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Octavio Arizmendi

Publications and source records attributed to Octavio Arizmendi.

At least 19 recordsLinked to original sources
arXiv2026

Freeness for the $G$-circulant Decomposition of the Partial Transpose of Random Matrices

We introduce a left $G$-circulant decomposition for matrices indexed by an arbitrary finite group $G$, extending the diagonal decomposition associated with cyclic groups. We show that, when $A\in M_{|G|}(\mathcal A)$ is free from $M_{|G|}(\mathbb C)$, the components arising from the left $G$-circulant decomposition of $A^t$ form a free family, with the components associated with inverse pairs forming $R$-diagonal pairs. We also describe the distributions of these components in terms of the distribution of $A$. Our results recover the cyclic case and show that different group structures of the same order may lead to different free decompositions of the same matrix.

math.PR↗
arXiv2026

An analytic approach to the finite R-transform

We revisit Marcus' finite free analogue of Voiculescu $R$-transform from an analytic viewpoint. By relating the finite free Fourier transform to the Laplace transform, we study the finite $R$-transform through logarithmic potentials and Legendre transforms. Under suitable assumptions, we prove that the finite $R$-transform of a polynomial differs from the Voiculescu $R$-transform of its empirical root distribution by $O(N^{-1})$. As an application, we obtain an analytic proof of the convergence of finite free additive convolution to free additive convolution.

math.PR↗
arXiv2026

Free multiplicative convolution with an arbitrary measure on the real line

We develop analytic tools for studying the free multiplicative convolution of any measure on the real line and any measure on the nonnegative real line. More precisely, we construct the subordination functions and the $S$-transform of an arbitrary probability measure. The important multiplicativity of $S$-transform is proved with the help of subordination functions. We then apply the $S$-transform to establish convolution identities for stable laws, which had been considered in the literature only for the positive and symmetric cases. Subordination functions are also used in order to extend Belinschi--Nica's semigroup of homomorphisms, and to establish regularity properties of free multiplicative convolution, in particular, the absence of singular continuous part and analyticity of the density.

math.PR↗
arXiv2026

Combinatorics of cyclic-conditional freeness

This work investigates the combinatorial structures underlying cyclic conditional freeness and introduces cumulants that serve to linearize the cyclic conditional additive convolution. In the process, we establish the notion of "cyclic freeness", demonstrating its equivalence to infinitesimal freeness in the presence of tracial states. Furthermore, we show that cyclic conditional freeness can be reduced to cyclic freeness through a multivariate extension of the inverse Markov-Krein transform.

math.OA↗
arXiv2025

Critical points of random polynomials and finite free cumulants

A result of Hoskins and Steinerberger [Int. Math. Res. Not., (13):9784-9809, 2022] states that repeatedly differentiating a random polynomials with independent and identically distributed mean zero and variance one roots will result, after an appropriate rescaling, in a Hermite polynomial. We use the theory of finite free probability to extend this result in two natural directions: (1) We prove central limit theorems for the fluctuations around these deterministic limits for the polynomials and their roots. (2) We consider a generalized version of the Hoskins and Steinerberger result by removing the finite second moment assumption from the roots. In this case the Hermite polynomials are replaced by a random Appell sequence conveniently described through finite free probability and an infinitely divisible distribution. We use finite free cumulants to provide compact proofs of our main results with little prerequisite knowledge of free probability required.

math.PR↗
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 Szegö 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

On matrices in finite free position

We study pairs $(A,B)$ of square matrices that are in additive (resp. multiplicative) finite free position, that is, the characteristic polynomial $χ_{A+B}(x)$ (resp. $χ_{AB}(x)$) equals the additive finite free convolution $χ_{A}(x) \boxplus χ_{B}(x)$ (resp. the multiplicative finite free convolution $χ_{A}(x) \boxtimes χ_{B}(x)$), which equals the expected characteristic polynomial $\mathbb{E}_U [ χ_{A+U^* BU}(x) ]$ (resp. $\mathbb{E}_U [ χ_{AU^* BU}(x) ]$) over the set of unitary matrices $U$. We examine the lattice of (non-irreducible) affine algebraic sets of matrices consisting of finite free complementary pairs with respect to the additive (resp. multiplicative) convolution. We show that these pairs include the diagonal matrices vs. the principally balanced matrices, the upper (lower) triangular matrices vs. the upper (lower) triangular matrices with constant diagonal, and the scalar matrices vs. the set of all square matrices.

math.RA↗
arXiv2025

Third Order Cumulants of products

We provide a formula for the third order free cumulants of products as entries. We apply this formula to find the third order free cumulants of various Random Matrix Ensambles including product of Ginibre Matrices and Wishart matrices, both in the Gaussian case.

math.PR↗
arXiv2025

$S$-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 $μ$ on $[0,\infty)$. Specifically, we relate the asymptotic behaviour of the ratio of consecutive coefficients to Voiculescu's $S$-transform of $μ$. 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--Möller 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

Central limit theorem for crossings in randomly embedded graphs

We consider the number of crossings in a random embedding of a graph, $G$, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of $G$. Using Stein's method and size-bias coupling, we establish an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. We also consider the case where $G$ is a random graph and obtain a Kolmogorov bound between the distribution of crossings and a Gaussian mixture distribution. As applications, we obtain central limit theorems with convergence rates for the number of crossings in random embeddings of matchings, path graphs, cycle graphs, disjoint union of triangles, random $d$-regular graphs, and mixtures of random graphs.

math.PR↗
arXiv2024

New combinatorial identity for the set of partitions and limit theorems in finite free probability theory

We provide a refined combinatorial identity for the set of partitions of $\{1,\dots, n\}$, which plays an important role in investigating several limit theorems related to finite free convolutions. Firstly, we present the finite free analogue of Sakuma and Yoshida's limit theorem. That is, we provide the limit of $\{D_{1/m}((p_d^{\boxtimes_d m})^{\boxplus_d m})\}_{m\in \mathbb{N}}$ as $m\rightarrow\infty$ in two cases: (i) $m/d\rightarrow t$ for some $t>0$, or (ii) $m/d\rightarrow0$. The second application presents a central limit theorem for finite free multiplicative convolution. We establish a connection between this theorem and the multiplicative free semicircular distributions through combinatorial identities. Our last result gives alternative proofs for Kabluchko's limit theorems concerning the unitary Hermite and the Laguerre polynomials.

math.PR↗
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

Free probability via entropic optimal transport

Let $μ$ and $ν$ be probability measures on $\mathbb{R}$ with compact support, and let $μ\boxplus ν$ denote their additive free convolution. We show that for $z \in \mathbb{R}$ greater than the sum of essential suprema of $μ$ and $ν$, we have \begin{equation*} \int_{-\infty}^\infty \log(z - x) μ\boxplus ν(\mathrm{d}x) = \sup_Π \left\{ \mathbf{E}_Π[\log(z - (X+Y)] - H(Π|μ\otimes ν) \right\}, \end{equation*} where the supremum is taken over all couplings $Π$ of the probability measures $μ$ and $ν$, and $H(Π|μ\otimes ν)$ denotes the relative entropy of a coupling $Π$ against product measure. We prove similar formulas for the multiplicative free convolution $μ\boxtimes ν$ and the free compression $[μ]_τ$ of probability measures, as well as for multivariate free operations. Thus the integrals of a log-potential against the fundamental measure operations of free probability may be formulated in terms of entropic optimal transport problems. The optimal couplings in these variational descriptions of the free probability operations can be computed explicitly, and from these we can then deduce the standard $R$- and $S$-transform descriptions of additive and multiplicative free convolution. We use our optimal transport formulations to derive new inequalities relating free and classical operations on probability measures, such as the inequality \begin{equation*} \int_{-\infty}^\infty \log(z - x) μ\boxplus ν(\mathrm{d}x) \geq \int_{-\infty}^{\infty} \log(z-x) μ\ast ν( \mathrm{d}x) \end{equation*} relating free and classical convolution. Our approach is based on applying a large deviation principle on the symmetric group to the quadrature formulas of Marcus, Spielman and Srivastava.

math.PR↗
arXiv2024

The change of vertex energy when joining trees

In this manuscript we study how the vertex energy of a tree is affected when joined with a bipartite graph. We find an alternating pattern with respect to the coalescence vertex: the energy decreases for vertices located at odd distances and increases for those located at even distances.

math.SP↗
arXiv2023

BMT Independence

We introduce the notion of BMT independence, allowing us to take arbitrary mixtures of boolean, monotone, and tensor independence and generalizing the notion of BM independence of Wysoczanski. Pair-wise independence relations are encoded through a directed graph, which in turn determines the way mixed moments must be computed. Corresponding Central and Poisson-Type Limit Theorems are provided along with an explicit construction to realize BMT independent random variables as bounded operators on certain Hilbert space.

math.OA↗
arXiv2023

Second Order Cumulants: second order even elements and R-diagonal elements

We introduce $R$-diagonal and even operators of second order. We give a formula for the second order free cumulants of the square $x^2$ of a second order even element in terms of the second order free cumulants of $x$. Similar formulas are proved for the second order free cumulants of $aa^*$, when $a$ is a second order $R$-diagonal operator. We also show that if $r$ is second order $R$-diagonal and $b$ is second order free from $r$, then $rb$ is also second order $R$-diagonal. We present a large number of examples, in particular the limit distribution of products of Ginibre matrices. We prove the conjectured formula of Dartois and Forrester for the fluctuations moments of the product of two independent complex Wishart matrices and generalize it to any number of factors.

math.OA↗