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Regularity and Convergence Properties of Finite Free Convolutions

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Pith's one-line read Finite free convolutions of real-rooted polynomials converge to free convolutions as the degree tends to infinity, with no compact-support assumption and with Kolmogorov-distance convergence when the inputs converge in that metric.

desk verdict Solid new convergence result with a repairable sign error in the key lemma; worth refereeing after fixes. read the letter →

arxiv 2505.15575 v1 pith:F3FZ7GBJ submitted 2025-05-21 math.PR math.OA

Katsunori Fujie This is my paper · ORCID
classification math.PRmath.OA MSC 46L5460B2026C10
keywords finitefreeconvolutionprobabilityreal-rootedpolynomialsKolmogorovdistanceweakconvergenceatomsofmeasuresinterlacingSchur–Szegőcomposition
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 argues that the finite free additive convolution of real-rooted degree-$d$ polynomials—a symmetrized polynomial operation on roots—approximates the free additive convolution of probability measures as $d \to \infty$, and that this approximation holds without the usual compact-support assumption on the limiting measures. It further claims that when the empirical root distributions converge in Kolmogorov distance, the convolved distributions converge in Kolmogorov distance as well, and it proves the analogous statements for the finite free multiplicative convolution with measures on $[0,\infty)$. Along the way it establishes finite-degree analogues of two known free-probability regularities: a contraction inequality for Kolmogorov and L\'evy distances under the convolution, and a complete description of the atoms (multiplicity-$m$ roots) of the convolved polynomial in terms of the input atoms. The proof is elementary, comparing root configurations through order and cut-up/cut-down approximations rather than through analytic transforms.

What carries the argument

The machinery combines three objects: the empirical root distribution $\mu_{[p]}$ of a polynomial, the partial order $\mu\le\nu$ defined by $F_\nu\le F_\mu$, and the cut-up and cut-down measures $\mu|_a$ and $\mu|^a$ that truncate a measure at a point $a$. The order and truncations let the paper compare root configurations before and after convolution, and interlacing of polynomials supplies the monotonicity that turns order comparisons into distance comparisons. The load-bearing mechanism is Lemma 4.1, which converts a one-sided closeness bound $F_q \le F_p + \ell/d$ into a chain of interlacing polynomials and then into the same closeness inequality after convolution; Theorems 1.2 and 1.3 are derived from this contraction step.

What would settle it

Apply Lemma 4.1 to $p(x)=x(x-2)$ and $q(x)=(x-1)^2$ with $d=2$ and $\ell=1$; the condition $F_q\le F_p+1/2$ holds, but replacing the left root of $q$ by a large number gives $q^{(1)}$ with roots $1$ and $3$, and then $F_p(2)=1$ exceeds $F_{q^{(1)}}(2)=1/2$, so $p\le q^{(1)}$ fails. This example would settle whether the lemma's order direction is correct as stated.

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Extended reading notes

Core claim

The central claim is that the empirical root distribution of $p_d \boxplus_d q_d$ converges weakly to $\mu \boxplus \nu$ whenever the root distributions of $p_d$ and $q_d$ converge weakly to $\mu$ and $\nu$, with no compact-support condition on $\mu$ or $\nu$; if the input distributions converge in Kolmogorov distance, so does the output. The same conclusion is proved for $\boxtimes_d$ when both measures are supported on $[0,\infty)$, and for a one-sided case in which one measure is compactly supported on $[0,\infty)$. The paper also establishes that every non-trivial root of $p_d \boxplus_d q_d$ is simple and that atom masses obey the free-convolution formula $\mu_{[p\boxplus_d q]}(\{\alpha+\beta\}) = \mu_{[p]}(\{\alpha\}) + \mu_{[q]}(\{\beta\}) - 1$ whenever $\alpha$ and $\beta$ are atoms of the inputs with multiplicities summing past the degree.

Load-bearing premise

Everything rests on Lemma 4.1 of Section 4.1, which says that if the roots of $q$ are close to the roots of $p$ from one side, then moving the leftmost roots of $q$ to the far right produces a polynomial $q^{(\ell)}$ with $p \le q^{(\ell)}$. In the written proof the moving produces the opposite ordering, so the claimed direction of this lemma is the premise on which the convergence theorem depends.

Editorial extensions

If this is right

  • Applying $\boxplus_d$ to degree-$d$ polynomials whose root distributions converge to measures with heavy tails still gives the correct free-convolution limit.
  • Kolmogorov-distance convergence of the input polynomials passes to the convolved polynomials, giving a quantitative upgrade of weak convergence.
  • The atom formula for $p_d \boxplus_d q_d$ matches the free atom formula, so atomic structure is preserved across the finite-to-infinite passage.
  • The multiplicative analogue now covers $[0,\infty)$-supported measures without compactness, and a one-sided generalization with one compact positive measure is included.

Reading between the lines

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

  • Editorial inference: The distance-contraction inequalities for $\boxplus_d$ and $\boxtimes_d$ are finite-dimensional versions of known contraction properties of free convolutions; if the underlying lemma survives, the same cut-and-paste route may yield contraction for other polynomial operations that preserve real-rootedness.
  • Editorial inference: The paper's approach suggests a recipe for proving 'no compact support needed' statements in other asymptotic settings: first prove a distance-contraction inequality, then approximate arbitrary measures by compactly supported truncations and compare.
  • Editorial inference: One could try to derive an explicit bound $\mathrm{d}_K(p_d\boxplus_d q_d,\mu\boxplus\nu) \le \mathrm{d}_K(\mu_{[p_d]},\mu) + \mathrm{d}_K(\mu_{[q_d]},\nu) + o(1)$ from Theorem 1.2; the paper proves convergence but does not state such a rate.
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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 / 4 minor

Summary. The paper develops regularity properties of the finite free additive and multiplicative convolutions ⊞_d and ⊠_d. The main results are: (1) a description of atoms of p⊞_d q and p⊠_d q, including a formula relating CDF values at atom triplets; (2) monotonicity of Kolmogorov and Lévy distances under finite free convolution; and (3) convergence of empirical root distributions of p_d ⊞_d q_d to the free additive convolution μ⊞ν under only weak convergence of the inputs, together with a strengthening to convergence in Kolmogorov distance when the input empirical measures converge in Kolmogorov distance. Multiplicative analogues are also stated. The proofs use root interlacing, truncation of measures, and prior results from free probability for compactly supported limits.

Significance. If the results are correct, they significantly extend the known approximation of free convolutions by finite free convolutions: previous moment-based proofs required compact support, whereas Theorem 1.3 removes that condition for the additive convolution and provides a quantitative Kolmogorov-distance transfer. The paper's elementary root-ordering approach is a genuine strength, as is the explicit use of truncation to reduce the unbounded-support case to the compact case. The atom identities in Propositions 2.10 and 2.12 and the finite-free analogues are also valuable. However, several proofs, especially Lemma 4.1 and the Lévy-distance part of Theorem 1.2, contain gaps or incorrect intermediate statements that must be repaired before the central claims can be regarded as fully established.

major comments (3)
  1. [§4.1, Lemma 4.1] The claimed reversal of the order in Lemma 4.1 is not correct under the paper's convention. By Definition 2.3, p≤q means λ_i(p)≤λ_i(q) for every i (equivalently F_p≥F_q). The construction gives q^(l) with ordered roots λ_{l+1}(q),...,λ_d(q),a,...,a, so (4.2) and a>λ_d(p) imply λ_i(p)≤λ_i(q^(l)) for all i, which is exactly p≤q^(l). What is genuinely wrong is the displayed pointwise inequality p(x)≤q^(l)(x): this is generally false and is not equivalent to the CDF order. For example, with roots 0,1,2 and 1,3,4, the inequality fails at x=2.5. The proof should be repaired by replacing the pointwise display with the root-order comparison. Because Lemma 4.1 is used in Theorems 1.2 and 1.3, this is a load-bearing gap, though I see no obstacle to a local fix.
  2. [§4.1, proof of Theorem 1.2(2)] The Lévy-distance part is not established by the sentence 'it is enough to note...'. The displayed implication from an ε-satisfying the Lévy definition to the two one-sided bounds with l/d requires the discrete fact that F_p and F_q take values in multiples of 1/d; this quantization step is not given. Moreover, even after obtaining those bounds, one must explain how they combine with Lemma 4.1 and the shift covariance of ⊞_d to yield d_L(p⊞_d r,q⊞_d r)≤l/d. The current text skips this essential chain, so the proof of part 2 of Theorem 1.2 is incomplete as written.
  3. [Appendix A, proof of Proposition 3.8] The identity (p+ε_0)⊞_d q = p⊞_d q+ε_0 is invoked and attributed to Proposition 2.16, but that proposition only states preservation of real-rootedness; it does not state this identity. The identity is true and follows directly from the coefficient definition (1.1), but it must be proved or given a correct reference. In addition, the assertion that p+ε remains in P_d(R) for all sufficiently small ε when p has only simple roots is standard but should be justified via continuous dependence of simple roots on coefficients. These are local gaps in the proof of Theorem 1.1(1).
minor comments (4)
  1. [§2.4 / §4.1] The notation for cut-up and cut-down measures is visually ambiguous in the text: both are typeset as μ| a in several places, although Definition 2.4 distinguishes μ|^a and μ|_a. Please ensure superscripts/subscripts are rendered consistently.
  2. [§4.2, Lemma 4.3] In the construction of p_d for K=[a,b], the polynomial uses λ_{d-1} twice rather than defining λ_d; the approximation estimate d_K≤1/d is correct, but the repeated root should be explicitly explained to avoid confusion.
  3. [§4.2, Theorem 4.5] The statement that the Kolmogorov-distance result 'might also hold' is imprecise; if it is not proved, it would be better phrased as an open problem or a conjecture rather than a tentative claim.
  4. [§3.2, Proposition 3.8] The final 'worst case' paragraph of the proof is very terse and uses several unstated identities, such as p_α⊞_d q = p_α⊞_{d-1} ∂^{(d-1)|d}q; these identities should be stated and proved or cited, since the contradiction argument depends on them.

Circularity Check

0 steps flagged · score 0.0 of 10

No meaningful circularity: the main convergence theorem is derived from independent free-probability and finite-free results; self-citations are not load-bearing.

full rationale

The paper's central claim, Theorem 1.3, is not obtained by assuming its own conclusion. The proof combines three independent ingredients: (i) the external compact-support convergence result Proposition 2.21, (ii) the paper's own distance-contraction Theorem 1.2 obtained from the constructive root-ordering Lemma 4.1, and (iii) truncation arguments using cut-up/cut-down measures. The only self-citation used in a supporting role is Proposition 2.19, attributed to the author's earlier paper [2], but that result concerns monotonicity of finite free convolutions and does not contain Theorem 1.2 or 1.3; moreover, the paper sketches how it follows from the interlacing-preservation Proposition 2.18, so it is not an unverified premise imported solely to force the conclusion. The finite-free atom computations (Propositions 3.6 and 3.7) are derived directly from the definition of boxplus_d and from coefficient identities, not from the target convergence statement. There is no fitted parameter later renamed as a prediction, no uniqueness theorem invoked from the authors' prior work, and no ansatz smuggled in through citation. The reader-flagged issue in Lemma 4.1 is a potential order/direction error in the proof ('p <= q^(l)' versus 'q^(l) <= p'), but that is a mathematical correctness issue, not a circularity: the lemma's construction and the interlacing chain do not assume the theorem being proved, and the stated consequence appears repairable. Since the derivation is self-contained apart from standard external results, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central convergence claims rest on standard free-probability inputs plus two ad-hoc assertions (the order direction in Lemma 4.1 and the affine identity in Appendix A) that the text does not justify. The first three axioms are standard background from the cited literature.

assumptions (5)
  • domain assumption Bercovici-Voiculescu monotonicity and atom results for free convolutions (Propositions 2.7, 2.9, 2.11)
    Used as black-box inputs to prove the finite-free analogues and to identify atoms in the limiting free convolution.
  • domain assumption Compact-support convergence of finite free additive convolution (Proposition 2.21 from [5])
    Used in the truncation step of Theorem 1.3 to handle the compactly supported pieces (mu)_a and (nu)_a.
  • domain assumption Real-rootedness and interlacing preservation under finite free convolutions (Propositions 2.16 and 2.18)
    Foundation for the distance-contraction argument in Lemma 4.1 and Theorem 1.2.
  • ad hoc to paper The order direction in Lemma 4.1: from F_q <= F_p + l/d one can construct q^{(l)} with p <= q^{(l)}
    The proof's root inequality lambda_i(p) <= lambda_{l+i}(q) and the construction by moving the smallest roots of q to a large value appear to give q^{(l)} <= p instead; Theorem 1.2 depends on this chain.
  • ad hoc to paper Identity (p+epsilon) boxplus_d q = p boxplus_d q + epsilon
    Used without proof in the first case of the proof of Proposition 3.8; not a standard consequence of the cited propositions.

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Pith. "Pith review of Regularity and Convergence Properties of Finite Free Convolutions." pith.science (2026). https://pith.science/paper/F3FZ7GBJ

@misc{pith2026250515575,
  author       = {Pith},
  title        = {Pith review of: Regularity and Convergence Properties of Finite Free Convolutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3FZ7GBJ}},
  note         = {Machine review of arXiv:2505.15575}
}
abstract

Finite free convolutions, $\boxplus_d$ and $\boxtimes_d$, are binary operations on polynomials of degree $d$ that are central to finite free probability, a developing field at the intersection of free probability and the geometry of polynomials. Motivated by established regularities in free probability, this paper investigates analogous regularities for finite free convolutions. Key findings include triangle inequalities for these convolutions and necessary and sufficient conditions regarding atoms of probability measures. Applications of these results include proving the weak convergence of $\boxplus_d$ and $\boxtimes_d$ to their infinite counterparts $\boxplus$ and $\boxtimes$ as $d \to \infty$, without compactness assumptions. Furthermore, this weak convergence is strengthened to convergence in Kolmogorov distance.

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