{"id":"37cadaac-b040-4a96-98cf-835358665709","arxiv_id":"2504.13833","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For d-regular random G_n-circulant matrices, the ESD converges in expectation iff the order distribution of a uniform group element converges, and in probability iff that order law is a point mass.","lead":"A new result identifies exactly when the eigenvalue distributions of random sparse circulant matrices on finite abelian groups converge, and ties the limits to the group order statistics. It also gives the exponential growth rate of the determinant when no exact zero eigenvalues occur.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1(6) is false as stated and is invoked at z=0 in the proof of Theorem 1.6; the determinant theorem's proof is incomplete as written.","rationale":"The reader correctly identifies a false lemma: Lemma 3.1(6) is contradicted by the d=2, m even, z=0 atom of mass 1/m. The concern is real and appears in the proof of Proposition 3.3 at exactly the point needed for the determinant theorem. I partially disagree with the reader's framing that the lemma is immediately load-bearing for Theorem 1.6 in its full force, because the theorem assumes 0∉dR_exp(G_n), which excludes the zero-sum atom; nevertheless, the paper invokes the false lemma verbatim and provides no restricted substitute, so the proof as written is incomplete. Theorems 1.2 and 1.3, the main spectral convergence results, do not depend on the false case: for λ-a.e. z, the preimage argument in Lemma 3.1(6) is valid away from z=0, and z=0 is a null set. Thus the central spectral claims are likely correct, but the determinant theorem needs a repaired small-ball estimate. This supports a CONDITIONAL verdict rather than rejection.","tokens_in":22151,"tokens_out":14931,"duration_ms":155854,"concrete_test":"Re-derive the z=0 estimate in Proposition 3.3 without invoking Lemma 3.1(6), using only the hypothesis 0∉dR_m. In particular, determine whether η_m^{*d}(B(0,m^{-3d})) ≤ C m^{-2} holds; as a numerical probe, enumerate all ordered d-tuples of m-th roots for d=3, m=100 and count sums with |sum|≤100^{-9}. If the count exceeds C/100^2, the proof of Theorem 1.6 fails in the theorem's regime; if the count is ≤C/100^2, the lemma should be replaced by a restricted version covering z=0 under 0∉dR_m, after which the rest of Proposition 3.3 and Section 6 go through.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 3.1(6) claims that η_m^{*d}(B(z,r)) is O((rm+1/m)^2) for every z. For d=2, m even, and z=0, the ball B(0,r) contains the atom at 0 with mass 1/m whenever r is smaller than the smallest nonzero sum of two m-th roots, so the claimed O(m^{-2}) bound fails. The geometric justification, that every point is the midpoint of at most one chord of the unit circle, is false for the center of the circle. This lemma is used in Proposition 3.3 in the 'complementary case' z=0, d≥2, to bound P(|S|≤m^{-3d}) by 4C/m^2, which feeds the third-summand estimate needed to show that 0 is good; Theorem 1.6 then relies on this through Section 6. Since the lemma as stated is false, the proof of Proposition 3.3 and hence of Theorem 1.6 does not go through as written. Note that Theorem 1.6's hypothesis 0∉dR_exp(G_n) rules out the exact-zero atom, so the result may still be true; but the paper neither states nor proves the restricted small-ball estimate needed at z=0, and Lemma 3.2's exponentially small lower bound does not by itself supply the polynomial estimate used in the proof. Theorems 1.2 and 1.3 appear unaffected, since the false case is a single point in the λ-a.e. statement and the z=0 case is only needed for the determinant theorem.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22397,"tokens_out":10875,"duration_ms":98399,"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":[{"comment":"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":"Section 3, Lemma 3.1(6)"},{"comment":"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.","section":"Section 3, Proposition 3.3; Section 6"}],"minor_comments":[{"comment":"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":"Section 3, Proposition 3.3"},{"comment":"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.","section":"Section 5, Lemma 5.1"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The reader's report correctly identifies the flaw in Lemma 3.1(6), and I agree with its assessment. The main spectral theorems 1.2 and 1.3 appear to be sound and novel, while the determinant theorem 1.6 is unproven as written. Since the flaw is localized to one main theorem and is plausibly repairable with a restricted small-ball estimate under the non-singularity hypothesis, I recommend major revision rather than rejection. The editor may wish to ask the author to state explicitly the restricted estimate used at z = 0, since it is not currently present in the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nNow, 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.\n\nTheorems 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.\n\nSo: 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.","headline":"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.","tokens_in":23000,"tokens_out":4244,"would_cite":true,"duration_ms":38183,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","15B52","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sparse random $G$-circulant spectra converge exactly when element-order distributions do, with explicit roots-of-unity limits.","keywords":["random circulant matrices","sparse random matrices","empirical spectral distribution","roots of unity","finite abelian groups","element order distribution","determinant","Hermitisation method"],"falsifier":"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.","tokens_in":21826,"feed_emoji":"🎲","tokens_out":14724,"duration_ms":115917,"temperature":0.7,"pith_summary":"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.","feed_headline":"Sparse random circulant spectra converge exactly when element orders do","feed_subtitle":"Empirical spectra limit to convolutions of roots-of-unity laws; point-mass order laws give convergence in probability.","key_machinery":"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 $m$.","core_discovery":"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 $c_{m,d}=\\int\\log|z|\\,d\\eta_m^{\\ast d}(z)$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Proposition 13.1, the Hermitisation criterion adopted here as Proposition 2.3, and the sparse i.i.d. circular-law problem that this paper answers in the circulant setting.","marker":"[41]"},{"why":"Provides the Hermitisation and uniform-integrability framework (Lemma 4.3) used to pass from shifted singular values to eigenvalue convergence.","marker":"[7]"},{"why":"Initiates the spectral theory of random circulant matrices and proves the non-sparse circular law for i.i.d. first rows.","marker":"[36]"},{"why":"Extends the circulant spectral result to arbitrary finite abelian $G$-circulant matrices, the setting generalised here.","marker":"[37]"},{"why":"Introduces the Hermitisation/log-potential method that underlies Proposition 2.3 and the determinant argument.","marker":"[24]"},{"why":"Supplies the separation estimate for sums of roots of unity used in the small-ball bound of Lemma 3.1.","marker":"[5]"},{"why":"Gives the lower bound for non-zero sums of $m$-th roots of unity used to prove uniform integrability at $z=0$.","marker":"[40]"},{"why":"Characterises when $d$ roots of unity can sum to zero, used in the determinant section to discuss the condition $0\\notin dR_{\\exp(G_n)}$.","marker":"[32]"},{"why":"Supplies the method-of-moments criterion (Theorem 3.9) behind Proposition A.3 for shifted singular value measures.","marker":"[23]"}],"fun_headline_variants":["Spectra of sparse circulants: order laws decide","Circulant spectral limit: order distribution is key","When element orders converge, so do spectra","Point-mass order gives probability convergence","Determinant exponent from roots-of-unity convolution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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 $O(1/m^2)$.","fun_headline_variants_meta":{"raw":{"variants":["Spectra of sparse circulants: order laws decide","Circulant spectral limit: order distribution is key","When element orders converge, so do spectra","Point-mass order gives probability convergence","Determinant exponent from roots-of-unity convolution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00062,"raw_usage":{"total_tokens":2962,"prompt_tokens":1119,"completion_tokens":1843,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":735,"completion_tokens_details":{"reasoning_tokens":1773}},"tokens_in":735,"tokens_out":1843,"duration_ms":12250,"temperature":1.0,"reasoning_tokens":1773,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T12:00:57.436790+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Around the circula r law","cited_arxiv_id":null,"evidence_quote":"Provides the Hermitisation and uniform-integrability framework (Lemma 4.3) used to pass from shifted singular values to eigenvalue convergence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Initiates the spectral theory of random circulant matrices and proves the non-sparse circular law for i.i.d. first rows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Extends the circulant spectral result to arbitrary finite abelian $G$-circulant matrices, the setting generalised here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Hermitisation/log-potential method that underlies Proposition 2.3 and the determinant argument."},{"cited_title":"Small sums of five roots of unity","cited_arxiv_id":null,"evidence_quote":"Supplies the separation estimate for sums of roots of unity used in the small-ball bound of Lemma 3.1."},{"cited_title":"Unsolved Problems: How Small Can a Sum o f Roots of Unity Be? Amer","cited_arxiv_id":null,"evidence_quote":"Gives the lower bound for non-zero sums of $m$-th roots of unity used to prove uniform integrability at $z=0$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Characterises when $d$ roots of unity can sum to zero, used in the determinant section to discuss the condition $0\\notin dR_{\\exp(G_n)}$."},{"cited_title":"Proof methods in r andom matrix theory","cited_arxiv_id":null,"evidence_quote":"Supplies the method-of-moments criterion (Theorem 3.9) behind Proposition A.3 for shifted singular value measures."}],"review_version":1}