Pith. sign in

REVIEW 3 major objections 5 minor 61 references

Spectral radius concentration for inhomogeneous random matrices with independent entries

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Inhomogeneous random matrices: spectral radius is governed by the variance profile, not the operator norm, down to optimal sparsity.

desk verdict The headline claim of Theorem 1.1 is false as stated; the proof only covers sigma/sigma* >> sqrt(log n), and a diagonal Gaussian example is a clean counterexample. read the letter →

arxiv 2501.01079 v2 pith:67WECZVA submitted 2025-01-02 math.PR

Yi Han This is my paper · ORCID
classification math.PR MSC 60B2060F1060F15
keywords spectralradiusinhomogeneousrandommatricesvarianceprofiletracemomentmethodsparsitythresholdsmalldeviationslargeheavy-tailedentries
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 tries to establish that for square random matrices with independent, mean-zero, non-identically distributed entries, the spectral radius $\rho(A)$ is controlled by the variance profile $S$, specifically by the maximal row/column variance sum $\sigma$, rather than by the operator norm bound $\|A\|$, which would carry an extra factor of 2. It proves that after normalization, $\rho(A) \leq (1+\epsilon)\sigma$ with high probability whenever the largest entry standard deviation $\sigma_*$ is $o((\log n)^{-1/2})$, and that this sparsity scale is optimal. For finer fluctuations it introduces a 'long-time control' condition on $S$ and proves small-deviation bounds at the almost optimal scale $\sigma_* \log n$, along with a large-deviation bound for Gaussian entries with doubly stochastic variance and a heavy-tailed bound under only $2+\epsilon$ moments. The motivation is that such inhomogeneous, often sparse matrices arise in neural networks, ecology, and non-Hermitian band matrices, where the variance profile is far from flat.

What carries the argument

The proof rests on the trace moment method: for even $p$, $\rho(A)^{2p}\le \operatorname{Tr}(A^p(A^*)^p)$, so bounding high moments bounds the spectral radius. The key is to compute $E[\operatorname{Tr}(A^p(A^*)^p)]$ for $p$ growing with $n$—up to $p=O(n)$ for Theorem 1.1 and $p\ll\sqrt{n}$ for the small-deviation bounds—by enumerating closed paths, with independence forcing the dominant contributions to be non-backtracking or doubled paths. Comparison with a homogeneous matrix of effective size $\lceil\sigma^2\rceil+p$ lets the variance profile enter only through $\sigma$; the 'long-time control' parameter from Definition 1.3, meaning all powers of $S$ grow at most like $\sigma^{2k}$, replaces $\sigma$ at smaller sparsity scales, and a free-probability resolvent/Dyson equation handles the large-deviation case.

What would settle it

Construct a variance profile that provably fails the long-time control condition for any $\sigma$ below the maximal row/column sum (for example a nonnegative matrix whose powers show transient amplification), scale the entries so that $\sigma_*\ll(\log n)^{-1}$, and check numerically whether $P(\rho(A)\ge\sigma(1+t\sigma_*))$ stays below $C_0 n e^{-Ct}$; a violation would show the small-deviation theorem does not extend beyond its stated assumption.

Watch

Extended reading notes

Core claim

The central claim is that for an $n\times n$ matrix $A=(b_{ij}g_{ij})$ with independent mean-zero entries, the spectral radius $\rho(A)$ is bounded, after normalization, by $(1+\epsilon)\sigma$, where $\sigma$ is the largest row or column sum of the variance matrix $S=(b_{ij}^2)$, as long as the largest entry scale $\sigma_*=\max b_{ij}$ satisfies $\sigma_*\ll(\log n)^{-1/2}$; in that regime almost surely no eigenvalues lie outside the disk of radius $\sigma$. The paper also claims small-deviation control at the almost optimal scale: if $S$ is 'long-time controlled' by $\sigma$, then $P(\rho(A)\ge\sigma(1+t\sigma_*))\le C_0 n e^{-Ct}$ for sub-exponential entries, and, when $S$ is flat, a bound by $\sqrt{\rho(S)}$ with fluctuations of order $n^{-1/2}\log n$. For Gaussian entries with doubly stochastic variance it claims a large-deviation inequality with the expected dependence $e^{-t^2/\sigma_*^2}$, and for symmetric heavy-tailed entries with only $2+\epsilon$ moments it claims boundedness by $\sigma$ or $\sqrt{\rho(S)}$.

Load-bearing premise

The load-bearing premise is that the entries have zero mean with sub-Gaussian (or sub-exponential) tails and that the variance profile admits a 'long-time control' parameter under which every power of $S$ grows at most like $\sigma^{2k}$; the refined small-deviation bounds collapse if only the trivial choice $\sigma$ equal to the maximal row/column sum is available.

Editorial extensions

If this is right

  • For any variance profile with $\sigma_*\ll(\log n)^{-1/2}$, the spectral radius of the inhomogeneous matrix is at most $(1+\epsilon)\sigma$ with high probability, so outliers are absent from the disk of radius $\sigma$.
  • The sparsity scale is sharp: if entries are concentrated in diagonal blocks of size $d_n=o(\log n)$, the spectral radius is eventually larger than any $a>1$, so $\sigma_*\ll(\log n)^{-1/2}$ cannot be relaxed.
  • When the variance profile is long-time controlled, fluctuations of $\rho(A)$ are at most of size $\sigma\sigma_*\log n$ with high probability, i.e., near the scale predicted for Gaussian matrices.
  • For flat variance profiles, $\rho(A)$ is bounded by $\sqrt{\rho(S)}$ plus fluctuations of order $n^{-1/2}(\log n)^{1+c}$, improving previous bounds in both scale and tail probability.
  • For Gaussian entries with doubly stochastic variance, the upper tail of $\rho(A)$ is sub-Gaussian with rate $\sigma_*^{-2}$, the same dependence one would get if $\rho(A)$ were Lipschitz in the entries.

Reading between the lines

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

  • If these bounds are correct, stability analyses of linear systems driven by inhomogeneous random matrices can be carried out using only the variance profile's row/column sums, which may simplify criteria for balanced neural networks and ecological networks.
  • The long-time control condition suggests a testable dichotomy: variance profiles with transient amplification may exhibit spectral radius closer to the maximal row/column sum than to the long-time parameter, and simulating such profiles would reveal whether a new parameter is needed.
  • The sharp sparsity threshold $\sigma_*\ll(\log n)^{-1/2}$ suggests a practical design rule: keep entry variances below $1/\log n$ to avoid outlier eigenvalues in applications that rely on spectral radius estimates.
  • The large-deviation result for Gaussian doubly stochastic variances might extend to sub-Gaussian entries, but the paper notes its current method gives suboptimal rates; a direct numerical test would compare simulated tail probabilities against the $e^{-t^2/\sigma_*^2}$ prediction.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the spectral radius ρ(A) of n×n random matrices with independent, mean-zero entries and variance profile S. It proves several upper bounds: Theorem 1.1 gives a non-asymptotic expectation bound and claims almost-sure absence of outliers (ρ(A) ≤ σ) when σ* = o(1/√log n); Theorem 1.6 establishes small-deviation bounds under a long-time control condition; Theorem 1.7 gives small-deviation bounds via ρ(S); Theorem 1.10 gives a large-deviation inequality for Gaussian doubly stochastic profiles; and Theorem 1.11 extends upper bounds to heavy-tailed entries with only 2+ε moments. The proofs rely on trace moment expansions, combinatorial counting, and comparisons to homogeneous random matrices.

Significance. The paper's technical core — the moment estimates in Section 2 and their inhomogeneous adaptation in Section 3 — is substantial and potentially useful for non-Hermitian random band matrices and inhomogeneous neural-network models. The long-time control parameter (Definition 1.3) is a noteworthy refinement over row/column variance sums. However, the main theorem as stated is false, and several proofs of load-bearing cases are only sketched or omitted. These issues must be addressed before the paper can be considered for publication.

major comments (3)
  1. [Theorem 1.1 and §3.1] The 'in particular' statement of Theorem 1.1 — that P(limsup ρ(A) > σ) = 0 whenever σ* = o(1/√log n) — is false as stated. Take A diagonal with b_ii = n^{-1/2} and b_ij = 0 for i ≠ j, with g_ij i.i.d. standard real Gaussians. Then σ = σ* = n^{-1/2}, σ/σ* = 1 ≤ n, and all assumptions on the entry distribution hold. But ρ(A) = n^{-1/2} max_i |g_i|, so ρ(A)/σ = max_i |g_i| ~ √(2 log n) → ∞ almost surely; hence P(limsup ρ(A) > σ) = 1. The proof in §3.1 explicitly derives the absence-of-outliers only under the additional condition σ/σ* ≫ √(log n) ('First suppose that σ/σ* ≫ √log n ... In the regime where C1√log n ≤ σ/σ* ≤ C2√log n ... only a bound with an extra factor is obtained'). Thus Theorem 1.1 needs an extra hypothesis such as σ/σ* ≫ √(log n) (or σ bounded below by a positive constant), and the abstract's claim (1) and the introduction's 'whenever σ*√log n → 0 then ρ(A) ≤ σ(1+ε)' must be amended accordingly. Without this correction, the central claim of the paper is incorrect.
  2. [§2.2.3 and §3.1 (complex case)] The complex case of Theorem 2.5 (and therefore of Theorem 1.1) is left unproved: the proof in §2.2.3 ends with 'The rest of the proof is the same and omitted,' and the proof of Theorem 1.1 for β = 2 in §3.1 says 'The details are omitted.' Since Theorem 1.1 is stated for both real and complex Gaussian entries, the complex case is part of the main result and requires a proof; referencing an argument that is 'essentially identical' is not sufficient for a load-bearing case.
  3. [Appendix C and §3.2] The removal of the symmetry assumption (Theorem 2.3) is deferred to Appendix C, and the adaptation to inhomogeneous profiles in Theorem 1.6 for non-symmetric distributions is only sketched ('The remaining computations are analogous', §3.2). The paper states that the gluing construction in Appendix C is lengthy, but since Theorem 1.6 is a claimed improvement over Theorem 1.1 (no symmetry needed), the non-symmetric case should be proved rather than indicated.
minor comments (5)
  1. [Abstract] The abstract states 'up to the optimal sparsity σ_* ≫ (\log n)^{-1/2}', but the condition in Theorem 1.1 is σ* = o((\log n)^{-1/2}); the inequality direction appears to be a typo and should be corrected.
  2. [Title] The title contains a typographical error: 'CONCENTRA TION' should read 'CONCENTRATION'.
  3. [Equation (3.3)] The notation E[g_i]^{n_i(s)} in (3.3) is not precise; it should be clarified that g_i denotes an i.i.d. copy of the entry distribution and n_i(s) is the multiplicity of the corresponding directed edge in the shape s.
  4. [Theorem 1.10] The condition '√σ∗(log n)^{3/4} ≤ t^{1.5}/500' mixes the parameters t and n in a way that is hard to interpret; stating the equivalent regime for t in terms of σ* and n would improve readability.
  5. [Definition 1.3] In Definition 1.3, the constant C is allowed to depend on σ, which is unusual for a definition of long-time control; the paper should specify whether the dependence is quantitative or merely existence of some C(σ).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: all bounds derive from variance-profile parameters via independent moment and free-probability inputs, with only minor non-load-bearing self-citations.

full rationale

The paper's central quantities σ, σ_*, and ρ(S) are defined directly from the deterministic variance profile S, not fitted to ρ(A). Theorem 1.1 obtains its bound by comparing E tr(A^p(A*)^p) to a homogeneous Ginibre-type matrix and applying the independent combinatorial moment bound (2.7), which is built on [28]'s diagram enumeration; Theorem 1.6 uses Definition 1.3 as a deterministic hypothesis and bounds path sums such as (3.7) by powers of σ; Theorem 1.7 and Theorem 1.11 likewise reduce to estimates on S and to [14]'s heavy-tail machinery; Theorem 1.10 is based on the free-probability concentration of [9] and the Matrix Dyson equation, whose existence is cited to [36]. The self-citations to [33] and [34] are contextual or auxiliary, for example 'by [33], Section 3.1' in the Dyson-equation step, and they do not supply the leading-order conclusion by themselves. A genuine correctness caveat, not a circularity reduction, is that the proof of Theorem 1.1's almost-sure absence-of-outliers claim explicitly restricts to 'σ^2 >> log n' after normalizing σ_*=1, via the passages 'First suppose that σ/σ_* ≫ sqrt(log n)' and 'By our choice p = α log n and the assumption σ^2 ≫ log n', whereas the theorem statement only assumes σ_* = o((log n)^{-1/2}); the diagonal Gaussian example in the skeptical analysis shows this regime issue is substantive. That is a mathematical gap in a stated regime, not a fitted-input or definitional circularity.

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

The results depend on the stated moment and structural assumptions on the entry distributions and variance profile, and on cited combinatorial and free-probability lemmas. No parameters are fitted to data; the bounds are explicit in terms of sigma and sigma_star.

assumptions (7)
  • domain assumption Sub-Gaussian tails and symmetry (or rotational invariance) of entries as in Theorem 1.1 conditions (1)-(3).
    Assumed on the model; the moment method requires these to control high moments of A.
  • domain assumption Sub-exponential tails (1.10) in Definition 1.2.
    Used in the small-deviation theorems to control moments up to p ~ sqrt(n).
  • domain assumption Long-time control of S: sup_i sum_j [S^k]_{ij} <= C sigma^{2k} for all k (Definition 1.3).
    Key structural condition for Theorem 1.6; failure of this condition can create transient non-normal growth.
  • domain assumption Flatness condition c/n <= b_ij^2 <= C/n (equation (1.12)).
    Needed in Theorem 1.7 to compare the inhomogeneous coefficients with the homogeneous case via (3.13).
  • standard math Diagram automaton and counting bounds from Feldheim-Sodin [28], Propositions 2.8 and 2.9.
    Used to count matched paths in the proof of Theorem 2.5; accepted as background.
  • standard math Free probability concentration result from Bandeira, Boedihardjo, van Handel [9], Theorem 2.1.
    Theorem 1.10 inherits the spectrum comparison from [9]; the Dyson equation solution assumes doubly stochastic profile.
  • standard math Heavy-tailed cycle statistics from Bordenave et al. [14], Lemma 5.1 and Proposition 5.2.
    Theorem 1.11 adapts the upper bound for the spectral radius without fourth moment from [14].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spectral radius concentration for inhomogeneous random matrices with independent entries." pith.science (2026). https://pith.science/paper/67WECZVA

@misc{pith2026250101079,
  author       = {Pith},
  title        = {Pith review of: Spectral radius concentration for inhomogeneous random matrices with independent entries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67WECZVA}},
  note         = {Machine review of arXiv:2501.01079}
}
abstract

Let $A$ be a square random matrix of size $n$, with mean zero, independent but not identically distributed entries, with variance profile $S$. When entries are i.i.d. with unit variance, the spectral radius of $n^{-1/2}A$ converges to $1$ whereas the operator norm converges to 2. Motivated by recent interest in inhomogeneous random matrices, in particular non-Hermitian random band matrices, we formulate general upper bounds for $\rho(A)$, the spectral radius of $A$, in terms of the variance $S$. We prove (1) after suitable normalization $\rho(A)$ is bounded by $1+\epsilon$ up to the optimal sparsity $\sigma_*\gg (\log n)^{-1/2}$ where $\sigma_*$ is the largest standard deviation of an individual entry; (2) a small deviation inequality for $\rho(A)$ capturing fluctuation beyond the optimal scale $\sigma_*^{-1}$; (3) a large deviation inequality for $\rho(A)$ with Gaussian entries and doubly stochastic variance; and (4) boundedness of $\rho(A)$ in certain heavy-tailed regimes with only $2+\epsilon$ finite moments and inhomogeneous variance profile $S$. The proof relies heavily on the trace moment method.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 52 canonical work pages

  1. [1]

    Norms of structured random matrices

    Rados law Adamczak et al. “Norms of structured random matrices”. In: Mathematische Annalen 388.4 (2024), pp. 3463–3527

    work page 2024

  2. [2]

    The stability–complexity relationship at age 40: a random matrix perspective

    Stefano Allesina and Si Tang. “The stability–complexity relationship at age 40: a random matrix perspective”. In: Population Ecology 57.1 (2015), pp. 63–75. 50 REFERENCES

    work page 2015

  3. [3]

    Spectral radius of random matri- ces with independent entries

    Johannes Alt, Laszlo Erdos, and Torben Kruger. “Spectral radius of random matri- ces with independent entries”. In: Probability and Mathematical Physics 2.2 (2021), pp. 221–280

    work page 2021

  4. [4]

    Local inhomogeneous circular law

    Johannes Alt, L´ aszl´ o Erd˝ os, and Torben Kr¨ uger. “Local inhomogeneous circular law”. In: Ann. Appl. Probab. 28.1 (2018), pp. 148–203. issn: 1050-5164,2168-8737

    work page 2018

  5. [5]

    The Dyson equation with linear self-energy: spectral bands, edges and cusps

    Johannes Alt, L´ aszl´ o Erd˝ os, and Torben Kr¨ uger. “The Dyson equation with linear self-energy: spectral bands, edges and cusps”. In: Documenta Mathematica 25 (2020), pp. 1421–1539

    work page 2020

  6. [6]

    On spectral outliers of inhomogeneous symmetric random matrices

    Dylan J Altschuler et al. “On spectral outliers of inhomogeneous symmetric random matrices”. In: arXiv preprint arXiv:2401.07852 (2024)

    arXiv 2024

  7. [7]

    Law of large numbers for the spectral radius of ran- dom matrix products

    Richard Aoun and Cagri Sert. “Law of large numbers for the spectral radius of ran- dom matrix products”. In: Amer. J. Math. 143.3 (2021), pp. 995–1010. issn: 0002- 9327,1080-6377

    work page 2021

  8. [8]

    Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang

    Z. D. Bai and Y. Q. Yin. “Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang”. In: Probability Theory and Related Fields 73 (1986), pp. 555–569

    work page 1986

Show all 61 references
  1. [9]

    Matrix concen- tration inequalities and free probability

    Afonso S Bandeira, March T Boedihardjo, and Ramon van Handel. “Matrix concen- tration inequalities and free probability”. In: Inventiones mathematicae 234.1 (2023), pp. 419–487

    2023

  2. [10]

    Sharp nonasymptotic bounds on the norm of random matrices with independent entries

    Afonso S. Bandeira and Ramon van Handel. “Sharp nonasymptotic bounds on the norm of random matrices with independent entries”. In: Ann. Probab. 44.4 (2016), pp. 2479–2506. issn: 0091-1798,2168-894X

    2016

  3. [11]

    The circular law for sparse non-Hermitian ma- trices

    Anirban Basak and Mark Rudelson. “The circular law for sparse non-Hermitian ma- trices”. In: The Annals of Probability 47.4 (2019), pp. 2359–2416

    2019

  4. [12]

    Spectral radii of sparse random matrices

    Florent Benaych-Georges, Charles Bordenave, and Antti Knowles. “Spectral radii of sparse random matrices”. In: Ann. Inst. H. Poincar´ e Probab. Statist 56.3 (2020), pp. 2141–2161

    2020

  5. [13]

    Convergence of the spec- tral radius of a random matrix through its characteristic polynomial

    Charles Bordenave, Djalil Chafai, and David Garcia-Zelada. “Convergence of the spec- tral radius of a random matrix through its characteristic polynomial”. In: Probability Theory and Related Fields (2021), pp. 1–19

    2021

  6. [14]

    On the spectral radius of a random matrix: an upper bound without fourth mosment

    Charles Bordenave et al. “On the spectral radius of a random matrix: an upper bound without fourth mosment”. In: Annals of Probability 46.4 (2018), pp. 2268–2286. issn: 0091-1798

    2018

  7. [15]

    Random band matrices

    Paul Bourgade. “Random band matrices”. In: Proceedings of the International Con- gress of Mathematicians: Rio de Janeiro 2018 (2018), pp. 2759–2783

    2018

  8. [16]

    Extremal random matrices with in- dependent entries and matrix superconcentration inequalities

    Tatiana Brailovskaya and Ramon van Handel. “Extremal random matrices with in- dependent entries and matrix superconcentration inequalities”. In: arXiv preprint arXiv:2401.06284 (2024)

    2024 arXiv

  9. [17]

    Progress on the study of the Ginibre ensembles I: GinUE

    Sung-Soo Byun and Peter J Forrester. “Progress on the study of the Ginibre ensembles I: GinUE”. In: arXiv preprint arXiv:2211.16223 (2022)

    2022 arXiv

  10. [18]

    On the spectral edge of non-Hermitian random matrices

    Andrew Campbell et al. “On the spectral edge of non-Hermitian random matrices”. In: arXiv preprint arXiv:2404.17512 (2024)

    2024 arXiv

  11. [19]

    Eigenvector statistics in non-Hermitian ran- dom matrix ensembles

    John T Chalker and Bernhard Mehlig. “Eigenvector statistics in non-Hermitian ran- dom matrix ensembles”. In: Physical review letters 81.16 (1998), p. 3367

    1998

  12. [20]

    Universality of extremal eigen- values of large random matrices

    Giorgio Cipolloni, L´ aszl´ o Erd˝ os, and Yuanyuan Xu. “Universality of extremal eigen- values of large random matrices”. In: arXiv preprint arXiv:2312.08325 (2023). REFERENCES 51

    2023

  13. [21]

    Lower bounds for the smallest singular value of structured random matrices

    Nicholas Cook. “Lower bounds for the smallest singular value of structured random matrices”. In: Ann. Probab. 46.6 (2018), pp. 3442–3500. issn: 0091-1798,2168-894X

    2018

  14. [22]

    Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs

    Nicholas Cook et al. “Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs”. In: Electronic Journal of Probability 23 (Jan. 2018)

    2018

  15. [23]

    Sparse matrices: convergence of the characteristic polynomial seen from infinity

    Simon Coste. “Sparse matrices: convergence of the characteristic polynomial seen from infinity”. In: Electronic Journal of Probability 28 (2023), pp. 1–40

    2023

  16. [24]

    Outliers in the spectrum for prod- ucts of independent random matrices

    Natalie Coston, Sean O’Rourke, and Philip Wood. “Outliers in the spectrum for prod- ucts of independent random matrices”. In: Annales de l’Institut Henri Poincar´ e, Prob- abilit´ es et Statistiques56 (Nov. 2017)

    2017

  17. [25]

    Power law decay for systems of randomly coupled differential equations

    Laszlo Erdos, Torben Kr¨ uger, and David Renfrew. “Power law decay for systems of randomly coupled differential equations”. In: SIAM Journal on Mathematical Analysis 50.3 (2018), pp. 3271–3290

    2018

  18. [26]

    Quantum diffusion and delocalization for band matrices with general distribution

    L´ aszl´ o Erd˝ os and Antti Knowles. “Quantum diffusion and delocalization for band matrices with general distribution”. In: Annales Henri Poincar´ e. Vol. 12. 7. Springer. 2011, pp. 1227–1319

    2011

  19. [27]

    Randomly coupled differential equations with elliptic correlations

    L´ aszl´ o Erd˝ os, Torben Kr¨ uger, and David Renfrew. “Randomly coupled differential equations with elliptic correlations”. In: The Annals of Applied Probability 33.4 (2023), pp. 3098–3144

    2023

  20. [28]

    A universality result for the smallest eigenvalues of certain sample covariance matrices

    Ohad N Feldheim and Sasha Sodin. “A universality result for the smallest eigenvalues of certain sample covariance matrices”. In: Geometric And Functional Analysis 20.1 (2010), pp. 88–123

    2010

  21. [29]

    The spectral radius of large random matrices

    Stuart Geman. “The spectral radius of large random matrices”. In: Annals of Proba- bility 14 (1986), pp. 1318–1328

    1986

  22. [30]

    What drives transient behavior in complex systems?

    Jacek Grela. “What drives transient behavior in complex systems?” In: Physical Re- view E 96.2 (2017), p. 022316

    2017

  23. [31]

    From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behav- ior

    Ewa Gudowska-Nowak et al. “From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behav- ior”. In: Neural Computation 32.2 (2020), pp. 395–423

    2020

  24. [32]

    Mixed moments for the product of Ginibre matri- ces

    Nick Halmagyi and Shailesh Lal. “Mixed moments for the product of Ginibre matri- ces”. In: arXiv preprint arXiv:2007.10181 (2020)

    2020 arXiv

  25. [33]

    Outliers and bounded rank perturbation for non-Hermitian random band matrices

    Yi Han. “Outliers and bounded rank perturbation for non-Hermitian random band matrices”. In: arXiv preprint arXiv:2408.00567 (2024)

    2024 arXiv

  26. [34]

    The circular law for random band matrices: improved bandwidth for general models

    Yi Han. “The circular law for random band matrices: improved bandwidth for general models”. In: arXiv preprint arXiv:2410.16457 (2024)

    2024

  27. [35]

    Edge universality of sparse Erd \H {o} sR\’enyi digraphs

    Yukun He. “Edge universality of sparse Erd \H {o} sR\’enyi digraphs”. In: arXiv preprint arXiv:2304.04723 (2023)

    2023 arXiv

  28. [36]

    Operator-valued semicir- cular elements: solving a quadratic matrix equation with positivity constraints

    J William Helton, Reza Rashidi Far, and Roland Speicher. “Operator-valued semicir- cular elements: solving a quadratic matrix equation with positivity constraints”. In: International Mathematics Research Notices 2007.9 (2007), rnm086–rnm086

    2007

  29. [37]

    Optimal control of transient dynamics in balanced networks supports generation of complex movements

    Guillaume Hennequin, Tim P Vogels, and Wulfram Gerstner. “Optimal control of transient dynamics in balanced networks supports generation of complex movements”. In: Neuron 82.6 (2014), pp. 1394–1406

    2014

  30. [38]

    Enumeration of non-crossing pairings on bit strings

    Todd Kemp et al. “Enumeration of non-crossing pairings on bit strings”. In: Journal of Combinatorial Theory, Series A 118.1 (2011), pp. 129–151. 52 REFERENCES

    2011

  31. [39]

    The dimension-free struc- ture of nonhomogeneous random matrices

    Rafa l Lata la, Ramon van Handel, and Pierre Youssef. “The dimension-free struc- ture of nonhomogeneous random matrices”. In: Inventiones mathematicae 214 (2018), pp. 1031–1080

    2018

  32. [40]

    Computing norms of free operators with matrix coefficients

    Franz Lehner. “Computing norms of free operators with matrix coefficients”. In: Amer- ican Journal of Mathematics 121.3 (1999), pp. 453–486

    1999

  33. [41]

    Will a large complex system be stable?

    Robert M May. “Will a large complex system be stable?” In: Nature 238.5364 (1972), pp. 413–414

    1972

  34. [42]

    Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks

    Dylan R Muir and Thomas Mrsic-Flogel. “Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks”. In: Physical Review E 91.4 (2015), p. 042808

    2015

  35. [43]

    Lectures on the combinatorics of free probability

    Alexandru Nica and Roland Speicher. Lectures on the combinatorics of free probability. Vol. 13. Cambridge University Press, 2006

    2006

  36. [44]

    Products of independent elliptic random matrices

    Sean O’Rourke et al. “Products of independent elliptic random matrices”. In: Journal of Statistical Physics 160.1 (2015), pp. 89–119

    2015

  37. [45]

    Wigner random matrices with non-symmetrically distributed entries

    Sandrine P´ ech´ e and Alexander Soshnikov. “Wigner random matrices with non-symmetrically distributed entries”. In: Journal of Statistical Physics 129 (2007), pp. 857–884

    2007

  38. [46]

    Eigenvalue spectra of random matrices for neural networks

    Kanaka Rajan and Larry F Abbott. “Eigenvalue spectra of random matrices for neural networks”. In: Physical review letters 97.18 (2006), p. 188104

    2006

  39. [47]

    A limit theorem at the edge of a non-Hermitian random matrix ensemble

    B. Rider. “A limit theorem at the edge of a non-Hermitian random matrix ensemble”. In: vol. 36. 12. Random matrix theory. 2003, pp. 3401–3409

    2003

  40. [48]

    Extremal laws for the real Ginibre ensem- ble

    Brian Rider and Christopher D. Sinclair. “Extremal laws for the real Ginibre ensem- ble”. In: Ann. Appl. Probab. 24.4 (2014), pp. 1621–1651. issn: 1050-5164,2168-8737

    2014

  41. [49]

    The sparse circular law under minimal assumptions

    Mark Rudelson and Konstantin Tikhomirov. “The sparse circular law under minimal assumptions”. In: Geometric and Functional Analysis 29 (2019), pp. 561–637

    2019

  42. [50]

    Central limit theorem for traces of large random sym- metric matrices with independent matrix elements

    Ya. Sinai and A. Soshnikov. “Central limit theorem for traces of large random sym- metric matrices with independent matrix elements”. In: Bol. Soc. Brasil. Mat. (N.S.) 29.1 (1998), pp. 1–24. issn: 0100-3569

    1998

  43. [51]

    A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices

    Ya. G. Sina ˘ ı and A. B. Soshnikov. “A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices”. In: Funktsional. Anal. i Prilozhen. 32.2 (1998), pp. 56–79, 96. issn: 0374-1990,2305-2899

    1998

  44. [52]

    The spectral edge of some random band matrices

    Sasha Sodin. “The spectral edge of some random band matrices”. In: Annals of math- ematics (2010), pp. 2223–2251

    2010

  45. [53]

    Chaos in random neural networks

    Haim Sompolinsky, Andrea Crisanti, and Hans-Jurgen Sommers. “Chaos in random neural networks”. In: Physical review letters 61.3 (1988), p. 259

    1988

  46. [54]

    Universality at the Edge of the Spectrum in Wigner Random Matrices

    Alexander Soshnikov. “Universality at the Edge of the Spectrum in Wigner Random Matrices”. In: Communications in mathematical physics 207 (1999), pp. 697–733

    1999

  47. [55]

    An introduction to the theory of large deviations

    Daniel W Stroock. An introduction to the theory of large deviations . Springer Science & Business Media, 2012

    2012

  48. [56]

    Random matrices: universality of ESDs and the circular law

    Terence Tao, Van Vu, and Manjunath Krishnapur. “Random matrices: universality of ESDs and the circular law”. In: Annals of Probability 38.5 (2010), pp. 2023–2065

    2010

  49. [57]

    On pseudospectrum of inhomogeneous non-Hermitian ran- dom matrices

    Konstantin Tikhomirov. “On pseudospectrum of inhomogeneous non-Hermitian ran- dom matrices”. In: arXiv preprint arXiv:2307.08211 (2023)

    2023 arXiv

  50. [58]

    On the spectral norm of Gaussian random matrices

    Ramon Van Handel. “On the spectral norm of Gaussian random matrices”. In: Trans- actions of the American Mathematical Society 369.11 (2017), pp. 8161–8178

    2017

  51. [59]

    Structured random matrices

    Ramon Van Handel. “Structured random matrices”. In: Convexity and concentration (2017), pp. 107–156. REFERENCES 53

    2017

  52. [60]

    High-dimensional probability: An introduction with applications in data science

    Roman Vershynin. High-dimensional probability: An introduction with applications in data science. Vol. 47. Cambridge university press, 2018

    2018

  53. [61]

    Universality and the circular law for sparse random matrices

    Philip Matchett Wood. “Universality and the circular law for sparse random matrices”. In: (2012). Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA Email address: hanyi16@mit.edu

    2012

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.