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$q$-deformed Perelomov-Popov measures and quantized free probability

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.03213 v1 pith:BOFBK4RI submitted 2025-01-06 math.PR

Panagiotis Zografos This is my paper · ORCID
classification math.PR MSC 60B2005E05
keywords q-deformedPerelomov-PopovmeasuresSchurgeneratingfunctionsfreeprobabilityR-transforminfinitesimalMarkov-Kreincorrespondenceasymptoticrepresentationtheoryrandomsignatures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $q$.

What would settle it

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.

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Extended reading notes

Core claim

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 $q$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

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.

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 (3)
  1. [§2.3, Theorem 4 and Remark 3] 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.
  2. [§3.2, Theorem 6] 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.
  3. [§2.2, proof of Theorem 3] 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.
minor comments (5)
  1. [Theorem 1, relation (3)] 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})).
  2. [Throughout] 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.
  3. [Definition 2 and eq. (17)] 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.
  4. [Example 1] 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.
  5. [Theorem 7, proof] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the q-deformed limit theorems are derived from Schur-generating-function asymptotics via differential operators; the heavy use of the companion preprint [14] is a dependence on prior work, not a self-definitional or fitted-input circularity.

full rationale

The paper's central claims are Theorems 3 and 4, which convert asymptotic conditions on log S_{\rho(N)} into moment formulas and then into an R-transform formula. The inputs are the Schur-generating-function asymptotics (conditions (9) and (19)); the outputs are moments and free cumulants of the q-deformed Perelomov-Popov limiting measure. These are not the same object by construction: the moments are obtained by applying the differential operators D_{U(N),q}^{(k)} and evaluating at the identity, and the R-transform is extracted from the moment sequence via Lemma 4 of [14]. No parameter of the limiting measure is fitted to the data, and the non-asymptotic relation (50) is a corollary of the symmetric-polynomial identity in Proposition 2 rather than an assumption. The main legitimate concern is structural: several technically load-bearing steps in the proofs of Theorems 3, 4, 5, and 6 are delegated to the same-author companion preprint [14], including the derivation of identity (23) 'using the same arguments as in Lemma 1 of [14]', the convergence analysis 'explained in detail in Theorem 8 of [14]', the moment-to-R-transform bridge (Lemma 4 of [14]), and the handling of mixed derivatives in Theorem 6 via Theorem 15 of [14]. This is reliance on prior work whose hypotheses are not restated or verified in the present text, and for q \neq 0 the paper itself notes that mixed derivatives do not vanish, so the transfer of estimates from [14] is not automatic. However, reliance on a companion preprint is not circular reasoning: the companion results are not the target conclusions of this paper (they concern q = 0 / Harish-Chandra-type asymptotics), and the present paper does not rename or assume the q-deformed R-transform formula as an input. The q-deformed convolution is explicitly introduced as an operation motivated by the proved free-cumulant formulas, not presented as a derived external object. Thus the appropriate finding is a modest score reflecting the heavy, unverified self-citation dependence, not a circularity score for a derivation that reduces to its own inputs.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central results depend on the assumed additive asymptotics of Schur generating functions, standard representation-theoretic tools, and the validity of technical lemmas in the companion preprint. The only explicit parameter is q, which is not fitted. No new physical or ontological entities are postulated; the new mathematical objects (q-deformed measures, quantized R-transform) are defined and their properties derived.

free parameters (1)
  • q = q in [-1,1]
    Deformation parameter of the Perelomov-Popov measures; it is not fitted to data but is the new parameter that interpolates between known q=0 and q=1 cases. All main formulas depend on it.
assumptions (4)
  • domain assumption The normalized log Schur generating function has uniform additive asymptotics (condition (9)/(19)): the limit of (1/N) log S_{rho(N)} is a sum of one-variable functions Psi, and similarly with Phi under condition (30).
    This is the hypothesis of Theorems 3-6; it is not proved. It is standard in the Schur-generating-function framework but it is the condition that selects which signature distributions are covered.
  • domain assumption The limiting objects are determined by their moment sequences and have compact support where R-transforms are used.
    Passing from moment formulas to the R-transform via Lemma 4 of [14], and speaking of a deterministic probability measure, requires moment determinacy or tightness; the theorems state convergence in the sense of moments but do not explicitly prove compact support or determinacy.
  • standard math Harish-Chandra/Itzykson-Zuber integral formula and Weyl dimension formula.
    Used in Proposition 3 to evaluate shifted characters at 1_N; standard representation-theoretic input.
  • standard math Speicher moment-cumulant relations, the linearizing property of free cumulants for free convolution, and Markov-Krein bijectivity from [1] and [31].
    Used throughout Sections 2.3 and 4 to identify R-transforms and define the q-deformed convolution; these are imported from existing literature.

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Pith. "Pith review of $q$-deformed Perelomov-Popov measures and quantized free probability." pith.science (2026). https://pith.science/paper/BOFBK4RI

@misc{pith2026250103213,
  author       = {Pith},
  title        = {Pith review of: $q$-deformed Perelomov-Popov measures and quantized free probability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BOFBK4RI}},
  note         = {Machine review of arXiv:2501.03213}
}
abstract

The asymptotic study of tuples of random non-increasing integers is crucial for probabilistic models coming from asymptotic representation theory and statistical physics. We study the global behavior of such tuples, introducing a new family of discrete probability measures, depending on a parameter $q \in [- 1, 1]$. We prove the Law of Large Numbers for these measures based on the asymptotics of the Schur generating functions and we provide explicit formulas for the moments and the free cumulants of the limiting measures. Our results provide an interpolation between the results of Bufetov and Gorin for $q = 0, 1$, who distinguished these two cases from the side of free probability theory. We show the connection with free probability theory and we introduce a deformation of free convolution, motivated by our formulas for the free cumulants. We also study the first order correction to the Law of Large Numbers and we make the connection with infinitesimal free probability, computing explicitly the infinitesimal moments and the infinitesimal free cumulants. Finally, we prove non-asymptotic relations between the limiting measures for different $q$, which are related to the celebrated Markov-Krein correspondence.

Figures

Figures reproduced from arXiv: 2501.03213 by the authors.

Figure 1
Figure 1. Graph of y = ffγ,q(x) for γ = 0.25 and different values of q ∈ [−1,1]. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.