Limiting spectral laws for sparse random circulant matrices

Adrian Beker

REVIEW 2 major objections 3 minor 48 references

Limiting spectral laws for sparse random circulant matrices

T0 review · 2 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read Sparse random $G$-circulant spectra converge exactly when element-order distributions do, with explicit roots-of-unity limits.

desk verdict Genuinely new sparse circulant model with a clean iff characterization; the main spectral theorems hold up, but Lemma 3.1(6) is false and the determinant proof needs repair. read the letter →

arxiv 2504.13833 v1 pith:DF53XYVV submitted 2025-04-18 math.PR math.CO

Adrian Beker This is my paper · ORCID

classification math.PRmath.CO MSC 60B2015B5260F05

keywords randomcirculantmatricessparseempiricalspectraldistributionrootsofunityfiniteabeliangroupselementorderdeterminantHermitisationmethod

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 treats uniformly random $G_n$-circulant matrices with entries in $\{0,1\}$ and exactly $d$ ones per row and column, for a growing sequence of finite abelian groups. It establishes an exact equivalence: the empirical spectral distribution converges weakly in expectation if and only if the distribution of the order of a uniform random element of $G_n$ converges weakly on $\mathbb{N}^\ast$, the one-point compactification of the natural numbers. When this happens, the limiting law is $\mu = \sum_m \rho(\{m\})\eta_m^{\ast d}$, a mixture of $d$-fold convolutions of the uniform distribution on $m$-th roots of unity (with $\eta_\infty$ the uniform law on the unit circle). Convergence in probability occurs if and only if that order distribution is a Dirac mass $\delta_m$, in which case the limit is simply $\eta_m^{\ast d}$. A further theorem identifies the log-determinant rate $c_{m,d} = \int \log|z|\,d\eta_m^{\ast d}(z)$ when no matrix in the family is singular, giving $\frac{1}{|G_n|}\log|\det C_n| \to c_{m,d}$ in probability.

What carries the argument

The central object is the $G$-circulant matrix for a finite abelian group $G$: a matrix $(A_{x,y})_{x,y\in G}$ with $A_{x,y}=a(xy^{-1})$, the matrix of convolution by $a$. These matrices are normal and are simultaneously diagonalised by the Fourier transform on $G$, so the eigenvalue multiset is $\{\widehat a(\gamma):\gamma\in\widehat G\}$. In the random sparse model, $\widehat{S_n}(\gamma)=\sum_{j=1}^d \gamma(X_{n,j})$ with $X_{n,j}$ independent uniform elements of $G_n$; this sum has law $\eta_{\mathrm{ord}(\gamma)}^{\ast d}$ and support in $dR_{\mathrm{ord}(\gamma)}$. The proof machinery is the Hermitisation method (as formulated in Proposition 2.3): eigenvalue convergence is deduced from weak convergence of shifted singular value measures plus uniform integrability of the logarithm, and the latter is reduced to small-ball estimates for sums of roots of unity, chiefly Lemma 3.1. The variance computation for even moments of shifted singular value measures (Lemma 5.4) forces the functional equation $\tau(\gcd(a,b))=\tau(a)\tau(b)$ on the limiting order distribution, whose $\{0,1\}$-valued solutions are exactly indicators of the multiples of a fixed $m$.

What would settle it

Take $d=2$, even $m$, $z=0$, and $r\to0$; directly count pairs of $m$-th roots of unity whose sum lies in $B(0,r)$. There is an atom of size $1/m$ at $0$, so $\eta_m^{\ast2}(B(0,r))$ does not decay as $1/m^2$, contradicting Lemma 3.1(6) as stated; the proof's claim that a point is the midpoint of at most one chord of the unit circle is false for the centre, which is the midpoint of every diameter.

Watch

Extended reading notes

Core claim

The paper's central claim is that the spectral law of a sparse random circulant matrix is completely encoded by a single group-theoretic statistic: the order of a uniformly random element. Concretely, Theorem 1.2 says $\mu_{C_n}$ converges weakly in expectation if and only if $\rho_{G_n}$ converges weakly to some $\rho$ on $\mathbb{N}^\ast$, with limit $\mu = \sum_{m\in\mathbb{N}^\ast} \rho(\{m\})\eta_m^{\ast d}$. Theorem 1.3 sharpens this: convergence in probability holds if and only if $\rho = \delta_m$, and then the limit is $\eta_m^{\ast d}$ -- the $d$-fold convolution of the uniform distribution on the $m$-th roots of unity, or on the whole unit circle when $m=\infty$. Because the eigenvalues of a $G$-circulant matrix are exactly the Fourier coefficients of its first row, each eigenvalue is a sum of $d$ independent uniform elements of the dual group, and its law is $\eta_{\mathrm{ord}(\gamma)}^{\ast d}$; the theorems follow by tracking how these laws mix as the group grows. Theorem 1.6 adds that, under the natural nonsingularity condition $0\notin dR_{\exp(G_n)}$ and $\rho_{G_n}\to\delta_m$, the normalized logarithm of the determinant converges in probability to $c_{m,d}=\int\log|z|\,d\eta_m^{\ast d}(z)$.

Load-bearing premise

The argument's control of the logarithm near zero rests on Lemma 3.1(6), the estimate $\eta_m^{\ast d}(B(z,r)) \ll ((rm+1)/m)^2$ for $d\ge2$; the proof's geometric justification, that every point is the midpoint of at most one chord of the circle, fails at the centre, where for $d=2$ and even $m$ the sum has an atom of size $1/m$, not $O(1/m^2)$.

Editorial extensions

If this is right

For $G_n=\mathbb{Z}/n\mathbb{Z}$, the element-order measure $\rho_{G_n}$ converges to $\delta_\infty$, so the ESD converges weakly in probability to $\eta_\infty^{\ast d}$, the $d$-fold convolution of the uniform distribution on the unit circle -- a commutative counterpart of the conjectured sparse i.i.d. limit.
For $G_n=(\mathbb{Z}/m\mathbb{Z})^n$ with $m$ fixed, the limiting law is $\eta_m^{\ast d}$, so the spectrum is governed by a finite root-of-unity convolution and is supported in the finite set $dR_m$.
If the order distribution converges to a non-Dirac measure, the ESD converges only in expectation and its random fluctuations do not disappear; the example $G_n=\mathbb{Z}/2\mathbb{Z}\oplus(\mathbb{Z}/3\mathbb{Z})^n$ has limit $\frac12\delta_3+\frac12\delta_6$.
When $0\notin dR_{\exp(G_n)}$ and $\rho_{G_n}\to\delta_m$, asymptotically almost surely $|\det C_n|=\exp((c_{m,d}+o(1))|G_n|)$; for $m=\infty$, $c_{\infty,d}=\frac12\log d-\frac{\gamma}{2}+o(1)$ with $\gamma\approx0.577$ the usual constant.

Reading between the lines

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

The if-and-only-if gives a recipe for constructing group sequences with no limiting spectrum in probability: mix groups whose exponents produce different root-of-unity limits, e.g. a proportion $p$ of cyclic groups of order $3^k$ and $1-p$ of order $5^k$, so that $\rho_{G_n}$ converges to a non-Dirac mixture; then the ESD converges in expectation but keeps a random component.
A repair of Lemma 3.1(6) appears necessary: for $d=2$, even $m$, and $z=0$, the probability $\eta_m^{\ast2}(\{0\})=1/m$ is not of order $1/m^2$, so the bound as stated cannot be correct; a corrected version that removes or separately handles the atom at zero would preserve the uniform-integrability conclusion used by Theorem 1.6.
The same Fourier-coefficient representation points to finer spectral statistics -- the spectral gap of the underlying random Cayley graph, or the proportion of eigenvalues near zero -- as natural next targets, since they reduce to concentration and small-ball properties of sums of $d$ roots of unity.
The non-abelian version (Problem 7.1) lacks simultaneous diagonalisation, so the present order-statistic characterisation cannot transfer verbatim; if a limiting law exists for symmetric-group circulants, it would need new tools to be identified.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Quick take: this is a real contribution, not a desk-reject. The model—d-regular G-circulant matrices—is new, and the characterization of weak convergence in expectation and in probability via the order distribution is clean and, as far as I can tell, correct. The explicit limits as convolutions of uniform root-of-unity distributions are exactly the right answer, and the mixture formula (2) is nice. The proofs of Theorems 1.2 and 1.3 use standard Hermitisation plus Smith normal form and they hang together.

The soft spot is Lemma 3.1(6). It claims η_m^{*d}(B(z,r)) = O(((rm+1)/m)^2) uniformly in z. That is false: for d=2, m even, z=0, the atom at zero has mass 1/m, which beats the claimed O(1/m^2) when r is small. The geometric justification, that every point is the midpoint of at most one chord of the unit circle, fails for the center. This lemma is invoked in Proposition 3.3 in the complementary case z=0, d≥2 to control P(|S| < ord^{-3d}), and Proposition 3.3 feeds Theorem 1.6.

Now, the damage is localized and probably repairable. Under the hypothesis 0 ∉ dR_exp(G_n), the exact-zero atom is absent, so the false case of the lemma never actually occurs in the application; one only needs a small-ball estimate for sums that are constrained to be nonzero, and that should follow from a corrected version of Lemma 3.1 or a separate argument using Lemma 3.2. But as written, the proof of Proposition 3.3 and hence of Theorem 1.6 does not go through; a referee should ask for this to be fixed. I would not be surprised if the determinant result is true, and the brief discussion of Lam–Leung and the density of F_d is a nice bonus.

Theorems 1.2 and 1.3 look solid to me; the false lemma is used at a single point, and the λ-a.e. statement in Proposition 3.3 only needs the small-ball bound off a null set. The variance computation in Section 5 is intricate and appears correct; the positivity of covariances and the reduction to τ satisfying (18) is elegant.

So: this deserves peer review, with a request to fix or restrict Lemma 3.1(6) and clean up the z=0 case in Proposition 3.3. I would cite it for the main spectral theorems once the determinant issue is squared away.

Referee Report

2 major / 3 minor

Summary. This paper studies the empirical spectral distribution (ESD) of d-regular G_n-circulant matrices for a sequence of finite abelian groups G_n with |G_n| tending to infinity. The main results are a complete characterization of weak convergence in expectation (Theorem 1.2) and in probability (Theorem 1.3) in terms of the limiting distribution rho of the order of a uniform element of G_n, with explicit limit mu = sum_m rho({m}) eta_m^{*d}. Theorem 1.6 gives an asymptotic log-determinant law under a non-singularity assumption. The proofs use Fourier diagonalisation, Hermitisation, moment computations, Smith normal form, and small-ball estimates for sums of roots of unity.

Significance. If Theorems 1.2 and 1.3 are correct, they provide a sharp and elegant counterpart to the sparse i.i.d. problem of Problem 1.1 for circulant (Cayley) matrices, identifying exactly when randomness in the group order produces fluctuation in the ESD. The limits are explicit, the characterization is necessary as well as sufficient (with Remark 1.4 showing that convergence in expectation and in probability genuinely differ), and the determinant asymptotics give a concrete constant c_{m,d}. The proofs are self-contained modulo standard results and contain no fitted parameters. Theorems 1.2 and 1.3 appear to be sound. However, the proof of Theorem 1.6 is incomplete because it relies on a false small-ball estimate, so the full set of claims in the abstract is not established as written.

major comments (2)

[Section 3, Lemma 3.1(6)] Lemma 3.1(6) is false as stated. Take d = 2, m even, z = 0, and r = c/m with c > 0 sufficiently small so that B(0,r) contains no nonzero element of 2R_m. Then the atom at 0 has eta_m^{*2}({0}) = m/m^2 = 1/m, while the claimed bound O((rm + 1/m)^2) is O(m^{-2}) in this regime. The proof's geometric assertion that every point is the midpoint of at most one chord of the unit circle fails at the center, which is the midpoint of m/2 chords when m is even.
[Section 3, Proposition 3.3; Section 6] The false estimate in Lemma 3.1(6) is load-bearing for Theorem 1.6. In the proof of Proposition 3.3, the 'complementary case' z = 0, d >= 2 uses Lemma 3.1(6) to bound the third summand by (4C/ord^2) * ord log(3d), which is what shows that 0 is good under the final hypothesis of the proposition. Since Lemma 3.1(6) is not available, the proof that 0 is good is incomplete, and consequently the derivation of Theorem 1.6 in Section 6 does not go through as written. Lemma 3.2 cannot repair this: its lower bound d^{1-phi(m)} is typically far smaller than the radius m^{-3d}, so it does not make the relevant event empty. The author should state and prove a restricted small-ball estimate for B^*(0,r) under the condition 0 notin dR_m, or give an alternative bound for P(|xSn(gamma)| < ord^{-3d}), and then rerun the argument of Proposition 3.3.

minor comments (3)

[Section 3, Proposition 3.3] The displayed lower bound |xSn(gamma) - z| >= (3d)^{-ord} is not a direct restatement of Lemma 3.2; please include the short derivation using phi(m) <= m.
[Section 5, Lemma 5.1] The notation rds^P (or [d]^P) for tuples indexed by P is not defined in the text; please define it explicitly before Lemma 5.1.
[Throughout] There are several typographical artifacts in the extracted text (e.g., 'f ollowing-up' in the first paragraph and 'matrice s' in the title on the first page); a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the spectral laws are derived from explicit Fourier diagonalization and standard external criteria, with no fitted inputs, no self-citation loops, and no definitional reduction.

full rationale

The derivation is self-contained and non-circular. The central identity in Section 4 expresses E∫f dμ_{C_n} as ∫_{N*} Φ(f) dρ_{G_n}, where Φ(f)(m)=∫f dη_m^{*d}; this identity is a direct consequence of the Fourier diagonalization of G-circulant matrices and the fact that the eigenvalue distribution attached to a character depends only on the character's order. It is not an input that assumes the target result. Theorem 1.2 is then proved by a genuine measure-theoretic separation argument using tent functions and the Möbius inversion lemma, while Theorem 1.3 is proved through the variance computation of Lemma 5.1 and a functional equation for τ, again without assuming the conclusion. The Hermitisation criterion (Proposition 2.3) is quoted from Sah–Sahasrabudhe–Sawhney, but it is a general external input and is not used to smuggle in the paper's specific limit. There are no fitted parameters, no parameter-free claim that is actually a renamed fit, and no load-bearing self-citations: the bibliography contains no prior work by the author, and the cited Lam–Leung and small-ball results are independent. The apparent defect in Lemma 3.1(6), which is invoked at z=0 in Proposition 3.3 and hence in the proof of Theorem 1.6, is a correctness gap rather than a circularity: the claimed O(m^{-2}) bound can fail at the atom of a sum of two roots of unity, but this does not make the derivation circular and does not affect the verdict for Theorems 1.2 and 1.3.

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

The proofs rely on standard background results plus one ad hoc small-ball lemma that is false as stated. There are no fitted parameters or invented physical/mathematical entities. The false lemma is the main unsupported axiom on which Theorem 1.6 depends.

assumptions (6)

standard math Hermitisation criterion (Proposition 2.3) from Sah-Sahasrabudhe-Sawhney [41]
Used to convert convergence of shifted singular value measures into convergence of ESDs; accepted external result cited with attribution.
standard math Smith normal form for integer matrices acting on finite abelian groups
Used in Section 5 to factor the probability of joint linear equations into products of torsion proportions.
domain assumption Contiguity of the with-replacement model to the exactly-d model via event (4)
Used throughout to pass from sampling d elements with replacement to the uniform d-regular matrix; valid because the event has probability tending to 1.
domain assumption Eigenvalue magnitudes are bounded by d
Each eigenvalue is a sum of d roots of unity, giving the deterministic support bound used for upper-tail uniform integrability.
ad hoc to paper Lemma 3.1(6) small-ball estimate for sums of roots of unity
This lemma as stated is false (d = 2, m even, z = 0). It is introduced in this paper and used to prove uniform integrability at z = 0 for Theorem 1.6, so it is an ad hoc assumption whose failure breaks that proof.
standard math Lam-Leung theorem on vanishing sums of roots of unity
Used in Remark 6.2 to characterize exactly when 0 belongs to dR_n, determining the density of n for which the determinant theorem's non-singularity condition holds.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Limiting spectral laws for sparse random circulant matrices." pith.science (2026). https://pith.science/paper/DF53XYVV

@misc{pith2026250413833,
  author       = {Pith},
  title        = {Pith review of: Limiting spectral laws for sparse random circulant matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DF53XYVV}},
  note         = {Machine review of arXiv:2504.13833}
}

abstract

Fix a positive integer $d$ and let $(G_n)_{n\geq1}$ be a sequence of finite abelian groups with orders tending to infinity. For each $n \geq 1$, let $C_n$ be a uniformly random $G_n$-circulant matrix with entries in $\{0,1\}$ and exactly $d$ ones in each row/column. We show that the empirical spectral distribution of $C_n$ converges weakly in expectation to a probability measure $\mu$ on $\mathbb{C}$ if and only if the distribution of the order of a uniform random element of $G_n$ converges weakly to a probability measure $\rho$ on $\mathbb{N}^*$, the one-point compactification of the natural numbers. Furthermore, we show that convergence in expectation can be strengthened to convergence in probability if and only if $\rho$ is a Dirac mass $\delta_m$. In this case, $\mu$ is the $d$-fold convolution of the uniform distribution on the $m$-th roots of unity if $m\in\mathbb{N}$ or the unit circle if $m = \infty$. We also establish that, under further natural assumptions, the determinant of $C_n$ is $\pm\exp((c_{m,d}+o(1))|G_n|)$ with high probability, where $c_{m,d}$ is a constant depending only on $m$ and $d$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 44 canonical work pages

[1]

Random non-Abelian G-circulant matrices

Radosław Adamczak. Random non-Abelian G-circulant matrices. Spectrum of random convolu- tion operators on large ﬁnite groups. Random Matrices Theory Appl. , 10(3):Paper No. 2250002, 40, 2021

work page 2021
[2]

Random Cayley graphs and ex panders

Noga Alon and Yuval Roichman. Random Cayley graphs and ex panders. Random Structures Algorithms, 5(2):271–284, 1994

work page 1994
[3]

The diameter of a ran dom Cayley graph of Zq

Gideon Amir and Ori Gurel-Gurevich. The diameter of a ran dom Cayley graph of Zq. Groups Complex. Cryptol. , 2(1):59–65, 2010

work page 2010
[4]

Patterned sparse ran dom matrices: a moment approach

Debapratim Banerjee and Arup Bose. Patterned sparse ran dom matrices: a moment approach. Random Matrices Theory Appl. , 6(3):1750011, 40, 2017

work page 2017
[5]

Small sums of ﬁve roots of unity

Ben Barber. Small sums of ﬁve roots of unity. Bull. Lond. Math. Soc. , 55(4):1890–1906, 2023

work page 1906
[6]

Salem-Zygmund ineq uality for locally sub-Gaussian random variables, random trigonometric polynomials, and r andom circulant matrices

Gerardo Barrera and Paulo Manrique. Salem-Zygmund ineq uality for locally sub-Gaussian random variables, random trigonometric polynomials, and r andom circulant matrices. Bol. Soc. Mat. Mex. (3) , 28(2):Paper No. 45, 29, 2022

work page 2022
[7]

Around the circula r law

Charles Bordenave and Djalil Chafaï. Around the circula r law. Probab. Surv., 9:1–89, 2012

work page 2012
[8]

Limiting spectral distribution of XX 1 ma- trices

Arup Bose, Sreela Gangopadhyay, and Arnab Sen. Limiting spectral distribution of XX 1 ma- trices. Ann. Inst. Henri Poincaré Probab. Stat. , 46(3):677–707, 2010

work page 2010

Show all 48 references

[9]

Spectra l norm of circulant-type matrices

Arup Bose, Rajat Subhra Hazra, and Koushik Saha. Spectra l norm of circulant-type matrices. J. Theoret. Probab., 24(2):479–516, 2011

2011
[10]

Limiting spectral distribu tion of a special circulant

Arup Bose and Joydip Mitra. Limiting spectral distribu tion of a special circulant. Statist. Probab. Lett., 60(1):111–120, 2002

2002
[11]

Random circulant matrices

Arup Bose and Koushik Saha. Random circulant matrices . CRC Press, Boca Raton, FL, 2019

2019
[12]

Another look at the moment metho d for large dimensional random matrices

Arup Bose and Arnab Sen. Another look at the moment metho d for large dimensional random matrices. Electron. J. Probab., 13:no. 21, 588–628, 2008. 21

2008
[13]

Patter ned random matrices and method of moments

Arup Bose, Rajat Subhra Hazra, and Koushik Saha. Patter ned random matrices and method of moments. In Proceedings of the International Congress of Mathematicians . Volume IV , pages 2203–2231. Hindustan Book Agency, New Delhi, 2010

2010
[14]

Uniform expansion boun ds for Cayley graphs of SL2pFpq

Jean Bourgain and Alex Gamburd. Uniform expansion boun ds for Cayley graphs of SL2pFpq. Ann. of Math. (2) , 167(2):625–642, 2008

2008
[15]

Spect ral measure of large random Hankel, Markov and Toeplitz matrices

Włodzimierz Bryc, Amir Dembo, and Tiefeng Jiang. Spect ral measure of large random Hankel, Markov and Toeplitz matrices. Ann. Probab., 34(1):1–38, 2006

2006
[16]

On the independence number of sparser random Cayley graphs

Marcelo Campos, Gabriel Dahia, and João Pedro Marciano . On the independence number of sparser random Cayley graphs. J. Lond. Math. Soc. (2) , 110(6):Paper No. e70041, 54, 2024

2024
[17]

Around the circular law: an update

Djalil Chafaï. Around the circular law: an update. https://djalil.chafai.net/blog/2018/11/04/around-th e-circular-law-an-update/

2018
[18]

The threshol ds for diameter 2 in random Cayley graphs

Demetres Christoﬁdes and Klas Markström. The threshol ds for diameter 2 in random Cayley graphs. Random Structures Algorithms , 45(2):218–235, 2014

2014
[19]

On the clique number of random Cayley graphs and related topics

David Conlon, Jacob Fox, Huy Tuan Pham, and Liana Yeprem yan. On the clique number of random Cayley graphs and related topics. arXiv:2412.21194

arXiv
[20]

Group representations in probability and statistics , volume 11 of Institute of Mathematical Statistics Lecture Notes—Monograph Series

Persi Diaconis. Group representations in probability and statistics , volume 11 of Institute of Mathematical Statistics Lecture Notes—Monograph Series . Institute of Mathematical Statistics, Hayward, CA, 1988

1988
[21]

Babai’s conjecture f or high-rank classical groups with random generators

Sean Eberhard and Urban Jezernik. Babai’s conjecture f or high-rank classical groups with random generators. Invent. Math. , 227(1):149–210, 2022

2022
[22]

Random circulant d eterminants

Sean Eberhard and Padraig Ó Catháin. Random circulant d eterminants. In preparation
[23]

Proof methods in r andom matrix theory

Michael Fleermann and Werner Kirsch. Proof methods in r andom matrix theory. Probab. Surv., 20:291–381, 2023

2023
[24]

V. L. Girko. The circular law. Teor. Veroyatnost. i Primenen. , 29(4):669–679, 1984

1984
[25]

Classical Fourier analysis , volume 249 of Graduate Texts in Mathematics

Loukas Grafakos. Classical Fourier analysis , volume 249 of Graduate Texts in Mathematics . Springer, New York, third edition, 2014

2014
[26]

Some applications of harmonic analysis to ar ithmetic combinatorics

Ben Green. Some applications of harmonic analysis to ar ithmetic combinatorics. 2001. Smith- Knight Prize essay, Cambridge University

2001
[27]

Counting sets with small sumset, and the cliq ue number of random Cayley graphs

Ben Green. Counting sets with small sumset, and the cliq ue number of random Cayley graphs. Combinatorica, 25(3):307–326, 2005

2005
[28]

On the chromatic number of random Cayley grap hs

Ben Green. On the chromatic number of random Cayley grap hs. Combin. Probab. Comput. , 26(2):248–266, 2017

2017
[29]

100 open problems

Ben Green. 100 open problems. 2024. manuscript

2024
[30]

Helfgott, Ákos Seress, and Andrzej Zuk

Harald A. Helfgott, Ákos Seress, and Andrzej Zuk. Rando m generators of the symmetric group: diameter, mixing time and spectral gap. J. Algebra, 421:349–368, 2015

2015
[31]

Basic algebra

Nathan Jacobson. Basic algebra. I . W. H. Freeman and Company, New York, second edition, 1985

1985
[32]

T. Y. Lam and K. H. Leung. On vanishing sums of roots of uni ty. J. Algebra , 224(1):91–109, 2000

2000
[33]

Norms of randomiz ed circulant matrices

Rafał Latała and Witold Świątkowski. Norms of randomiz ed circulant matrices. Electron. J. Probab., 27:Paper No. 80, 23, 2022. 22

2022
[34]

Diameters of r andom circulant graphs

Jens Marklof and Andreas Strömbergsson. Diameters of r andom circulant graphs. Combina- torica, 33(4):429–466, 2013

2013
[35]

V. A. Marčenko and L. A. Pastur. Distribution of eigenva lues in certain sets of random matrices. Mat. Sb. (N.S.) , 72(114):507–536, 1967

1967
[36]

Mark W. Meckes. Some results on random circulant matric es. In High dimensional probability V: the Luminy volume , volume 5 of Inst. Math. Stat. (IMS) Collect. , pages 213–223. Inst. Math. Statist., Beachwood, OH, 2009

2009
[37]

Mark W. Meckes. The spectra of random abelian G-circulant matrices. ALEA Lat. Am. J. Probab. Math. Stat. , 9(2):435–450, 2012

2012
[38]

Ein Beitrag zur analytischen Zahlenthe orie

Franz Mertens. Ein Beitrag zur analytischen Zahlenthe orie. J. Reine Angew. Math. , 78:46–62, 1874
[39]

One-point concentration of the clique a nd chromatic numbers of the random Cayley graph on Fn 2

Rudi Mrazović. One-point concentration of the clique a nd chromatic numbers of the random Cayley graph on Fn 2 . SIAM J. Discrete Math. , 31(1):143–154, 2017

2017
[40]

Unsolved Problems: How Small Can a Sum o f Roots of Unity Be? Amer

Gerald Myerson. Unsolved Problems: How Small Can a Sum o f Roots of Unity Be? Amer. Math. Monthly , 93(6):457–459, 1986

1986
[41]

A. Sah, J. Sahasrabudhe, and M. Sawhney. The limiting sp ectral law for sparse iid matrices. arXiv:2310.17635

arXiv
[42]

How small can a sum of a few roots of unity be? https://mathoverflow.net/questions/46068/how-small- can-a-sum-of-a-few-roots-of-unity-be

Terence Tao. How small can a sum of a few roots of unity be? https://mathoverflow.net/questions/46068/how-small- can-a-sum-of-a-few-roots-of-unity-be
[43]

Topics in random matrix theory , volume 132 of Graduate Studies in Mathematics

Terence Tao. Topics in random matrix theory , volume 132 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2012

2012
[44]

Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics

Terence Tao and Van Vu. Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2006

2006
[45]

Random matrices: the circular la w

Terence Tao and Van Vu. Random matrices: the circular la w. Commun. Contemp. Math. , 10(2):261–307, 2008

2008
[46]

Random matrices: universality o f ESDs and the circular law

Terence Tao and Van Vu. Random matrices: universality o f ESDs and the circular law. Ann. Probab., 38(5):2023–2065, 2010. With an appendix by Manjunath Kris hnapur

2023
[47]

Tikhomirov

K. Tikhomirov. Quantitative invertibility of non-Her mitian random matrices. arXiv:2206.00601

arXiv
[48]

Eugene P. Wigner. On the distribution of the roots of cer tain symmetric matrices. Ann. of Math. (2) , 67:325–327, 1958. 23

1958

Pith tools

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

[1] [1]

Random non-Abelian G-circulant matrices

Radosław Adamczak. Random non-Abelian G-circulant matrices. Spectrum of random convolu- tion operators on large ﬁnite groups. Random Matrices Theory Appl. , 10(3):Paper No. 2250002, 40, 2021

work page 2021

[2] [2]

Random Cayley graphs and ex panders

Noga Alon and Yuval Roichman. Random Cayley graphs and ex panders. Random Structures Algorithms, 5(2):271–284, 1994

work page 1994

[3] [3]

The diameter of a ran dom Cayley graph of Zq

Gideon Amir and Ori Gurel-Gurevich. The diameter of a ran dom Cayley graph of Zq. Groups Complex. Cryptol. , 2(1):59–65, 2010

work page 2010

[4] [4]

Patterned sparse ran dom matrices: a moment approach

Debapratim Banerjee and Arup Bose. Patterned sparse ran dom matrices: a moment approach. Random Matrices Theory Appl. , 6(3):1750011, 40, 2017

work page 2017

[5] [5]

Small sums of ﬁve roots of unity

Ben Barber. Small sums of ﬁve roots of unity. Bull. Lond. Math. Soc. , 55(4):1890–1906, 2023

work page 1906

[6] [6]

Salem-Zygmund ineq uality for locally sub-Gaussian random variables, random trigonometric polynomials, and r andom circulant matrices

Gerardo Barrera and Paulo Manrique. Salem-Zygmund ineq uality for locally sub-Gaussian random variables, random trigonometric polynomials, and r andom circulant matrices. Bol. Soc. Mat. Mex. (3) , 28(2):Paper No. 45, 29, 2022

work page 2022

[7] [7]

Around the circula r law

Charles Bordenave and Djalil Chafaï. Around the circula r law. Probab. Surv., 9:1–89, 2012

work page 2012

[8] [8]

Limiting spectral distribution of XX 1 ma- trices

Arup Bose, Sreela Gangopadhyay, and Arnab Sen. Limiting spectral distribution of XX 1 ma- trices. Ann. Inst. Henri Poincaré Probab. Stat. , 46(3):677–707, 2010

work page 2010