an erroneous printed recurrence would not make the final 'prediction' equivalent to an input by construction.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central method introduces no new physical or probabilistic entities; it relies on standard Pfaffian machinery, stated-but-unproved recurrences, and an external numerical routine. The only fitted constants are exploratory asymptotic parameters.

free parameters (2)
  • c1, c2, c3 in variance ansatz (3.6) = c1 approximately 2.4229, c2 approximately 0.006, c3 approximately -0.52
    Fitted to the GOE variance table (Table 1) by choosing combinations of three rows. This is an exploratory asymptotic fit, not part of the central computational method; c1 matches literature [53].
  • rate alpha(N) in bulk central limit correction (3.13) = alpha(N) = 1/log N
    Inferred by comparing the difference (3.14) for N=21 and N=41; used to propose the form of the leading correction, not a derived rate.
assumptions (3)
  • standard math Pfaffian expressions for Xi_N (Prop. 2.1) remain valid for complex zeta and for both even and odd N, with odd N requiring bordering by nu_j.
    Uses de Bruijn integration over alternate variables and the Mehta Pfaffian structure; standard in random matrix theory.
  • domain assumption Recurrences in Prop. 2.2 (Gaussian) and Prop. 2.3 (Laguerre) correctly compute all matrix elements H_0, H_1, H_2 and nu_j.
    The propositions are stated without proof, described as straightforward integration by parts. If any recurrence or initial condition is wrong, the computed marginals are wrong.
  • domain assumption Wimmer's algorithm computes the Pfaffian of a complex skew-symmetric matrix to sufficient accuracy when implemented with increased working precision.
    The paper relies on this external code for numerical Pfaffians and counters ill-conditioning by increasing digits, without a rigorous error analysis.

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Pith. "Pith review of Computing marginal eigenvalue distributions for the Gaussian and Laguerre orthogonal ensembles." pith.science (2026). https://pith.science/paper/MDVEFTNG

@misc{pith2026241115635,
  author       = {Pith},
  title        = {Pith review of: Computing marginal eigenvalue distributions for the Gaussian and Laguerre orthogonal ensembles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MDVEFTNG}},
  note         = {Machine review of arXiv:2411.15635}
}
abstract

The Gaussian and Laguerre orthogonal ensembles are fundamental to random matrix theory, and the marginal eigenvalue distributions are basic observable quantities. Notwithstanding a long history, a formulation providing high precision numerical evaluations for $N$ large enough to probe asymptotic regimes, has not been provided. An exception is for the largest eigenvalue, where there is a formalism due to Chiani which uses a combination of the Pfaffian structure underlying the ensembles, and a recursive computation of the matrix elements. We augment this strategy by introducing a generating function for the conditioned gap probabilities. A finite Fourier series approach is then used to extract the sequence of marginal eigenvalue distributions as a linear combination of Pfaffians, with the latter then evaluated using an efficient numerical procedure available in the literature. Applications are given to illustrating various asymptotic formulas, local central limit theorems, and central limit theorems, as well as to probing finite size corrections. Further, our data indicates that the mean values of the marginal distributions interlace with the zeros of the Hermite polynomial (Gaussian ensemble) and a Laguerre polynomial (Laguerre ensemble).

Figures

Figures reproduced from arXiv: 2411.15635 by the authors.

Figure 1
Figure 1. Plotted parametrically are the values in the complex plane of [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. [color online] Plot of the difference (3.14) for [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.