{"id":"ba7e9892-52a7-475e-9e97-8809b3ba3c86","arxiv_id":"2412.14070","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For generalized Wigner matrices, the characteristic function of linear spectral statistics is expanded around its Gaussian limit with error O(N^{-1}) and an explicit N^{-1/2} correction.","lead":"This paper proves an O(N^{-1}) expansion for the characteristic function of linear spectral statistics of generalized Wigner random matrices, including the N^{-1/2} non-Gaussian correction. The result extends a bound previously available only for Wigner matrices to the whole generalized Wigner class, with an application to the maximum of the log-characteristic polynomial.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The explicit cubic coefficient B(f) in Theorem 1.6 appears to be wrong by a factor of 2 in an exactly solvable admissible case, so the stated central expansion is false as written.","rationale":"The reader's conditional verdict focuses on the local law input and on the unproved extension of Theorem 1.6 to logarithm test functions in Section 9. I find a more direct, load-bearing problem in the central theorem itself: the explicit subleading cubic coefficient B(f) is inconsistent with an exactly solvable admissible case. The counterexample uses only the paper's own normalization: T1(x)=x/2, so for a function equal to x on [-3,3] we have t1(f)=2; LSS(f) equals trH with overwhelming probability because all eigenvalues are in [-3,3]; and the exact log-characteristic function gives the λ^3 term as iλ^3ŝ3/6, requiring B=ŝ3/2 rather than B=ŝ3. The claimed O(N^{-1}) error is far too small to conceal the difference at λ=N^{-1/8}. Moreover, the paper's own Proposition 8.3, using a_{1,a}=-t1/2, produces the magnitude ŝ3 t1^3/16, which is the correct value for f=x; this suggests the displayed formulae (1.18) and (8.46) contain a factor error, not merely an application gap. The local-law dependence remains a secondary robustness concern, and the Section 9 application is additionally conditional, but neither is the primary obstacle: the stated central expansion is not correct as written.","tokens_in":73828,"tokens_out":38021,"duration_ms":356176,"concrete_test":"Analytically compute the characteristic function of LSS(f) for the admissible f above with Wigner variance and non-Gaussian diagonal third moments. Since f(H)=H with overwhelming probability, expand ∏_i E[e^{iλH_ii}] in cumulants; the coefficient of iλ^3 in the log-characteristic function is ŝ3/6. Compare with Theorem 1.6, which with t1(f)=2 gives iλ^3B/3=iλ^3ŝ3/3. If the factor-of-two discrepancy reproduces, the definition of B(f) in (1.18) and the corresponding computations in Lemma 8.7 and (8.46) need correction before the main theorem can be accepted.","verdict_should_be":"REJECT","load_bearing_attack":"Let f be an admissible compactly supported function with f(x)=x on [-3,3] and a smooth cutoff to zero on [3,4]; let H be a real symmetric generalized Wigner matrix with Wigner variance profile S_ij=1/N and diagonal entries having nonzero third cumulant, say κ3(H_ii)=cN^{-3/2} with c≠0. With overwhelming probability all eigenvalues of H lie in [-3,3], so LSS(f)=trH up to exponentially small events. Therefore the exact characteristic function factorizes: E[e^{iλLSS(f)}]=∏_i E[e^{iλH_ii}], and its log expansion has cubic term iλ^3 ŝ3/6, where ŝ3=Σ_i κ3(H_ii). Under the paper's own Chebyshev convention (Definition 1.3, T1(x)=x/2) one has t1(f)=2 for f=x on [-3,3]. Hence Definition 1.4 gives B(f)=ŝ3/8·t1(f)^3=ŝ3, whereas matching iλ^3B/3 to iλ^3ŝ3/6 requires B=ŝ3/2. This is not absorbed by the error term: at λ=N^{-1/8} the discrepancy is O(N^{-3/8}), while the claimed error in Theorem 1.6 is O(N^{-1}(1+λ)+N^{-1}λ^2)=O(N^{-1}). The internal derivation also points to the same defect: Proposition 8.3 and Lemma 8.4 give B_a=(~s3/(2N^{1/2}))a_{1,a}^3 with a_{1,a}=-t1(f)/2, i.e. magnitude ŝ3 t1(f)^3/16=ŝ3/2 for f=x, not the ŝ3 t1(f)^3/8 stated in (8.46) and (1.18).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the characteristic function of linear spectral statistics (LSS) for generalized Wigner matrices. It develops a graphical cumulant-expansion framework, proves estimates for loop and line hypergraphs, and derives an expansion of the characteristic function with error O(N^{-1}) around the Gaussian limit, with explicit sub-leading terms of size O(N^{-1/2}) controlled by functionals V_beta(f), E_beta(f), and B(f). The main theorem (Theorem 1.6) states this expansion in two regimes depending on whether V_beta(f) is bounded below. As an application, the paper extends results of Bourgade-Lopatto-Zeitouni on the maximum of the log-characteristic polynomial to generalized Wigner matrices.","tokens_in":74236,"tokens_out":16815,"duration_ms":129132,"significance":"The paper addresses an important open step in random matrix theory: an O(N^{-1}) characteristic-function expansion for LSS of generalized Wigner matrices, previously available only for Wigner matrices. The graphical loop/line estimates and the Stein-method implementation are serious technical advances, and the proposed applications to log-characteristic polynomial maxima are natural. However, the explicit cubic coefficient B(f) in the central theorem is incorrect as stated: it is off by a factor of 2 and has the wrong sign relative to both the paper's own internal computation and an exactly solvable example. This makes Theorem 1.6 false as written, though the error appears localized and correctable. The paper's framework remains potentially valuable once the coefficient is fixed and the consequences re-derived.","major_comments":[{"comment":"The cubic coefficient B(f) is internally inconsistent and incorrect as stated. Substituting a_{1,a} = -t_1(f)/2 from Lemma 8.4 into B_a(f) = tilde{s}_3/(2 N^{1/2}) (a_{1,a})^3 of (8.19) gives B_a = -tilde{s}_3 t_1(f)^3/(16 N^{1/2}) = -hat{s}_3 t_1(f)^3/16, whereas (8.46) and Definition 1.4 assert +hat{s}_3 t_1(f)^3/8. These formulas differ by a factor of 2 and a sign. This is not a minor typo: in the exactly solvable case f(x)=x on [-3,3] with a real symmetric generalized Wigner matrix having S_ij=1/N and diagonal third cumulants, LSS(f)=tr H with overwhelming probability and the exact log-characteristic function has cubic term -i lambda^3 hat{s}_3/6, so Theorem 1.6 would require B(f) = -hat{s}_3/2. The stated B(f) is +hat{s}_3. At lambda = N^{-1/8} the discrepancy in the exponent is O(N^{-7/8}), which exceeds the claimed O(N^{-1}) error in Theorem 1.6. The derivation in (8.19) appears to yield the correct coefficient, so the error likely lies in the transcription to (8.46) and (1.18); the author should correct these formulas, fix the sign conventions, and re-verify Lemma 8.9 and the bounds in Section 9 that use B(f).","section":"Definition 1.4 (1.18); Proposition 8.3 (8.19); Lemma 8.4 (8.26); Lemma 8.7 (8.46)"},{"comment":"The application of Theorem 1.6 in Proposition 9.3 uses the functions f_z(x) = Re log(z-x) and Im log(z-x), which are not compactly supported and are not admissible under Definition 1.2. The proof says it uses the coefficients V_z, E_z, B_z of Theorem 1.6 for these functions and then applies (9.8), but no truncation or approximation argument is given. Since the spectrum lies in a compact interval with overwhelming probability, a truncation to a fixed interval should be possible, but the needed uniformity in eta >= exp(-K (log log N)^2) is not demonstrated. As written, the passage from Theorem 1.6 to (9.8) is unjustified.","section":"Section 9, Proposition 9.3"}],"minor_comments":[{"comment":"After correcting the cubic coefficient, the author should ensure that all displayed formulas for B(f) match the deterministic computation in Section 8.1 and that the sign convention for the third cumulant is stated consistently.","section":"Section 1, Eq. (1.18); Section 8, Eq. (8.46)"},{"comment":"The relationship between tilde{s}_3 in (8.14) and hat{s}_3 in (1.16) should be made explicit (tilde{s}_3 = N^{1/2} hat{s}_3 up to the conventions for s^{(3)}_{aa}), as the current notation invites confusion in the verification of the coefficient.","section":"Section 8, Eq. (8.14)"},{"comment":"The proof of Theorem 9.2 relies on the arguments of [11], including the derivation of (6.7) there. The author notes that (6.7) is not fully proven in [11] and asserts that rigidity estimates suffice, but no proof is given in the present paper. This gap should be closed or the dependence made explicit.","section":"Section 9.1"},{"comment":"There are many small typos and missing differentials (e.g., in (2.20) and (8.18)); a careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The exactly solvable check described in the stress-test note is a good diagnostic: the author should be asked to include it or an equivalent consistency check after correcting B(f). The main theorem should not appear in its current form. I would not recommend reject, because the derivations in Sections 3-7 and the structure of the expansion appear sound; the error seems to be in the final assembly of the cubic coefficient. However, given the centrality of the coefficient, the author must re-verify all deterministic computations in Section 8.1 and the proof of Lemma 8.9."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth a serious look, but the headline result is not correct as written. Take an admissible f that equals x on [-3,3] and is smoothly cut off to zero. With overwhelming probability all eigenvalues lie in [-3,3], so LSS(f)=tr H exactly. The exact characteristic function factorizes over the diagonal entries, giving a log-cubic term of -iλ³ŝ3/6. Theorem 1.6, with B(f)=ŝ3 t1(f)^3/8 and t1(f)=2, predicts +iλ³ŝ3/3. That is a factor of 2 and a sign error, and it is not absorbed by the error term at, say, λ=N^{-1/8}.\n\nThat said, the paper has real substance. The loop estimate (Proposition 3.2) and line estimate (Proposition 4.2) are genuinely new and are the key to closing the expansion at order N^{-1} for non-constant variance profiles. The iterative cumulant-expansion argument is intricate and, as far as I can verify, coherent. The internal derivation actually points to the correct answer: combining (8.19) with Lemma 8.4 gives B_a = -ŝ3 t1^3/16, which reproduces the exact solvable case. So the mistake is localized to the statement of the coefficient in Definition 1.4 and the corresponding line in Lemma 8.7, probably a sign/factor slip in translating a_{1,a}= -t1/2 into the final formula.\n\nThe other soft spot is Section 9: the application to the logarithm of the characteristic polynomial goes beyond the compactly supported admissibility condition, and the paper relies on sketched fixes to gaps in [11]. That part is rougher, but it is secondary to the main technical contribution.\n\nWho gets value from this: anyone working on mesoscopic linear spectral statistics, generalized Wigner matrices, or extremal eigenvalue statistics. The loop/line estimates will likely be cited even after the B(f) issue is fixed. This deserves a serious referee, but the paper should not be accepted in its current form. The author needs to correct the cubic coefficient, re-check the deterministic coefficient computations, and tighten the application section. With that done, the core result should be publishable.","headline":"Main theorem as stated is false: the cubic coefficient B(f) has the wrong sign and is off by a factor of 2; the paper's real value is the loop/line machinery, which looks sound.","tokens_in":74774,"tokens_out":8535,"would_cite":false,"duration_ms":73928,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15B52","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For generalized Wigner matrices, the eigenvalue-statistic law matches a Gaussian to order 1/N, with explicit non-Gaussian corrections.","keywords":["generalized Wigner matrices","linear spectral statistics","characteristic function expansion","local semicircle law","cumulant expansion","Stein's method","mesoscopic CLT","log-characteristic polynomial"],"falsifier":"Choose a generalized Wigner matrix with a doubly stochastic two-block variance profile $S$ that is not constant, set $f(x)=x^3$ (or another smooth compactly supported admissible function), and compute $E[e^{i\\lambda(\\mathrm{LSS}(f))}]$ for $\\lambda=1$ at increasing $N$; compare the difference with the right-hand side of Theorem 1.6. If the discrepancy after removing the explicit $N^{-1/2}$ corrections does not decay like $N^{-1}$, the expansion is wrong.","tokens_in":73626,"feed_emoji":"🔢","tokens_out":8169,"duration_ms":73712,"temperature":0.7,"pith_summary":"The paper proves a sharp expansion for the law of linear spectral statistics of generalized Wigner matrices. For an admissible test function $f$, the characteristic function is shown to equal $\\exp(-\\lambda^2 V_\\beta(f)/2 + i\\lambda^3 B(f)/3 + i\\lambda E_\\beta(f))$ with an error of order $N^{-1}$ (up to factors $(1+|\\lambda|)$ and $\\|f''\\|_{1,w}$), provided the Gaussian variance $V_\\beta(f)$ is bounded below, and with a slightly weaker error otherwise. Previously, such $O(N^{-1})$ accuracy was known only for Wigner matrices, whose constant variance profile allows a direct simplification; only a polynomial error rate was available for general variance profiles. The corrections $B(f)$ and $E_\\beta(f)$ are explicit: they depend on the low-order cumulants of the matrix entries and the Chebyshev coefficients of $f$, and are of order $N^{-1/2}$. In effect, the paper supplies the missing optimal-rate input needed to extend optimal CLT and extremal-statistics results from Wigner to generalized Wigner ensembles.","feed_headline":"Generalized Wigner spectra reach the 1/N Gaussian limit","feed_subtitle":"An O(1/N) error expansion, with explicit non-Gaussian 1/√N corrections, now covers non-constant variance profiles.","key_machinery":"The argument is carried by an iterated cumulant expansion applied to monomials of resolvent entries, organized graphically as 'loops' and 'lines'. A loop is a product $G_{i_1i_2}(z_1)\\cdots G_{i_k i_1}(z_k)$ over distinct indices; a line is the analogue with distinct endpoints $G_{i_1i_2}(z_1)\\cdots G_{i_k i_{k+1}}(z_k)$. The key estimates (Propositions 3.2 and 4.2) show that the expectation of a loop with a distinguished spectral parameter $z$ gains a factor $\\Psi_1(z)=(N|\\operatorname{Im}z|)^{-1/2}$ over the naive entry-wise local-law size, and that the expectation of a line gains $N^{-1/2}$, with the gain coming from cancellation in the cumulant expansion. The resolvent entries are packaged with the variance matrix $S=A+N^{-1}ee^{\\mathsf T}$, where $A$ has spectral norm at most $1-c$; inverting the resulting self-consistent equation for vectors such as $v_j=E[e_a G_{j2}(z)G_{2j}(w)]$ uses bounds on $(1-m_{sc}(z)m_{sc}(w)(A-B))^{-1}$, the identity $(m_{sc}(z)-m_{sc}(w))/(z-w)=m_{sc}(z)m_{sc}(w)/(1-m_{sc}(z)m_{sc}(w))$, and Sherman–Morrison. The expansion is then fed into the Helffer–Sjostrand representation and Stein's method to derive the differential equation for the characteristic function.","core_discovery":"The central claim is Theorem 1.6: for a real symmetric or complex Hermitian generalized Wigner matrix $H$ with doubly stochastic variance matrix $S_{ij}=\\mathbb E[|H_{ij}|^2]$, and an admissible test function $f$, the characteristic function of the centered linear spectral statistic $\\mathrm{LSS}(f)=\\mathrm{tr}\\,f(H)-N\\int f(x)\\rho_{sc}(x)\\,dx$ obeys $$E[$e^{{i\\lambda\\mathrm{LSS}}$(f)}]=\\exp\\left(-\\frac{\\$lambda^{{2}}$V_{\\$\\beta$}(f)}2+\\frac{i\\$lambda^{{3}}$B(f)}3+i\\$\\lambda$ E_{\\$\\beta$}(f)\\right)+O\\left(\\|f''\\|_{1,w}$N^{{-1}}$(1+|\\$\\lambda$|)+$N^{{-1}}$|\\$\\lambda$|^{2}\\right)$$ when $V_\\beta(f)\\geq c>0$ and $|\\lambda|\\leq N^{-\\varepsilon}\\sqrt{N\\|f''\\|_1^{-1}}$; a related estimate holds without the lower-bound condition. The paper's own way of stating the result is that this is an expansion around the Gaussian limit with error $O(N^{-1})$ and an explicit sub-leading non-Gaussian correction $N^{-1/2}P_f(\\lambda)$. The functionals $V_\\beta$, $E_\\beta$ and $B$ are deterministic and explicit in the cumulants $\\kappa_k(H_{ij})$ and the Chebyshev coefficients $t_j(f)$ of $f$; $B(f)$ is a cubic term controlled by the third cumulant of the diagonal entries. This is the first such optimal-rate expansion for full linear spectral statistics of generalized Wigner matrices.","pith_inferences":["Extension (not in the paper): the same loop/line machinery should apply to any ensemble whose local law, fluctuation averaging, and variance matrix with a spectral gap hold at the stated rates—most directly to band matrices and Wigner-type profiles—so the optimal characteristic-function expansion is likely a benchmark for the whole family.","Extension (not in the paper): because $B(f)$ is proportional to the third cumulant of diagonal entries, one could use the formula to engineer variance profiles that suppress or amplify skewness in the limiting fluctuations; the paper does not discuss this design consequence.","Extension (not in the paper): numerically testing the two-point resolvent estimate (Corollary 6.7) on a two-block variance profile would isolate the new technical step; a clean $N^{-1}$ decay there would corroborate the full expansion."],"forward_implications":["If Theorem 1.6 is correct, the CLT for full linear spectral statistics of generalized Wigner matrices holds with the same $O(N^{-1})$ characteristic-function accuracy as for Wigner matrices, closing the gap left by polynomial-error estimates.","The explicit expressions for $V_\\beta(f)$, $E_\\beta(f)$ and $B(f)$ give a computationally usable formula for the leading and sub-leading moments of $\\mathrm{LSS}(f)$, including the non-Gaussian cubic correction.","The same expansion applies to mesoscopic test functions that vary on scales $N^{\\alpha-1}$, yielding CLT accuracy there rather than only at global scale.","Theorem 9.2 extends the recent Wigner-matrix result [11] on the maximum of the log-characteristic polynomial, and the associated optimal rigidity of extreme eigenvalues, to generalized Wigner matrices."],"supporting_citations":[{"why":"Supplies the cumulant expansion formula (Lemma 2.6) that every iterative argument in the paper rests on.","marker":"[28]"},{"why":"Supplies the isotropic local semicircle law used to estimate sums of resolvent entries such as $\\sum_j s_{1j}G_{j2}$.","marker":"[9]"},{"why":"Supplies the fluctuation-averaging estimate (2.7) used to truncate cumulant expansions and control diagonal resolvent terms.","marker":"[2]"},{"why":"Provides the Stein's-method-plus-local-law approach to the characteristic function that the present paper adapts.","marker":"[26]"},{"why":"Gives the prior generalized-Wigner expansion with only polynomial error, the baseline the paper improves to $O(N^{-1})$.","marker":"[30]"},{"why":"Establishes the optimal $O(N^{-1})$ rate for Wigner matrices, which is the target the generalized case must match.","marker":"[8]"},{"why":"Prior work establishing the $O(N^{-1})$ expansion for Wigner matrices, cited as the motivation and technical starting point.","marker":"[27]"},{"why":"The application: its maximum-of-log-characteristic-polynomial and optimal rigidity results are extended to generalized Wigner matrices.","marker":"[11]"},{"why":"Develops the self-consistent method for variance-profile terms of the form $\\sum s_{ij}G_{ij}G_{ji}$, which the paper's loops/lines machinery builds on.","marker":"[18]"},{"why":"Provides the rigidity estimates for generalized Wigner matrices used in the extremal-statistics application.","marker":"[20]"}],"fun_headline_variants":["Generalized Wigner: O(1/N) Gaussian limit, 1/√N corrections","Optimal-rate expansion for generalized Wigner spectral stats","Beyond Wigner: 1/N error for generalized matrix spectra","Sharp Gaussian limit for generalized Wigner ensembles","First O(1/N) expansion for generalized Wigner linear statistics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire expansion rests on the quoted local semicircle law for generalized Wigner matrices holding with sharp entry-wise, isotropic, and fluctuation-averaging rates; if any of those rates degrades for a particular variance profile, the $O(N^{-1})$ error estimate would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Generalized Wigner: O(1/N) Gaussian limit, 1/√N corrections","Optimal-rate expansion for generalized Wigner spectral stats","Beyond Wigner: 1/N error for generalized matrix spectra","Sharp Gaussian limit for generalized Wigner ensembles","First O(1/N) expansion for generalized Wigner linear statistics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000797,"raw_usage":{"total_tokens":3513,"prompt_tokens":960,"completion_tokens":2553,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":2465}},"tokens_in":576,"tokens_out":2553,"duration_ms":18822,"temperature":1.0,"reasoning_tokens":2465,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:31:23.202112+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a generalized Wigner matrix with a doubly stochastic two-block variance profile $S$ that is not constant, set $f(x)=x^3$ (or another smooth compactly supported admissible function), and compute $E[e^{i\\lambda(\\mathrm{LSS}(f))}]$ for $\\lambda=1$ at increasing $N$; compare the difference with the right-hand side of Theorem 1.6. If the discrepancy after removing the explicit $N^{-1/2}$ corrections does not decay like $N^{-1}$, the expansion is wrong.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the cumulant expansion formula (Lemma 2.6) that every iterative argument in the paper rests on."},{"cited_title":"Bloemendal, L","cited_arxiv_id":null,"evidence_quote":"Supplies the isotropic local semicircle law used to estimate sums of resolvent entries such as $\\sum_j s_{1j}G_{j2}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fluctuation-averaging estimate (2.7) used to truncate cumulant expansions and control diagonal resolvent terms."},{"cited_title":"Landon and P","cited_arxiv_id":null,"evidence_quote":"Provides the Stein's-method-plus-local-law approach to the characteristic function that the present paper adapts."},{"cited_title":"Li and Y","cited_arxiv_id":null,"evidence_quote":"Gives the prior generalized-Wigner expansion with only polynomial error, the baseline the paper improves to $O(N^{-1})$."},{"cited_title":"Bao and Y","cited_arxiv_id":null,"evidence_quote":"Establishes the optimal $O(N^{-1})$ rate for Wigner matrices, which is the target the generalized case must match."},{"cited_title":"Erd˝ os, A","cited_arxiv_id":null,"evidence_quote":"Develops the self-consistent method for variance-profile terms of the form $\\sum s_{ij}G_{ij}G_{ji}$, which the paper's loops/lines machinery builds on."},{"cited_title":"Erd˝ os, H.-T","cited_arxiv_id":null,"evidence_quote":"Provides the rigidity estimates for generalized Wigner matrices used in the extremal-statistics application."}],"review_version":1}