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Daniel Perales

Publications and source records attributed to Daniel Perales.

17 recordsLinked to original sources
arXiv2026

Repeated differentiation of deterministic polynomials with asymptotically radial root distributions

Recent works of Galligo, Najnudel, and Vu (2025) and Najnudel and Vu (2026) study repeated differentiation for polynomials of the form $P(z)=p(z^m)$, where $p$ is a deterministic polynomial of degree $n$ with real, non-negative roots, in the regime where $m$ and $n$ are large. If $m\gg \log(n)$ and the root distribution of $P$ converges to a compactly supported, radial probability measure $\mu_0$, these works show that for $0\le t<1$, the root distribution of the $\lfloor nmt\rfloor$-th derivative of $P$ converges to a compactly supported probability measure $\mu_t$ given by an explicit formula for its radial quantile function. We give a substantially simplified proof of this result and also extend the result from repeated differentiation to repeated applications of the differential operator $z^a(d/dz)^b$. We also compute the limiting root distribution in the case when $m$ is fixed and $n$ tends to infinity.

math.PR↗
arXiv2026

Multiplicative and Additive Finite Free Convolutions for q-Polynomials

We study $q$-analogs of finite free convolutions and their interaction with families of $q$-hypergeometric polynomials. First, we revisit the $q$-multiplicative finite free convolution, previously introduced in the literature, and show that it acts naturally on $q$-hypergeometric polynomials: the convolution of two such polynomials remains within the same class, with parameters obtained by concatenation. This observation provides a simple mechanism for constructing large families of $q$-hypergeometric polynomials whose zeros are real and whose logarithmic mesh is controlled. We illustrate it with an example of multiple little $q$-Jacobi polynomials of the first kind. A result of independent interest is also an alternative definition of the $q$-multiplicative convolution in terms of $q$-differential operators. Motivated by the additive finite free convolution, we introduce a $q$-additive finite free convolution and study its algebraic and analytic properties. Although this convolution does not preserve real-rootedness in general, we show that a natural modification involving a $q$-multiplicative convolution restores the preservation of real roots and interlacing for polynomials with bounded logarithmic mesh. Finally, we develop a systematic method to translate product identities of $q$-hypergeometric functions into convolution identities for $q$-hypergeometric polynomials. This approach yields several explicit formulas for $q$-additive convolutions and produces new families of real-rooted $q$-hypergeometric polynomials.

math.CA↗
arXiv2025

Asymptotic root distribution of polynomials under repeated polar differentiation

Given a sequence of real rooted polynomials $\{p_n\}_{n\geq 1}$ with a fixed asymptotic root distribution, we study the asymptotic root distribution of the repeated polar derivatives of this sequence. This limiting distribution can be seen as the result of fractional free convolution and pushforward maps along M\"obius transforms for distributions. This new family of operations on measures forms a semigroup and satisfy some other nice properties. Using the fact that polar derivatives commute with one another, we obtain a non-trivial commutation relation between these new operations. We also study a notion of polar free infinite divisibility and construct Belinschi-Nica type semigroups. Finally, we provide some interesting examples of distributions that behave nicely with respect to these new operations, including the Marchenko-Pastur and the Cauchy distributions.

math.PR↗
arXiv2025

Second order free cumulants: product, commutator, and anti-commutator

Given two second order free random variables $a$ and $b$, we study the second order free cumulants of their product $ab$, their commutator $ab-ba$, and their anti-commutator $ab+ba$. Let $(\kappa_n^a)_{n\geq 1}$ and $(\kappa_{n,m}^a)_{n,m\geq 1}$ denote the sequence of free cumulants of first and second order, respectively, of a random variable $a$ in a second order non-commutative probability space $(\mathcal{A},\varphi,\varphi^2)$. Given $a$ and $b$ two second order freely independent random variables, we provide formulas to compute each of the cumulants $(\kappa_{n,m}^{ab})_{n,m\geq 1}$, $(\kappa_{n,m}^{ab-ba})_{n,m\geq 1}$, and $(\kappa_{n,m}^{ab+ba})_{n,m\geq 1}$ in terms of the individual cumulants $(\kappa_{n}^{a})_{n\geq 1}$, $(\kappa_{n,m}^{a})_{n,m\geq 1}$, $(\kappa_{n}^{b})_{n\geq 1}$, and $(\kappa_{n,m}^{b})_{n,m\geq 1}$. For $n=m=1$ our formulas read: \begin{align*} \kappa_{1,1}^{ab} &= \kappa_{2}^{a}\kappa_{2}^{b} +\kappa_{1,1}^{a}(\kappa_{1}^{b})^2+\kappa_{1,1}^{b}(\kappa_{1}^{a})^2,\\ \kappa_{1,1}^{ab-ba} &= 2\kappa_{2}^{a}\kappa_{2}^{b},\\ \kappa_{1,1}^{ab+ba} &= 2\kappa_{2}^{a}\kappa_{2}^{b} +4\kappa_{1,1}^{a}(\kappa_{1}^{b})^2+4\kappa_{1,1}^{b}(\kappa_{1}^{a})^2. \end{align*} In general, our formulas express the cumulants $\kappa_{n,m}^{ab}$, $\kappa_{n,m}^{ab-ba}$, and $\kappa_{n,m}^{ab+ba}$ as sums indexed by special subsets of non-crossing partitioned permutations. The formulas for the commutator and anti-commutator where not studied before, while the formula for the product was only known in the case the where the individual second order free cumulants vanish. As an application, we compute explicitly the cumulants of the anti-commutator and product of two second order free semicircular variables.

math.OA↗
arXiv2025

Zeros of orthogonal little q-Jacobi polynomials: interlacing and monotonicity

We investigate the distribution of zeros of the little q-Jacobi polynomials and related q-hypergeometric families. We prove that the zeros of these orthogonal polynomials exhibit strong interlacing properties and obey natural monotonicity rules with respect to the parameters. A key tool in our approach is the logarithmic mesh, which quantifies the relative spacing of the positive real zeros and allows us to classify families of polynomials with prescribed interlacing patterns. Our results include new interlacing relations, monotonicity with respect to parameters, and structural decompositions in non-orthogonal regimes. Several classical families of q-hypergeometric polynomials, including q-Bessel and Stieltjes-Wigert polynomials, are treated as limit cases. The methods rely on a combination of classical orthogonality theory and q-difference equations.

math.CA↗
arXiv2025

Finite Free Convolution: Infinitesimal Distributions

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.

math.PR↗
arXiv2025

Even Hypergeometric Polynomials and Finite Free Commutators

We study in detail the class of even polynomials and their behavior with respect to finite free convolutions. To this end, we use some specific hypergeometric polynomials and a variation of the rectangular finite free convolution to understand even real-rooted polynomials in terms of positive-rooted polynomials. Then, we study some classes of even polynomials that are of interest in finite free probability, such as even hypergeometric polynomials, symmetrizations, and finite free commutators. Specifically, we provide many new examples of these objects, involving classical families of special polynomials (such as Laguerre, Hermite, and Jacobi). Finally, we relate the limiting root distributions of sequences of even polynomials with the corresponding symmetric measures that arise in free probability.

math.CA↗
arXiv2024

$S$-transform in Finite Free Probability

We present a simplified explanation of why free fractional convolution corresponds to the differentiation of polynomials, by finding how the finite free cumulants of a polynomial behave under differentiation. This approach allows us to understand the limiting behaviour of the coefficients $\widetilde{\mathsf{e}}_k(p_d)$ of $p_d$ when the degree $d$ tends to infinity and the empirical root distribution of $p_d$ has a limiting distribution $\mu$ on $[0,\infty)$. Specifically, we relate the asymptotic behaviour of the ratio of consecutive coefficients to Voiculescu's $S$-transform of $\mu$. This prompts us to define a new notion of finite $S$-transform, which converges to Voiculescu's $S$-transform in the large $d$ limit. It also satisfies several analogous properties to those of the $S$-transform in free probability, including multiplicativity and monotonicity. This new insight has several applications that strengthen the connection between free and finite free probability. Most notably, we generalize the approximation of $\boxtimes_d$ to $\boxtimes$ and prove a finite approximation of the Tucci--Haagerup--M\"oller limit theorem in free probability, conjectured by two of the authors. We also provide finite analogues of the free multiplicative Poisson law, the free max-convolution powers and some free stable laws.

math.OA↗
arXiv2024

Zeros of generalized hypergeometric polynomials via finite free convolution. Applications to multiple orthogonality

We address the problem of the weak asymptotic behavior of zeros of families of generalized hypergeometric polynomials as their degree tends to infinity. The main tool is the representation of such polynomials as a finite free convolution of simpler elements; this representation is preserved in the asymptotic regime, so we can formally write the limit zero distribution of these polynomials as a free convolution of explicitly computable measures. We derive a simple expression for the S-transform of the limit distribution, which turns out to be a rational function, and a representation of the Kampé de Fériet polynomials in terms of finite free convolutions. We apply these tools, as well as those from [arXiv:2309.10970], to the study of some well-known families of multiple orthogonal polynomials (Jacobi-Piñeiro and multiple Laguerre of the first and second kinds), obtaining results on their zeros, such as interlacing, monotonicity, and asymptotics.

math.CA↗
arXiv2024

Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions

Given two polynomials $p(x), q(x)$ of degree $d$, we give a combinatorial formula for the finite free cumulants of $p(x)\boxtimes_d q(x)$. We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that $\boxtimes_d$ converges to $\boxtimes$ as $d$ goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building off our results we give a new short and conceptual proof of a recent result [Steinerberger (2020), Hoskins and Kabluchko (2020)] that connects root distributions of polynomial derivatives with free fractional convolution powers.

math.CO↗
arXiv2024

Real roots of hypergeometric polynomials via finite free convolution

We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a hypergeometric polynomial as a finite free convolution of more elementary blocks, combined with the preservation of the real zeros and interlacing by the free convolutions, is an effective tool that allows us to analyze when all roots of a specific hypergeometric polynomial are real. Moreover, the known limit behavior of finite free convolutions allows us to write the asymptotic zero distribution of some hypergeometric polynomials as free convolutions of Marchenko-Pastur, reversed Marchenko-Pastur, and free beta laws, which has an independent interest within free probability.

math.CA↗
arXiv2022

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

We consider the group $(\mathcal{G},*)$ of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where ``$*$'' denotes the convolution operation. We introduce a larger group $(\widetilde{\mathcal{G}},*)$ of unitized functions from the same incidence algebra, which satisfy a weaker condition of being ``semi-multiplicative''. The natural action of $\widetilde{\mathcal{G}}$ on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of $\widetilde{\mathcal{G}}$ in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of $t$-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of Bozejko and Wysoczanski. It is known that the group $\mathcal{G}$ can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that $\widetilde{\mathcal{G}}$ can also be identified as group of characters of a Hopf algebra $\mathcal{T}$, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion of $\mathcal{G}$ into $\widetilde{\mathcal{G}}$ turns out to be the dual of a natural bialgebra homomorphism from $\mathcal{T}$ onto Sym.

math.CO↗
arXiv2021

Relations between infinitesimal non-commutative cumulants

Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Our approach is based on the use of Grassmann algebra. Formulas involving infinitesimal cumulants can be obtained by applying a formal derivation to known formulas. The relations between the various types of cumulants turn out to be captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. In this formulation, (free, Boolean and monotone) cumulants are represented as elements of the Lie algebra of infinitesimal characters over a particular combinatorial Hopf algebra. The latter consists of the graded connected double tensor algebra defined over a non-commutative probability space and is neither commutative nor cocommutative. In this note it is shown how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space. The basic step consists in replacing the base field as target space of linear Hopf algebra maps by the Grassmann algebra over the base field. We also consider the infinitesimal analog of the Boolean Bercovici--Pata map.

math.CO↗
arXiv2021

On operator-valued infinitesimal Boolean and monotone independence

We introduce the notion of operator-valued infinitesimal (OVI) independence for the Boolean and monotone cases. Then show that OVI Boolean (resp. monotone) independence is equivalent to the operator-valued Boolean (resp. monotone) independence over an algebra of $2\times 2$ upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (resp. monotone) additive convolution by reducing it to the operator-valued case. We also define OVI Boolean and monotone cumulants and study its basic properties. Moreover, for each notion of OVI independence, we construct the corresponding OVI Central Limit Theorem. The relations among free, Boolean and monotone cumulants are extended to this setting. Besides, in the Boolean case we deduce that the vanishing of mixed cumulants is still equivalent to independence, and use this to connect scalar-valued with matrix-valued infinitesimal Boolean independence. Finally we study two random matrix models that are asymptotically Boolean independent but turn out to not be infinitesimally Boolean independent.

math.OA↗
arXiv2021

On the anti-commutator of two free random variables

Let $(\kappa_n(a))_{n\geq 1}$ denote the sequence of free cumulants of a random variable $a$ in a non-commutative probability space $(\mathcal{A},\varphi)$. Based on some considerations on bipartite graphs, we provide a formula to compute the cumulants $(\kappa_n(ab+ba))_{n\geq 1}$ in terms of $(\kappa_n(a))_{n\geq 1}$ and $(\kappa_n(b))_{n\geq 1}$, where $a$ and $b$ are freely independent. Our formula expresses the $n$-th free cumulant of $ab+ba$ as a sum indexed by partitions in the set $\mathcal{Y}_{2n}$ of non-crossing partitions of the form \[ \sigma=\{B_1,B_3,\dots, B_{2n-1},E_1,\dots,E_r\}, \quad \text{with }r\geq 0, \] such that $i\in B_{i}$ for $i=1,3,\dots,2n-1$ and $|E_j|$ even for $j\leq r$. Therefore, by studying the sets $\mathcal{Y}_{2n}$ we obtain new results regarding the distribution of $ab+ba$. For instance, the size $|\mathcal{Y}_{2n}|$ is closely related to the case when $a,b$ are free Poisson random variables of parameter 1. Our formula can also be expressed in terms of cacti graphs. This graph theoretic approach suggests a natural generalization that allows us to study quadratic forms in $k$ free random variables.

math.OA↗
arXiv2017

Cumulants for finite free convolution

In this paper we define cumulants for finite free convolution. We give a moment-cumulant formula and show that these cumulants satisfy desired properties: they are additive with respect to finite free convolution and they approach free cumulants as the dimension goes to infinity.

math.CO↗