{"id":"c99fc8d7-169b-46c6-beb8-429bec3a876c","arxiv_id":"2505.01705","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper gives explicit moment-cumulant formulas and functional equations for the 1/d corrections (infinitesimal distributions) of finite-free polynomial convolutions.","lead":"This paper derives explicit formulas for the order 1/d fluctuations, called infinitesimal distributions, of finite-free additive and multiplicative convolutions of real-rooted polynomials, and connects them to infinitesimal free probability. A reader might care because these fluctuations describe how eigenvalue statistics of randomly rotated matrices approach their free probability limits, and the new transforms offer a systematic way to compute next-order corrections.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the central convolution formulas are internally consistent and the cited AGVP23 expansion is used within its stated scope.","rationale":"The reader's weakest-assumption remarks point to the same peripheral soft spots: reliance on the AGVP23 expansion and the analytic subordination switch in Theorem 4.8. I agree these deserve attention, but I do not find them load-bearing for the central claim. Lemma 3.1 is the hinge of the paper, and its proof checks out: the induction isolates kappa_n correctly, and the annular term only contributes at leading order because of the explicit 1/d prefactor. Theorem 3.4 is then a direct consequence of additivity of finite-free cumulants and Lemma 3.1. Theorem 3.5's formula (48) is obtained by expanding the products in (37); every term of order 1/d is accounted for, and no hidden o(1/d) term can affect the coefficient because fixed n and d -> infinity are used. The multiplicative moment formula (50) was independently spot-checked against the identity and zero-root cases, where it reduces to the known infinitesimal moments. The analytic subordination theorem is the only genuine caveat: as stated it assumes analytic Cauchy transforms in a neighborhood of infinity, which is not automatic for non-compactly supported measures with only finite moments. However, the equality (60) can be read as an identity of formal power series using formal subordination functions, and the paper's main convolution formulas do not depend on this analytic interpretation. Therefore the ACCEPT verdict remains appropriate.","tokens_in":40455,"tokens_out":22650,"duration_ms":225964,"concrete_test":"Perform a direct symbolic check of (45) and (48) for small degrees: take d = 3 or 4, prescribe p_d and q_d with cumulants of the form kappa_n(p_d) = r_n(mu) + r-hat_n(p)/d and similarly for q, compute kappa_n of p_d ⊞_d q_d and p_d ⊠_d q_d using (7) and (35), form the corresponding root moments, and compare the 1/d coefficient with the right-hand sides of (45), (46), and (48). If the coefficients match for all n ≤ d and for several choices of the r-hat sequences, the central formulas pass this check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No load-bearing concern found. The main theorems (3.4, 3.5, 4.6, 4.9) are supported by Lemma 3.1, whose equivalence of moment and cumulant expansions is proved by induction and correctly handles the annular term. Theorem 3.5's expansion of (37) accounts for all 1/d contributions: the NC sum contributes the two r-hat terms in (48), and the prefactor 1/d on the annular sum makes only the leading r_sigma * r_Kr(sigma) term contribute. Theorem 3.6 was checked against the identity case q_d = (x-1)^d and q_d = x^d, where the formula reduces correctly. The analytic subordination part (Theorem 4.8) is the only place where the paper moves from formal power series to analytic functions; for distributions without compact support the Cauchy transforms may lack a common neighborhood of infinity. This is a domain/presentation issue rather than a defect in the central formal-power-series claim, since formulas (45), (48), (57), and (61) stand independently of the analytic interpretation. Reliance on the AGVP23 expansion (40) is explicit, and the arguments only need fixed n as d grows, so no hidden uniformity in n is required.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":40685,"tokens_out":10465,"duration_ms":104580,"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.","major_comments":[],"minor_comments":[{"comment":"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":"Section 3.3, Eq. (46)"},{"comment":"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).","section":"Section 4.1, proof of Lemma 4.3"},{"comment":"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.","section":"Equations (8) and (40)"},{"comment":"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.","section":"Theorem 4.8 and Corollary 4.11"},{"comment":"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.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is mathematically sound in its central claims. The only mathematical issue I found is a missing n/2 factor in equation (46), which is clearly typographical because the proof and all later uses carry the correct factor. I therefore recommend minor revision rather than accept, mainly to fix displayed formulas and the analytic-domain caveat in Theorem 4.8. The paper fits the journal's scope and should be published after these local corrections."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nWhere 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.\n\nCitation 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.\n\nBottom 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.","headline":"Solid, genuinely new treatment of 1/d fluctuations for finite-free convolutions; the H-transform is a useful bridge to infinitesimal free probability.","tokens_in":41240,"tokens_out":1829,"would_cite":true,"duration_ms":17734,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L54","60B20","05A18","30C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives explicit order-1/d fluctuation formulas for finite-free additive and multiplicative convolutions of real-rooted polynomials.","keywords":["finite-free convolution","infinitesimal distribution","finite-free cumulants","non-crossing partitions","annular non-crossing permutations","free probability","real-rooted polynomials","random matrices"],"falsifier":"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.","tokens_in":40274,"feed_emoji":"🧮","tokens_out":8312,"duration_ms":81710,"temperature":0.7,"pith_summary":"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.","feed_headline":"Order-1/d fluctuations of finite-free convolutions are explicit","feed_subtitle":"Additive convolution adds cumulant fluctuations; multiplicative convolution needs a new formula.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It introduces finite-free cumulants and their linearization of additive convolution, which is the foundation of the cumulant-fluctuation argument.","marker":"[AP18]"},{"why":"It supplies the $1/d$ moment-cumulant expansion (40) and the multiplicative cumulant formulas that the new theorems build on.","marker":"[AGVP23]"},{"why":"It defines infinitesimal free convolution and the type-B subordination relations used for the comparison in Theorems 4.8 and 4.11.","marker":"[BS12]"},{"why":"It provides the additive and multiplicative subordination functions used to express the final relations with infinitesimal freeness.","marker":"[Bia98]"},{"why":"It supplies the additive subordination functions for free convolution used in the proof of Theorem 4.8.","marker":"[Voi93]"},{"why":"It provides the inverse Markov-Krein transform used to identify the $H$-transform and to write the one-derivative formula.","marker":"[Ker98]"},{"why":"It supplies the differentiation formula for finite-free cumulants used in Proposition 5.4 on repeated differentiation.","marker":"[AFPU24]"},{"why":"It supplies the finite-rank perturbation distributions that are recovered and generalized as special cases of the new formulas.","marker":"[Shl18]"}],"fun_headline_variants":["Explicit order-1/d fluctuations for finite-free convolutions","Finite-free convolution fluctuations: additive adds, multiplicative doesn't","Infinitesimal distributions for finite-free convolution now explicit","New cumulant formula for multiplicative finite-free convolution"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Explicit order-1/d fluctuations for finite-free convolutions","Finite-free convolution fluctuations: additive adds, multiplicative doesn't","Infinitesimal distributions for finite-free convolution now explicit","New cumulant formula for multiplicative finite-free convolution"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000868,"raw_usage":{"total_tokens":3777,"prompt_tokens":977,"completion_tokens":2800,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":2733}},"tokens_in":593,"tokens_out":2800,"duration_ms":21999,"temperature":1.0,"reasoning_tokens":2733,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:11:11.248599+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}