Octavio Arizmendi

Relations between cumulants in noncommutative probability

Abstract We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.

On the asymptotic distribution of block-modified random matrices

We study random matrices acting on tensor product spaces which have been transformed by a linear block operation. Using operator-valued free probability theory, under some mild assumptions on the linear map acting on the blocks, we compute the asymptotic eigenvalue distribution of the modified matrices in terms of the initial asymptotic distribution. Moreover, using recent results on operator-valued subordination, we present an algorithm that computes, numerically but in full generality, the limiting eigenvalue distribution of the modified matrices. Our analytical results cover many cases of interest in quantum information theory: we unify some known results and we obtain new distributions and various generalizations.

Asymptotic Spectral Distributions of Distance-k Graphs of Star Product Graphs

Let G be a finite connected graph and let G[⋆N, k] be the distance-k graph of the N-fold star power of G. For a fixed k ≥ 1, we show that the large N limit of the spectral distribution of G[⋆N, k] converges to a centered Bernoulli distribution, \(1/2\delta _{-1} + 1/2\delta _{1}\). The proof is based in a fourth moment lemma for convergence to a centered Bernoulli distribution.

Free Subordination and Belinschi-Nica Semigroup

We realize the Belinschi–Nica semigroup of homomorphisms as a free multiplicative subordination. This realization allows to define more general semigroups of homomorphisms with respect to free multiplicative convolution. For these semigroups we show that a differential equation holds, generalizing the complex Burgers equation. We give examples of free multiplicative subordination and find a relation to the Markov–Krein transform, Boolean stable laws and monotone stable laws. A similar idea works for additive subordination, and in particular we study the free additive subordination associated to the Cauchy distribution and show that it is a homomorphism with respect to monotone, Boolean and free additive convolutions.

Cumulants for finite free convolution

Abstract 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.

A Berry-Esseen type limit theorem for Boolean convolution

We give estimates on the rate of convergence on the Boolean central limit theorem for the Lévy distance. In the case of measures with bounded support, we obtain a sharp estimate by giving a qualitative description of this convergence.

On the spectral distribution of distance-k graph of free product graphs

We calculate the distribution with respect to the vacuum state of the distance-

On the Law of Free Subordinators

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On a class of explicit Cauchy–Stieltjes transforms related to monotone stable and free Poisson laws

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Classical scale mixtures of Boolean stable laws

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