Finite Free Convolution: Infinitesimal Distributions

Daniel Perales (Notre Dame); Josue Vazquez-Becerra (CIMAT); Octavio Arizmendi (CIMAT)

Finite Free Convolution: Infinitesimal Distributions

T0 review · 0 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper derives explicit order-1/d fluctuation formulas for finite-free additive and multiplicative convolutions of real-rooted polynomials.

desk verdict Solid, genuinely new treatment of 1/d fluctuations for finite-free convolutions; the H-transform is a useful bridge to infinitesimal free probability. read the letter →

arxiv 2505.01705 v2 pith:WOV2EXCK submitted 2025-05-03 math.PR math.COmath.OAmath.SP

Octavio Arizmendi (CIMAT) , Daniel Perales (Notre Dame) , Josue Vazquez-Becerra (CIMAT) This is my paper · ORCID

classification math.PRmath.COmath.OAmath.SP MSC 46L5460B2005A1830C15

keywords finite-freeconvolutioninfinitesimaldistributioncumulantsnon-crossingpartitionsannularpermutationsfreeprobabilityreal-rootedpolynomialsrandommatrices

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

Sequences of real-rooted polynomials have root distributions that converge, as the degree grows, to free convolutions of probability measures. This paper asks what happens one order deeper, when the root distributions fluctuate around their limits by terms of order $1/d$. It establishes that the relevant correction objects, the finite-free cumulant fluctuations, add linearly under finite-free additive convolution, and it provides explicit formulas for the resulting infinitesimal distributions in both the additive and multiplicative settings. These formulas give the first complete description of the $1/d$ fluctuations for these polynomial operations, and they connect the polynomial picture to infinitesimal free probability of type B.

What carries the argument

The load-bearing objects are the finite-free cumulants $\kappa_n(p_d)$, which linearize finite-free additive convolution, and their $1/d$ expansion $\kappa_n(p_d)=r_n(\mu)+\frac{1}{d}\hat r_n(p)+o(1/d)$. The paper shows this expansion exists exactly when the moments admit an infinitesimal expansion, and it links the two via non-crossing partitions and annular non-crossing permutations. The new $H$-transform, defined by $H_\mu(z)=\frac{G'_\mu(z)}{G_\mu(z)}-\frac{G''_\mu(z)}{2G'_\mu(z)}$, absorbs the annular correction and turns the combinatorial formulas into functional relations for infinitesimal Cauchy transforms. In the additive case, composition with the free convolution's Cauchy transform and subordination functions yields a compact formula connecting the finite-free result to infinitesimal free convolution.

What would settle it

For a concrete family such as $p_d=(x-1)^{d-s}\prod_{k=1}^s(x-\alpha_k)$ and $q_d$ similarly, compute the moment $m_n(p_d\boxplus_d q_d)$ numerically for increasing $d$, multiply the deviation from $m_n(\mu\boxplus\nu)$ by $d$, and compare with formula (45); a systematic disagreement after extrapolating in $1/d$ would falsify the claimed infinitesimal distribution.

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Extended reading notes

Core claim

If two sequences of real-rooted polynomials $p_d$ and $q_d$ have infinitesimal distributions $(\mu,\mu')$ and $(\nu,\nu')$, then the finite-free additive convolution $p_d \boxplus_d q_d$ has infinitesimal distribution $(\mu\boxplus\nu,\rho')$, with $\rho'$ given by the moment formula (45) and by the functional relation (57). The cumulant fluctuations add: $\hat r_n(p\boxplus q)=\hat r_n(p)+\hat r_n(q)$, equivalently $\hat R_{p\boxplus q}=\hat R_p+\hat R_q$. For the multiplicative convolution, the paper proves that $p_d \boxtimes_d q_d$ has infinitesimal distribution $(\mu\boxtimes\nu,\tau')$, where $\tau'$ is determined by the cumulant-fluctuation formula (48) and the functional relation (61); unlike the additive case, the multiplicative fluctuation term does not simply add, because annular non-crossing permutations contribute. In the special cases where one sequence degenerates to $\delta_0$ (additive) or $\delta_1$ (multiplicative), the finite-free infinitesimal convolution agrees with the infinitesimal free convolution, recovering known finite-rank perturbation distributions.

Load-bearing premise

Everything rests on assuming each sequence's finite-free cumulants differ from their free limit by a stable $1/d$ term (equivalently, its moments do), and the prior $1/d$ moment-cumulant expansion used in the proofs holds uniformly.

Editorial extensions

If this is right

For additive convolution, the order-$1/d$ fluctuation term is completely determined by the two input cumulant fluctuations and the limiting free convolution; no new fluctuation data are created by the operation.
For multiplicative convolution, even inputs with zero cumulant fluctuations produce a nontrivial fluctuation term coming from annular non-crossing partitions, so multiplicative convolution genuinely mixes fluctuations.
When one sequence converges to $\delta_0$ (additive) or $\delta_1$ (multiplicative), the finite-free infinitesimal convolution coincides with the infinitesimal free convolution of Belinschi and Shlyakhtenko, recovering finite-rank perturbation formulas.
Differentiating a polynomial sequence once changes the infinitesimal distribution by $\mu'\mapsto \mu'+\mu-\Xi(\mu)$, where $\Xi$ is the inverse Markov-Krein transform; repeated differentiation has an explicit transform formula.
The results give a systematic way to compute $1/d$ corrections for concrete families such as Hermite, Bernoulli, Laguerre, and finite-rank perturbations of identity.

Reading between the lines

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

The linear additivity of cumulant fluctuations suggests that, at first order, finite-free independence closely resembles infinitesimal freeness; one could test whether higher-order ($1/d^2$) terms obey a similar factorization involving genus expansions.
The $H$-transform appears as a universal correction term, which hints that the same functional relation may control fluctuations of orthogonal and $\beta$-ensembles, where annular non-crossing permutations already play a role.
A natural extension is to let the polynomial coefficients be random and ask whether the $1/d$ fluctuations converge in probability to the same deterministic formulas; the paper's combinatorial core would likely carry over unchanged.
The subordination formula (60) suggests an object interpolating between free convolution and infinitesimal free convolution; inverting that relation could define an explicit finite-free counterpart to the type-B infinitesimal convolution operation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a solid, genuinely new contribution to finite free probability. It works out the 1/d (infinitesimal) fluctuations for finite-free additive and multiplicative convolutions, gives explicit moment-cumulant formulas, and introduces a useful H-transform that links the finite-free fluctuations to Belinschi–Shlyakhtenko infinitesimal free convolution. The formulas check out; I could not find a load-bearing gap.

What is actually new: Theorems 3.4 and 3.5 give the first explicit formulas for the infinitesimal distribution of p⊞q and p⊠q in terms of cumulant fluctuations. The additive case is clean: the cumulant fluctuations of the convolution are just the sum of the individual ones, leading to the functional relation (57). The multiplicative case is messier but explicit via (48). Section 4's H-transform and the relation (13) are also new, and Theorem 4.8's subordination comparison with infinitesimal free convolution is a nice touch. The examples are genuinely useful: differentiation, finite-rank perturbations, and recovery of Shlyakhtenko's type-B results.

Where it is soft: mostly at the edges. The proofs rely on the AGVP23 expansion (40) as a black box; that is a deep theorem, but the authors use it within its stated scope and the stress-test confirms the logic holds. The analytic portion (Theorem 4.8 and Corollary 4.11) moves to analytic functions and implicitly assumes compact support or at least well-defined Cauchy transforms near infinity; for general infinitely supported measures the subordination functions still exist in the upper half-plane, but the formal series identities stand on their own, so this is a presentation issue rather than a defect. Some computations are left as 'similar arguments', but they are routine and I spot-checked a couple. There are a few typos (e.g., 'dirstributions' in the intro), but nothing substantive.

Citation pattern looks fine: the heavy reliance on AGVP23 and AP18 is legitimate because those are the prior theorems being extended; self-citation is appropriate here.

Bottom line: this is a paper for people working in finite free probability and infinitesimal freeness. It deserves serious refereeing. My recommendation: send it to review; the referee should check the multiplicative formula (48) carefully against examples, but I expect it to pass.

Referee Report

0 major / 5 minor

Summary. The paper investigates the order-1/d fluctuations (infinitesimal distributions) of sequences of real-rooted polynomials under finite-free additive and multiplicative convolution. Starting from the known AGVP23 expansion (40), the authors prove in Lemma 3.1 that a sequence has infinitesimal moments of all orders if and only if its finite-free cumulants admit a matching 1/d expansion, with explicit conversion formulas (43) and (44). They then derive the infinitesimal distribution of the finite-free additive convolution (Theorem 3.4), of the multiplicative convolution (Theorems 3.5 and 3.6), and translate these combinatorial identities into functional equations involving a new H-transform (Theorems 4.6 and 4.9). A substantial part of the paper compares these formulas with infinitesimal free convolution from the Belinschi-Shlyakhtenko theory, including a subordination formula (Theorem 4.8) and special cases recovering known results. The final section contains applications to repeated differentiation, finite-rank perturbations, and Laguerre-type examples.

Significance. If correct, the paper gives a complete and explicit description of the next-to-leading-order behavior of finite-free convolutions, a topic that connects finite free probability with infinitesimal/type-B freeness and with fluctuation results for random matrices. The main strengths are the clean reduction of the fluctuation problem to cumulant fluctuations, the invertible moment-cumulant formulas, and the identification of the H-transform as the formal object separating the finite-free fluctuation from its infinitesimal-free counterpart. The paper also recovers and extends previously known examples from AGVP23 and Shlyakhtenko, and it provides many concrete signed-measure computations, which makes the theory falsifiable and usable. The results are derived from established prior theorems (AP18, AGVP23) with detailed proofs for the new combinatorial steps; the remaining reliance on the AGVP23 expansion (40) is explicit and only used for fixed n as d grows. Overall this is a valuable contribution to finite free probability and to the infinitesimal fluctuation literature.

minor comments (5)

[Section 3.3, Eq. (46)] Equation (46) is missing the factor n/2 in the annular correction term. The displayed formula reads - sum_{...} r_sigma(μ⊠ν)/(ts), whereas Lemma 3.1 and Theorem 3.6 both require - (n/2) sum_{...} r_sigma(μ⊠ν)/(ts). The subsequent proof and Theorem 4.9 use the correct factor, so this appears to be a local typographical error, but the displayed statement should be corrected.
[Section 4.1, proof of Lemma 4.3] In the proof of Lemma 4.3, the displayed expansion for m'_n(p) and the displayed expression for H_mu(z) omit the factor n/2 (and the definition of h_n). The statement of the lemma, equation (54), is correct, but the intermediate display should match formulas (43) and (12).
[Equations (8) and (40)] The summation index in equations (8) and (40) is written as t,s=n in two places; this should be t+s=n. The same notational slip appears in the surrounding text.
[Theorem 4.8 and Corollary 4.11] The statement of Theorem 4.8 (and the proof of Corollary 4.11) invokes analytic subordination functions for arbitrary probability distributions, while the rest of the paper works with formal power series. For distributions without compact support, the formal Cauchy series and the analytic Cauchy transform may not coincide in a common neighborhood of infinity. The core formulas (45), (48), (57), and (61) are independent of this analytic interpretation, but the hypotheses of Theorem 4.8 should state explicitly what regularity is assumed.
[Throughout] There are several typographical slips that should be fixed in a final version: 'Infinitesimal dirstributions' in the introduction, 'infintesimal' in Section 2.2.3, the stray 's' in the citation in Proposition 4.1, and 'Beronulli' in Example 5.9. None of these affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the infinitesimal-convolution formulas are derived from prior published theorems and standard free probability, not from their own conclusions.

full rationale

The paper's central results (Theorems 3.4, 3.5, 3.6, 4.6, 4.9) are genuine derivations from earlier published results, principally the AGVP23 expansion (40) and the AP18 cumulant linearization, together with standard free-probability moment-cumulant relations. No parameter is fitted to a subset of data and then renamed a prediction: the cumulant fluctuations r-hat_n(p) are explicitly defined from the input sequence p in Lemma 3.1 and Definition 3.2, and the output infinitesimal moments are computed from those inputs. The additive formula (45) follows from additivity of finite-free cumulants plus Lemma 3.1; the multiplicative formulas (48) and (50) follow by substituting the assumed 1/d expansions into the prior AGVP23 expansion (37), which is an independent published theorem. The functional relations (57) and (61) are translations of these combinatorial formulas, not assumptions. The self-citations to AP18 and AGVP23 are load-bearing, but they are citations to published, parameter-free theorems with stated hypotheses that do not include the target results; they are therefore independent support under the applicable standard. Proposition 4.1 cites AGVP23 for the H-transform identity, but this is a prior proved identity, not an ansatz introduced to force the conclusions. The analytic subordination part (Theorem 4.8) assumes enough regularity for Cauchy transforms and subordination functions, which is a domain/presentation condition rather than a circular step. No equation is equivalent by construction to the claimed prediction, and no uniqueness claim is imported from the authors' own prior work to exclude alternatives. Accordingly, the derivation chain is self-contained relative to its cited external results, and no significant circularity is present.

Assumptions & free parameters 0 free parameters · 4 assumptions · 2 invented entities

The paper introduces no fitted parameters. The H-transform and ̂R_p are new mathematical objects but are explicitly constructed from known quantities (Cauchy transforms and cumulant fluctuations). The axioms and background results are standard theorems from free probability and finite free probability, which the paper cites and relies upon.

assumptions (4)

standard math Finite-free cumulant-moment formula of AP18 (Corollary 4.6): κ_n(p_d) is expressed as a d-dependent sum over partitions with Möbius weights.
Used in the proof of Lemma 3.1 to connect finite-free cumulants to moments and to establish the equivalence of 1/d expansions.
standard math Expansion (40) from AGVP23 (Theorem 1.3): m_n(p_d) = Σ_{π∈NC(n)} κ_π(p_d) - (n/(2d)) Σ_{σ∈SNC(t,s)} κ_σ(p_d)/(ts) + o(1/d).
This prior theorem is the cornerstone that links the finite-free moment and cumulant expansions; the new proofs expand this formula to first order in 1/d.
standard math Free moment-cumulant formula: m_n(μ) = Σ_{π∈NC(n)} r_π(μ) for any probability measure μ with finite moments.
Used to pass from cumulant expansions to moment expansions of the limiting measures and to define the R-transform and the free convolutions.
domain assumption Existence and properties of subordination functions for free additive and multiplicative convolutions (Voiculescu, Biane).
Used in Theorem 4.8 and Corollary 4.11 to relate the finite-free infinitesimal distribution to the infinitesimal free convolution via analytic functions, assuming the measures are such that these functions are well-defined.

invented entities (2)

H-transform H_μ(z)
purpose: Formal power series defined as H_μ(z) = Σ_{n≥2} h_n(μ) z^{-n-1} with h_n(μ) = (n/2) Σ_{t+s=n} Σ_{σ∈SNC(t,s)} r_σ(μ)/(ts). It captures the annular correction term in the 1/d moment expansion and appears in the functional relation (54).
Introduced in Equation (12) and shown in Proposition 4.1 to equal G_μ'/G_μ - G_μ''/(2G_μ'), a closed expression in terms of the Cauchy transform G_μ. It is not an unexplained postulate but rather a derived quantity.
Cumulant fluctuation transform ̂R_p(z)
purpose: Generating function of the 1/d fluctuations ̂r_n(p) of the finite-free cumulants of a sequence p. It is central to the functional relations (54), (57), (61) and to the comparison with infinitesimal free cumulants.
Defined by ̂R_p(z) = Σ_{n≥1} ̂r_n(p) z^{n-1} in Lemma 3.1; the coefficients ̂r_n(p) are uniquely determined by the infinitesimal moments via (44). It is not an additional free parameter.

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Cite this review

Pith. "Pith review of Finite Free Convolution: Infinitesimal Distributions." pith.science (2026). https://pith.science/paper/WOV2EXCK

@misc{pith2026250501705,
  author       = {Pith},
  title        = {Pith review of: Finite Free Convolution: Infinitesimal Distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WOV2EXCK}},
  note         = {Machine review of arXiv:2505.01705}
}

abstract

Finite-free additive and multiplicative convolutions are operations on the set of polynomials with real roots, introduced independently by Szeg\"{o} and Walsh in the 1920s. These operations have regained some interest, in the last decade, after being rediscovered by Marcus, Spielman, and Srivastava as the expected characteristic polynomial of randomly rotated matrices. They converge, as the degree $d$ of the polynomials increases, to the additive and multiplicative convolution of measures from free probability of Voiculescu. In this paper, we investigate the fluctuations of order $1/d$ -- also known as infinitesimal distributions -- related to these two operations and their limiting behavior, providing a detailed description of their convergence. Our approach relies on understanding the infinitesimal moment-cumulant formulas and the corresponding functional relations. We also establish several applications and examples, including instances related to the infinitesimal free convolution of Belinschi and Shlyakhtenko, as well as the computation of infinitesimal distributions after differentiation of polynomials.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

S-transform in finite free probability

Octavio Arizmendi, Katsunori Fujie, Daniel Perales, and Yuki Ueda. S-transform in finite free probability. Preprint arXiv:2408.09337 , 2024

arXiv 2024

[2] [2]

Finite free cumulants: multiplicative convolutions, genus expansion and infinitesimal distributions

Octavio Arizmendi, Jorge Garza-Vargas, and Daniel Perales. Finite free cumulants: multiplicative convolutions, genus expansion and infinitesimal distributions. Transactions of the American Mathematical Society , 376(06):4383--4420, 2023

work page 2023

[3] [3]

Cumulants for finite free convolution

Octavio Arizmendi and Daniel Perales. Cumulants for finite free convolution. Journal of Combinatorial Theory, Series A , 155:244--266, 2018

work page 2018

[4] [4]

Non-crossing cumulants of type B

Philippe Biane, Frederick Goodman, and Alexandru Nica. Non-crossing cumulants of type B . Transactions of the American Mathematical Society , 355(6):2263--2303, 2003

work page 2003

[5] [5]

Processes with free increments

Philippe Biane. Processes with free increments. Mathematische Zeitschrift , 227:143--174, 1998

work page 1998

[6] [6]

Free probability of type B : analytic interpretation and applications

Serban Belinschi and Dimitri Shlyakhtenko. Free probability of type B : analytic interpretation and applications. American Journal of Mathematics , 134(1):193--234, 2012

work page 2012

[7] [7]

Free convolution of measures with unbounded support

Hari Bercovici and Dan Voiculescu. Free convolution of measures with unbounded support. Indiana University Mathematics Journal , 42(3):733--773, 1993

work page 1993

[8] [8]

Infinitesimal freeness for orthogonally invariant random matrices

Guillaume C \'e bron and James A Mingo. Infinitesimal freeness for orthogonally invariant random matrices. arXiv preprint arXiv:2506.05139 , 2025

work page arXiv 2025