{"id":"48f6d5c0-1772-4ffe-8cba-6a46bd487ba3","arxiv_id":"2509.03252","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a class of unitarily invariant random matrices, eigenvalues of a rank-one perturbation converge to zeros of the Gaussian analytic function a minus sum c_k z^k, which also makes the critical outlier time scale optimal.","lead":"This paper proves that the eigenvalue cloud of a rank-one perturbed random unitary matrix converges, in a critical scaling, to the zeros of a Gaussian analytic function, and uses this to show that the time scale at which an outlier emerges is optimal for a broader family of unitary matrices. A generalist reader may care because it connects random matrix spectra to random analytic function zeros and settles an open question about outlier dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 15's proof, not the moment condition, is the load-bearing soft spot: it conflates M^n with (M^*)^n and miscounts the stabilizer, so the Gaussian coefficient limit is not established as written.","rationale":"The central claim is the GAF convergence and no-outlier conclusion for the i.i.d.-eigenvalue model. The proof's keystone is Lemma 15, which supplies the Gaussian coefficients. I agree with the reader that the super-polynomial moment condition is the key hypothesis making the coefficients asymptotically uncorrelated; without it the Gaussian limit could fail. But the more immediate, load-bearing weakness is that Lemma 15's proof as written does not establish the claimed convergence: the false identity between M^n and (M^*)^n means the moments computed are for the wrong object, and the stabilizer count typo (2p vs 2^p) would, if taken literally, give the wrong Gaussian variance. Both issues are fixable and do not appear to affect the underlying mathematics, hence the verdict remains CONDITIONAL: the manuscript should be corrected before acceptance. The proposed test would settle whether the fix goes through and whether any hidden dependence on the direction of conjugation (i.e., an asymmetric moment condition) remains.","tokens_in":17431,"tokens_out":37301,"duration_ms":306836,"concrete_test":"Reprove Lemma 15 by direct moment computation of S_n := sqrt(N/2) e1^*(M^*)^n e1 = sqrt(N/2) sum_i |v_{1i}|^2 \\bar{Z}_i^n, using E[\\bar{Z}^k] in place of E[Z^k] and the correct stabilizer count 2^p. If the joint mixed moments still converge to \\prod_k p_k! 1_{l=r}1_{p_k=q_k}1_{n_k=n'_k}, then the lemma is valid and the proof gap is purely presentational; if the limit changes, Theorem 17 and Theorem 21 fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3, the entire finite-dimensional convergence argument rests on Lemma 15, which claims sqrt(N/2)(e1^*U^*e1, e1^*(U^*)^2e1,...) converges to independent standard complex Gaussians. The proof defines S_n = sqrt(N/2) e1^* U^n e1 = sqrt(N/2) sum_i |v_{1i}|^2 Z_i^n, then asserts \"Therefore S_n = sqrt(N/2) e1^*(U^*)^n e1.\" This equality is false in general (U^n and (U^*)^n differ unless D is real). Since the coefficients of g_N in Lemma 14 are v^*(M^*)^k v, it is the (M^*)^k object that must be shown Gaussian. The moment proof actually computes moments of e1^* M^n e1, not of e1^*(M^*)^n e1. This is fixable because the moment condition also holds for the conjugate variables, but as written Lemma 15 does not prove what it states. Additionally, in the final Weingarten computation the stabilizer size of the repeated-index map m in (8) is 2^p (each of the p distinct indices appears exactly twice), but the text writes \"2p\". The subsequent line drops this factor without explanation; with the correct 2^p factor the (N/2)^p prefactor cancels and the Gaussian moment p1!...pl! results. With the printed \"2p\" the limit would be wrong. The paper therefore currently contains an unproven keystone lemma; the convergence of the whole GAF (Theorem 17) and the no-outlier theorem (Theorem 21) rest on it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the point process of eigenvalues of rank-one multiplicative perturbations A = I_N - (1 - a N^{-1/2}) vv^* of random unitary matrices. For Haar-distributed U it recovers the Forrester–Ipsen result that the eigenvalue process converges vaguely to the zero set of phi_a(z) = a - sum_{k>=1} c_k z^k (Theorems 2 and 12), allowing v to be random with a density. It then extends this to the model M = V D V^* with V Haar and D i.i.d. on the unit circle satisfying sup_{k>=1} |E[Z_1^k]| = o(N^{-b}) for every b, where the limit is phi_{a/sqrt(2)} (Theorems 3 and 17). Finally, it uses this GAF description to prove that at the critical timescale t = a N^{-1/2} a strongly separated outlier cannot occur (Theorem 21), extending the Haar-only optimality result of Dubach and Reker. The proof strategy combines a determinant identity for the characteristic polynomial, coefficient convergence by the method of moments, and a direct L^2-based tightness criterion.","tokens_in":17768,"tokens_out":15978,"duration_ms":146511,"significance":"If the results are correct, they establish a valuable universality statement: both the Gaussian analytic function describing the eigenvalue scaling and the critical outlier timescale persist for a broad class of non-Haar unitarily invariant ensembles with i.i.d. eigenvalues, beyond the integrable CUE setting. The proof method is attractive and mostly elementary; in particular, the tightness argument via a locally integrable second-moment bound avoids the integrability machinery used by Dubach and Reker. The paper also gives a clean Rouché-theorem argument showing that the limiting GAF has positive probability of having no zero, and positive probability of having at least two zeros, in a fixed disk, which is exactly what is needed to rule out strong separation. However, the coefficient convergence for the M = V D V^* model is the keystone of Theorems 17 and 21, and in its present form Lemma 15 does not prove what it states; the proof needs repair.","major_comments":[{"comment":"The proof defines S_n := sqrt(N/2) e_1^* U^n e_1 = sqrt(N/2) sum_i |v_{1i}|^2 Z_i^n and then states \"Therefore S_n = sqrt(N/2) e_1^*(U^*)^n e_1.\" This equality is false unless the diagonal entries are real: with U = V D V^*, one has e_1^* U^n e_1 = sum_i |v_{1i}|^2 Z_i^n, while e_1^*(U^*)^n e_1 = sum_i |v_{1i}|^2 \\bar Z_i^n. The coefficients of g_N in Lemma 14 are built from (U^*)^k, so the moment computation actually establishes convergence for the wrong sequence. The gap is repairable because sup_k |E[Z^k]| and sup_k |E[\\bar Z^k]| coincide, so the same proof with \\bar Z_i in place of Z_i gives the stated limit; nevertheless, as printed the proof of this keystone lemma is not valid, and Theorems 17 and 21 rest on it.","section":"Section 3, Lemma 15"},{"comment":"In the final Weingarten computation, the map m defined around equation (8) has p distinct values, each with exactly two preimages, so its stabilizer in S_{2p} has size 2^p, not 2p. The displayed claim \"#{sigma in S_{2p} : m = m circle sigma} = 2p\" is therefore wrong. With the printed 2p, the final Gaussian moment would be off by a factor; with the correct 2^p, the prefactor (N/2)^p cancels against N^{-2p} #T and the stated limit p_1! ... p_l! follows. This arithmetic error must be corrected for the moment computation to be valid.","section":"Proof of Lemma 15, equation (8)"}],"minor_comments":[{"comment":"The notation U is used in Lemmas 14, 15, and 16 even though only M = V D V^* is defined in Section 3; presumably U denotes M. This undefined notation is closely connected to the conjugacy error in Lemma 15 and should be fixed by writing M (or M^*) throughout.","section":"Section 3, Lemmas 14-16"},{"comment":"After expanding (I_N - z D^*)^{-1}, the displayed series should contain \\bar Z_k rather than Z_k; the two are not interchangeable unless D is real.","section":"Lemma 16"},{"comment":"In the computation of P(B), the event |c_2| q^2 > 2s gives P(|c_2| q^2 > 2s) = exp(-4s^2/q^4), not exp(-s^2/q^4); the factor is irrelevant for positivity but should be corrected.","section":"Lemma 20"},{"comment":"There is a typo in the final estimate: \"=<=pm max ...\" should read \"<= pm max ...\".","section":"Proposition 4"},{"comment":"The heading contains a typo: \"unitarily inv ariant\" should be \"unitarily invariant\".","section":"Section 3 heading"}],"recommendation":"major_revision","confidential_remarks":"The two defects in Lemma 15 appear to be mechanical and repairable within the scope of the manuscript: one replaces Z_i by \\bar Z_i (or defines S_n with U^*) and corrects 2p to 2^p. I therefore recommend major revision rather than rejection, but the published version must contain a corrected proof of Lemma 15, since Theorems 17 and 21 depend on it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper has two genuinely new results — the VDV* version of Forrester–Ipsen (Theorem 3) and the Dubach–Reker critical-time optimality for that model (Theorem 21) — and the overall strategy is sound. But the keystone coefficient lemma, Lemma 15, is not proved as written. I would send it to a serious referee, but I would not accept it before that lemma is rewritten.\n\nWhat is good: the direct proof of Theorem 1 is a real improvement in exposition; tightness via Proposition 6 avoids Krishnapur's more elaborate argument. The extension to M = VDV* under the moment decay condition is natural, and the examples (uniform, wrapped normal, wrapped Cauchy) are honest. Theorem 21's argument — ruling out strong separation by showing that both the zero-count and the multiple-zero events in a fixed disk have positive probability in the GAF limit — is clean and plausible. The citation pattern is fine; there is no fitting, no invented entity, no circularity. Theorem 2, as the reader says, is just Theorem 1 conjugated by a unitary sending e1 to v; presenting it as a separate generalization overstates novelty.\n\nSoft spots, in order. Lemma 15 is the real one. The proof defines S_n = sqrt(N/2) e1^(*) U^n e1 = sqrt(N/2) sum_i |v1i|^2 Z_i^n and then asserts equality to sqrt(N/2) e1^(*) (U*)^n e1. That is false unless D is real. The actual coefficient in g_N involves (M*)^n, i.e. V \\bar{D}^n V*, so the object needed is the conjugate of what is computed. The fix is easy — the moment condition holds for \\bar{Z}_1 as well — but as printed Lemma 15 does not prove the convergence it states. The stress-test note is right. Also, in the final Weingarten computation, the stabilizer size of the map m in (8) is 2^p, not 2p; with 2^p the (N/2)^p prefactor cancels exactly, while with 2p the limiting moment would be wrong. That is a small typo in a place that matters.\n\nMinor: Section 3 writes v*(U*)^k v without ever defining U; M is the matrix and M* is clearly intended. Lemma 14 also has a typesetting issue around sqrt(N/2 - a') that should be cleaned. Not substantive.\n\nThe load-bearing assumption sup_k |E[Z_1^k]| = o(N^{-b}) is indeed a modeling condition, not a consequence of unitary invariance. It is strong but not unreasonable, and the paper is explicit about it.\n\nBottom line: the central argument holds up after a corrected Lemma 15. The paper is for RMT people working on rank-one perturbations and GAF zeros; it deserves refereeing. I would accept it for peer review with the expectation of a minor-to-moderate revision.","headline":"A useful extension of the Forrester–Ipsen/GAF result to VDV* models plus a clean critical-time optimality proof; the main theorem is sound but Lemma 15 needs a corrected proof before acceptance.","tokens_in":18295,"tokens_out":3788,"would_cite":true,"duration_ms":35672,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","60G55","30C15","15B52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that, for a broad class of random unitary matrices, the eigenvalues of a rank-one multiplicative perturbation at critical size converge to the zeros of a Gaussian analytic function, and that at the critical time no…","keywords":["random unitary matrices","rank-one perturbation","Gaussian analytic function","point process convergence","eigenvalue outliers","Haar measure","Weingarten calculus","critical timescale"],"falsifier":"Choose $D$ with $Z_1$ uniform on a set of $M_N$ roots of unity so that $\\sup_k |E[Z_1^k]| \\sim N^{-2}$, and check numerically or analytically whether the coefficients $\\sqrt{N/2}\\, v^*(U^*)^k v$ are asymptotically uncorrelated Gaussians; if the covariance of the first two coefficients has a non-vanishing off-diagonal term, or if at $t = aN^{-1/2}$ an isolated eigenvalue persists with probability tending to one, the central claim fails.","tokens_in":17222,"feed_emoji":"🎲","tokens_out":9922,"duration_ms":90218,"temperature":0.7,"pith_summary":"This paper studies $N\\times N$ random unitary matrices of the form $M = VDV^{*}$, where $V$ is Haar-distributed and $D$ has i.i.d. diagonal entries on the unit circle, perturbed by $A = I - (1 - aN^{-1/2})vv^{*}$. The main result is that the eigenvalues of $MA$ converge, as a point process, to the zeros of the Gaussian analytic function $\\varphi_{a/\\sqrt{2}}(z)=a/\\sqrt{2} - \\sum_{k\\geq 1} c_k z^k$ with $c_k$ i.i.d. standard complex Gaussians. The same conclusion is proved for a Haar matrix $U$ with a possibly random unit vector $v$, recovering and extending the earlier Haar result [4]. The paper then answers a question from [2]: at time $t = aN^{-1/2}$ the dynamics has no strongly separated outlier, so the critical timescale is universal across these ensembles. The payoff is that eigenvalue statistics near the critical perturbation inherit the universal statistics of Gaussian analytic function zeros.","feed_headline":"Perturbed unitary spectra match Gaussian zeros at critical scale","feed_subtitle":"The result extends the Gaussian-zero description beyond Haar matrices and fixes the outlier threshold.","key_machinery":"The central object is the random analytic function whose zeros are the eigenvalues: for the $VDV^{*}$ model, $g_N(z) = a' - (\\sqrt{N/2} - a')\\sum_{k\\geq 1} v^*(U^*)^k v\\, z^k$ with $a' = a/\\sqrt{2}$, while for the Haar model it is $f_N(z) = a - (\\sqrt{N} - a)\\sum_{k\\geq 1} v^*(U^*)^k v\\, z^k$. Three load-bearing parts carry the argument: Sylvester's determinant identity converts $\\det(UA - z)$ into a scalar equation whose zeros are exactly the eigenvalues; unitarily invariant moment asymptotics make the random coefficients $\\sqrt{N/2}\\, v^*(U^*)^k v$ converge to independent complex Gaussians; and a second-moment bound using $E|\\langle e_1,r_{k_1}\\rangle|^2|\\langle e_1,r_{k_2}\\rangle|^2$ together with $E[|\\mathrm{tr}(U^l)|^2] = \\min(l,N)$ bounds $\\sup_N E[|g_N(z)|^2]$ by a locally integrable function, yielding tightness in the space of analytic functions. These pieces feed a general theorem that convergence in law of random analytic functions implies vague convergence of their zero processes.","core_discovery":"On the paper's own terms, the discovery is that the Gaussian-analytic-function description of the perturbed unitary spectrum is not an accident of the Haar measure alone. The spectral point process of $MA$, for $M = VDV^{*}$ with i.i.d. eigenphases satisfying $\\sup_{k\\geq 1}|E[Z_1^k]| = o(N^{-b})$ for every integer $b$, converges vaguely to the zero set of $\\varphi_{a'}$ with $a' = a/\\sqrt{2}$. The proof obtains the eigenvalues as zeros of the explicit random analytic function $g_N(z) = a' - (\\sqrt{N/2} - a')\\sum_{k\\geq 1} v^*(U^*)^k v\\, z^k$, shows finite-dimensional coefficient convergence by the moment method with Weingarten asymptotics, and proves tightness by a uniform $L^2$ bound from a locally integrable dominating function. Consequently, at the critical timescale $t = aN^{-1/2}$, the eigenvalue configuration retains a positive-probability chance of having zero or at least two eigenvalues in a suitable inner disk, which excludes the simultaneous inner and outer separation needed for a strongly separated outlier.","pith_inferences":["A natural testable extension is that the same convergence holds for rank-$r$ perturbations, with the limiting function acquiring an extra polynomial factor of degree $r$; the Sylvester-identity route should adapt, though the coefficient correlations become more intricate.","If the Fourier-moment condition fails polynomially, one expects a non-universal intermediate regime: the limiting coefficients would inherit correlations from the diagonal law, and an outlier might persist at $t \\sim N^{-1/2}$ with probability governed by that law.","The no-outlier conclusion at critical time is equivalent to saying the limiting Gaussian analytic function itself typically has no strongly separated zero near zero; computing the positive probabilities in the paper's Lemma 20 explicitly as functions of $a$ could give quantitative bounds on how fast the outlier emerges as $t$ crosses $N^{-1/2}$.","Because the limit is a Gaussian-analytic-function zero process, finer statistics such as the variance of eigenvalue counts in a disk, or the spacing between the innermost eigenvalues, can in principle be transferred to the $VDV^{*}$ spectra at critical scaling; the paper does not compute these."],"forward_implications":["The eigenvalue process of the perturbed model converges to the Gaussian-analytic-function zero process for every ensemble satisfying the Fourier-moment decay, so local statistics near the critical scale are universal.","At time $t = aN^{-1/2}$ no strongly separated outlier exists: with high probability there is no eigenvalue isolated in a small disk around zero while the rest stay near the unit circle, so the critical timescale is optimal for this wider class.","The proof gives a checkable route for other unitary ensembles: derive the eigenfunction by Sylvester's identity, prove coefficient moments converge to Gaussians, and prove tightness by a second-moment bound.","The limiting Gaussian analytic function has constant term $a' = a/\\sqrt{2}$, so the perturbation strength selects a one-parameter family of limiting zero processes interpolating between a deterministic constant-dominated regime and a Kac-type process as $a \\to 0$."],"supporting_citations":[{"why":"Supplies the model and the base Gaussian-analytic-function limit that the paper reproves and generalizes.","marker":"[4]"},{"why":"Introduces the dynamical trajectory model and proves critical-time optimality for Haar matrices; the paper extends that result and answers the question raised there.","marker":"[2]"},{"why":"Provides the lemma on Gaussian fluctuations of $\\sqrt{N}\\, e_1^* U^k e_1$ and the moment result used for the coefficients of the $VDV^{*}$ model.","marker":"[13]"},{"why":"Gives the Weingarten formula and its large-$N$ asymptotics for joint moments of Haar-distributed entries, the backbone of the coefficient convergence proof.","marker":"[15]"},{"why":"Supplies the expectation of $|\\langle e_1,r_{k_1}\\rangle|^2|\\langle e_1,r_{k_2}\\rangle|^2$ used in the tightness bound.","marker":"[8]"},{"why":"Supplies the identity $E[|\\mathrm{tr}(U^l)|^2] = \\min(l,N)$ used to estimate eigenvalue correlations in the Haar tightness proof.","marker":"[14]"},{"why":"Provides the limit theorems for random analytic functions, including the tightness criterion and the continuity of zero empirical measures.","marker":"[16]"},{"why":"Provides the lemma that finite-dimensional coefficient convergence plus tightness implies convergence in law of the random analytic functions.","marker":"[1]"}],"fun_headline_variants":["Gaussian zeros describe perturbed unitary spectra at critical scale","Direct proof extends spectral zeros to general invariant unitary models","Outlier threshold fixed for rank-one perturbed unitaries","Eigenvalues converge to GAF zeros beyond Haar measures","Critical timescale for outlier emergence determined"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the diagonal entries' Fourier moments vanish faster than any polynomial in $N$, since weaker decay would let different powers of the matrix stay correlated and break the Gaussian limit that yields both the zero convergence and the no-outlier conclusion.","fun_headline_variants_meta":{"raw":{"variants":["Gaussian zeros describe perturbed unitary spectra at critical scale","Direct proof extends spectral zeros to general invariant unitary models","Outlier threshold fixed for rank-one perturbed unitaries","Eigenvalues converge to GAF zeros beyond Haar measures","Critical timescale for outlier emergence determined"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000548,"raw_usage":{"total_tokens":2574,"prompt_tokens":855,"completion_tokens":1719,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":471,"completion_tokens_details":{"reasoning_tokens":1643}},"tokens_in":471,"tokens_out":1719,"duration_ms":11755,"temperature":1.0,"reasoning_tokens":1643,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:36:26.245832+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose $D$ with $Z_1$ uniform on a set of $M_N$ roots of unity so that $\\sup_k |E[Z_1^k]| \\sim N^{-2}$, and check numerically or analytically whether the coefficients $\\sqrt{N/2}\\, v^*(U^*)^k v$ are asymptotically uncorrelated Gaussians; if the covariance of the first two coefficients has a non-vanishing off-diagonal term, or if at $t = aN^{-1/2}$ an isolated eigenvalue persists with probability tending to one, the central claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the model and the base Gaussian-analytic-function limit that the paper reproves and generalizes."},{"cited_title":"Dubach and J","cited_arxiv_id":null,"evidence_quote":"Introduces the dynamical trajectory model and proves critical-time optimality for Haar matrices; the paper extends that result and answers the question raised there."},{"cited_title":"Krishnapur,From random matrices to random analytic functions, The Annals of Probability37 (2009), no","cited_arxiv_id":null,"evidence_quote":"Provides the lemma on Gaussian fluctuations of $\\sqrt{N}\\, e_1^* U^k e_1$ and the moment result used for the coefficients of the $VDV^{*}$ model."},{"cited_title":"Nica and R","cited_arxiv_id":null,"evidence_quote":"Gives the Weingarten formula and its large-$N$ asymptotics for joint moments of Haar-distributed entries, the backbone of the coefficient convergence proof."},{"cited_title":"Hiai and D","cited_arxiv_id":null,"evidence_quote":"Supplies the expectation of $|\\langle e_1,r_{k_1}\\rangle|^2|\\langle e_1,r_{k_2}\\rangle|^2$ used in the tightness bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the identity $E[|\\mathrm{tr}(U^l)|^2] = \\min(l,N)$ used to estimate eigenvalue correlations in the Haar tightness proof."},{"cited_title":"Shirai,Limit theorems for random analytic functions and their zeros, Bulletin of the American Mathe- matical Society 32 (1995), 1–37","cited_arxiv_id":null,"evidence_quote":"Provides the limit theorems for random analytic functions, including the tightness criterion and the continuity of zero empirical measures."},{"cited_title":"Bordenave, D","cited_arxiv_id":null,"evidence_quote":"Provides the lemma that finite-dimensional coefficient convergence plus tightness implies convergence in law of the random analytic functions."}],"review_version":1}