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Quantitative Tracy-Widom laws for sparse random matrices

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Pith's one-line read The paper proves that, for sparse random matrices with sparsity parameter $q\ge N^{1/6+\delta}$, the shifted largest eigenvalue converges to the Tracy–Widom law with error $N^\omega(N^{-1/3}+N^{2/3}/q^4)$.

desk verdict First quantitative Tracy-Widom rate for sparse random matrices, with the real risk concentrated in one unpinned computer-assisted linear algebra step. read the letter →

arxiv 2507.19340 v1 pith:WK74LSE3 submitted 2025-07-25 math.PR

Teodor Bucht , Kevin Schnelli , Yuanyuan Xu This is my paper · ORCID
classification math.PR MSC 60B2015B52
keywords Tracy–WidomlawsparserandommatricesErdős–RényigraphslargesteigenvalueconvergencerateGreenfunctioncomparisonedgecorrectioncomputer-assistedproof
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 establishes the first quantitative edge-universality result for sparse random matrices, the model class that contains the normalized adjacency matrix of an Erdős–Rényi graph. The main theorem states that when the mean-degree parameter satisfies $q\ge N^{1/6+\delta}$, the largest eigenvalue $N^{2/3}(\lambda_N-\widehat L)$ converges in distribution to the Tracy–Widom law $\mathrm{TW}_1$, uniformly over tail events, with error at most $N^\omega(N^{-1/3}+N^{2/3}/q^4)$ after subtracting the corrected edge $\widehat L=2+6\kappa_4/q^2+\chi$. This is a rate problem, not just a limit problem: it says how large $N$ must be before the limiting law is a trustworthy approximation. Quantitative statements of this kind are what turn edge universality into a tool for finite-$N$ inference, such as hypothesis tests based on the largest eigenvalue.

What carries the argument

The load-bearing mechanism is a Green function comparison theorem for fine spectral scales, built on averaged products of Green function entries $G_{xy}(z)$, with $z=\widehat L_t+x+i\eta$. Terms are encoded as rational combinations of such products up to permutation of summation indices, and the comparison is driven by three operations: the cumulant expansion formula for the interpolation flow between the sparse matrix and a Gaussian Wigner matrix; resolvent identities that generate algebraic relations among the leading terms; and iterative expansion of unmatched indices, which turns odd-order terms into negligible ones. The fourth-order cancellation is verified by computer algebra: 4,288 identities, obtained from the resolvent identities, are assembled into a 13,852-by-14,246 rational linear system whose exact solution expresses the leading terms as a combination of the identities, proving that only non-leading terms remain.

What would settle it

Re-run the exact rational Gaussian elimination for the system in equation (6.19) in independent software and verify that the claimed solution, with its 4,288 non-zero entries, has zero residual; a non-zero residual would invalidate Lemma 6.7 and remove the proof of the Green function comparison theorem.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.3: under Assumption 1.1 with $q\ge N^{1/6+\delta}$, for every fixed $r_0$ and small $\omega>0$, the bound $\sup_{r>r_0}|P(N^{2/3}(\lambda_N-2-6\kappa_4/q^2-\chi)\le r)-\mathrm{TW}_1(r)|\le N^\omega(N^{-1/3}+N^{2/3}/q^4)$ holds for all large $N$. The proof flows from a long-time Green function comparison theorem (Theorem 1.5) that compares the sparse matrix with a Gaussian Wigner matrix at spectral scales $\eta\gg N^{-1}+q^{-4}$ and time $t\asymp\log N$; the comparison error is the same $N^{-1/3}+N^{2/3}/q^4$ rate. What makes the comparison possible is a cancellation principle: all leading fourth-order cumulant terms in the time derivative are shown to cancel against the terms generated by the edge correction, leaving only non-leading terms that can be bounded by the local law and the Ward identity.

Load-bearing premise

The proof rests on the correctness of a large computer-generated algebraic database, namely the rational solution of the 13,852-by-14,246 linear system and the 4,288 identities used to cancel the leading terms, and no machine-checked certificate or independent formal verification of that computation is supplied.

Editorial extensions

If this is right

  • For Erdős–Rényi adjacency matrices with $p\gg N^{-2/3}$, the largest eigenvalue after the corrected edge has Tracy–Widom fluctuations at the stated rate, giving the first quantitative edge law for this sparse model class.
  • In the dense limit $q\asymp\sqrt{N}$, the bound reduces to $N^\omega N^{-1/3}$, matching the sharpest known convergence rate for Wigner matrices up to the $N^\omega$ factor.
  • The edge shift $2+6\kappa_4/q^2+\chi$ is part of the statement, so centering at the unshifted semicircle edge $2$ would leave a systematic $q^{-2}$ bias in any finite-$N$ comparison.
  • The Green function comparison at spectral scales below the eigenvalue spacing is a reusable tool for other fine-scale edge statistics of sparse matrices, such as counting eigenvalues in $N^{-2/3}$ windows.
  • Quantitative Kolmogorov-type bounds provide finite-$N$ error control for p-values in tests based on the largest eigenvalue, the application to community detection in stochastic block models mentioned in the introduction.

Reading between the lines

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

  • The threshold $q\ge N^{1/6+\delta}$ and the constraint $\eta\ge N^{2\epsilon}q^{-4}$ come from truncating the cumulant expansion at sixth order; the authors themselves indicate that higher-order edge corrections should extend the result down to $q\ge N^\delta$, but that extension is not proved here.
  • The large sparse linear system has only about 0.07% non-zero entries, which suggests the cancellation may have a compact structural explanation; finding an explicit combinatorial identity for the fourth-order terms would remove the reliance on computer verification.
  • The same identity-generation pipeline, resolvent identities plus cumulant expansions encoded as a rational linear system, looks transferable to other high-order edge statistics such as joint fluctuations of the largest eigenvalues, where explicit cancellation formulas are not known.
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Editorial analysis

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

Referee Report

4 major / 6 minor

Summary. The paper proves a quantitative Tracy–Widom law for the largest eigenvalue of sparse random matrices satisfying Assumption 1.1, in the regime q ≥ N^{1/6+δ}. The main result, Theorem 1.3, states that after subtracting the corrected edge 2 + 6κ4/q^2 + χ, the Kolmogorov distance between the rescaled largest eigenvalue distribution and TW1 is bounded by N^ω(N^{-1/3} + N^{2/3}/q^4). The proof follows Schnelli–Xu's long-time Green function comparison strategy: Theorem 1.5 gives a GFT for fine spectral scales, whose proof via Proposition 3.1 and Proposition 4.1 reduces to cancellation among fourth-order cumulant terms and edge-correction terms. A computer-assisted symbolic computation generates identities (Rules 1–3) and solves sparse rational linear systems (110×138 and 13,852×14,246) to verify the cancellations. The paper includes Section 7 on implementation, appendices with proofs of the supporting lemmas, and a GitHub repository with code and identities.

Significance. If correct, Theorem 1.3 is a substantial advance: it gives the first quantitative edge-universality rate for sparse random matrices with q as small as N^{1/6+δ}, recovers the Wigner rate N^{-1/3} from [51] when q ≍ √N, and provides an explicit additional O(N^{2/3}/q^4) error that is natural in view of Remark 1.4. The proof strategy—unmatched-index expansions combined with a computer-generated cancellation between fourth-order cumulant terms and the edge correction—is well matched to the difficulty. The paper is unusually transparent about the computer-assisted component: Section 7 describes the term representation, equivalence checking, and exact rational Gaussian elimination, and the full code and identities are deposited on GitHub [16]. No fitted parameters enter the theorem, and the Wigner baseline [51] is prior peer-reviewed work. These are genuine strengths. The main weakness is the lack of a machine-checked or independently audited certificate for the 13,852×14,246 rational linear system and the equivalence-checking implementation; this is a verifiability concern rather than a demonstrated mathematical error, and it can be addressed within the manuscript's scope.

major comments (4)
  1. [§6.2, Eq. (6.19)] The load-bearing step in the proof of Lemma 6.7 is the assertion that the rational linear system (6.19), with dimensions 13,852×14,246 and a 4,288-entry solution, exactly expresses the leading terms of d/dt E[F(X(t))] as a linear combination of the identities (6.17). The paper states that the solution is found by a custom C++ sparse Gaussian elimination but does not provide a machine-checkable certificate of the elimination, a proof that the identity-generation rules were implemented without error, or a proof that the equivalence test (Section 7.2) correctly distinguishes all non-equivalent basis terms. The GitHub repository [16] supplies code and data, but the version is unpinned (no commit hash), and the equivalence check for i0 > 1 is a recursive search whose correctness is not formally verified. Since an error in any of these three components—identity generation, equivalence detection, or exact solve—would invalidate Lemma 6.7 and hence Theorem 1.5 and Theorem 1.3, this is a genuine verification gap. Please provide an independent re-verification, a symbolic certificate with a documented verification procedure, or at least a precise description of how a reader can reproduce the full pipeline from the supplied scripts without trusting the same code path that produced the answer.
  2. [§3.2, proof of Theorem 1.5] The perturbation argument bounding |E[F(X(10 log N))] − E[F(X(∞))]| uses ∥G(10 log N, z(10 log N)) − G(∞, z(∞))∥₂ ≺ 1/(N^{7/2}η²). The stated bound appears to be missing the factor N^{1/2} from the difference of the H-terms in (3.5), and the displayed inequality is not checked against the stated ranges of η (η can be as small as N^{-1+ϵ}+N^{2ϵ}q^{-4}, so N^{7/2}η² may be far larger than N). The subsequent claim |E[F(X(10 log N))] − E[F(X(∞))]| ≤ N^{-1} is therefore not justified as written. This step is load-bearing because it converts the integrated bound from Proposition 3.1 into the full GFT of Theorem 1.5; please write out the complete estimate and verify it for all allowed η.
  3. [§5.3, proof of Lemma 5.9, case 2] In the case-2 bound of Lemma 5.9, the manuscript claims that for ph ≥ 1 with an index v distinct from a and b occurring at least twice in the off-diagonal entries, an additional cumulant expansion gains 1/N. The premise is needed to ensure the resulting terms have degree at least 2 after the expansion, but the proof does not verify the degree condition for all subcases (e.g., when the derivative hits different Green function factors in (4.16) and produces terms such as G_{ja}G_{bv}). The bound may be correct, but the argument as written does not check it; a short explicit verification would close the gap.
  4. [Appendix A.1, proof of Lemma 2.6] The proof of (A.1) extends the averaged local law from Theorem 2.4 to all η ≫ N^{-1} and uses the assertion that y ↦ y Im m_N(E + iy) is strictly increasing in y. That monotonicity is not proved and is used to bound the first term in the extension argument. This is a supporting lemma, but (A.1) enters Lemma 2.6 and hence Theorem 1.3. Please either prove the monotonicity on a high-probability event or replace the argument with a bound that does not rely on it.
minor comments (6)
  1. [§3.2 and §6.2] The notation ∆ Im is defined in (6.2) but used earlier in the main proof of Proposition 3.1; please reorder so that the definition precedes first use.
  2. [§1, Theorem 1.3] The statement says sup_{r > r0}, but the proof works with r ∈ (r0, N^ϵ) and uses rigidity for |r| ≥ N^ϵ; please clarify whether r0 may be negative and whether the supremum should be over r0 < r < ∞ with the small-r behavior covered by rigidity.
  3. [§5.2.3 and §6.2] The counts of basis terms and identities (M = 138, L = 110 for the small system; M_F = 14,246, L_F = 13,852 for the large system) would be much easier to verify if the paper included a small table of the distribution of type-0, type-A, and type-AB terms by number of summation indices, together with the counts of identities generated by each rule.
  4. [§7.2] The description of the almost-unique identifier AUID is heuristic; while the manuscript states that collisions are checked by exact equivalence, it would help to state the maximum number of summation indices occurring in the computations and why the recursive equivalence check terminates quickly in practice.
  5. [§6.2, Remark 6.10] The reported runtime of roughly one hour for the large system would be more informative with the exact software versions, hardware, and a description of how the final solution was verified in exact arithmetic (e.g., by re-multiplication and checking zero residues).
  6. [Throughout] There are a few typos and undefined notations: '[GOE' appears without definition; 'κ4' in (1.8) is used before Assumption 1.1 states that κ4 = κ4(N); and the line 'd/dt E[m(t,z(t))] = =' in Section 4 has a double equals sign.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the proof is self-contained, with only a minor non-load-bearing self-citation to prior Wigner convergence work.

full rationale

Theorem 1.3 is derived from the Green function comparison theorem (Theorem 1.5), whose proof reduces to Proposition 3.1 and then to the symbolic cancellation claims in Lemmas 5.8 and 6.7. These cancellations are verified by generating identities from the resolvent identity and cumulant expansions using Rules 1–3 and solving sparse rational linear systems (5.28) and (6.19). This is algebraic verification, not a parameter fit: no quantity in the theorem is fitted to data, and the random shift chi is defined from the matrix entries, not chosen to force the Tracy–Widom conclusion. The edge correction 2 + 6*kappa_4/q^2 + chi is inherited from the prior edge-correction analyses [28,42], not introduced ad hoc in this paper, and the proof does not define the correction in terms of the target distribution. The main self-citation is [51], a peer-reviewed result by two of the present authors, used for the baseline GOE convergence rate and the initial bound in Proposition 4.1; its assumptions do not include Theorem 1.3, so it constitutes independent external support rather than circular reasoning. The computer-aided cancellation is flagged by the authors as implementation-dependent and is supplied with code, but a possible implementation error would be a verification or correctness gap, not circularity. Overall, the derivation chain is self-contained, and no load-bearing step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters are fitted to data; the theorem is a universal bound with constants depending only on model parameters (N,q,δ,ε,ω). No new physical or mathematical entities are postulated. The random shift χ and corrected edge bL are functions of the matrix entries inherited from prior edge-correction literature.

assumptions (7)
  • domain assumption Entrywise local law for sparse matrices (Theorem 2.1 of [19], Eq. (2.4)-(2.5))
    The proof bounds Green function entries via this law, a previously established result for the sparse ensemble satisfying Assumption 1.1.
  • domain assumption Averaged local law and optimal rigidity (Theorems 2.4-2.5 from [32,42])
    Used for the eigenvalue count (2.12) and rigidity (2.11), and for the small-η extension in Appendix A.1.
  • domain assumption Existence and properties of the corrected semicircle-like density eρ with edge eL (Proposition 2.3 from [32])
    The shifted edge bL and the local law comparison for the trace Green function rest on this prior result.
  • standard math Convergence rate for the Gaussian Wigner baseline to Tracy-Widom (Schnelli-Xu [51])
    Used in the proof of Theorem 1.3 to identify the TW1 limit for the GOE comparison matrix; a published peer-reviewed theorem.
  • standard math Cumulant expansion formula (Lemma 2.7 from [26])
    The backbone of all expansions; a standard tool in random matrix theory.
  • standard math Right-tail asymptotics of TW1 (Baik-Buckingham-DiFranco [7])
    Used to extend the supremum over r>r0 to the full tail in Theorem 1.3.
  • domain assumption The corrected edge shift bL = 2 + 6κ4/q^2 + χ from [28,32,42]
    The theorem statement and the cancellation in Section 5 are built around this previously derived shift; the paper proves the needed cancellation rather than postulating it.

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Pith. "Pith review of Quantitative Tracy-Widom laws for sparse random matrices." pith.science (2026). https://pith.science/paper/WK74LSE3

@misc{pith2026250719340,
  author       = {Pith},
  title        = {Pith review of: Quantitative Tracy-Widom laws for sparse random matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WK74LSE3}},
  note         = {Machine review of arXiv:2507.19340}
}
abstract

We consider the fluctuations of the largest eigenvalue of sparse random matrices, the class of random matrices that includes the normalized adjacency matrices of the Erd\H{o}s-R\'enyi graph $G(N, p)$. We show that the fluctuations of the largest eigenvalue converge to the Tracy-Widom law at a rate almost $O(N^{-1/3 } + p^{-2} N^{-4/3})$ in the regime $p \gg N^{-2/3 }$. Our proof builds upon the Green function comparison method initiated by Erd\H{o}s, Yau, and Yin [22]. To show a Green function comparison theorem for fine spectral scales, we implement algorithms for symbolic computations involving averaged products of Green function entries.

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