Repeated singular values of a random symmetric matrix and decoupled singular value estimates

Yi Han

REVIEW 3 major objections 5 minor 37 references

Repeated singular values of a random symmetric matrix and decoupled singular value estimates

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Joint small singular values of a random symmetric matrix decouple into a product at separated bulk locations.

desk verdict A strong, genuinely new two-point decoupling result with a load-bearing gap where the uniform one-point estimate is asserted rather than proved; worth serious refereeing if the gap is filled. read the letter →

arxiv 2504.15992 v1 pith:O5XAAL3C submitted 2025-04-22 math.PR

Yi Han This is my paper · ORCID

classification math.PR MSC 60B2015B52

keywords randomsymmetricmatrixleastsingularvaluegapsWignersmallballprobabilityinverseLittlewood-OfforddecouplingVu'sconjecture

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's central claim is that, for a symmetric random matrix with independent mean-zero variance-one subgaussian entries, the event that the least singular value of $A_n-\lambda_1 I$ is small and the same event at a separated bulk location $\lambda_2$ are essentially independent: the joint probability is bounded by $C\delta_1\delta_2$ plus an exponentially small error. If correct, this decoupling lets the author prove that all singular values in $[\kappa\sqrt{n},(2-\kappa)\sqrt{n}]$ are distinct with probability $1-e^{-cn}$, and that the minimal gap between them is at least order $n^{-3/2}$ with high probability. A sympathetic reader would care because this is a strong quantitative resolution of Vu's conjecture on the absence of repeated singular values, for any fixed fraction of the bulk spectrum away from the edge, and it supplies a two-point version of the recent one-point least-singular-value result. The mechanism is a new two-location 'inversion of randomness' argument replacing the Cauchy interlacing tricks that had blocked earlier attempts.

What carries the argument

The argument's central object is a two-vector version of the essential least common denominator (LCD), defined for a pair of vectors $(A_n-\lambda_1 I)^{-1}\tilde{X}$ and $(A_n-\lambda_2 I)^{-1}\tilde{X}$ restricted to a random subset of coordinates; it measures how close any linear combination can come to a lattice point. Around this the paper builds a two-location inversion-of-randomness scheme: a zeroed-out matrix $M_n$ with independent blocks replaces the dependent symmetric matrix, threshold functions $\tau_{L,\varepsilon_1}$ stratify vector pairs by their anti-concentration, and a double-counting argument bounds the size of the corresponding nets. The second pillar is spectral decoupling: the local semicircle law (or the super-exponential concentration from a log-Sobolev inequality) shows that at separated locations the $k$-th singular value at one location dominates the correlated singular value at the other, so small-ball estimates at the two locations multiply. A bootstrap lemma then iterates the product exponent up to the optimal $\delta_1\delta_2$.

What would settle it

Simulate symmetric Bernoulli matrices at $n\approx 2000$ and, for many bulk pairs $\lambda_1,\lambda_2$ with $|\lambda_1-\lambda_2|\geq\sqrt{n}$, estimate $\mathbb{P}(\sigma_{\min}(A_n-\lambda_i I)\leq\delta_i n^{-1/2}, i=1,2)$ for $\delta_1=\delta_2=e^{-n^{1/4}}$; a value above $C\delta_1\delta_2+2e^{-cn}$ would falsify Theorem 1.2. Separately, to test the hinge, estimate the one-point probability at a fine grid of bulk locations; if for some $\lambda$ it exceeds $C\delta+e^{-cn}$, then the uniform Proposition 2.9 fails.

Watch

Extended reading notes

Core claim

Theorem 1.2 states that for locations $\lambda_1,\lambda_2$ in the bulk $[-(2-\kappa)\sqrt{n},(2-\kappa)\sqrt{n}]$ with $|\lambda_1-\lambda_2|\geq \Delta\sqrt{n}$, one has $\mathbb{P}(\sigma_{\min}(A_n-\lambda_i I)\leq \delta_i n^{-1/2}, i=1,2)\leq C\delta_1\delta_2+2e^{-cn}$, under a finite log-Sobolev constant. Theorem 1.3 achieves the same product structure for mesoscopic separation $|\lambda_1-\lambda_2|\geq \Delta n^{\sigma-1/2}$ with error $e^{-c n^{\sigma/2}}$ and no log-Sobolev assumption. As corollaries, the singular values in $[\kappa\sqrt{n},(2-\kappa)\sqrt{n}]$ are all distinct with probability $1-e^{-cn}$, and the minimal gap among them satisfies $\mathbb{P}(\text{gap}\leq \varepsilon n^{-3/2})\leq C\varepsilon+e^{-cn}$ (or $C\varepsilon+e^{-c n^{\sigma/2}}$ in the mesoscopic case). This is what the paper means by a strong quantitative form of Vu's conjecture up to a $(1-\kappa)$-fraction of the spectrum.

Load-bearing premise

The load-bearing premise is Proposition 2.9: the one-location least-singular-value bound of [5] extends to every bulk location $\lambda$ uniformly with the same exponential error $e^{-cn}$; the paper gives this as a stated but omitted 'straightforward generalization', and both the two-point reduction and the log-factor-removal bootstrap use it at arbitrary bulk locations.

Editorial extensions

If this is right

For symmetric Bernoulli matrices, with probability $1-e^{-cn}$ all singular values in $[\kappa\sqrt{n},(2-\kappa)\sqrt{n}]$ are distinct; the proof is direct from Theorem 1.2 and a covering argument.
The minimal gap between these bulk singular values is at least $n^{-3/2}$ up to constants: the probability of a gap below $\varepsilon n^{-3/2}$ is at most $C\varepsilon+e^{-cn}$.
Extreme small-ball events of the least singular value at two separated bulk locations factor, up to a constant and an exponentially small error, exactly as if the two locations were independent.
Linear statistics of distant eigenvalues are controlled: the probability that two distant bulk eigenvalues satisfy $a_1 x_1+a_2 x_2=D$ within $\varepsilon n^{-3/2}$ is $O(\varepsilon)$ plus an exponential error.
For mesoscopic separations $|\lambda_1-\lambda_2|\geq \Delta n^{\sigma-1/2}$, the same decoupling holds with exponential error $e^{-c n^{\sigma/2}}$ for any subgaussian entry distribution without log-Sobolev.

Reading between the lines

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

Extension: numerically, one expects the joint tail $\mathbb{P}(\sigma_{\min}(A_n-\lambda_1 I)\leq\delta_1 n^{-1/2},\sigma_{\min}(A_n-\lambda_2 I)\leq\delta_2 n^{-1/2})$ to track $C\delta_1\delta_2$ at separations much larger than $n^{-1/2}$; Monte Carlo at very small $\delta$ would be a direct check.
Extension: Proposition 2.9 is the hinge; testing one-point probabilities over a fine grid of bulk locations separates the paper's theorem from its assumed uniform generalization.
Extension: the $n^{-3/2}$ gap scale suggests that $n^{3/2}$ times the minimal bulk gap should converge in distribution; the paper does not assert this, but its bound is consistent with it.

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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

First thing you should know: this is a credible and significant extension of the Campos et al. one-point estimate to two separated bulk locations, with the right product form C δ1δ2 + e^{-cn}, and it delivers a quantitative n^{-3/2} minimal-gap statement for most of the bulk spectrum. The novelty is real: the inversion-of-randomness step is upgraded to vector pairs, with angle-dependent threshold functions, and the bootstrap in §10 is a genuine machine. Section 12 contains substantial proof detail for the conditioned inverse Littlewood–Offord theorem. If the main theorems are correct, this is a major within-field result.

What is actually new: Theorem 1.2 is a two-point decoupling at macroscopic separation; Theorem 1.3 extends to mesoscopic separation with e^{-c n^{σ/2}} error without LSI. Corollaries 1.8 and 1.9 follow by covering arguments and are plausible.

Now the soft spots, in proportion. The biggest is Proposition 2.9: the uniform one-location estimate for arbitrary bulk λ is asserted with 'we omit the proof of this straightforward generalization' of [5]. That is not a stylistic omission. For λ ≠ 0 the matrix A_n − λI has diagonal entries with mean −λ, so the mean-zero structure used in [5] is broken, and the local-law part does not transfer verbatim. The proof of Lemma 2.1, the reduction in §5, and the bootstrap in §10 all lean on this proposition. It is probably true—one expects P(σmin(A_n−λI) ≤ ε n^{-1/2}) ≲ ε + e^{-cn} uniformly in the bulk—but the paper does not show it, and the claim that the shift is harmless needs an actual argument or a citation to a published uniform version. Second, Theorem 1.3's proof is dismissed as 'identical' with details omitted; given the parameter changes, that is a lot to wave through. The rest of the proof is long and, as far as I can verify, carefully put together; the main architecture is coherent and the cited building blocks are real.

Who it's for: specialists in random matrix theory, especially people working on least singular values and eigenvalue gaps. It deserves a serious referee: the potential payoff is large and the approach is new. But I would not accept it in the current form. Send it out with a request to supply the missing uniform one-point estimate and a detailed proof for the mesoscopic case, or to point to a verifiable source. My verdict: conditional accept after major revision.

Referee Report

3 major / 5 minor

Summary. The paper claims a two-point decoupling estimate for the least singular value of a symmetric random matrix: for bulk locations λ1, λ2 separated by at least a fixed macroscopic multiple of n^{1/2}, the probability that both σmin(A_n − λ_i I) are small is bounded by Cδ1δ2 plus an exponentially small error. A mesoscopic version is claimed with separation Δ n^{σ−1/2} and error e^{−c n^{σ/2}} under only subgaussian assumptions. From this the author derives that all singular values in a bulk interval are distinct with probability 1 − e^{−cn}, with minimal gap at least order n^{−3/2} up to a constant, giving a quantitative form of Vu's conjecture for a (1−κ)-fraction of the spectrum. The proof develops a two-location quasirandomness framework based on an inversion-of-randomness method, a decoupling of quadratic forms, and a bootstrap argument.

Significance. If the proof is completed, the result would be a significant advance in the quantitative invertibility of symmetric random matrices: it gives exponential-type distinctness of singular values and an n^{−3/2} gap bound, and it extends the one-location estimate of Campos et al. to a genuine two-location decoupling. The paper also contains a serious new technical apparatus for studying joint arithmetic structure of two random vectors associated with shifted Wigner matrices. The main caveat is that several load-bearing statements are asserted with proofs omitted or only sketched, so the significance currently rests on unverified technical claims rather than on a fully checkable argument.

major comments (3)

[Section 2.2, Proposition 2.9 (Eq. (2.12))] The uniform one-location estimate P(σmin(A_n − λI) ≤ ε n^{−1/2}) ≤ cε + e^{−cn} for all λ in the bulk is stated without proof. The text says the proof of [5] generalizes straightforwardly, but [5] treats λ = 0 for a mean-zero Wigner matrix, while A_n − λI has nonzero diagonal mean for λ ≠ 0; in particular the eigenvector LCD event E3 in Lemma 2.1 is not established for A_n − λI by a verbatim application of [5]. This proposition is used in Proposition 5.3, Lemma 9.2, Lemma 10.2, and the bootstrap in Section 10, so the product form Cδ1δ2 in Theorem 1.2 depends on it. A complete proof, or a precise citation of a theorem that gives this uniform bulk estimate with the same exponential error, is required.
[Section 11, Theorem 1.3 and Lemmas 11.2, 11.3] The proof of Theorem 1.3 is omitted: the final line says 'The proof here is identical to the proof of Theorem 1.2... The details are omitted.' Lemma 11.2 and Lemma 11.3, which are the mesoscopic analogues of Lemma 8.1 and Lemma 9.1, are only sketched. Since Theorem 1.3 is a stated main result with a different parameter range (n^σ separation), a different error e^{−c n^{σ/2}}, and without the log-Sobolev assumption, this is not a cosmetic omission. In particular, the analogue of the quadratic-form decoupling at mesoscopic separation needs a careful check that all constants and error scales behave as claimed. Please provide full proofs of Lemmas 11.2 and 11.3 and a detailed derivation of Theorem 1.3 from them.
[Section 3, Lemmas 3.3, 3.4, 3.13 and Proposition 3.9] Several statements that are load-bearing for Theorem 2.6 are asserted without proof or with only a sketch. Lemma 3.3 (box covering of Λ_{ε,ε1}(c)) is stated with 'proof omitted'; Lemma 3.4 (random grid vectors have no rigid arithmetic structure) is stated with 'the proof is essentially the same as [18], Lemma 3.16' and details omitted; Lemma 3.13 (Fourier replacement) is stated with 'we omit the straightforward proof'; Proposition 3.9 is only sketched. These results feed directly into the net cardinality bounds and hence into the verification of the quasirandomness event E4 and Theorem 2.6. Without complete proofs or exact references that cover the two-vector statements, the central bootstrap cannot be verified. Please add the missing arguments.

minor comments (5)

[Abstract and Theorem 1.2] The separation parameter is denoted κ in the abstract and ∆ in Theorem 1.2; please unify the notation to avoid confusion.
[Corollary 1.9] The phrase 'for any (n-dependent) constants a1, a2, D with a1 = 1' is awkward; a1 is a fixed coefficient, not an n-dependent constant. Please rephrase.
[Section 2.3] There are numerous typographical issues, including 'F act 2.11' and undefined notation such as |sin(v_D, r_D)| before it is introduced. These should be cleaned up.
[Lemma 10.2 proof] In the proof, p is fixed to be 1.01, while Proposition 5.1 is stated for arbitrary p > 1. The dependence of the constants on p and on the choice p near 1 should be tracked explicitly.
[Lemma 8.1 proof, Part 2/Part 4] The inequality |θ_I| ≥ (θ1θ2)^{1/2} appears to require an absolute value and a fixed normalization of θ; please clarify the normalization used in the two cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two-location estimate is proved from an external one-location estimate and self-contained quasi-randomness reductions.

full rationale

The paper's central claim (Theorem 1.2) is a product-form two-location least singular value bound. The derivation is an explicit reduction chain: Proposition 5.1 reduces the two small singular values to small random-distance events, Sections 3-4 establish the quasi-randomness theorem 2.6 from elementary box/Littlewood-Offord estimates, Theorem 6.1 and Lemma 8.1 estimate the decoupled quadratic forms, and Section 10 bootstraps these into the final δ1δ2 bound. The one-location estimate Proposition 2.9 is used as an input, and its proof is explicitly omitted ('we omit the proof of this straightforward generalization'); this is a genuine load-bearing external-input gap, but it is not circularity because Proposition 2.9 is a strictly weaker one-point estimate taken from the external paper [5], and Theorem 1.2 is not used to prove it. The author's own preprint [19] is cited only to disclose that earlier computations are merged and rewritten, and those computations are contained in the manuscript; [18] is cited only for comparison. No parameter is fitted to a subset of data and renamed a prediction, no uniqueness theorem is imported from the author's own prior work, and no known result is merely relabelled. Thus the derivation does not reduce to its own conclusion.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof rests on standard random matrix tools and a handful of external estimates. It does not fit any parameter to data. The main new mathematical objects (angle-dependent threshold functions, subvector LCD) are internal proof devices rather than invented physical entities. The most delicate unproved input is Proposition 2.9, a uniform bulk generalization of the one-point estimate of [5].

assumptions (6)

domain assumption The entry distribution zeta is subgaussian with finite subgaussian moment B.
Used throughout for concentration and Littlewood-Offord estimates (Theorem 1.2, 1.3).
domain assumption For Theorem 1.2, zeta has a finite, n-independent log-Sobolev constant.
Used in Proposition 7.8 and Lemma 7.4 to get super-exponential concentration of the empirical spectral measure, which is critical for the macroscopic decoupling.
domain assumption The one-location least singular value estimate of [5] extends uniformly to all bulk lambda.
Proposition 2.9 is stated without proof and is used in the bootstrap and in the reduction to distance estimates.
standard math Local semicircle law holds with exponential error for bulk windows of length at least K/n.
The paper invokes Theorem 7.5 from [11] to prove eigenvalue rigidity estimates.
standard math The empirical spectral measure concentrates super-exponentially under the log-Sobolev inequality.
Proposition 7.8 from [17] is a known concentration result used for Lemma 7.4.
standard math No-gaps delocalization of eigenvectors holds for Wigner matrices.
Theorem 5.4 from [27] is used in the proof of Proposition 5.1 and Proposition 5.3.

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Pith. "Pith review of Repeated singular values of a random symmetric matrix and decoupled singular value estimates." pith.science (2026). https://pith.science/paper/O5XAAL3C

@misc{pith2026250415992,
  author       = {Pith},
  title        = {Pith review of: Repeated singular values of a random symmetric matrix and decoupled singular value estimates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5XAAL3C}},
  note         = {Machine review of arXiv:2504.15992}
}

abstract

Let $A_n$ be a random symmetric matrix with Bernoulli $\{\pm 1\}$ entries. For any $\kappa>0$ and two real numbers $\lambda_1,\lambda_2$ with a separation $|\lambda_1-\lambda_2|\geq \kappa n^{1/2}$ and both lying in the bulk $[-(2-\kappa)n^{1/2},(2-\kappa)n^{1/2}]$, we prove a joint singular value estimate $$ \mathbb{P}(\sigma_{min}(A_n-\lambda_i I_n)\leq\epsilon n^{-1/2};i=1,2)\leq C\epsilon^2+2e^{-cn}. $$ For general subgaussian distribution and a mesoscopic separation $|\lambda_1-\lambda_2|\geq \kappa n^{-1/2+\sigma},\sigma>0$ we prove the same estimate with $e^{-cn}$ replaced by an exponential type error. This means that extreme behaviors of the least singular value at two locations can essentially be decoupled all the way down to the exponential scale when the two locations are separated. As a corollary, we prove that all the singular values of $A_n$ in $[\kappa n^{1/2},(2-\kappa)n^{1/2}]$ are distinct with probability $1-e^{-cn}$, and with high probability the minimal gap between these singular values has order at least $n^{-3/2}$. This justifies, in a strong quantitative form, a conjecture of Vu up to $(1-\kappa)$-fraction of the spectrum for any $\kappa>0$.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [5]

The least singular value of a rando m symmetric matrix

Marcelo Campos et al. “The least singular value of a rando m symmetric matrix”. In: Forum of Mathematics, Pi . Vol. 12. Cambridge University Press. 2024, e3. REFERENCES 75

work page 2024

[2] [18]

Simplicity of singular value spectrum of rando m matrices and two-point quantitative invertibility

Yi Han. “Simplicity of singular value spectrum of rando m matrices and two-point quantitative invertibility”. In: arXiv preprint arXiv:2502.13819 (2025)

arXiv 2025

[3] [1]

Extreme gaps between eigenvalues of Wig ner matrices

Paul Bourgade. “Extreme gaps between eigenvalues of Wig ner matrices”. In: Journal of the European Mathematical Society 24.8 (2021), pp. 2823–2873

work page 2021

[4] [2]

Fixed energy universality for gene ralized Wigner matrices

Paul Bourgade et al. “Fixed energy universality for gene ralized Wigner matrices”. In: Communications on Pure and Applied Mathematics 69.10 (2016), pp. 1815–1881

work page 2016

[5] [3]

On the singularity of random symme tric matrices

Marcelo Campos et al. “On the singularity of random symme tric matrices”. In: Duke Math. J. 170.5 (2021), pp. 881–907. issn: 0012-7094,1547-7398

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