{"id":"506f7c40-ea1b-41a9-a68d-6e370170df41","arxiv_id":"2501.08039","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the complex Ginibre ensemble, the W1 and uniform distances from the spectral radius to the Gumbel law decay as 2 log log n / log n and 2 log log n / (e log n).","lead":"This paper calculates the exact speed at which the spectral radius of a complex Ginibre random matrix converges to the Gumbel distribution. It provides sharp Wasserstein and Kolmogorov error rates for a central model in non-Hermitian random matrix theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3.4's proof of the key estimate (3.2) is invalid: two different substitutions are combined into one integral and the positive tail \\(Y_{(n)}>n\\) is discarded.","rationale":"The reader's weakest assumption was the uniformity of the Gaussian tail expansion in Lemma 2.2. I do not find a concrete failure there; the Petrov-type bound has the stated \\(u^{-2}\\) relative error in the range \\(u=\\tilde O(n^{1/10})\\). The more dangerous spot is Section 3.4, where the proof of (3.2) – a step needed for Theorem 1 – appears to combine two different changes of variables. The claim that (4.2) follows from \\(W_1\\ll\\gamma_n^{-1}\\) in Theorem 2 is also unsupported, but that is secondary because (3.2) itself is not proved. The exact coupling formula shows the true \\(W_1(L(W_n),F_n)\\) is \\(O(\\gamma_n^{3/2}/\\sqrt n)\\), which is still \\(o(\\gamma_n^{-1})\\), so the theorem is likely true but the written derivation is wrong. This does not change the reader's CONDITIONAL verdict; it sharpens the reason. The paper needs a corrected proof of (3.2) before either theorem can be considered established.","tokens_in":14959,"tokens_out":53798,"duration_ms":505478,"concrete_test":"Recompute (3.2) directly from the coupling representation \\(W_1(L(W_n),F_n)=\\sqrt{\\gamma_n/n}\\,\\mathbb E[(\\sqrt{Y_{(n)}}-\\sqrt n)^2]\\), or equivalently evaluate the missing tail \\(\\sqrt{\\gamma_n}\\int_n^\\infty(1/\\sqrt n-1/\\sqrt t)\\mathbb P(Y_{(n)}>t)\\,dt\\). If the missing term is positive and of order \\(\\gamma_n^{3/2}/\\sqrt n\\), then the printed bound \\(\\sqrt{\\gamma_n}n\\mathbb P(Y_{(n)}\\le n)\\) cannot be an upper bound for \\(W_1\\), and Section 3.4 must be rewritten. The test is analytic: derive the exact two-term identity by integration by parts and compare the omitted tail with \\(\\gamma_n^{-1}\\); the hoped-for bound holds iff the tail is \\(o(\\gamma_n^{-1})\\).","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.4, to prove the transfer estimate \\(W_1(L(W_n),F_n)\\ll\\gamma_n^{-1}\\), the paper substitutes \\(t=x/\\sqrt{4\\gamma_n}+\\sqrt n+\\sqrt{\\gamma_n}/4\\) in the first CDF integral and \\(t=a_n+b_nx\\) in the second, then combines them into \\(\\sqrt{\\gamma_n}\\int_0^\\infty(1/\\sqrt t-1/\\sqrt n)F(t)\\,dt\\). These two \\(t\\)'s are different coordinates, so the combination is not an equality. Since \\(X_n\\) and \\(W_n\\) are increasing functions of \\(Y=R_n^2\\), the exact Wasserstein distance is \\(\\sqrt{\\gamma_n/n}\\,\\mathbb E[(\\sqrt{Y_{(n)}}-\\sqrt n)^2]\\). Integration by parts gives the paper's \\(\\int_0^n\\) term plus a positive tail \\(\\sqrt{\\gamma_n}\\int_n^\\infty(1/\\sqrt n-1/\\sqrt t)\\mathbb P(Y_{(n)}>t)\\,dt\\), which is omitted. The tail is of order \\(\\gamma_n^{3/2}/\\sqrt n\\), so the paper's bound \\(\\sqrt{\\gamma_n}n\\mathbb P(Y_{(n)}\\le n)\\) is not an upper bound for \\(W_1\\); in fact \\(\\mathbb P(Y_{(n)}>n)\\to1\\). Thus (3.2) is not established by the written argument. Because \\(\\gamma_n^{3/2}/\\sqrt n=o(\\gamma_n^{-1})\\), the estimate is probably repairable, but Theorem 1 currently rests on this incorrect identity.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the spectral radius R_n of an n by n complex Ginibre matrix. With X_n = sqrt(4 gamma_n)(R_n - sqrt(n) - (1/2) sqrt(gamma_n)) and gamma_n = log n - 2 log(sqrt(2 pi) log n), it claims the sharp Wasserstein rate lim_{n->infty} (log n / log log n) W_1(F_n, Lambda) = 2 and the Kolmogorov rate lim_{n->infty} (log n / log log n) sup_x |F_n(x) - exp(-exp(-x))| = 2/e, where F_n is the distribution function of X_n and Lambda is the standard Gumbel distribution. The proof uses Kostlan's representation of R_n^2 as the maximum of independent Gamma variables, a uniform Gaussian-tail expansion for Gamma tails, and detailed integral and sum estimates to extract the first-order constants. The paper also introduces an intermediate variable W_n and reduces the problem to estimates on W_1(L(W_n), Lambda) and W_1(L(W_n), F_n).","tokens_in":15238,"tokens_out":21753,"duration_ms":179881,"significance":"If correct, the two limits are sharp convergence-rate results for the extremal statistics of the complex Ginibre ensemble, going beyond Rider's distributional limit. The method is transparent: it reduces the problem to classical Gamma tail bounds and explicitly derives the constants 2 and 2/e from asymptotic expansions rather than fitting them. The paper states explicit error terms and does not rely on unproved heuristics. However, the current written proof contains several internal definitional inconsistencies and at least two invalid logical steps, so the claimed theorems are not yet established by the manuscript as written.","major_comments":[{"comment":"The auxiliary variable W_n is defined by W_n = (sqrt(gamma_n)/sqrt(n))(R_n^2 - a_n) with a_n := n + sqrt(n) gamma_n. Then P(W_n >= x) = P(R_n^2 >= a_n + (sqrt(n)/sqrt(gamma_n))x), so the correct coefficient b_n is sqrt(n)/sqrt(gamma_n), not sqrt(n)/gamma_n as stated. Moreover, the quantity u_n(k,x) = k/sqrt(n) + sqrt(gamma_n) + x/sqrt(gamma_n) used throughout Lemma 2.2 and Section 3 corresponds to a_n = n + sqrt(n) sqrt(gamma_n) and b_n = sqrt(n)/sqrt(gamma_n), not to the stated a_n and b_n. This is not a purely notational issue: the expansions of P(Y_{n-k} > a_n + b_n x) in Lemma 2.2 are for a different centering than the W_n actually defined, so the proof of the main estimates does not apply to the quantity the paper calls W_n. The definitions should be corrected and carried through consistently.","section":"Section 2, definitions of W_n, a_n, b_n and Lemma 2.2"},{"comment":"The proof of the transfer estimate W_1(L(W_n), F_n) << gamma_n^{-1} is invalid. After substituting t = x/sqrt(4 gamma_n) + sqrt(n) + sqrt(gamma_n)/4 in the first integral and t = a_n + b_n x in the second, the two integrals are combined as if the integration variables were the same, but they are different coordinates, so the displayed equality is not justified. A correct expression for the L1-Wasserstein distance between the two increasing functions of Y_(n) is sqrt(gamma_n/n) E[(sqrt(Y_(n)) - sqrt(n))^2]; integration by parts yields the paper's integral from 0 to n plus a positive tail term sqrt(gamma_n) integral_n^infty (1/sqrt(n) - 1/sqrt(t)) P(Y_(n) > t) dt, which is omitted. Since P(Y_(n) > n) -> 1, this tail is not negligible in general, and the subsequent bound sqrt(gamma_n) n P(Y_(n) <= n) does not establish (3.2). The estimate may be repairable by a direct analysis of the omitted tail, but as written Theorem 1 lacks a valid transfer estimate.","section":"Section 3.4, Eq. (3.2)"},{"comment":"The assertion that sup_x |F_n(x) - bar(F_n)(x)| << gamma_n^{-1} log log n follows from W_1(L(W_n), F_n) << gamma_n^{-1} is false: a small L1-Wasserstein distance between two probability distributions does not control the uniform sup-norm difference of their distribution functions without additional assumptions. No extra regularity or direct argument is supplied. Since (4.1) and (4.2) together imply Theorem 2, this is a load-bearing gap in the proof; a separate uniform estimate for |F_n - bar(F_n)| is needed.","section":"Section 4, Eq. (4.2)"},{"comment":"The centering of X_n is inconsistent across the manuscript. The Abstract and Sections 2 and 3 define X_n = sqrt(4 gamma_n)(R_n - sqrt(n) - (1/2) sqrt(gamma_n)), whereas the Introduction and the computation in Section 3.4 use the shift sqrt(gamma_n)/4. These two variables are not asymptotically equivalent on the Gumbel scale: with the 1/4 shift, sqrt(4 gamma_n)(R_n - sqrt(n) - sqrt(gamma_n)/4) diverges in probability to +infty along the Rider scaling. Since Section 3.4's transfer estimate is computed with the 1/4 shift, the proof of Theorems 1 and 2 is not for the X_n stated in those theorems.","section":"Introduction and Section 3.4"}],"minor_comments":[{"comment":"The definition 't_n = O(z_n) if lim t_n/z_n = c != 0' is nonstandard and conflicts with the usual meaning of O; elsewhere in the paper O is used with its standard interpretation, which is likely to confuse readers.","section":"Section 1, notation"},{"comment":"The precise asymptotic in the second part of Lemma 2.2 should state explicitly that the error is uniform in k and x in the indicated ranges, since this uniformity is used implicitly in the summations in Section 3.2 and in the proof of (3.9).","section":"Lemma 2.2"},{"comment":"The symbol tilde(O) is defined as 'limit exists' in the introduction but is later used to indicate a bounded relative error with possible logarithmic factors; the paper should adopt a single, clearly defined convention for tilde(O).","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The results are likely correct and the topic is suitable for a probability journal, but the current manuscript has several load-bearing inconsistencies and invalid proof steps. The authors should be asked to correct the definitions of W_n, a_n, b_n and X_n, provide a valid proof of (3.2), and prove (4.2) directly instead of deriving it from a Wasserstein bound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Name],\n\nBottom line: the paper claims the exact W1 and Kolmogorov rates for the complex Ginibre spectral radius to a Gumbel, with constants 2 and 2/e. That is a natural question and the result is plausible. But as written the proof has several load-bearing errors, and I wouldn't trust the current version without major revision.\n\nWhat's new and good: Rider's paper only gives weak convergence, and Cipolloni et al. give a rate for the largest real part, not the spectral radius. So the exact constants are genuinely new. The strategy—using Kostlan's Gamma representation and then doing a careful tail analysis via non-uniform Berry–Esseen—is the right route. The paper does not appear to fit constants; the asymptotic machinery is standard.\n\nNow the problems. The most serious is an internal scaling inconsistency. They define an = n + √n γn, bn = √n/γn, yet Lemma 2.2 and all subsequent asymptotics use un(k,x) = k/√n + √γn + x/√γn. The ratio (an+bnx−(n−k))/√(n−k) is actually k/√n + γn + x/γn, not un. Lemma 2.2 as stated is therefore false. The fix is likely to replace γn with √γn in an and bn, but that change propagates through everything. This is not a typo; the definitions contradict the expansions used.\n\nSecond, Section 3.4's proof of (3.2) drops the absolute value in the W1 integral. The paper claims (x/√(4γn)+√n+√γn/4)^2 ≥ an+bnx for all x, which is false—already at x=0 (and with the correct shift, for many negative x). So the equality W1 = ∫(F_U−F_V) is not established. The tail issue mentioned in the stress-test is secondary; the main problem is the missing absolute value. The bound itself may be repairable, but the present argument doesn't prove it.\n\nThird, Theorem 2's step from W1(L(Wn),Fn) << γn^{-1} to the sup-norm bound (4.2) is asserted. A small L1 distance between distribution functions does not imply a small sup-norm distance. That needs a direct proof.\n\nThe paper also has a shift inconsistency: the abstract uses 1/2√γn, while Section 3.4 uses √γn/4. That's minor compared to the above.\n\nWho is this for: specialists in non-Hermitian random matrices and extreme value statistics. The question is worth answering, and the constants are likely correct. But the current version is not coherent enough to be published as is.\n\nMy recommendation: send it to peer review, but with the expectation of heavy revision. The core idea is sound and the result is plausible; the authors need to fix the scaling definitions, give a correct proof of (3.2), and handle the sup-norm step directly.","headline":"Plausible and likely correct rates, but the written proof has a scaling inconsistency in the core definitions and an invalid absolute-value drop in Section 3.4.","tokens_in":15818,"tokens_out":22733,"would_cite":false,"duration_ms":181573,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60B20","60G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the spectral radius of the complex Ginibre ensemble reaches the Gumbel distribution at sharp logarithmic rates: the Wasserstein distance satisfies $\\lim_{n\\to\\infty}\\frac{\\log n}{\\log\\log n}W_1=2$, and the…","keywords":["Ginibre ensemble","spectral radius","Gumbel distribution","Wasserstein distance","Berry-Esseen bound","convergence rate","extreme eigenvalues"],"falsifier":"Compute, for increasing $n$ and for many choices of $k$ and $x$, the exact tail probability $P(Y_{n-k}>a_n+b_nx)$ for the Gamma variables in Lemma 2.1 and compare it with $e^{-u_n^2/2}/(\\sqrt{2\\pi}u_n)$ over the range $1\\ll u_n\\ll n^{1/10}$; a relative error larger than $O(u_n^{-2})$ at any point would falsify the uniformity on which the rates rest. Alternatively, estimate $W_1$ and $\\sup_x|F_n(x)-e^{-e^{-x}}|$ by simulation for $n$ up to $10^6$; the quantities $(\\log n/\\log\\log n)W_1$ and $(\\log n/\\log\\log n)\\sup_x|F_n-\\Lambda|$ should approach $2$ and $2/e$, and any clear drift away from these constants would disprove the theorem.","tokens_in":14708,"feed_emoji":"📈","tokens_out":11227,"duration_ms":99617,"temperature":0.7,"pith_summary":"This paper proves that the largest eigenvalue modulus of an $n\\times n$ complex Ginibre random matrix approaches its Gumbel limiting distribution at a sharp, explicitly known speed. The Wasserstein distance between the distribution of the scaled spectral radius and the Gumbel law satisfies $\\lim_{n\\to\\infty}\\frac{\\log n}{\\log\\log n}W_1(F_n,\\Lambda)=2$, and the Kolmogorov (uniform) distance satisfies $\\lim_{n\\to\\infty}\\frac{\\log n}{\\log\\log n}\\sup_x|F_n(x)-e^{-e^{-x}}|=\\frac2e$. The paper supplies exact rate constants for this edge statistic, turning the earlier weak-convergence theorem into a quantitative statement. The rates are slow, of order $\\frac{\\log\\log n}{\\log n}$, so the Gumbel approximation becomes accurate only after $n$ is exponentially large in the desired precision.","feed_headline":"Ginibre spectral radius hits Gumbel at sharp rate 2","feed_subtitle":"Wasserstein and Kolmogorov distances both scale as log log n / log n, with constants 2 and 2/e.","key_machinery":"The argument rests on Lemma 2.1, which identifies $R_n^2$ in distribution with the maximum of $n$ independent random variables $Y_j$ having density proportional to $y^{j-1}e^{-y}$ on $y>0$, i.e. Gamma variables. This reduces the two-dimensional eigenvalue problem to a one-dimensional extreme-value problem. The proof then uses Lemma 2.2, a uniform Gaussian tail expansion for $P(Y_{n-k}>a_n+b_nx)$ in terms of $u_n(k,x)=\\frac{k}{\\sqrt n}+\\sqrt{\\gamma_n}+\\frac{x}{\\sqrt{\\gamma_n}}$, with relative error $O(u_n^{-2})$, valid for $1\\ll u_n\\ll n^{1/10}$. Lemma 2.3 converts the resulting sums and integrals over $k$ and $x$ into leading-order exponentials; the cutoffs $\\ell_1(n)=\\frac12\\log\\log n$ and $\\ell_2(n)=\\log(\\sqrt{2\\pi}\\,\\log n)$ isolate the interval that contributes the constants $2$ and $2/e$.","core_discovery":"Let $G_n$ be an $n\\times n$ matrix with i.i.d. standard complex Gaussian entries, and let $R_n=\\max_{1\\le k\\le n}|\\lambda_k|$ be its spectral radius. With $\\gamma_n=\\log n-2\\log(\\sqrt{2\\pi}\\,\\log n)$ and $X_n=\\sqrt{4\\gamma_n}(R_n-\\sqrt n-\\tfrac12\\sqrt{\\gamma_n})$, the paper proves two sharp limits. If $F_n$ is the distribution function of $X_n$ and $\\Lambda(x)=e^{-e^{-x}}$ is the Gumbel distribution, then $\\lim_{n\\to\\infty}\\frac{\\log n}{\\log\\log n}W_1(F_n,\\Lambda)=2$, and $\\lim_{n\\to\\infty}\\frac{\\log n}{\\log\\log n}\\sup_{x\\in\\mathbb R}|F_n(x)-e^{-e^{-x}}|=\\frac2e$. In words, both natural distances between the edge statistic and its Gumbel limit are of order $\\frac{\\log\\log n}{\\log n}$, with the Wasserstein constant exactly $2$ and the Kolmogorov constant exactly $2/e$.","pith_inferences":["Beyond the paper, the same constants should transfer to any ensemble whose squared spectral radius reduces to independent Gamma-type variables and satisfies the same uniform tail bound; the paper only claims the complex Ginibre case.","A practical check is to simulate the Gamma maxima of Lemma 2.1 for $n=10^3,\\dots,10^6$; the ratios $(\\log n/\\log\\log n)W_1$ and $(\\log n/\\log\\log n)\\sup|F_n-\\Lambda|$ should creep toward $2$ and $2/e$, but so slowly that small-$n$ deviations are expected.","The method suggests an extension to the joint distribution of the largest few moduli, with constants depending on the order statistic; this is a natural next step, not contained in the paper.","If a future universality result for general iid matrices supplied the same uniform tail estimate, the proof strategy would transfer directly; the constants obtained here would then serve as the benchmark."],"forward_implications":["For large $n$, $W_1(F_n,\\Lambda)=(2+o(1))\\frac{\\log\\log n}{\\log n}$ and $\\sup_x|F_n(x)-e^{-e^{-x}}|=(\\frac2e+o(1))\\frac{\\log\\log n}{\\log n}$.","The worst-case approximation error is localized to $x\\in(-\\frac12\\log\\log n,\\,\\log(\\sqrt{2\\pi}\\,\\log n))$; outside this window both the tail and the Gumbel term are exponentially negligible.","Since $W_1(L(W_n),F_n)\\ll \\gamma_n^{-1}$, the simpler centered variable $W_n$ can be substituted for $X_n$ in rate computations without changing the leading constant.","The rate is a property of the maximum of independent Gamma variables, so after the reduction of Lemma 2.1 the proof bypasses the correlations among eigenvalues entirely."],"supporting_citations":[{"why":"Establishes the weak convergence of the scaled spectral radius to the Gumbel distribution, the limit refined here.","marker":"[32]"},{"why":"Supplies the reduction of the squared spectral radius to a maximum of independent Gamma variables, stated as Lemma 2.1.","marker":"[26]"},{"why":"Supplies the classical large-deviation expansion for tails of sums of independent exponentials used in Lemma 2.2.","marker":"[31]"},{"why":"Supplies the non-uniform Berry-Esseen bound used for the matching upper tail in Lemma 2.2.","marker":"[14]"},{"why":"Derives the joint eigenvalue density of the complex Ginibre ensemble on which the whole reduction is based.","marker":"[22]"}],"fun_headline_variants":["Ginibre spectral radius to Gumbel: exact rate 2","Exact Wasserstein rate 2, Kolmogorov 2/e for edge","Ginibre edge: Wasserstein rate 2, Kolmogorov 2/e","Exact rate 2 and 2/e for Ginibre edge to Gumbel"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the uniform Gaussian tail approximation of Lemma 2.2: the relative error in $P(Y_{n-k}>a_n+b_nx)$ is $O(u_n(k,x)^{-2})$ for every $k$ and $x$ in the summation range, a uniformity imported from classical Berry-Esseen bounds; if that error is larger anywhere, the constants $2$ and $2/e$ are not forced.","fun_headline_variants_meta":{"raw":{"variants":["Ginibre spectral radius to Gumbel: exact rate 2","Exact Wasserstein rate 2, Kolmogorov 2/e for edge","Ginibre edge: Wasserstein rate 2, Kolmogorov 2/e","Exact rate 2 and 2/e for Ginibre edge to Gumbel"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001343,"raw_usage":{"total_tokens":5483,"prompt_tokens":998,"completion_tokens":4485,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":4398}},"tokens_in":614,"tokens_out":4485,"duration_ms":30627,"temperature":1.0,"reasoning_tokens":4398,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:30:44.745296+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, for increasing $n$ and for many choices of $k$ and $x$, the exact tail probability $P(Y_{n-k}>a_n+b_nx)$ for the Gamma variables in Lemma 2.1 and compare it with $e^{-u_n^2/2}/(\\sqrt{2\\pi}u_n)$ over the range $1\\ll u_n\\ll n^{1/10}$; a relative error larger than $O(u_n^{-2})$ at any point would falsify the uniformity on which the rates rest. Alternatively, estimate $W_1$ and $\\sup_x|F_n(x)-e^{-e^{-x}}|$ by simulation for $n$ up to $10^6$; the quantities $(\\log n/\\log\\log n)W_1$ and $(\\log n/\\log\\log n)\\sup_x|F_n-\\Lambda|$ should approach $2$ and $2/e$, and any clear drift away from these constants would disprove the theorem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the weak convergence of the scaled spectral radius to the Gumbel distribution, the limit refined here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the reduction of the squared spectral radius to a maximum of independent Gamma variables, stated as Lemma 2.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical large-deviation expansion for tails of sums of independent exponentials used in Lemma 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the non-uniform Berry-Esseen bound used for the matching upper tail in Lemma 2.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the joint eigenvalue density of the complex Ginibre ensemble on which the whole reduction is based."}],"review_version":1}