Zeros Of Random Analytic Functions And Spectral Properties Of Perturbed Unitary Matrices

Aniss Fares

REVIEW 2 major objections 5 minor 17 references

Zeros Of Random Analytic Functions And Spectral Properties Of Perturbed Unitary Matrices

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2509.03252 v1 pith:7TBXRFMX submitted 2025-09-03 math.PR

Aniss Fares This is my paper · ORCID

classification math.PR MSC 60B2060G5530C1515B52

keywords randomunitarymatricesrank-oneperturbationGaussiananalyticfunctionpointprocessconvergenceeigenvalueoutliersHaarmeasureWeingartencalculuscriticaltimescale

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $a \to 0$.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

What 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.

Soft 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.

Minor: 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.

The 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.

Bottom 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.

Referee Report

2 major / 5 minor

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.

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 (2)

[Section 3, Lemma 15] 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.
[Proof of Lemma 15, equation (8)] 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.

minor comments (5)

[Section 3, Lemmas 14-16] 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.
[Lemma 16] 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.
[Lemma 20] 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.
[Proposition 4] There is a typo in the final estimate: "=<=pm max ..." should read "<= pm max ...".
[Section 3 heading] The heading contains a typo: "unitarily inv ariant" should be "unitarily invariant".

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the GAF limit is derived from coefficient moments, not assumed; no fitted inputs and no load-bearing self-citations.

full rationale

The paper's central derivation is not circular. The target Gaussian analytic function appears as a limit object, not as an input: the eigenvalues are first characterized as zeros of explicit random analytic functions (Lemmas 8 and 14), and the coefficients are then shown to converge to independent standard complex Gaussians via Krishnapur's external lemma in the Haar case and via a self-contained moment/Weingarten computation in the VDV* model (Lemma 15). The super-polynomial decay condition on E[Z_1^k] is a stated modeling assumption, not a fitted parameter, and it is used to make cross terms vanish in the moment computation rather than to force the GAF conclusion. Tightness is obtained from an explicit L^2 bound and Proposition 6, again without importing the target result. Theorem 21's no-outlier conclusion follows by combining the derived convergence with Lemma 20, which gives strictly positive probabilities for zero-count events of the limiting GAF; this is independent content, not a restatement of the conclusion. There are no self-citations, and the cited external results (Krishnapur, Weingarten calculus, Dubach-Reker, Forrester-Ipsen) are used as tools or as comparisons, not as substitutes for the proof. Even if a referee questions the correctness of the moment computation in Lemma 15, that would be a mathematical-error issue, not a circularity issue.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data. The central claims rest on the model assumptions (independent i.i.d diagonal entries with super-polynomial moment decay and an independent perturbation vector) and on standard random matrix facts such as Weingarten calculus and Haar eigenvalue moments. The GAF phi_a is from prior literature, not newly postulated.

assumptions (5)

standard math A Haar-distributed unitary matrix has eigenvectors independent of eigenvalues, with Weingarten moments E[|v_11|^4]=2/(N(N+1)) and E[|v_11|^2|v_12|^2]=1/(N(N+1)).
Used in Lemmas 11 and 16 to compute second moments of the spectral coefficients; cited to [8].
standard math E[|tr(U^l)|^2]=min(l,N) for a Haar-distributed unitary U.
Used in Lemma 11 to evaluate eigenvalue pair correlations; cited to [14].
standard math The moment method determines the law of the infinite Gaussian coefficient sequence from finite mixed moments.
Used in Lemma 15 to prove finite-dimensional coefficient convergence.
domain assumption The i.i.d diagonal entries Z_i are on the unit circle with sup_{k>=1}|E[Z^k]|=o(N^{-b}) for every integer b and have a simple spectrum almost surely.
This is the defining model assumption of Theorem 3; it ensures cross-correlations between different powers vanish.
domain assumption The perturbation vector v is a unit vector independent of the unitary matrix, with a non-vanishing density on the unit sphere.
Used to replace v by e1 via unitary invariance; the density condition is stronger than needed.

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Cite this review

Pith. "Pith review of Zeros Of Random Analytic Functions And Spectral Properties Of Perturbed Unitary Matrices." pith.science (2026). https://pith.science/paper/7TBXRFMX

@misc{pith2026250903252,
  author       = {Pith},
  title        = {Pith review of: Zeros Of Random Analytic Functions And Spectral Properties Of Perturbed Unitary Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TBXRFMX}},
  note         = {Machine review of arXiv:2509.03252}
}

read the original abstract

We study the spectral properties of a rank-one multiplicative perturbation of a unitary matrix, a model introduced by Fyodorov. Building upon earlier results by Forrester and Ipsen, we provide a direct proof that the eigenvalues converge to the zeros of a specific Gaussian analytic function. Our approach extends these results to other unitarily invariant models. This method enables us to address a question raised by Dubach and Reker concerning the critical timescale at which an outlier emerges.

Figures

Figures reproduced from arXiv: 2509.03252 by the authors.

**Figure 1.** Trajectories of the eigenvalues of the UA model of size 100 × 100. The evolution of the time parameter t is represented by shades of red. We say that a sequence of matrix GN (t) has a strongly separated outlier towards the origin for t ∈ T ⊂ [−1, 1] if there exists α1 < α2, d1, d2 ∈ (0, 1) such that w.h.p, for every t ∈ T, • D(0, d1 Nα1 1+Nα1 ) contains exactly one eigenvalue of GN (t); • D\D(0, d2 Nα2 1+Nα2 ) conta… view at source ↗

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Bordenave, D

C. Bordenave, D. Chafaï, and D. García-Zelada,Convergence of the spectral radius of a random matrix through its characteristic polynomial, Probability Theory and Related Fields (2022), 1–19

work page 2022

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work page 2024

[3] [3]

D. E. Edmunds and W. D. Evans,Elliptic differential operators and spectral analysis, Springer Monographs in Mathematics, Springer, New York, 2018

work page 2018

[4] [4]

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work page 2019

[5] [5]

Y. V. Fyodorov,Spectra of random matrices close to unitary and scattering theory for discrete-time systems, in: Disordered and complex systems, AIP Conference Proceedings, vol. 553, American Institute of Physics, Melville, NY, 2001

work page 2001

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Y. V. Fyodorov, B. A. Khoruzhenko, and T. Prellberg,Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series(2024), available at2412.06086

work page Pith review arXiv 2024

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Y. V. Fyodorov and D. V. Savin,Resonance Scattering of Waves in Chaotic Systems, in: The Oxford Handbook of Random Matrix Theory (G. Akemann, J. Baik, and P.Di Francesco, eds.), Oxford Handbooks in Mathematics, Oxford University Press, Oxford, 2011

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Hiai and D

F. Hiai and D. Petz,The semicircle law, free random variables and entropy, in: Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2000

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