{"id":"53586056-e290-4ee7-ac59-32615ffc8ba3","arxiv_id":"2607.11547","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Optimal global rigidity of order (log N)/N holds for eigenvalues of the Laguerre unitary ensemble, proved via GMC convergence of the counting-function measure after RH asymptotics of Hankel determinants.","lead":"The paper proves an optimal global eigenvalue rigidity bound for the Laguerre unitary ensemble: the largest deviation of any eigenvalue from its classical location, weighted by the equilibrium density, is of order (log N)/N with high probability. This matches the known optimal bounds for GUE, JUE and CUE and is obtained by showing that a random measure built from the eigenvalue counting function converges to Gaussian multiplicative chaos.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Coefficient mismatch in Prop. 4.3 N-term blocks cancellation needed for exponential moments of h_N (and thus GMC).","rationale":"The reader's weakest-assumption call (one-cut regular V) is correct but external to the argument; the internal inconsistency in the leading coefficient of Prop. 4.3 is more immediate and load-bearing, because it directly falsifies the exponential-moment bounds that feed GMC convergence. The rest of the RH analysis, edge refinements, and logical structure appear solid and match the GUE/JUE templates, so the claim is almost certainly true once the coefficient is fixed. Hence CONDITIONAL rather than REJECT or UNCHANGED: a one-line correction restores the proof, but as written the central chain is broken. The reader missed the mismatch (partial agreement).","tokens_in":35384,"tokens_out":671,"duration_ms":60474,"concrete_test":"Re-derive the O(N) term in d/dx log D_N from the differential identity (2.7) after the hard-edge R-transformation (3.69) and the local parametrix of Lemma 3.12; extract the contribution of (P_{-1}^{-1} P_{-1}')_{21} on the jump. If the prefactor is √(2π) ψ_V rather than √2, correct Prop. 4.3; if it is √2, the normalization of h_N is incompatible with the CLT of [6].","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Prop. 4.3 states log(D_N(x;γ;0)/D_N(x;0;0)) = √2 γ N ∫_{-1}^x √((1-s)/(1+s)) ds + (γ²/2)log N + (γ²/4)log(1+x) + O(1) (hard edge; analogous soft-edge formula). For the standard case ψ_V = 1/π used in §4–5 this is √2 γ N · π F(x). But E[e^{γ h_N(x)}] = exp(-√(2π) γ N F(x)) · (D_N ratio) by (1.26) and (5.4), so the linear terms cancel if and only if the coefficient equals √(2π) γ N F = (√(2π)/π) γ N ∫ √\theta ds. Numerically √2 ≉ √(2π)/π. (Prop. 4.2 correctly writes the matching factor √(2π) N γ ∫ ψ_V \rho.) Without cancellation the moments are exp(Θ(N)), Prop. 5.1 fails, the GMC assumptions of [7] are not verified, and both Thm. 1.1 and the bulk part of Thm. 1.2 collapse. The discrepancy is almost certainly a transcription error (copying a GUE coefficient), but the written argument is inconsistent at the precise place that feeds the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.5","summary":"The paper proves an optimal global eigenvalue rigidity result for the Laguerre unitary ensemble (LUE) with a one-cut regular real-analytic potential V. The main claim (Theorem 1.2) is that for any ε>0, lim P((1-ε)log N/N < max_j F'(κ_j)|λ_j-κ_j| < (1+ε)log N/N)=1, where λ_j are the ordered eigenvalues and κ_j the classical percentiles of the equilibrium measure μ_L. The argument proceeds by constructing the random measure dμ_γ^N from the eigenvalue counting function h_N, verifying the GMC sufficient conditions of Claeys et al. [7] via asymptotics of the associated Hankel determinants (obtained by RH steepest-descent analysis in the separated, merging, and edge regimes), and then refining the bound near the hard and soft edges by an iterative argument.","tokens_in":35722,"tokens_out":1265,"duration_ms":12110,"significance":"If correct, the result places LUE on the same footing as CUE, GUE and JUE with respect to optimal O(log N/N) global rigidity, supporting the emerging universality picture for classical unitary ensembles. The technical contribution is a complete RH analysis of Hankel determinants with Fisher-Hartwig singularities for Laguerre-type weights, including new local parametrices near the hard edge and a careful refinement procedure that handles the singularity of F' at -1. The derivation is essentially parameter-free once the one-cut regular assumption is granted, and the reduction to the GMC framework of [7] is explicit.","major_comments":[{"comment":"Proposition 4.3 (hard-edge formula) writes the linear term as √2 γ N ∫_{-1}^x √((1-s)/(1+s)) ds. For the standard density ψ_V=1/π used throughout §4–5 this equals √2 γ N · π F(x). Equation (1.26)/(5.4) multiplies by exp(-√(2π) γ N F(x)), so the linear terms cancel if and only if the coefficient is √(2π) γ N F = (√(2π)/π) γ N ∫ √((1-s)/(1+s)) ds. Numerically √2 ≈ 1.414 while √(2π)/π ≈ 0.798; the two do not match. (Proposition 4.2 correctly uses the factor √(2π) N γ ∫ ψ_V \rho.) Without cancellation the exponential moments are exp(Θ(N)), Proposition 5.1 fails, the GMC assumptions of [7] are not verified, and both Theorem 1.1 and the bulk part of Theorem 1.2 collapse. This is almost certainly a transcription error (copying a GUE coefficient), but as written the argument is inconsistent at the precise place that feeds the central claim. The soft-edge formula in the same proposition has the an","section":null},{"comment":"Several intermediate statements that are load-bearing for the edge refinement (Lemmas 5.7, 5.8, 5.12 and the bulk-iteration Proposition 5.6) are declared “similar to [9]” and the proofs are omitted. While the hard-edge density singularity is of the same type as in the Jacobi case, the soft-edge refinement (Lemma 5.12 and Proposition 5.13) uses a different scaling (N^{-2/3} log log N) and a different Markov estimate; a self-contained sketch of at least the soft-edge argument is needed for the paper to be independently verifiable.","section":null}],"minor_comments":[{"comment":"Author affiliations and e-mail addresses appear swapped (first author listed with second author’s address and vice versa).","section":null},{"comment":"In (1.8) the index is written λ_k while the maximum is over j; the same slip appears in a few other places.","section":null},{"comment":"The four cases listed after (3.13) are labelled (I)–(IV) but the introductory sentence says “three cases”.","section":null},{"comment":"Notation for the equilibrium density switches between \rho, \rhõ and ψ_V \rho without a single consistent definition; a short glossary would help.","section":null},{"comment":"Several model RH problems in the appendix are stated with contours whose orientations are reversed relative to the classical literature; a one-sentence remark that existence still holds would remove ambiguity.","section":null}],"recommendation":"major_revision","confidential_remarks":"The coefficient mismatch is almost certainly a copy-paste error from the GUE literature and should be fixable by a short recalculation of the differential identity / global parametrix contribution. Once that is corrected and the omitted edge lemmas are sketched, the paper is a solid contribution that fits the journal well. I would be happy to re-review a revised version quickly."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The paper fills the last classical unitary gap: optimal global rigidity for LUE at the (log N)/N scale, matching GUE/JUE/CUE. The strategy is the familiar CLT-to-GMC route of Claeys et al., but they supply the missing hard-edge local parametrix (model RH A.4) and the iterative refinement lemmas that push the bound all the way to κ_j near -1. That technical work is real and carefully written; the three-regime steepest-descent analysis (separated, merging, edge) is complete, the differential identities are standard, and the verification of the GMC hypotheses is explicit once the moments are in hand.\n\nThe soft spot is not minor. Prop. 4.3 writes the leading N-term of log(D_N(x;γ;0)/D_N(x;0;0)) as √2 γ N ∫ √((1-s)/(1+s)) ds. For the standard density ψ_V=1/π used in §§4–5 this is √2 γ N · π F(x). But the prefactor coming from the definition of h_N is exp(-√(2π) γ N F(x)), so the linear terms cancel if and only if the coefficient is √(2π)/π. Numerically √2 ≉ √(2π)/π. (Prop. 4.2 has the matching √(2π) factor, so the authors know the right constant.) Without cancellation the exponential moments are exp(Θ(N)), Prop. 5.1 fails, the GMC assumptions of [7] are not verified, and both Theorem 1.1 and the bulk half of Theorem 1.2 collapse. This is almost certainly a transcription error (copying a GUE coefficient), but the written argument is inconsistent exactly where it feeds the central claim.\n\nEverything else looks clean: no free parameters, no circularity, citations are appropriate, and the hard-edge analysis is new. The paper is for people who already work on RH asymptotics and GMC rigidity; they will get value from the parametrices and the refinement method even if the coefficient needs a one-line fix. I would send it to referees—they will catch the mismatch immediately and the rest is solid enough to survive revision. Engage with it once the moments are corrected.","headline":"Solid optimal LUE rigidity via GMC, but a coefficient mismatch in Prop. 4.3 breaks the linear cancellation that feeds the exponential moments and thus the whole GMC argument.","tokens_in":36349,"tokens_out":580,"would_cite":true,"duration_ms":6747,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","41A60","47B35","60G15","60G57"],"pacs":[],"model":"grok-4.5","headline":"Laguerre unitary eigenvalues fluctuate from their classical locations by at most order (log N)/N, and the bound is sharp.","keywords":["eigenvalue rigidity","Laguerre unitary ensemble","Gaussian multiplicative chaos","log-correlated fields","Hankel determinants","Riemann-Hilbert analysis","Fisher-Hartwig singularities"],"falsifier":"Compute the maximal deviation max_j F'(κ_j)|λ_j−κ_j| for large-N samples of the standard Laguerre unitary ensemble (V(x)=2(x+1)); if for some fixed ε>0 the probability that this quantity exceeds (1+ε)log N/N fails to tend to zero, or falls below (1−ε)log N/N with positive probability, the claimed rigidity fails.","tokens_in":36272,"feed_emoji":"📐","tokens_out":750,"duration_ms":6300,"temperature":0.7,"pith_summary":"The paper proves that the ordered eigenvalues of a generalized Laguerre unitary ensemble stay within a distance of order (log N)/N of their classical percentiles, measured against the derivative of the equilibrium distribution. Both the upper and lower bounds hold with probability tending to one, so the estimate is optimal. The argument first turns the eigenvalue counting function into a random measure, shows that this measure converges to a Gaussian multiplicative chaos measure built from the known log-correlated field of the ensemble, and then reads the maximum of the counting function off that convergence. Near the hard and soft edges a separate refinement step is needed because the density of the equilibrium measure vanishes or blows up; once that refinement is in place the same (log N)/N scale governs the entire spectrum. The result places the Laguerre ensemble on the same footing as the Gaussian and Jacobi ensembles, suggesting that optimal global rigidity of this order is universal for classical unitary ensembles.","feed_headline":"Laguerre eigenvalues stay within (log N)/N of classical spots","feed_subtitle":"Optimal global rigidity matches the scale already known for GUE and JUE, via GMC.","key_machinery":"The random measure dμ_N^γ = exp(γ h_N(x))/E[exp(γ h_N(x))] constructed from the centered eigenvalue counting function h_N; once this measure is shown to converge to the Gaussian multiplicative chaos of the limiting log-correlated field, the maximal size of h_N (and therefore the rigidity scale) follows from known GMC tail estimates.","core_discovery":"For any ε>0 the probability that the maximum, over all indices j, of F'(κ_j)|λ_j−κ_j| lies between (1−ε)log N/N and (1+ε)log N/N tends to one as N tends to infinity, where λ_j are the ordered Laguerre eigenvalues, κ_j their classical locations under the equilibrium measure μ_L, and F its cumulative distribution function.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Laguerre eigenvalues rigid within (log N)/N of classical spots","Optimal LUE rigidity: max deviation scales as (log N)/N","LUE eigenvalues stick to equilibrium locations at (log N)/N","Global rigidity for Laguerre ensemble matches GUE (log N)/N scale","Laguerre spectrum locks to classical positions within (log N)/N"],"cache_read_input_tokens":32768,"weakest_assumption_plain":"The external potential must be real-analytic and one-cut regular with a growth condition at infinity, so that the equilibrium measure has a single interval of support with square-root vanishing at the soft edge and inverse-square-root blow-up at the hard edge.","fun_headline_variants_meta":{"raw":{"variants":["Laguerre eigenvalues rigid within (log N)/N of classical spots","Optimal LUE rigidity: max deviation scales as (log N)/N","LUE eigenvalues stick to equilibrium locations at (log N)/N","Global rigidity for Laguerre ensemble matches GUE (log N)/N scale","Laguerre spectrum locks to classical positions within (log N)/N"]},"model":"grok-4.5","effort":"low","cost_usd":0.005638,"raw_usage":{"total_tokens":1359,"prompt_tokens":629,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":56380000,"prompt_tokens_details":{"text_tokens":629,"audio_tokens":0,"image_tokens":0,"cached_tokens":0},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":653,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":629,"tokens_out":77,"duration_ms":5933,"temperature":1.0,"reasoning_tokens":653,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T04:49:48.599715+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the maximal deviation max_j F'(κ_j)|λ_j−κ_j| for large-N samples of the standard Laguerre unitary ensemble (V(x)=2(x+1)); if for some fixed ε>0 the probability that this quantity exceeds (1+ε)log N/N fails to tend to zero, or falls below (1−ε)log N/N with positive probability, the claimed rigidity fails.","supporting_citations":[],"review_version":1}