{"id":"48a8accd-6edb-4e68-a117-f2f4707ab9b2","arxiv_id":"2411.15635","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A generating function and finite Fourier series turn the Pfaffian structure of GOE and LOE into a practical algorithm for all marginal eigenvalue distributions, tested up to N=100.","lead":"This paper introduces a fast numerical method to compute the probability distribution of each individual eigenvalue of two classic random matrix ensembles, the Gaussian and Laguerre orthogonal ensembles. It reaches matrix sizes large enough to test theoretical formulas about eigenvalue fluctuations and finite-size corrections.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.2's Gaussian recurrence for I^G has the wrong gamma-function denominator; as written it gives I^G(2,1;0)=0.2929 instead of -0.2071 from the definition.","rationale":"The reader's weakest assumption was that the recurrences in Props. 2.2 and 2.3 are stated without derivation, so a sign or index error would invalidate the numerical results. A careful derivation shows that Prop. 2.2 contains precisely such an error: the denominator Γ(k/2+1) should be Γ(k/2). A numerical check at j=0, k=1, x=0 is decisive. This does not overturn the reader's conditional verdict, but it converts a suspected gap into a concrete defect. The paper should not be accepted until the recurrence is corrected and the Gaussian tables and notebooks are cross-checked. The Laguerre recurrences, by contrast, appear consistent with direct integration, so the LOE results are less suspect. The central claim is still plausibly correct if the notebooks implement the correct recurrence, which is why the verdict remains conditional rather than reject.","tokens_in":21889,"tokens_out":20313,"duration_ms":148854,"concrete_test":"Re-derive Prop. 2.2 from (2.7) and compare both versions against direct numerical integration for small j,k, e.g. compute I^G(2,1;0) and I^G(3,1;0) by the definition and by the printed recurrence. Then inspect the provided notebook GOEnumeric1.nb or GOEsymbolic.nb to determine which denominator is actually coded: if the code uses Γ(k/2), the paper's formula must be corrected and the GOE tables re-verified; if the code also uses Γ(k/2+1), the GOE numerical results are invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The entire GOE computation rests on the recurrences in Proposition 2.2, yet the second and third recurrences in (2.10) are not merely unproved: as written they are false. Integration by parts on I^G(j,k;x) = (1/Γ(j/2))∫_{-∞}^x t^{j-1}e^{-t^2/2}Ψ^G(k;t)dt gives I^G(j+2,k;x) = 2I^G(j,k;x) - [x^j e^{-x^2/2}/Γ(j/2+1)]Ψ^G(k;x) + [2^{-(j+k)/2}Γ((j+k)/2)/(Γ(j/2+1)Γ(k/2))]Ψ^G(j+k;√2x). The printed denominator Γ(k/2+1) is incorrect; the correct denominator is Γ(k/2), matching the Laguerre analogue Prop. 2.3 (2.23), which uses Γ(β). A concrete failure: for j=0, k=1, x=0, the printed recurrence gives I^G(2,1;0)=2I^G(0,1)-Ψ^G(1;0)+2^{-1/2}Γ(1/2)/(Γ(1)Γ(3/2))·Ψ^G(1;0)=0-1/√2+1=0.2929, whereas direct evaluation of (2.7) gives I^G(2,1;0)=∫_{-∞}^0 t e^{-t^2/2}Ψ^G(1;t)dt = 1/2-1/√2 = -0.2071. Even the sign is wrong. Since the H^G matrix elements (2.8) are built from I^G and its tilde counterpart, a reader implementing the printed formulas will not reproduce the GOE tables or any of the Gaussian results. The Mathematica notebooks may contain the correct recurrence, but as submitted the paper's stated algorithm is incorrect.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22323,"tokens_out":14398,"duration_ms":121505,"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":[{"comment":"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.","section":"§2.1, Eq. (2.10)"},{"comment":"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.","section":"§2, Props. 2.2–2.3"},{"comment":"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.","section":"§3, Tables 1–6"}],"minor_comments":[{"comment":"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.","section":"§2.1, Eq. (2.11)"},{"comment":"The caption of Table 5 refers to âvariances (3.5)â but the displayed formula is (3.19); the cross-reference should be corrected.","section":"§3.2, Table 5 caption"},{"comment":"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.","section":"Appendix A, Table A.1"},{"comment":"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.","section":"§3.1, Eq. (3.6)"}],"recommendation":"major_revision","confidential_remarks":"The Gaussian recurrence error is likely fixable, and the Laguerre part of the paper appears internally consistent. Nevertheless, the printed Proposition 2.2 is a load-bearing false statement, so the manuscript cannot be accepted in its current form. The numerical section also needs explicit convergence checks before the high-precision claims can be assessed. With those corrections, the paper would be a useful contribution to the computational random-matrix literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I'll get straight to it. The core idea is good and this paper deserves a serious referee after a fix. The generating function (2.1) plus finite Fourier extraction (3.1) is a genuine extension of Chiani's largest-eigenvalue scheme to every marginal and to the conditioned gap probabilities. That closes a practical gap.\n\nBut there is a concrete error in the text. The recurrence for I^G in Proposition 2.2 (second line of (2.10)) has Γ(k/2+1) in the coefficient of the last term. Integration by parts from the definition (2.7) gives Γ(k/2), exactly as in the Laguerre analogue Prop. 2.3 (2.23). As printed, plugging in j=0, k=1, x=0 yields I^G(2,1;0)=0.2929, while direct evaluation of (2.7) gives -0.2071. Even the sign is wrong. Since the GOE matrix elements are all built from these I^G and ~I^G, a reader implementing the printed formulas will not reproduce any Gaussian table. The authors' Mathematica notebooks may contain the correct recurrence, but the submitted algorithm is incorrect.\n\nThe other soft spots are secondary but worth naming. Propositions 2.2 and 2.3 are stated without derivation; for a computational paper that is tolerable, but when one of them is false the absence of proof becomes a real burden. The claimed 8-10 decimal accuracies are given with no error analysis or convergence checks; the paper mentions the Pfaffian ill-conditioning and increasing working precision, but does not back it up. These are fixable with a few paragraphs.\n\nTo give credit where it is due: the internal consistency checks are genuinely reassuring. The variance of the number of positive GOE eigenvalues matches the known constant 3log2+γ+1-π^2/8, and the large-deviation formula (3.10) is checked to decreasing error. The interlacing of means with Hermite/Laguerre zeros is a nice empirical observation, and the Laguerre side appears to use the correct recurrence.\n\nBottom line: the central method is probably sound, and the numerical results are likely right, but the text has a load-bearing typo (if it is a typo) that must be corrected and cross-checked before anyone can build on it. I'd send it out, with instructions to the referee to verify the recurrences and to ask the authors for a short derivation or an independent implementation check.\n\nThis is a paper for specialists in random matrix theory and numerical special functions. With the recurrence fixed and a modest error analysis added, it would be a solid contribution.","headline":"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.","tokens_in":22890,"tokens_out":7625,"would_cite":false,"duration_ms":58014,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper computes every marginal eigenvalue distribution for the Gaussian and Laguerre orthogonal ensembles to high precision at large matrix size.","keywords":["Gaussian orthogonal ensemble","Laguerre orthogonal ensemble","marginal eigenvalue distribution","Pfaffian","finite Fourier series","gap probability","local central limit theorem","counting statistics"],"falsifier":"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.","tokens_in":21676,"feed_emoji":"🧮","tokens_out":7583,"duration_ms":64670,"temperature":0.7,"pith_summary":"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.","feed_headline":"Formula yields every marginal eigenvalue distribution for GOE, LOE","feed_subtitle":"A generating function plus finite Fourier sum extends largest-eigenvalue Pfaffian methods to all k at N=100.","key_machinery":"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 $\\mathrm{Pf}(A)^2=\\det A$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $3\\log 2 + \\gamma + 1 - \\pi^2/8$."],"supporting_citations":[{"why":"Supplies the Pfaffian recurrence strategy and numerical scheme that the paper generalizes from the largest eigenvalue to all marginals.","marker":"[18]"},{"why":"Provides the efficient numerical Pfaffian evaluation routine used for all complex $\\zeta$ values in the Fourier sum.","marker":"[59]"},{"why":"Supplies the Pfaffian formulas for the generating function and the underlying eigenvalue density theory for GOE and LOE.","marker":"[28]"},{"why":"Gives the integration-over-alternate-variables method behind the Pfaffian structure of the eigenvalue averages.","marker":"[13]"},{"why":"Provides prior exact functional forms and cumulant tables for GOE marginals that the paper extends and uses for validation.","marker":"[25]"},{"why":"Supplies the rigorous leading variance of the number of positive GOE eigenvalues used to calibrate the numerical data.","marker":"[52]"},{"why":"Provides the constant $3\\log 2 + \\gamma + 1 - \\pi^2/8$ used to verify the variance expansion against known results.","marker":"[53]"}],"fun_headline_variants":["New formula yields every marginal eigenvalue density for GOE and LOE","Generating function plus Fourier sum computes all GOE and LOE marginal densities","Efficient Pfaffian algorithm gives all marginals for GOE and LOE","High-precision GOE and LOE eigenvalue densities up to N=100"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New formula yields every marginal eigenvalue density for GOE and LOE","Generating function plus Fourier sum computes all GOE and LOE marginal densities","Efficient Pfaffian algorithm gives all marginals for GOE and LOE","High-precision GOE and LOE eigenvalue densities up to N=100"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000946,"raw_usage":{"total_tokens":4051,"prompt_tokens":972,"completion_tokens":3079,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":2997}},"tokens_in":588,"tokens_out":3079,"duration_ms":19443,"temperature":1.0,"reasoning_tokens":2997,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:04:31.523743+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Chiani, Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution, J","cited_arxiv_id":null,"evidence_quote":"Supplies the Pfaffian recurrence strategy and numerical scheme that the paper generalizes from the largest eigenvalue to all marginals."},{"cited_title":"Wimmer,Algorithm 923: Efficient numerical computation of the Pfaffian for dense and banded skew-symmetric matrices, ACM Trans","cited_arxiv_id":null,"evidence_quote":"Provides the efficient numerical Pfaffian evaluation routine used for all complex $\\zeta$ values in the Fourier sum."},{"cited_title":"de Bruijn,On some multiple integrals involving determinants, J","cited_arxiv_id":null,"evidence_quote":"Gives the integration-over-alternate-variables method behind the Pfaffian structure of the eigenvalue averages."},{"cited_title":"Eckert,Distributions of the individual ordered roots of random matrices, PhD","cited_arxiv_id":null,"evidence_quote":"Provides prior exact functional forms and cumulant tables for GOE marginals that the paper extends and uses for validation."},{"cited_title":"Theory and Applications in Mathematical Physics","cited_arxiv_id":null,"evidence_quote":"Supplies the rigorous leading variance of the number of positive GOE eigenvalues used to calibrate the numerical data."},{"cited_title":"Smith, P","cited_arxiv_id":null,"evidence_quote":"Provides the constant $3\\log 2 + \\gamma + 1 - \\pi^2/8$ used to verify the variance expansion against known results."}],"review_version":1}