{"id":"be202e0d-5f9e-435c-9f86-7ba6d2a84e69","arxiv_id":"2501.03213","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A q-deformed family of Perelomov-Popov measures is introduced and its large-N limits, free cumulants, 1/N corrections, and Markov-Krein relations are computed for all q in [-1,1].","lead":"This paper introduces a one-parameter family of random measures on integer partitions and proves that, as the size grows, they converge to explicit limiting laws that interpolate between known cases. The limits are described through free probability, giving a q-deformed free convolution and new links to the Markov-Krein transform.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central R-transform formula is supported by outsourced analytic lemmas from the companion preprint [14]; the most load-bearing step is the unverified applicability of Lemma 1 and Theorem 8 of [14] to the q-deformed generating functions T^{(q)}.","rationale":"The reader's conditional verdict identifies the external dependence on [14] as the weakest point, and I agree: the proofs of Theorems 4, 5, and 6 import the core analytic machinery, including Lemma 1, Lemma 4, Lemma 5, Theorem 8, and Theorem 15 of [14], without stating their hypotheses or checking them for the q-deformed generating functions. This is load-bearing because it is exactly at this point that the differential-operator expansion is converted into moment asymptotics and then into the R-transform formula (17). I found no direct algebraic contradiction in the moment formulas; for k=1,2, formula (20) and formula (10) agree, and the q=0,1 limits reduce to the known Bufetov-Gorin results. However, the non-vanishing of mixed derivatives for q≠0, mentioned explicitly in the proof of Theorem 6, shows that the q=0 estimates of [14] cannot be assumed verbatim without verification. Minor issues such as the 'Theorem 13' reference in Example 1 and the absence of an explicit tightness or moment-determinacy discussion do not change the assessment. The appropriate action is to keep the reader's CONDITIONAL verdict: acceptance should await confirmation that the companion preprint is available and that its lemmas apply to the q-deformed setting.","tokens_in":29727,"tokens_out":11392,"duration_ms":105822,"concrete_test":"Independently re-derive equality (23) and the leading asymptotics (24) for q ≠ 0, k = 2 and k = 3, with Ψ(u) = γ(u-1), verifying that the symmetrized sums are independent of b0,...,bm at u1=...=uN=0 and that the leading coefficient M^{(q)}_{0,k,N}(Ψ) is exactly as stated in Theorem 4. If the leading term differs, the moment formula collapses; if it matches, the remaining gap is the availability and correctness of [14]'s Lemma 4, whose conclusion should additionally be checked directly by verifying that the moment sequence defined by (20) has free cumulants κ_n = (1/(n-1)!) d^{n-1}/du^{n-1} F_q(u)|_{u=0}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4's conclusion (20) and the R-transform formula (17) both depend on the assertion that the differential-operator expansion can be analyzed exactly as in [14]. In the proof, equality (23) is obtained 'using the same arguments as in Lemma 1 of [14]', and the convergence of the symmetrized sums (24) is deferred to 'Theorem 8 of [14]'. The companion [14] is written for the q=0 case, where the operator has the simpler weight (1 - q u_i)^{k+1} with q=0 and where mixed derivatives of the logarithmic limit vanish. For q ≠ 0, the paper itself notes in the proof of Theorem 6 that mixed derivatives such as ∂_{u1}∂_{u2} log T^{(q)}_{ρ(N)} do not vanish (see the discussion before (45)), so it is not automatic that the estimates in [14] apply unchanged. Lemma 4 of [14] is also the only bridge from moment asymptotics to the R-transform; if its hypotheses require compact support, moment determinacy, or a particular form of the moment sequence, those hypotheses are not checked in the present text. The same holds for Lemma 5 and Theorem 15, which are used in Theorem 6. Thus the central claim is conditional on a companion preprint whose precise hypotheses are neither stated nor verified here.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a one-parameter family of discrete random measures on signatures, the q-deformed Perelomov-Popov measures m_{N,PP(q)}[λ(N)] for q ∈ [−1,1], and studies their global asymptotics as the dimension N tends to infinity. Under asymptotic additivity assumptions on the logarithm of the Schur generating function (conditions (9) and (19)), the author proves convergence of moments in expectation and in probability, giving explicit formulas (10) and (20) for the limiting moments. The main structural result is the R-transform formula (17), R^{(q)}(z) = e_q(z)Ψ'(e_q(z)) + e_q(z)/(e_q(z)-1) - 1/z, which interpolates between the known q=0 and q=1 cases of Bufetov and Gorin. The paper further analyzes the 1/N correction to the law of large numbers via infinitesimal free probability (Theorems 5 and 6), introduces a q-deformed quantized free convolution, and proves non-asymptotic relations among the limiting measures for different q, connecting them to the Markov-Krein correspondence. A concrete example gives a one-parameter family of densities interpolating between semicircle, Marchenko-Pastur, and one-sided Plancherel distributions.","tokens_in":29996,"tokens_out":9518,"duration_ms":90126,"significance":"If the main theorems are correct, the paper provides a genuinely unifying q-family: the two previously separate regimes q=0 and q=1 are recovered from a single R-transform formula, and the deformation of free convolution and of the infinitesimal free convolution are natural and explicitly computable. The non-asymptotic Markov-Krein-type relations and the explicit densities in Example 1 are attractive and potentially useful. The paper also gives credit where due by clearly attributing the q=0,1 results to Bufetov-Gorin and by spelling out the role of the companion preprint [14]. However, the central analytic theorems are not self-contained: the proofs of Theorems 4 and 6 import several load-bearing lemmas from the companion preprint [14] without stating their hypotheses or verifying them for the q-deformed objects. The significance of the paper is therefore conditional on the validity and applicability of those external results.","major_comments":[{"comment":"The central R-transform formula (17) and the moment formula (20) rest on imported results that are neither stated nor proved here: equality (23) is obtained 'using the same arguments as in Lemma 1 of [14]', the convergence of the symmetrized sums (24) is deferred to 'Theorem 8 of [14]', and the passage from the moment formula to the R-transform is attributed to 'Lemma 4 of [14]'. Since [14] is written for the q=0 case, where mixed derivatives vanish, its hypotheses are not automatic for T^{(q)}_{ρ(N)} with q≠0; the text itself notes before (45) that mixed derivatives such as ∂_{u1}∂_{u2} log T^{(q)} do not vanish. The manuscript should either prove these lemmas in the q-deformed setting or state and verify the precise hypotheses under which they apply to the functions considered here.","section":"§2.3, Theorem 4 and Remark 3"},{"comment":"The infinitesimal free cumulant formulas (38)–(39) and the computation of the 1/N correction depend on Lemma 5 and Theorem 15 of [14], which are invoked for the treatment of mixed derivative contributions around (45). For q≠0, the non-vanishing of mixed second derivatives is exactly the point that may invalidate a direct transfer of the q=0 estimates from [14]. Without a statement and verification of the hypotheses of these external results, Theorem 6 is unsupported. This is particularly important because Theorem 2/Theorem 6 are presented as new infinitesimal-level results, not merely as routine extensions.","section":"§3.2, Theorem 6"},{"comment":"Convergence in probability is not actually proved in the text. The proof says that convergence in probability for q=0 was proved in [10] via relation (15), and that 'by Proposition 2 all the moments of m_{N,PP(q)}[ρ(N)] converge in probability if all the moments of m_{N,PP(0)}[ρ(N)] converge in probability'. The latter transfer is only sketched, and it does not address the higher powers E[(moment)^l] that are needed for convergence in probability of moments. Since Theorem 1 and Theorem 4 assert convergence in probability, this step should be written out or replaced by a precise reference that covers the q-deformed measures.","section":"§2.2, proof of Theorem 3"}],"minor_comments":[{"comment":"Formula (3) contains the expression (1 - q Σ μ_k z^{k+1})^{1/q}, which is undefined at q=0; the statement should explicitly say that q=0 is understood by continuity (the limit giving exp(-Σ μ_k z^{k+1})).","section":"Theorem 1, relation (3)"},{"comment":"There are numerous typos and small presentation issues: 'whith moments' in Theorem 1, 'Marcheno-Pastur' in §2.4, 'Lebesque' in several places, 'conections' in §1.1, 'Or results provide' in §1.1, and inconsistent spacing in 'V oiculescu'. These should be corrected.","section":"Throughout"},{"comment":"The function e_q(u) = (1 - q u)^{-1/q} is multivalued for q<0 unless a branch is specified. The paper should specify that the principal branch is taken in a sufficiently small neighborhood of 0, and similarly for the fractional powers appearing in the non-asymptotic relations.","section":"Definition 2 and eq. (17)"},{"comment":"The displayed densities f_{γ,q}(t) and f̃_{γ,q}(t) contain factors 1/q and are not directly meaningful at q=0; since the q=0 case is one of the main interpolation endpoints, the limiting interpretation should be stated explicitly.","section":"Example 1"},{"comment":"The proof is a single sentence ('The claim is a corollary of Proposition 2'). Since the relation (50) involves a limit as q→0, it would be helpful to spell out the continuity argument that justifies interchanging the limit in q with the limit in N.","section":"Theorem 7, proof"}],"recommendation":"major_revision","confidential_remarks":"The main risk is structural: the paper's core analytic results are conditional on the companion preprint [14], which is by the same author and not yet peer-reviewed as far as the manuscript indicates. The presented paper is not independently checkable as written. I would ask the editor to require that the companion preprint be made available and that the hypotheses of Lemmas 1, 4, 5 and Theorems 8, 15 of [14] be stated and verified for the q-deformed Schur generating functions, or that proofs be included in the present paper. The interpolation formulas are attractive and worth publishing if the dependence can be resolved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. This is real new mathematics: q-deformed Perelomov-Popov measures for q in [-1,1], an LLN, and an R-transform R^{(q)}(z) = e_q(z)Psi'(e_q(z)) + e_q(z)/(e_q(z)-1) - 1/z that at q=0 and q=1 lands on Bufetov-Gorin and in between is new. The q-deformed convolution, the infinitesimal corrections, and the Markov-Krein-style relations (50)/Theorem 8 are also new. Proposition 2, the combinatorial core, is self-contained and clean, and the small-n cumulant checks (kappa_1 = Psi'(1)+(1-q)/2, kappa_2 = Psi''(1)+Psi'(1)+(1-q^2)/12) are a good sanity pass. Example 1's closed density, interpolating semicircle, Marchenko-Pastur and one-sided Plancherel, is striking; I had not seen it.\n\nThe soft spot is structural. Theorems 3-6 import the analytic machinery (Lemma 1, Lemma 4, Lemma 5, Theorem 8, Theorem 15) from the companion preprint [14], which is written at q=0. For q != 0 the paper concedes mixed derivatives of the log limit no longer vanish (see the discussion before (45)), so the transfer is not automatic. The present text never states the hypotheses of those lemmas nor verifies them. That is a load-bearing gap, though not a visible error; the author flags the q != 0 complication in the text, which makes me think the fix is within reach: either move the lemmas in or state and check the exact hypotheses. Two smaller issues: the moment formulas are asserted as probability measures without an explicit tightness or moment-determinacy check, and the introduction references a 'Theorem 13' that does not exist (the paper ends at Theorem 8). Minor, but sloppy.\n\nThe central derivations are coherent and I found nothing circular. The paper deserves a serious referee; acceptance should be conditional on the companion preprint and its hypotheses. Worth citing and worth a reading-group slot if anyone in your group does Schur generating functions or quantized free probability.","headline":"New q-interpolation between tiling and free-probability limits with elegant formulas and a striking explicit density; core proofs lean on a companion preprint whose hypotheses go unverified.","tokens_in":30537,"tokens_out":3351,"would_cite":true,"duration_ms":29168,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60B20","05E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that a single q-dependent R-transform formula gives the law of large numbers, moments, and free cumulants for q-deformed Perelomov-Popov measures on random signatures for every q in [-1,1], interpolating the known q=0…","keywords":["q-deformed Perelomov-Popov measures","Schur generating functions","free probability","R-transform","infinitesimal free probability","Markov-Krein correspondence","asymptotic representation theory","random signatures"],"falsifier":"Evaluate formula (17) for a concrete $\\Psi$, for instance the extreme-character example $\\Psi(u)=u-1$, and compute the first three free cumulants from the moment formula (10); if the two routes disagree, or if a direct simulation of $m_{N,\\mathrm{PP}(q)}$ for moderately large $N$ does not approach the predicted moments, the central claim fails.","tokens_in":29532,"feed_emoji":"🎲","tokens_out":13918,"duration_ms":101069,"temperature":0.7,"pith_summary":"This paper introduces a one-parameter family of discrete probability measures on random signatures—non-increasing integer tuples that label irreducible representations of the unitary group—and asks how they behave as the dimension $N$ grows. It proves a law of large numbers: under a natural asymptotic-additivity condition on the Schur generating functions of the underlying signatures, the $q$-deformed Perelomov-Popov measures converge in probability, in the sense of moments, to a deterministic limit whose moments and free cumulants are given by explicit formulas. The centerpiece is a single $R$-transform formula valid for every $q\\in[-1,1]$ that reduces at $q=0$ and $q=1$ to the two previously known boundary results, so the claim is that the whole interval is governed by one interpolating formula. The paper also derives the first-order correction as explicit infinitesimal free cumulants and proves non-asymptotic relations between limits for different $q$ that specialize to the Markov-Krein correspondence. A general reader should care because this connects asymptotic representation theory, symmetric functions, and free probability through one parameter, with concrete formulas in each regime.","feed_headline":"All q-deformed limits obey one explicit moment formula","feed_subtitle":"New law of large numbers and free-cumulant formulas connect the q=0 and q=1 regimes with infinitesimal free probability.","key_machinery":"The central objects are the $q$-deformed Perelomov-Popov measures $m_{N,\\mathrm{PP}(q)}[\\lambda^{(N)}]=\\frac1N\\sum_{i=1}^N \\prod_{j\\ne i}\\frac{(\\lambda_i-i)-(\\lambda_j-j)-q}{(\\lambda_i-i)-(\\lambda_j-j)}\\,\\delta_{\\frac{\\lambda_i+N-i}{N}}$, defined for signatures $\\lambda^{(N)}$ (non-increasing integer $N$-tuples), and the $q$-exponential $e_q(u)=(1-qu)^{-1/q}$ that replaces the ordinary exponential when $q\\ne0$. The argument is carried by differential operators $D^{U(N),q}_k$ applied to Schur generating functions—the expectation of normalized characters of $U(N)$—and to their $q$-deformed versions $T^{(q)}_{\\rho^{(N)}}$; these operators turn the asymptotic additivity of $\\frac1N\\log S_{\\rho^{(N)}}$ into explicit moment formulas. A companion-preprint lemma extracts the $R$-transform from those moment expansions, and the symmetric-polynomial identity (7) rewrites the moments through super-symmetric complete homogeneous polynomials, which yields the non-asymptotic relations between different $q$.","core_discovery":"The paper's central claim is Theorem 4: under the asymptotic additivity condition (19) on the normalized logarithms of the Schur generating functions of the random signatures, the $k$-th moment of $m_{N,\\mathrm{PP}(q)}[\\rho^{(N)}]$ converges in probability to $\\sum_{m=0}^{k-1} \\frac{k!}{m!(m+1)!(k-m)!} \\frac{d^m}{du^m}\\left(e_q(u)\\Psi'(e_q(u)) + \\frac{e_q(u)}{e_q(u)-1} - \\frac{1}{u}\\right)^{k-m}\\Big|_{u=0}$, where $e_q(u)=(1-qu)^{-1/q}$. Consequently the limiting measure has $R$-transform $R^{(q)}(z)=e_q(z)\\Psi'(e_q(z)) + \\frac{e_q(z)}{e_q(z)-1} - \\frac{1}{z}$. For $q=0$ and $q=1$ this reduces to the two previously known boundary formulas, so the claim is that a single formula governs the whole interval $q\\in[-1,1]$. An equivalent moment formula in Theorem 3 writes the moments as derivatives of $u^{k-q}(\\Psi'(u))^{k-m}$; when $\\Psi\\equiv0$ the limit is the $\\beta$ distribution $\\beta(1-q,1+q)$. The paper also claims explicit formulas for the $1/N$ correction in terms of infinitesimal free cumulants, and a non-asymptotic relation $\\exp\\left(-q\\sum_{k\\ge0}\\mu_k^{(0)}z^{k+1}\\right)=1-q\\sum_{k\\ge0}\\mu_k^{(q)}z^{k+1}$ linking different values of $q$.","pith_inferences":["If the moment formula (10) extends beyond the paper's hypotheses, the family could interpolate between the semicircle, Marchenko-Pastur, and one-sided Plancherel laws; the paper's Example 1 already shows such an interpolation of densities, but a random-matrix realization is left open.","The non-asymptotic relation is proved for limiting measures; testing whether a finite-$N$ analogue holds for the expectations would give a direct numerical check of the framework.","The $q$-deformed quantized $R$-transform suggests reading $\\otimes_q$ as ordinary free convolution conjugated by the Markov-Krein bijection, which could lead to $q$-deformed Lévy processes or a $q$-analogue of beta-free infinite divisibility—directions the paper does not explore.","For $q=-1$ and $q=1$, the infinitesimal formulas make explicit predictions for outlier eigenvalues of finite-rank perturbations; comparing those with known random-matrix results would test the infinitesimal branch."],"forward_implications":["For $q=0$ and $q=1$, formula (17) reduces to the boundary relations $R^{(0)}(z)=e^z\\Psi'(e^z)+\\frac{e^z}{e^z-1}-\\frac1z$ and $R^{(1)}(z)=\\frac{1}{1-z}\\Psi'\\left(\\frac{1}{1-z}\\right)$, so the theorem continuously interpolates the two known regimes.","When $\\Psi\\equiv0$, the limiting measure is the beta distribution $\\beta(1-q,1+q)$; for general $\\Psi$, its free cumulants are the beta cumulants plus $\\frac{1}{(n-1)!}\\frac{d^{n-1}}{du^{n-1}}e_q(u)\\Psi'(e_q(u))|_{u=0}$.","The operation $\\otimes_q$ defined via the $q$-deformed quantized $R$-transform satisfies $(\\mu_1\\otimes_q\\mu_2)\\boxplus\\beta(1-q,1+q)=\\mu_1\\boxplus\\mu_2$, giving a deformed free convolution for compactly supported measures in the appropriate class.","The non-asymptotic relation $\\exp(-q\\sum_{k\\ge0}\\mu_k^{(0)}z^{k+1})=1-q\\sum_{k\\ge0}\\mu_k^{(q)}z^{k+1}$ links different $q$; at $q=-1$ it is the Markov-Krein correspondence, and the paper shows the $q$-version is a bijection for densities bounded by $1/|q|$.","The $1/N$ correction is described by explicit infinitesimal free cumulants, extending the framework to infinitesimal free probability and outlier-type phenomena."],"supporting_citations":[{"why":"It establishes the q=0 and q=1 boundary cases that this paper interpolates and supplies the differential-operator method for Schur generating functions.","marker":"[10]"},{"why":"It is the companion preprint that provides Lemma 1, Lemma 4, Lemma 5, Theorem 8, and Theorem 15, including the lemma that extracts the R-transform from moment asymptotics.","marker":"[14]"},{"why":"It supplies the Fourier-analytic framework on high-dimensional unitary groups and the convergence conditions for Schur generating functions used by the main theorems.","marker":"[12]"},{"why":"It develops the Schur generating function techniques for fluctuations of particle systems that the proof reuses.","marker":"[11]"},{"why":"It is the origin of the Perelomov-Popov measure for q=1 and motivates the algebraic interpretation of the family.","marker":"[38]"},{"why":"It defines free cumulants and the moment-cumulant relation that identifies the R-transform of the limiting measure.","marker":"[35]"},{"why":"It gives the Markov-Krein correspondence used to interpret the non-asymptotic relations between different values of q.","marker":"[31]"},{"why":"It defines infinitesimal free cumulants, which are used to state the 1/N correction theorems.","marker":"[22]"}],"fun_headline_variants":["Single formula unifies q-deformed random measures","q-deformed laws obey one explicit moment formula","One R-transform bridges q=0 and q=1 regimes","New moment formula covers whole q-interval","Quantized free probability: one formula for all q"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main theorems import several technical lemmas from a companion preprint, including the lemma that turns moment asymptotics into the $R$-transform formula; if those lemmas are wrong or require hypotheses not verified in this paper, the central formulas are unsupported.","fun_headline_variants_meta":{"raw":{"variants":["Single formula unifies q-deformed random measures","q-deformed laws obey one explicit moment formula","One R-transform bridges q=0 and q=1 regimes","New moment formula covers whole q-interval","Quantized free probability: one formula for all q"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1756,"prompt_tokens":1126,"completion_tokens":630,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":742,"completion_tokens_details":{"reasoning_tokens":556}},"tokens_in":742,"tokens_out":630,"duration_ms":17514,"temperature":1.0,"reasoning_tokens":556,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:55:38.398558+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate formula (17) for a concrete $\\Psi$, for instance the extreme-character example $\\Psi(u)=u-1$, and compute the first three free cumulants from the moment formula (10); if the two routes disagree, or if a direct simulation of $m_{N,\\mathrm{PP}(q)}$ for moderately large $N$ does not approach the predicted moments, the central claim fails.","supporting_citations":[{"cited_title":"Representations of classical Lie groups and quantized free convolution","cited_arxiv_id":null,"evidence_quote":"It establishes the q=0 and q=1 boundary cases that this paper interpolates and supplies the differential-operator method for Schur generating functions."},{"cited_title":"Fourier transform on high-dimensional unitary groups with applications to random tilings","cited_arxiv_id":null,"evidence_quote":"It supplies the Fourier-analytic framework on high-dimensional unitary groups and the convergence conditions for Schur generating functions used by the main theorems."},{"cited_title":"Fluctuations of particle systems determined by Schur generating functions","cited_arxiv_id":null,"evidence_quote":"It develops the Schur generating function techniques for fluctuations of particle systems that the proof reuses."},{"cited_title":"Perelomov and Vladimir S","cited_arxiv_id":null,"evidence_quote":"It is the origin of the Perelomov-Popov measure for q=1 and motivates the algebraic interpretation of the family."},{"cited_title":"Lectures on the combinatorics of free probability, volume 335 of London Mathematical Society Lecture Note Series","cited_arxiv_id":null,"evidence_quote":"It defines free cumulants and the moment-cumulant relation that identifies the R-transform of the limiting measure."},{"cited_title":"Krein and Adolf A","cited_arxiv_id":null,"evidence_quote":"It gives the Markov-Krein correspondence used to interpret the non-asymptotic relations between different values of q."},{"cited_title":"Infinitesimal non-crossing cumulants and free probability of type B","cited_arxiv_id":null,"evidence_quote":"It defines infinitesimal free cumulants, which are used to state the 1/N correction theorems."}],"review_version":1}