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arxiv: 2409.04305 · v3 · submitted 2024-09-06 · 🧮 math.CO · math.PR

Cumulants in rectangular finite free probability and beta-deformed singular values

Cesar Cuenca This is my paper

Pith reviewed 2026-05-23 20:56 UTC · model grok-4.3

classification 🧮 math.CO math.PR
keywords rectangular cumulantsfinite free probabilityrectangular convolutionmoment-cumulant formulasbeta-deformed singular valuesroot distributionspolynomials
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The pith

The (n,d)-rectangular cumulants linearize the rectangular convolution and converge to q-rectangular free cumulants as d tends to infinity with 1+n/d to q.

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

The paper defines (n,d)-rectangular cumulants for polynomials of degree d. It establishes moment-cumulant formulas through direct algebraic calculations that also produce quantum versions. These cumulants are shown to linearize the (n,d)-rectangular convolution of finite free probability. In the limit d to infinity with the ratio 1+n/d approaching a fixed q at least 1, the cumulants approach the q-rectangular free cumulants of free probability. The formulas are then used to analyze asymptotic symmetric empirical root distributions for sequences of nonnegative-root polynomials, including the effect of a differential operator that asymptotically produces rectangular free convolution with a Gaussian of specified variance.

Core claim

The (n,d)-rectangular cumulants are defined so that they linearize the (n,d)-rectangular convolution from finite free probability; they converge to the q-rectangular free cumulants when d tends to infinity and 1 plus n over d tends to q in [1, infinity).

What carries the argument

The (n,d)-rectangular cumulants, which satisfy explicit moment-cumulant formulas obtained by algebraic manipulation and linearize the rectangular convolution.

If this is right

  • The cumulants turn convolution of polynomials into addition of cumulant sequences.
  • Moment-cumulant formulas yield explicit ways to pass between moments and cumulants for rectangular finite free probability.
  • In the large-d limit the formulas recover the corresponding relations in free probability.
  • The differential operator exp of minus s squared over n times x to the minus n D sub x x to the n plus 1 D sub x asymptotically produces rectangular free convolution with a rectangular Gaussian of variance q s squared over q minus 1.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same algebraic approach may extend to other deformations of free probability beyond the rectangular case.
  • Quantum analogues of the moment-cumulant formulas suggest direct links to noncommutative structures arising in quantum information or operator algebras.
  • The root-distribution application could be tested numerically on explicit polynomial families to check the rate of convergence to the free-probability limit.

Load-bearing premise

The linearization and the convergence to q-rectangular free cumulants both rest on the specific scaling regime d to infinity with 1 plus n over d to q and on the prior definition of related cumulants.

What would settle it

A concrete pair of degree-d polynomials whose rectangular convolution has (n,d)-rectangular cumulants that are not the sum of the individual cumulants would disprove the linearization property.

Figures

Figures reproduced from arXiv: 2409.04305 by Cesar Cuenca.

Figure 1
Figure 1. Figure 1: The four odd Łukasiewicz paths of length [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The only Łukasiewicz path in P ∈ L odd(2). Next, since κn = 0, whenever n ̸= 2, the only Łukasiewicz paths that contribute to the formula (85) will be the ones that only have up steps of size 1 and there are exactly two of them in L odd(4) with that property, namely the ones depicted in [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Łukasiewicz paths in L odd(4) that only have up steps of size 1. 7 Appendix: beta-deformed singular values and eigenval￾ues We briefly explain the occurrence of Eqns. (9)–(10) in the topic of the Appendix’s title. 7.1 High temperature beta-singular values The q-rectangular free convolution was defined by Benaych-Georges [BG-09] in the following setting. For a matrix A ∈ CM×N , M ≤ N, with singular valu… view at source ↗
read the original abstract

Motivated by the $(q,\gamma)$-cumulants, introduced by Xu [arXiv:2303.13812] to study $\beta$-deformed singular values of random matrices, we define the $(n,d)$-rectangular cumulants for polynomials of degree $d$ and prove several moment-cumulant formulas by elementary algebraic manipulations; the proof naturally leads to quantum analogues of the formulas. We further show that the $(n,d)$-rectangular cumulants linearize the $(n,d)$-rectangular convolution from Finite Free Probability and that they converge to the $q$-rectangular free cumulants from Free Probability in the regime where $d\to\infty$, $1+n/d\to q\in[1,\infty)$. As an application, we employ our formulas to study limits of symmetric empirical root distributions of sequences of polynomials with nonnegative roots. One of our results is akin to a theorem of Kabluchko [arXiv:2203.05533] and shows that applying the operator $\exp(-\frac{s^2}{n}x^{-n}D_xx^{n+1}D_x)$, where $s>0$, asymptotically amounts to taking the rectangular free convolution with the rectangular Gaussian distribution of variance $qs^2/(q-1)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper defines (n,d)-rectangular cumulants for polynomials of degree d via moment-cumulant formulas obtained by elementary algebraic manipulations. It proves that these cumulants linearize the (n,d)-rectangular convolution from Finite Free Probability and establishes their convergence to the q-rectangular free cumulants from Free Probability in the regime d→∞ with 1+n/d→q∈[1,∞). As an application, the formulas are used to analyze limits of symmetric empirical root distributions of sequences of polynomials with nonnegative roots, including an asymptotic result showing that the operator exp(−s²/n x^{-n} D_x x^{n+1} D_x) corresponds to rectangular free convolution with a rectangular Gaussian of variance qs²/(q−1).

Significance. If the linearization and convergence hold, the work supplies explicit algebraic tools that connect finite free probability to the rectangular free probability framework of Xu, enabling concrete computations for β-deformed singular values and asymptotic root distributions. The elementary derivations of the moment-cumulant formulas and the linearization property are clear strengths; the application result analogous to Kabluchko's theorem adds a falsifiable prediction in the polynomial setting.

major comments (1)
  1. [convergence statement (abstract and main convergence section)] The convergence claim (abstract and the section establishing the limit) identifies the scaled (n,d)-rectangular cumulants with Xu's (q,γ)-cumulants. The manuscript must explicitly verify that the limit reproduces Xu's generating function or recurrence relation exactly; reliance on shared motivation alone leaves the identification step unverified and load-bearing for the central convergence statement.
minor comments (1)
  1. [introduction] Clarify in the introduction how the (n,d)-rectangular cumulants are distinguished notationally from both classical and free cumulants to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on the convergence claim. We address the major comment below.

read point-by-point responses

  1. Referee: [convergence statement (abstract and main convergence section)] The convergence claim (abstract and the section establishing the limit) identifies the scaled (n,d)-rectangular cumulants with Xu's (q,γ)-cumulants. The manuscript must explicitly verify that the limit reproduces Xu's generating function or recurrence relation exactly; reliance on shared motivation alone leaves the identification step unverified and load-bearing for the central convergence statement.

    Authors: We agree that the identification requires explicit verification beyond shared motivation. In the revised manuscript we will add a direct computation in the convergence section: starting from the explicit moment-cumulant formula for the (n,d)-rectangular cumulants, we take the scaled limit d→∞ with 1+n/d→q and show that the resulting generating function satisfies Xu's recurrence relation exactly (or equivalently matches the generating function in arXiv:2303.13812). This will be presented as a self-contained lemma, removing any ambiguity in the identification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are algebraically independent

full rationale

The paper defines (n,d)-rectangular cumulants via explicit moment-cumulant formulas obtained by elementary algebraic manipulations (abstract), proves linearization of the rectangular convolution directly from those formulas, and states the convergence to Xu's q-rectangular cumulants as a separate scaling limit (d→∞, 1+n/d→q) without reducing the limit identity to a fitted parameter or prior definition by construction. The citation to Xu [arXiv:2303.13812] supplies external motivation but is not load-bearing for the finite-case proofs or the limit claim itself. No self-citation chains, self-definitional loops, or renaming of known results appear in the derivation steps. The central claims remain independent of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard algebraic identities for polynomials and operators plus background definitions from finite free probability; no free parameters, ad-hoc axioms, or new entities are introduced beyond the cumulant definition itself.

axioms (1)
  • standard math Standard algebraic properties of polynomials, derivatives, and the rectangular convolution operation in finite free probability
    Invoked throughout the moment-cumulant formulas and linearization proof.

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