{"id":"1c40fb50-be09-487f-a8f1-6f1cf4b386e4","arxiv_id":"2412.09290","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves new asymptotic expansion formulas for Harish-Chandra and Schur generating functions, links them to (quantized) infinitesimal freeness, and demonstrates a BBP-type phase transition in domino tilings.","lead":"This paper develops general tools for extracting the next terms in the large-size limit of random matrix and random tiling models, going beyond the law of large numbers. It connects these corrections to infinitesimal free probability, introduces a quantized version, and computes explicit examples including a phase transition in domino tilings of the Aztec diamond.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main theorems are conditional on a uniform two-term Harish-Chandra asymptotic (e.g., (79)); the paper asserts rather than fully derives this input for finite-rank perturbation examples, so the soft spot is applicability, not the conditional implication itself.","rationale":"The reader's weakest_assumption identifies exactly the analytic input that is outside the conclusion of the main theorems, and I agree that this is the point on which the paper's applicability hinges. However, this is an explicit hypothesis of a conditional theorem, not a hidden or circular assumption. The paper is clear that the results are conditional on such uniform asymptotics, and for the exactly solvable examples (e.g., the domino-tiling model with explicit Schur generating function) the hypothesis is immediately checkable from the displayed formulas. The finite-rank perturbation examples are the place where the verification is sketched rather than fully shown, and Example 7's central computation is omitted, which justifies the CONDITIONAL verdict. My read does not weaken the central theorem or the proof strategy; it only underscores that the advertised applications inherit a nontrivial analytic condition that should be verified explicitly before the examples are used. Therefore I keep the reader's verdict unchanged.","tokens_in":65184,"tokens_out":10370,"duration_ms":122469,"concrete_test":"For B = N G + N theta E_{1,1} with G a GUE matrix, start from the Itzykson-Zuber determinant formula (34) and Lemma 6 to derive the full two-term asymptotic N[(1/N) log E[HC(x_1,...,x_r,0; lambda(B))] - (1/2) sum x_i^2] -> -sum log(1 - theta x_i), uniformly in a fixed complex neighborhood of 0^r, for every fixed r. If this derivation cannot be carried out, the BBP example is not covered by Theorem 12; if it can, the load-bearing assumption is verified for the flagship example, and the remaining issue is exposition. A second, independent check is to recompute the correction measure (123) of Example 7 from Theorem 14 with Psi(x) = (1/alpha - 1) log((1+x)/2) and Phi(x) = log((1+A x)/(1+x)), verifying the delta location, threshold, and continuous density.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result, Theorem 12, is an implication: if the averaged Harish-Chandra transform admits the two-term expansion (79), uniformly in a complex neighborhood of 0^r for every fixed r, then the explicit correction formula (81) follows. The theorem itself is not internally inconsistent; the proofs of Theorems 8 and 12 are detailed and the algebra can be checked line by line. The load-bearing weakness is that the paper does not supply a general criterion or derivation of this analytic hypothesis from the random matrix model. In particular, for the finite-rank perturbation examples (Examples 4, 5, 9, 11), the paper moves from condition (40) for an ergodic unitarily invariant A to condition (79) for B = A + sum_i N theta_i E_{i,i} with only a brief appeal to Lemma 6, without displaying the full asymptotic verification. If the rank-one term introduces non-uniformity in the complex neighborhood, or if the two-term expansion is not analytic in the required region, then the claimed correction moments and the BBP-type phase transitions drawn from them are unsupported. Example 7 also explicitly omits the lengthy computation that produces the correction measure (123). None of this contradicts the conditional statements, but it means the advertised applicability to finite-rank perturbations and to the Aztec diamond model rests on asserted rather than fully demonstrated analytic inputs.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general asymptotic calculus for averaged Harish-Chandra transforms and Schur generating functions. Its main results are conditional theorems: if the logarithm of the averaged transform admits a prescribed multi-term expansion in a complex neighborhood of the origin (or of 1^r), uniformly in r, then the moments of the averaged empirical measure have explicit expansions, with correction terms expressed through derivatives of the functions Ψ and Φ. Theorem 12 gives the first-order correction, Theorem 14 its Schur-generating analog, Theorems 16 and 17 extend to second and higher order corrections, and Theorem 11 covers an intermediate scaling regime. The paper also connects the correction formulas to infinitesimal free cumulants, introduces quantized infinitesimal cumulants, and applies the results to finite-rank perturbations of random matrices and to a perturbed Aztec diamond tiling model, where it exhibits BBP-type phase transitions.","tokens_in":65441,"tokens_out":4840,"duration_ms":53792,"significance":"If the main theorems are correct, they provide a useful general toolbox: a parameter-free formula for the 1/N correction of averaged empirical measures from two-term Harish-Chandra asymptotics, a quantized analog for Schur generating functions, and explicit higher-order corrections. The proofs are detailed and largely self-contained, built on differential operators and symmetrization arguments, and the connections to infinitesimal freeness are explicit and concrete. The paper also gives credit to prior work and reproduces known results such as Shlyakhtenko's correction measure for finite-rank perturbations, which lends credibility to the method. The advertised applications, including the Aztec-diamond outlier effect, are potentially significant if the underlying asymptotic assumptions are fully justified.","major_comments":[{"comment":"The application to finite-rank perturbations assumes that for B = A + Σ_i N θ_i E_{i,i} the averaged Harish-Chandra transform satisfies the two-term asymptotic (79) with the required uniformity in a complex neighborhood of 0^r. The paper proves that (79) implies the moment formula (80), and Lemma 6 provides the asymptotic for the deterministic rank-one term, but it never verifies the combined condition (79) for the average over A. In particular, the uniformity in a complex neighborhood of 0^r is not established, and the possible non-commutativity of taking logarithms and averaging is not addressed. Since the BBP-type formulas (96) in Examples 4 and 5 rest on this unproved input, the advertised phase transition for finite-rank perturbations is not fully derived. The authors should either prove the required expansion for B from the standing assumptions on A or state as a separate lemma the precise analytic conditions under which (96) holds.","section":"Section 5, eq. (96) and Examples 4–5"},{"comment":"The explicit correction measure for the Aztec-diamond perturbation is stated after 'somewhat lengthy computations (we omit them for brevity)'. This is one of the two new applications advertised in the introduction, and formula (123) contains several nontrivial ingredients: a delta measure with a critical threshold, a sign change at α = (A+1)^2/(2(A^2+1)), and a rational density inside the arctic circle. Because the computation is omitted, the claim (17)–(19) is not verifiable from the manuscript. The authors should include the computation, or at least a detailed outline sufficient for an expert to reproduce it independently.","section":"Example 7, eq. (123)"},{"comment":"The statement of Theorem 17 is not well-formed as written. In the Harish-Chandra transform on the left-hand side of (141), the second argument is written as (λ_1(A), ..., λ_n(A)); it should be the full N-tuple of eigenvalues, presumably λ_1(A), ..., λ_N(A). Also, the inner sum over i is written as running from i = 0 to r, but the terms Ψ_j(x_i) are evaluated at the r variables, so the sum should run from i = 1 to r. The condition on ε should also be clarified: the proof and the intended scaling require 0 < ε < n^{-1}, and the printed '0 < ε < n−1' is ambiguous. These are notational typos, but they affect a central theorem statement and must be corrected.","section":"Theorem 17, eq. (141)"}],"minor_comments":[{"comment":"There are stray '− −' artifacts in the displayed evaluation notation, e.g., '((Ψ′(x))^{n−m}) − − x=0'; these should read '|_{x=0}' for clarity.","section":"Eq. (7) and eq. (54)"},{"comment":"The limit condition 'M,N→∞, M−λN→0' should presumably be 'M/N → λ'; as written, the condition involving M−λN is dimensionally inconsistent with the notation.","section":"Example 3, eq. (94)"},{"comment":"The variable y in (123) is not explicitly defined; the introduction formula (19) uses t, and the geometric discussion later introduces y˜ = α y. The authors should define the coordinate y used in the correction measure.","section":"Example 7, paragraph after eq. (123)"},{"comment":"There is a typo 'normaizing constant' in the description of the perturbed Aztec diamond measure; it should be 'normalizing constant'.","section":"Example 7"}],"recommendation":"major_revision","confidential_remarks":"The paper is long and contains many conditional results; the conditional nature of the theorems is an accepted framing, but the advertised applications require the analytic hypotheses to be verified in each example. The main gap is the verification of the two-term expansion for finite-rank perturbations, and the omitted computation in Example 7. If these are supplied, the paper would be a solid contribution to the toolbox of Harish-Chandra and Schur-generating asymptotics. The fit with the journal is good, though the manuscript is quite long and could benefit from a more prominent statement of what is proven versus what is assumed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is the take: this paper does real work. It takes the Bufetov--Gorin method for laws of large numbers from Harish-Chandra transforms and extends it to the next terms of the asymptotic expansion of averaged empirical measures, connecting those corrections to infinitesimal free probability and a quantized variant. The general Theorems 12, 14, 16, 17 are new and proven in detail via the differential operator D_k and symmetrization lemmas. The algebra is checkable, the hypotheses are stated cleanly, and the framework is genuinely more general than what existed. The examples, especially the discrete BBP-type transition and the Aztec diamond perturbation, are new predictions and look like the right kind of application. There is no circularity: the theorems are conditional implications from the asymptotic assumptions to explicit moment formulas, with no fitted parameters.\n\nThe soft spots are at the boundary between the general theorems and the examples. Every main theorem is conditional on a uniform multi-term asymptotic expansion of the log Harish-Chandra transform in a complex neighborhood (e.g., (79)). That input is not derived inside the paper. For the finite-rank perturbation examples, the step from the ergodic condition to the two-term expansion is a brief appeal to Lemma 6; the required uniformity and analyticity in the complex neighborhood are not displayed. If that expansion is not uniform in the needed domain, the correction formulas, and the BBP phase transitions drawn from them, lose their support. Example 7 also omits the 'lengthy computations' that produce the correction measure (123); that is a real gap in the paper's own example section. Theorem 17 has apparent typos in the statement (the argument list of the Harish-Chandra transform is inconsistent). None of this contradicts the conditional implications, but it means the advertised applicability is asserted rather than fully demonstrated.\n\nWho gets value: anyone working on Harish-Chandra transforms, Schur generating functions, or infinitesimal free probability. The general theorems deserve a serious referee, and the examples should be checked or the omitted computations supplied. I would accept it for peer review, with a request that the analytic hypotheses be verified directly for the finite-rank and Aztec diamond models before publication.","headline":"The general theorems are a genuine extension of the BG toolbox and the proofs hold up; the applications lean on asserted analytic inputs and at least one omitted computation, so referee time is justified but the examples need checking.","tokens_in":65986,"tokens_out":3831,"would_cite":true,"duration_ms":37180,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","46L54"],"pacs":[],"model":"deepseek-v4-flash","headline":"The 1/N correction to the empirical spectral measure of a unitarily invariant random matrix is determined by an explicit moment formula from the two leading terms of its averaged Harish-Chandra transform.","keywords":["Harish-Chandra transform","infinitesimal free probability","infinitesimal free cumulants","Schur generating functions","quantized free convolution","BBP phase transition","random matrix asymptotics","empirical measure"],"falsifier":"Take the explicit model $A=N\\cdot\\mathrm{GUE}+N\\theta E_{11}$ with fixed $\\theta$, as in Example 4, compute the limit $N\\big(N^{-(k+1)}\\mathbb{E}[\\sum_i\\lambda_i^k]-\\mu_k\\big)$ for $k=1,2,3$ by exact diagonalization, and compare with formula (96); any mismatch would falsify Theorem 12, while agreement at several $k$ would support the claimed correction mechanism.","tokens_in":64988,"feed_emoji":"📊","tokens_out":10237,"duration_ms":96610,"temperature":0.7,"pith_summary":"This paper studies what the Harish-Chandra transform—the unitary-group average of $\\exp(\\operatorname{Tr}(AUBU^*))$—can reveal about the eigenvalues of a large random Hermitian matrix $A$. Its central result is a general theorem: if the logarithm of the averaged transform has a two-term asymptotic expansion with leading function $\\Psi$ and $1/N$ correction $\\Phi$, then the average empirical measure of $N^{-1}A$ has a computable first-order correction whose $k$-th moment is an explicit polynomial in derivatives of $\\Psi'$ and $\\Phi'$. This connects the second term of the Harish-Chandra expansion to infinitesimal free cumulants, turning analytic information about the transform into distributional information about eigenvalue outliers. The same mechanism is developed for Schur generating functions, giving a quantized version of infinitesimal free probability with explicit outlier formulas, and it is iterated to second- and higher-order $1/N$ corrections.","feed_headline":"Explicit formula gives the 1/N correction of random-matrix spectra","feed_subtitle":"If the log Harish-Chandra transform has a two-term expansion, the whole first-order correction is computable.","key_machinery":"The load-bearing object is the averaged Harish-Chandra transform $f_N(x_1,\\ldots,x_N)=\\mathbb{E}\\int_{U(N)}e^{\\operatorname{Tr}(AUBU^*)}\\,dU$ together with the differential operator $D_kf=(\\prod_{i<j}(x_i-x_j))^{-1}\\sum_i\\partial_i^k\\big(\\prod_{i<j}(x_i-x_j)f\\big)$, for which $D_kf_N|_{x=0}=\\mathbb{E}[\\operatorname{Tr}(A^k)]$. Writing $f_N=\\exp(N\\cdot\\frac1N\\log f_N)$, the chain rule and a symmetrization lemma show that, after taking $x_i=i\\varepsilon\\to0$, only certain derivatives of $\\frac1N\\log f_N$ survive; assumption (79) then converts those derivatives into derivatives of $\\Psi$ and $\\Phi$, producing formula (81). The discrete analogue uses the operator $(u_i\\partial_i)^k$ on Schur generating functions around $u_i=1$, which yields the quantized infinitesimal cumulants.","core_discovery":"On the paper's own terms, the discovery is Theorem 12: under uniform convergence of $N\\big(\\frac{1}{N}\\log \\mathbb{E}[\\mathrm{HC}(x_1,\\ldots,x_r,0^{N-r};\\lambda_1(A),\\ldots,\\lambda_N(A))]-\\sum_{i=1}^r\\Psi(x_i)\\big)\\to\\sum_{i=1}^r\\Phi(x_i)$ in a complex neighborhood of $0^r$ for every fixed $r$, the $1/N$ correction of the averaged empirical measure of $N^{-1}A$ has moments $\\mu'_k=\\sum_{m=0}^{k-1}\\frac{k!}{m!(m+1)!(k-m-1)!}\\frac{d^m}{dx^m}\\big((\\Psi'(x))^{k-m-1}\\Phi'(x)\\big)\\big|_{x=0}$. The leading moments are the same expression with $\\Phi$ absent, and $\\Psi'$ is the $R$-transform of the limiting measure, so $\\Phi'$ supplies the infinitesimal free cumulants. With the replacement $\\Phi'(x)\\mapsto \\Phi'(x)-\\frac{1}{2x}$ and evaluation at $x=1$, the same theorem holds for Schur generating functions, and iterating the machinery yields explicit second- and higher-order formulas.","pith_inferences":["Editorial inference: the moment formula is effectively a Taylor-coefficient machine: models whose log Harish-Chandra transform is analytic in the scaling parameter can be fed directly into the Stieltjes-transform reduction of Remark 5, giving explicit signed densities for corrections without moment-by-moment computation.","Editorial inference: if the authors' belief that all odd terms in the $1/N$ expansion vanish is correct, then this class of models has a parity gap in its correction expansion; checking the third $1/N^3$ term in a concrete ensemble would test that belief directly.","Editorial inference: the quantized infinitesimal cumulants introduced for Schur generating functions should linearize the $1/N$ correction to tensor-product operations in the same way the quantized $R$-transform linearizes the leading-order quantized free convolution; this is testable by expanding characters of tensor products of large $U(N)$ representations."],"forward_implications":["For any ensemble whose log-averaged Harish-Chandra transform satisfies (79), the full first-order correction to the empirical spectral measure is computable from $\\Psi'$ and $\\Phi'$ without knowing the transform otherwise.","The correction moments coincide with infinitesimal free cumulants $\\kappa'_n=\\Phi^{(n)}(0)/(n-1)!$, so the theorem places finite-rank perturbations of unitarily invariant ensembles inside infinitesimal free probability.","In the Schur generating function setting, the analogous theorem gives quantized infinitesimal cumulants and predicts outliers at threshold values; concretely, it yields a discrete BBP phase transition in representation-theoretic models and in a perturbed Aztec diamond tiling model.","Under the stronger $N^2$-scale expansion with an additional function $T$, the $1/N^2$ correction is explicit and splits into a second-order infinitesimal freeness term plus a model-specific term, with similar formulas for all higher orders on the $N^{n\\varepsilon}$ scale."],"supporting_citations":[{"why":"Supplies the base Law of Large Numbers result via Harish-Chandra transforms and the differential-operator proof that Theorem 12 refines, as well as the Schur generating function framework.","marker":"[BG]"},{"why":"Provides the second (log-scale) limit regime and the finite-rank Harish-Chandra asymptotics used for the correction examples.","marker":"[OV]"},{"why":"Introduces infinitesimal free probability, the framework that Theorem 12 connects to through the function $\\Phi$.","marker":"[BS]"},{"why":"Defines infinitesimal non-crossing cumulants, which are used in Lemma 5 and in the cumulant interpretation of the correction formula.","marker":"[FN]"},{"why":"Describes the phase transition for the largest eigenvalue under finite-rank perturbation, whose discrete and higher-order analogs the paper derives.","marker":"[BBP]"},{"why":"Gives finite-rank perturbation results through type B free probability, matching the corrections computed in Example 4.","marker":"[Sh]"},{"why":"Introduces higher-order infinitesimal freeness, which the second- and higher-order correction formulas are matched to.","marker":"[F]"},{"why":"Provides the non-crossing partition moment-cumulant formalism underlying the combinatorial calculations.","marker":"[Sp]"}],"fun_headline_variants":["Explicit 1/N correction for random-matrix spectra","Infinitesimal freeness from Harish-Chandra transform","First-order correction computed via Harish-Chandra","Beyond law of large numbers: explicit 1/N term","Harish-Chandra transform yields infinitesimal cumulants"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole chain of explicit correction formulas rests on one analytic input that the paper does not prove: the averaged Harish-Chandra transform (or Schur generating function) must have the stated two-term asymptotic expansion uniformly near the zero vector (or, in the discrete case, near the all-ones vector) for every fixed number of nonzero arguments.","fun_headline_variants_meta":{"raw":{"variants":["Explicit 1/N correction for random-matrix spectra","Infinitesimal freeness from Harish-Chandra transform","First-order correction computed via Harish-Chandra","Beyond law of large numbers: explicit 1/N term","Harish-Chandra transform yields infinitesimal cumulants"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000311,"raw_usage":{"total_tokens":1774,"prompt_tokens":949,"completion_tokens":825,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":565,"completion_tokens_details":{"reasoning_tokens":746}},"tokens_in":565,"tokens_out":825,"duration_ms":7808,"temperature":1.0,"reasoning_tokens":746,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:06:10.041038+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the explicit model $A=N\\cdot\\mathrm{GUE}+N\\theta E_{11}$ with fixed $\\theta$, as in Example 4, compute the limit $N\\big(N^{-(k+1)}\\mathbb{E}[\\sum_i\\lambda_i^k]-\\mu_k\\big)$ for $k=1,2,3$ by exact diagonalization, and compare with formula (96); any mismatch would falsify Theorem 12, while agreement at several $k$ would support the claimed correction mechanism.","supporting_citations":[],"review_version":1}