{"id":"66d30188-17c7-46d5-8672-f1394643a782","arxiv_id":"1908.06713","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the spherical and truncated unitary ensembles, diagonal eigenvector overlaps decompose into products of independent variables and converge, conditional on one eigenvalue, to the inverse of a gamma-2 distribution.","lead":"Diagonal overlap statistics for eigenvectors are derived exactly for the spherical and truncated unitary ensembles of random matrices, conditional on the spectrum. The same heavy-tailed limit, the inverse of a gamma-2 distribution, is shown to govern scaled diagonal overlaps, matching the complex Ginibre case and strengthening the case for universality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Off-diagonal conditional-expectation formulas in Theorems 2.11 and 3.8 (and Table 1) are internally inconsistent with the trace identity (1.10); the Schur-entry prefactor (1+|λ1|^2)(1+|λ2|^2) is missing.","rationale":"The reader's weakest_assumption concerned the imported Schur densities and Lemma 2.9; neither shows a defect. However, the full text contains a concrete internal inconsistency in a stated result: the quenched off-diagonal overlap expectations are incompatible with the trace identity (1.10). Tracing the error, the recurrence for O_{1,2} in §1.4.1 drops conjugates and the proof of Theorem 2.11 drops the (1+|λ_1|^2)(1+|λ_2|^2) scale of the Schur entry u_2. The same missing prefactor appears in the truncated unitary analogue. The diagonal product decompositions, the limit to 1/γ_2, and Theorem 1.1 remain supported by the surrounding argument, so this is not grounds for rejection; it is grounds for a conditional acceptance requiring corrected off-diagonal formulas and a re-checked Table 1. The trace identity provides a fast, unambiguous numerical check, so the concern is settled rather than speculative.","tokens_in":18888,"tokens_out":47669,"duration_ms":447480,"concrete_test":"For Sph(N=2), fix λ_1=1, λ_2=1+i. (1) Use Proposition 2.12: E_Λ((1/2)Tr GG^*) = (1+(1+1)/2)(1+(1+2)/2)-2 = 3, so E_Λ Tr GG^* = 6. (2) Use Theorem 2.6: E_Λ O_{1,1} = E_Λ O_{2,2} = 1+(2·3)/(2·1)=4. (3) Use the printed Theorem 2.11: E_Λ O_{1,2} = -1/(2·1)=-1/2. Then the trace identity (1.10) gives E_Λ Tr GG^* = 4·1+4·2+(-1/2)(λ_1\\barλ_2+λ_2\\barλ_1) = 12-1 = 11, contradicting step (1). Replacing -1/2 with -(1+|λ_1|^2)(1+|λ_2|^2)/(2|λ_1-λ_2|^2)=-3 gives 12-6=6, matching step (1). This single two-line check settles whether the off-diagonal formulas are missing the prefactor.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The advertised formulas for quenched off-diagonal overlaps are wrong as printed. In §1.4.1, O_{1,2} is written as -b_2 ∑ b_i d_i, but the overlap definition (1.2) gives O_{1,2} = (L_1 L_2^*)(R_2^* R_1) = -\\bar b_2 ∑ b_i \\bar d_i. The proof of Theorem 2.11 then asserts that |u_2|^2 d= X_N, whereas the proof of Theorem 2.6 (via Lemma 2.16 with S_1^2=(1+|λ_2|^2)(1+|λ_1|^2)) gives |u_2|^2 = (1+|λ_1|^2)(1+|λ_2|^2) X_N. This prefactor is exactly what is missing from Theorems 2.11 and 3.8 and from Table 1, where the initial term is printed as -1/(N|λ_1-λ_2|^2) instead of -(1+|λ_1|^2)(1+|λ_2|^2)/(N|λ_1-λ_2|^2). Consequently the table entries labelled E_Λ(O_{1,2}) are inconsistent with the trace identity (1.10) and with the paper's own trace formulas (Propositions 2.12 and 3.9). For Sph(2), λ_1=1, λ_2=1+i, the stated formulas give E_Λ Tr GG^* = 11, while Proposition 2.12 gives 6; the missing prefactor restores 6. The analogous TUE formulas carry the same error, with (1-|λ_1|^2)(1-|λ_2|^2) missing from the prefactor. The diagonal theorems (2.6, 2.8, 3.5, 3.7, 1.1) appear unaffected, but a stated contribution of the paper is false as written.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the diagonal and off-diagonal overlaps between left and right eigenvectors for the spherical ensemble Sph(N) and the truncated unitary ensemble TUE(N,M). Conditionally on the full spectrum, the diagonal overlap O_{1,1} is shown to factor as a product of independent random variables (Theorem 2.6 for Sph, Theorem 3.5 for TUE), and this yields quenched expectation formulas. At the origin, the conditional expectation of O_{1,1} is exactly N (Propositions 2.7 and 3.6), and the scaled overlap N^{-1}O_{1,1} converges in distribution to 1/γ_2 (Propositions 2.8 and 3.7). For the spherical ensemble the same inverse-gamma limit is obtained conditionally on any eigenvalue z, and E(O_{1,1}|λ_1=z)=N for every z (Theorem 1.1). The paper also gives formulas for quenched expectations of off-diagonal overlaps and for E_Λ Tr GG* (Theorems 2.11 and 3.8, Propositions 2.12 and 3.9, and Table 1). The diagonal-overlap results are the core of the paper; the off-diagonal expectation formulas, however, are incorrect as printed, as detailed below.","tokens_in":19167,"tokens_out":14391,"duration_ms":134086,"significance":"If corrected, the diagonal-overlap results are significant: they extend the known Ginibre overlap decomposition to two further integrable non-Hermitian ensembles, establish an exact N-independent conditional expectation at the origin via a telescoping product, and support the emerging universality of the heavy-tailed inverse-γ_2 limit for scaled diagonal overlaps. The proof method is clean and self-contained given three imported inputs: the Schur-form densities from [9], the TUE eigenvalue density from [14], and the Kostlan property from [11,12]. The exact identity E_{\\lambda_1=0}O_{1,1}=N and the z-independence in the spherical case are particularly attractive. These central diagonal claims appear sound. The off-diagonal formulas in Theorem 2.11, Theorem 3.8 and Table 1 are inconsistent with the trace identity (1.10) and with the paper's own trace formulas, so the paper's secondary advertised contribution needs correction before the manuscript is complete.","major_comments":[{"comment":"The off-diagonal quenched expectations are incorrect as printed. In the proof of Theorem 2.11 the base step states that |u_2|^2 \\stackrel{d}{=} X_N and concludes E_Λ O^{(2)}_{1,2} = -1/(N|λ_1-λ_2|^2). But the proof of Theorem 2.6, specifically equation (2.15) with d=1, gives |u_2|^2 \\stackrel{d}{=} (1+|λ_1|^2)(1+|λ_2|^2) X_N, not X_N. The correct initial value is therefore -(1+|λ_1|^2)(1+|λ_2|^2)/(N|λ_1-λ_2|^2), and the analogous factor (1-|λ_1|^2)(1-|λ_2|^2)/M is missing from Theorem 3.8. As a consistency check, for Sph(2) with λ_1=1, λ_2=1+i the printed formulas together with the correct trace identity ∑ λ_i \\bar λ_j O_{i,j} and the diagonal expectations O_{11}=O_{22}=4 give E_Λ Tr GG* = 11, whereas Proposition 2.12 gives Tr = 6; inserting the missing prefactor restores 6. Because the abstract advertises formulas for off-diagonal overlaps, this is a load-bearing error. The defect appears localized: the induction after the base step needs only the corrected initial value (together with the conjugates required by (1.2)), so the diagonal theorems are not affected.","section":"Theorem 2.11, Theorem 3.8, Table 1"}],"minor_comments":[{"comment":"The expression O_{1,2}=-b_2\\sum b_i d_i is inconsistent with the definition (1.2); the correct form should be -\\bar b_2\\sum b_i \\bar d_i (or an explicitly stated equivalent convention). As printed, the base value -b_2^2 is not equal to -|b_2|^2, which is used immediately afterwards in the proof of Theorem 2.11.","section":"Section 1.4.1, Eq. (1.14)"},{"comment":"The trace identity should read Tr GG* = \\sum_{i,j} λ_i \\bar λ_j O_{i,j}; the missing conjugates also appear in the product factors of (2.19), (3.12) and Table 1, where the printed (λ_2-λ_k) and (1+λ_1λ_2) should be (\\bar λ_2-\\bar λ_k) and (1+λ_1\\bar λ_2). Please make the overline convention uniform.","section":"Eq. (1.10) and Table 1"},{"comment":"The selection of the cut-off k_n via Lemma 2.9 is non-constructive and gives no rate; the proof is valid, but it would be helpful to state explicitly that Proposition 2.8 and its TUE analogue inherit this non-quantitative feature.","section":"Lemma 2.9 and Proposition 2.8"}],"recommendation":"major_revision","confidential_remarks":"The off-diagonal error is real but localized, and the diagonal results appear sound. I would not reject the paper; with the base-step prefactor corrected and the conjugates made consistent, the manuscript should be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth reading for its diagonal overlap results, but the off-diagonal expectation formulas are not correct as printed.\n\nWhat is new and good: Theorem 2.6 and Theorem 3.5 give exact product decompositions for the diagonal overlap O_1,1 conditional on the spectrum, for Sph(N) and TUE(N,M). These are natural analogues of the Ginibre case and are proved by a clean induction using the Schur form and explicit integration lemmas. Propositions 2.8 and 3.7 establish the inverse-gamma-2 heavy tail at the origin, and Theorem 1.1 extends this to all z in the spherical case using rotational invariance. The paper is honest about scope: TUE is treated in the N <= M regime and the bulk limit is labelled as expected. The computations are reproducible and the imported inputs (Forrester-Krishnapur densities, Kostlan property) are prior published work.\n\nThe soft spot is serious and localized. Theorems 2.11 and 3.8, and the corresponding entries in Table 1, give quenched expectations for off-diagonal overlaps. The initial step in the proof of Theorem 2.11 states that |u_2|^2 d= X_N. But the proof of Theorem 2.6, via Lemma 2.16, gives |u_2|^2 = (1+|\\lambda_1|^2)(1+|\\lambda_2|^2) X_N. That prefactor is missing from the printed off-diagonal formula. The paper's own trace identity (1.10) exposes the problem: for Sph(2) with \\lambda_1=1, \\lambda_2=1+i, the table formula gives E_\\Lambda Tr GG^* = 11, while Proposition 2.12 gives 6. Adding the missing prefactor restores 6. The TUE formulas have the analogous missing (1-|\\lambda_1|^2)(1-|\\lambda_2|^2). There are also apparent conjugate issues in (1.10) and (1.14) that are likely part of the same bookkeeping problem.\n\nThe diagonal theorems appear unaffected by this. The non-constructive Lemma 2.9 is delicate but does not seem to hide a defect. This is a paper that should go to peer review, not be desk rejected: the main results are new and the flaw is a fixable algebraic slip. The referee should ask the author to correct the off-diagonal formulas and re-check the trace identity.","headline":"The diagonal overlap results are a real contribution, but the off-diagonal formulas are wrong as printed and fail the paper's own trace identity.","tokens_in":19906,"tokens_out":10914,"would_cite":true,"duration_ms":93352,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","60F05","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"In the spherical and truncated unitary ensembles, the diagonal eigenvector overlap, conditioned on the spectrum, is a product of independent random variables, and its scaled limit is the inverse of a gamma-2 distribution.","keywords":["eigenvector overlaps","non-Hermitian random matrices","spherical ensemble","truncated unitary ensemble","diagonal overlaps","inverse gamma distribution","Schur form","radial independence property"],"falsifier":"Numerically sample $\\mathrm{Sph}(N)$ and $\\mathrm{TUE}(N,N)$ for large $N$, collect eigenvalues in a small disk around 0, and compare $O_{1,1}/N$ for those eigenvalues with the inverse-gamma-2 law, checking the $x^{-3}$ tail; for $\\mathrm{Sph}(N)$ repeat in a small disk around a nonzero $z$. If the limit differs between ensembles, has a different tail exponent, or changes with $z$ in the spherical case, Theorem 1.1 is wrong.","tokens_in":18489,"feed_emoji":"🎲","tokens_out":20666,"duration_ms":183708,"temperature":0.7,"pith_summary":"The paper aims to show that, in two non-Hermitian random matrix ensembles, the overlap between left and right eigenvectors of a single eigenvalue is, conditionally on the whole spectrum, exactly a product of independent random factors---one factor per other eigenvalue. These factors involve only pairwise eigenvalue distances and one ensemble-specific scalar variable. From that product structure the paper derives exact conditional expectations and proves that the overlap of an eigenvalue pinned at 0, divided by $N$, converges in distribution to the inverse of a gamma-2 variable, the same heavy-tailed law known for the complex Ginibre ensemble. In the spherical ensemble the limit is independent of the pinning point $z$, and the conditional expectation is exactly $N$ at every $z$. A sympathetic reader should care because this reduces a high-dimensional eigenvector statistic to independent one-dimensional variables and extends a candidate universal heavy-tail law beyond Ginibre.","feed_headline":"Overlaps in two random matrix ensembles share a heavy-tailed law","feed_subtitle":"Conditioned on the spectrum, diagonal overlaps factor into independent pieces and converge to the inverse gamma-2 limit of Ginibre.","key_machinery":"The mechanism is an induction on the Schur form $T = U^*GU$, the upper-triangular matrix with the spectrum on its diagonal. Biorthogonality fixes the left-eigenvector coefficients by the recurrence $b_{n+1} = (\\lambda_1-\\lambda_{n+1})^{-1}B_n u_{n+1}$, where $u_{n+1}$ collects the first $n$ entries of column $n+1$ of $T$, and the diagonal overlap is $O_{1,1} = \\sum_{i=1}^N |b_i|^2$. The load-bearing identity is the determinant decomposition $\\det(I_n + T_nT_n^*) = (1+|\\lambda_n|^2)\\,\\det(I_{n-1}+T_{n-1}T_{n-1}^*)\\,\\left(1 + (1+|\\lambda_n|^2)^{-1} u_n^* H_{n-1}^{-1}u_n\\right)$, with a minus-sign analogue for the truncated unitary case. Integrating out $u_{n+1}$ against the Schur density turns $|B_n u_{n+1}|^2/\\|B_n\\|^2$ into an independent scalar variable, so each overlap increment is an independent multiplicative factor and the factors telescope into (2.13) and (3.9). The limit proofs then use a radial independence property of such eigenvalue densities, under which the squared moduli conditioned on $\\lambda_1=0$ are independent generalized-gamma or $\\beta$ variables, and a diagonal selection lemma that finds a cutoff $k_N$ after which the tail factors collapse to 1 while the early factors converge to the Ginibre product with limit $\\gamma_2$.","core_discovery":"The central discovery is a distributional factorization. For the spherical ensemble, conditionally on $\\Lambda = (\\lambda_1,\\dots,\\lambda_N)$, one has\n$$O_{1,1} \\stackrel{d}{=} \\prod_{k=2}^N \\left(1 + \\frac{(1+|\\lambda_1|^2)(1+|\\lambda_k|^2)}{|\\lambda_1-\\lambda_k|^2} $X_N^{{(k)}}$\\right),$$\nwith the $X_N^{(k)}$ i.i.d. with density $(N+1)/(1+x)^{N+2}$ on $\\mathbb{R}_+$. The truncated unitary analogue replaces $1+|\\lambda|^2$ by $1-|\\lambda|^2$, the $X$'s by $\\mathrm{Beta}(1,M-1)$ variables, and the prefactor $N$ by $M$. Theorem 1.1 then establishes that for the spherical ensemble, conditionally on $\\lambda_1 = z$ for any $z\\in\\mathbb{C}$, $\\mathbb{E}\\,O_{1,1} = N$ and $N^{-1}O_{1,1} \\to 1/\\gamma_2$ in distribution; Propositions 2.8 and 3.7 give the same $1/\\gamma_2$ limit at the origin for both ensembles. The same machinery yields exact quenched expectations for off-diagonal overlaps and for $N^{-1}\\operatorname{Tr} G^*G$, displayed in Table 1.","pith_inferences":["Because $O_{1,1} = \\|L_1\\|^2\\|R_1\\|^2$, the theorem translates into a universal statement about squared eigenvalue condition numbers: for an eigenvalue pinned at 0, $N^{-1}\\kappa_1^2$ converges to $1/\\gamma_2$ in all three ensembles.","The proof-by-comparison with the Ginibre ensemble suggests a broader universality class: any ensemble whose Schur form has the same rank-one determinant structure after conditioning should show the same $x^{-3}$ tail, which could be tested on elliptic or weakly non-Hermitian deformations.","For $\\mathrm{TUE}(N,M)$, the origin limit is insensitive to the growth rate of $M$ relative to $N$; in the bulk one expects the scaling factor to involve the local density and the distance to the unit-circle boundary, a prediction that numerical simulation of $\\mathrm{TUE}(N,cN)$ with several values of $c$ could check."],"forward_implications":["For both $\\mathrm{Sph}(N)$ and $\\mathrm{TUE}(N,M)$ with $N\\le M$, the diagonal overlap of an eigenvalue pinned at 0, divided by $N$, converges in distribution to $1/\\gamma_2$, the same law as in the complex Ginibre ensemble.","In the spherical ensemble, conditioning on any other fixed $z\\in\\mathbb{C}$ gives the same limit and the exact identity $\\mathbb{E}(O_{1,1}\\mid\\lambda_1=z)=N$, so the conditional law is rotationally invariant on the sphere.","The quenched formulas for diagonal overlaps, off-diagonal overlaps, and $N^{-1}\\operatorname{Tr}G^*G$ are exact for every $N$ (and, in the TUE case, every $M\\ge N$), with the three columns of Table 1 linked by the identity $\\operatorname{Tr}GG^* = \\sum_{i,j}\\lambda_i\\overline{\\lambda_j}O_{i,j}$.","The diagonal overlap is deterministically at least 1 and typically of order $N$, so the normalized variable has finite low moments but a heavy tail, signalling large fluctuations in eigenvector non-orthogonality even in the bulk."],"supporting_citations":[{"why":"Supplies the Schur-form densities of both ensembles (2.1 and 3.1) and the determinant-recursion integration identities that make the product decomposition possible.","marker":"[9]"},{"why":"Provides the complex Ginibre analogue and the comparison scheme used to transfer the inverse-gamma limit to the new ensembles.","marker":"[3]"},{"why":"Gives the joint eigenvalue density of the truncated unitary ensemble (Theorem 3.4), used for the conditioning computation in the TUE case.","marker":"[14]"},{"why":"Supplies the radial independence result for squared eigenvalue moduli, which turns the quenched product into expectations over independent variables.","marker":"[12]"}],"fun_headline_variants":["Diagonal overlaps hit heavy tail in two ensembles","Conditional overlaps factor into i.i.d. pieces, then go heavy-tailed","Inverse gamma-2 limit for eigenvector overlaps in spherical and truncated unitary","Factorization yields heavy-tailed overlap statistics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole computation assumes the stated joint density for the upper-triangular coordinates (the Schur form) of the spherical and truncated unitary matrices, including the support condition for the truncation; if either density is wrong or has an unstated support restriction, the product decompositions and the inverse-gamma limits fail.","fun_headline_variants_meta":{"raw":{"variants":["Diagonal overlaps hit heavy tail in two ensembles","Conditional overlaps factor into i.i.d. pieces, then go heavy-tailed","Inverse gamma-2 limit for eigenvector overlaps in spherical and truncated unitary","Factorization yields heavy-tailed overlap statistics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1588,"prompt_tokens":949,"completion_tokens":639,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":572}},"tokens_in":565,"tokens_out":639,"duration_ms":6460,"temperature":1.0,"reasoning_tokens":572,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:36:47.775480+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically sample $\\mathrm{Sph}(N)$ and $\\mathrm{TUE}(N,N)$ for large $N$, collect eigenvalues in a small disk around 0, and compare $O_{1,1}/N$ for those eigenvalues with the inverse-gamma-2 law, checking the $x^{-3}$ tail; for $\\mathrm{Sph}(N)$ repeat in a small disk around a nonzero $z$. If the limit differs between ensembles, has a different tail exponent, or changes with $z$ in the spherical case, Theorem 1.1 is wrong.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Schur-form densities of both ensembles (2.1 and 3.1) and the determinant-recursion integration identities that make the product decomposition possible."},{"cited_title":"The distribution of overlaps between eigenvectors of Ginibre matrices","cited_arxiv_id":"1801.01219","evidence_quote":"Provides the complex Ginibre analogue and the comparison scheme used to transfer the inverse-gamma limit to the new ensembles."},{"cited_title":"Życzkowski and H.-J","cited_arxiv_id":null,"evidence_quote":"Gives the joint eigenvalue density of the truncated unitary ensemble (Theorem 3.4), used for the conditioning computation in the TUE case."},{"cited_title":"Kostlan, On the spectra of Gaussian matrices , Linear Algebra Appl.162/164 (1992), 385–388","cited_arxiv_id":null,"evidence_quote":"Supplies the radial independence result for squared eigenvalue moduli, which turns the quenched product into expectations over independent variables."}],"review_version":1}