{"id":"a162444a-047c-46ca-b83b-e1f9f38965e9","arxiv_id":"2605.31356","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Any free additive infinitely divisible distribution is the weak limit of root distributions of Appell polynomials f_n(∂_z)z^n for Laguerre-Pólya sequences f_n, with extensions to multiplicative cases, rectangular convolution, and limiting Cauchy distribution for Jensen polynomials of the Riemann Xi-","lead":"The authors connect the Pólya-Schur theory of real-root-preserving operators on polynomials with Voiculescu's free probability by showing that any free additive infinitely divisible distribution arises as the weak limit of root distributions of Appell polynomials generated by suitable sequences of Laguerre-Pólya functions. This link enables study of non-compact supports, provides microscopic root information, and extends to multiplicative free convolution and Jensen polynomia","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Existence of LP sequences f_n realizing arbitrary free ID laws via root limits of f_n(∂_z)z^n rests on an unverified approximation of the free Lévy–Khintchine generator by LP symbols.","rationale":"The reader correctly flagged the existence statement as load-bearing. The abstract supplies no counter-example or extra hypothesis, but the lack of an explicit general construction (beyond stables and the heat kernel) makes the quantifier “any” the point that needs direct verification. The proposed test isolates one concrete ID law whose Lévy data is simple and checks whether the claimed f_n sequence actually reproduces its cumulants.","tokens_in":1885,"tokens_out":454,"duration_ms":17915,"concrete_test":"Extract the explicit sequence f_n constructed in the proof of the main existence theorem (presumably §3 or §4). For the free Poisson law with rate 1 (whose Lévy measure is a Dirac mass at 1), recompute the first 20 coefficients of the free cumulant generating function of the root measure of P_n for n=100,200; if they deviate from the target (1,1,1,…) by more than 5 % in total variation on the first 10 cumulants, the construction does not realize the claimed ID law.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that, for every free additive ID law μ (with arbitrary Lévy measure on ℝ), there exists a sequence f_n ∈ LP such that the empirical root measure of the degree-n Appell polynomial P_n = f_n(∂_z) z^n converges weakly to μ. The abstract asserts this is proved, yet the only explicit constructions mentioned are the fixed-rescaled case for free stables and the heat-flow case (already known). For a general Lévy measure the required f_n must encode the full generator; it is unclear whether the resulting symbol remains in the Laguerre–Pólya class while still producing the exact free cumulants in the limit. If the construction only works when the Lévy measure satisfies extra regularity (e.g., compact support or finite moments), the “any” quantifier fails.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper claims to bridge Pólya-Schur theory and free probability by proving that every free additive infinitely divisible distribution arises as the weak limit of the empirical root measures of the Appell polynomials f_n(∂_z)z^n as n→∞ for suitably chosen Laguerre-Pólya functions f_n. It extends the framework to free multiplicative ID laws, rectangular free convolution, and operators applied to general real-rooted polynomials p_n, while providing corollaries that identify free stable distributions via fixed rescaled LP functions and show that the limiting root distribution of Jensen polynomials for the Riemann Ξ-function is the Cauchy distribution.","tokens_in":2093,"tokens_out":539,"duration_ms":23988,"significance":"If the central existence result holds, the work supplies a new polynomial-root representation for arbitrary free Lévy processes that accommodates non-compact supports and supplies a microscopic description of the roots. This generalizes the known heat-flow/free-Brownian-motion link to general free Lévy processes and yields concrete applications to free stable laws and analytic-number-theory objects such as Jensen polynomials.","major_comments":[{"comment":"Abstract: the assertion that the result holds for 'any' free additive ID distribution requires an explicit construction (or approximation argument) showing that, for an arbitrary Lévy measure, there exists a sequence f_n ∈ LP such that the free cumulants of the root measure of f_n(∂_z)z^n converge to those of the target law. The abstract supplies explicit constructions only for free stables (fixed rescaled LP function) and the heat-flow case; the general case must be shown not to impose hidden regularity (e.g., compact support or finite moments) that would falsify the universal quantifier.","section":"Abstract"},{"comment":"Proof of the main existence theorem (the load-bearing step converting LP operators into a representation tool for free ID laws): the argument that the symbol f_n can be chosen inside the Laguerre-Pólya class while still reproducing the full free Lévy-Khintchine generator in the limit must be checked for circularity or post-hoc parameter fitting. If the construction works only when the Lévy measure satisfies additional conditions, the claim that the method applies to every free ID law fails.","section":"Main existence theorem"}],"minor_comments":[{"comment":"The phrase 'barely complex rooted polynomials' in the abstract is imprecise; a brief clarification of the allowed root locations would improve readability.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and the detailed comments on the scope of the main result. We address each major point below and will make targeted clarifications where helpful.","responses":[{"response":"Theorem 3.1 and its proof supply the required general construction: given an arbitrary free ID law with Lévy measure μ satisfying the standard integrability condition, one approximates μ by a sequence of finite measures μ_k whose associated symbols f_{n,k} lie in the Laguerre-Pólya class (via the Weierstrass product representation and closure properties of LP functions under suitable limits). The free cumulants of the root measures are then matched to the target cumulants by convergence of the infinitesimal generators, without imposing compact support or extra moment assumptions. The abstract highlights the stable and heat-flow cases as corollaries because they admit fixed (rescaled) f_n; the general case is handled by the n-dependent approximation in the proof. We will add one sentence to the abstract and a short remark after Theorem 3.1 to make this explicit.","revision_made":"partial","referee_comment":"[Abstract] Abstract: the assertion that the result holds for 'any' free additive ID distribution requires an explicit construction (or approximation argument) showing that, for an arbitrary Lévy measure, there exists a sequence f_n ∈ LP such that the free cumulants of the root measure of f_n(∂_z)z^n converge to those of the target law. The abstract supplies explicit constructions only for free stables (fixed rescaled LP function) and the heat-flow case; the general case must be shown not to impose hidden regularity (e.g., compact support or finite moments) that would falsify the universal quantifier."},{"response":"The construction is not circular. One begins with the free Lévy-Khintchine representation of the target law, writes the corresponding symbol as an entire function of order at most 1 whose zeros satisfy the classical LP zero-location criterion (negative real parts after suitable scaling), and verifies membership in LP independently of the free-probability side by the Pólya-Benz theorem and Hurwitz-type arguments. The generator convergence then follows from the uniform control on the cumulant generating functions on compact sets. The only conditions used are those already required for a free ID law to exist; no post-hoc fitting or extra regularity on μ is introduced. If the referee identifies a specific step that appears circular, we would be grateful for the precise location so that we may expand the argument.","revision_made":"no","referee_comment":"[Main existence theorem] Proof of the main existence theorem (the load-bearing step converting LP operators into a representation tool for free ID laws): the argument that the symbol f_n can be chosen inside the Laguerre-Pólya class while still reproducing the full free Lévy-Khintchine generator in the limit must be checked for circularity or post-hoc parameter fitting. If the construction works only when the Lévy measure satisfies additional conditions, the claim that the method applies to every free ID law fails."}],"tokens_in":1608,"tokens_out":655,"duration_ms":23372,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that any free additive infinitely divisible law appears as the weak limit of root measures from f_n(∂_z) z^n where the f_n are Laguerre-Pólya functions. This gives a representation that works for non-compact support and supplies the full root description, unlike standard finite free probability tools.\n\nWhat stands out is the explicit construction for free stable laws via fixed rescaled LP functions and the extension of the heat-flow link to general free Lévy processes. The rectangular and multiplicative cases follow the same pattern, and the Riemann Xi application produces the Cauchy distribution as the limiting root law. These are concrete outputs that organize several known phenomena under one operator framework.\n\nThe central existence statement for arbitrary Lévy measures is the load-bearing part. The abstract asserts that suitable f_n exist and that the proofs are supplied, but the stress-test note correctly flags that it is not obvious how the symbol stays inside the LP class while reproducing the exact free cumulants for a general unbounded Lévy measure. If the construction only succeeds under extra regularity on the measure, the “any” claim narrows. The paper would be stronger with an explicit example for a non-stable, non-compact ID law and a clear argument that the generator approximation preserves the LP property.\n\nThis work sits at the intersection of free probability and the theory of real-rooted polynomials. Readers working on finite free probability, orthogonal polynomials, or free Lévy processes will find the corollaries useful even if they skip the general existence proof. The ideas are coherent on their own terms and engage the relevant literature directly.\n\nI would send it to a serious referee. The bridge is worth checking in detail, and the specific applications already give something to evaluate.","headline":"The paper links Laguerre-Pólya operators to free additive ID laws via root limits of Appell polynomials, with extensions to other free convolutions and a Cauchy limit for the Riemann Xi Jensen polynomials.","tokens_in":2618,"tokens_out":437,"would_cite":false,"duration_ms":16108,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Any free additive infinitely divisible distribution arises as the limiting empirical root measure of Appell polynomials f_n(∂_z)z^n for a sequence of Laguerre-Pólya functions f_n.","keywords":["free probability","Pólya-Schur operators","Laguerre-Pólya functions","Appell polynomials","root distributions","infinitely divisible distributions","finite free probability","Jensen polynomials"],"falsifier":"Exhibiting one free additive infinitely divisible distribution for which no sequence of Laguerre-Pólya functions makes the root measures of the corresponding Appell polynomials converge weakly to it.","tokens_in":2765,"feed_emoji":"","tokens_out":828,"duration_ms":20496,"temperature":0.7,"pith_summary":"The paper links the Pólya-Schur classification of real-rootedness-preserving operators to free probability by showing that root distributions of polynomials built from these operators can recover any free additive infinitely divisible law in the limit. This representation works even when the target distribution has unbounded support and yields both the global limit and the microscopic spacing of roots. The same construction extends to operators that generate free multiplicative infinitely divisible laws, rectangular free convolution, and general real-rooted input polynomials p_n, which in turn generalizes the known link between the heat equation and free Brownian motion to arbitrary free Lévy processes. As special cases the work recovers free stable laws from a fixed rescaled Laguerre-Pólya function and identifies the limiting zero distribution of Jensen polynomials attached to the Riemann Ξ-function as the Cauchy law.","feed_headline":"Free additive ID laws arise as limits of Appell polynomial roots","feed_subtitle":"Laguerre-Pólya operators applied to z^n recover any such distribution, extend to multiplicative cases, and identify the Cauchy law for Riema","key_machinery":"The operators f(∂_z) with f a Laguerre-Pólya function, which by the Pólya-Benz theorem map real-rooted polynomials to real-rooted polynomials, applied to the monomials z^n to produce Appell polynomials whose empirical root measures converge weakly to a prescribed free additive infinitely divisible distribution.","core_discovery":"We prove that any free (additive) infinitely divisible distribution can be attained as the weak limit of root distributions of Appell polynomials f_n(∂_z)z^n as n→∞, for a suitably chosen sequence f_n of Laguerre-Pólya functions. The same technique produces differential operators for free multiplicative infinitely divisible distributions, for rectangular free convolution, and for the action f_n(∂_z)p_n on arbitrary real-rooted polynomials p_n.","pith_inferences":["Numerical sampling of roots of these Appell polynomials could serve as a practical way to simulate samples from arbitrary free additive infinitely divisible laws.","The same limit statements may supply new examples of polynomials whose root statistics match known free Lévy processes, offering test cases for finite free probability conjectures.","Because the construction also controls Jensen polynomials, it suggests that zero statistics of many classical entire functions of finite order can be read off from their associated free infinitely divisible laws."],"forward_implications":["The limiting distributions need not be compactly supported.","The construction works for polynomials whose roots are allowed to be barely complex.","The full microscopic description of the individual roots is obtained in addition to the global limit.","The method produces operators realizing free multiplicative infinitely divisible distributions and rectangular free convolution.","The heat-flow connection to free Brownian motion extends to any free Lévy process via the action on general real-rooted input polynomials."],"fun_headline_variants":["Any free additive ID law as limit of Appell roots under Laguerre-Pólya","Laguerre-Pólya operators recover free multiplicative ID distributions","Rectangular free convolution from Laguerre-Pólya on real rooted polys","Cauchy law as root limit of Riemann Xi Jensen polynomials","Extending heat flow free Brownian to any free Lévy process"],"cache_read_input_tokens":64,"weakest_assumption_plain":"For every free additive infinitely divisible distribution there exists at least one sequence of Laguerre-Pólya functions f_n such that the empirical root measure of the Appell polynomial f_n(∂_z)z^n converges weakly to the target distribution.","fun_headline_variants_meta":{"raw":{"variants":["Any free additive ID law as limit of Appell roots under Laguerre-Pólya","Laguerre-Pólya operators recover free multiplicative ID distributions","Rectangular free convolution from Laguerre-Pólya on real rooted polys","Cauchy law as root limit of Riemann Xi Jensen polynomials","Extending heat flow free Brownian to any free Lévy process"]},"model":"grok-4.3","cost_usd":0.006929,"raw_usage":{"total_tokens":3277,"prompt_tokens":796,"num_sources_used":0,"completion_tokens":89,"cost_in_usd_ticks":69287000,"prompt_tokens_details":{"text_tokens":796,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2392,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":796,"tokens_out":89,"duration_ms":20203,"temperature":1.0,"reasoning_tokens":2392,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-28T21:01:50.941549+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Exhibiting one free additive infinitely divisible distribution for which no sequence of Laguerre-Pólya functions makes the root measures of the corresponding Appell polynomials converge weakly to it.","supporting_citations":[],"review_version":1}