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Mihai Popa

Publications and source records attributed to Mihai Popa.

At least 19 recordsLinked to original sources
arXiv2026

Freely infinitely divisible $R$-diagonal elements and Brown measure

We study freely infinitely divisible $R$-diagonal elements in the unbounded setting and Brown measures for free additive perturbations by such elements. This class includes circular elements, circular Cauchy elements, and other previously studied $R$-diagonal models. We construct examples and prove stability under several algebraic operations, including homogeneous noncommutative polynomials in bounded, freely independent elements from this class. Using results for general $R$-diagonal perturbations, together with several analytic estimates specific to freely infinitely divisible $R$-diagonal elements, we prove that, in the bounded case, the support of the Brown measure coincides with the spectrum, and we obtain a criterion for property (H) in this non-normal setting. Finally, we study the free convolution semigroup associated with the symmetrized law of the modulus and derive a Hamilton--Jacobi equation for the regularized logarithmic potential.

math.OA↗
arXiv2025

On some properties of free commutators with semicircular variables

We investigate commutators of free variables of the form \( i[x, s] \), where \( s \) is a semicircular element. We show that although \( s \) and \( i[x, s] \) are not free, their sum nevertheless satisfies the free additive convolution identity \[ \mu_{s + i[x, s]} = \mu_s \boxplus \mu_{i[x, s]}. \] Furthermore, we prove that the polynomial \( x + i[x, s] \) is freely infinitely divisible whenever \( x \) itself is freely infinitely divisible.

math.OA↗
arXiv2024

On partial transposes of unitarily invariant random matrices

We compute the limit distribution of partial transposes (when both the number and the size of blocks tends to infinity) for a large class of ensembles of unitarily invariant random matrices. Furthermore, it is shown the asymptotic freeness relation between the ensembles of random matrices, their transposes and their left and right partial transposes.

math.PR↗
arXiv2021

Answer to a question by A. Mandarino, T. Linowski and K. Życzkowski

A recent work by A. Mandarino, T. Linowski and K. Życzkowski left open the following question. If $ μ_N $ is a certain permutation of entries of a $ N^2 \times N^2 $ matrix ("mixing map") and $ U_N $ is a $ N^2 \times N^2 $ Haar unitary random matrix, then is the family $ U_N, U_N^{μ_N}, ( U_N^2 )^{μ_N}, \dots , ( U_N^m)^{μ_N} $ asymptotically free? (here by $A^{ μ}$ we understand the matrix resulted by permuting the entries of $ A $ according to the permutation $ μ$). This paper presents some techniques for approaching such problems. In particular, one easy consequence of the main result is that the question above has an affirmative answer.

math.PR↗
arXiv2021

On the Partial Transpose of a Haar Unitary Matrix

We consider the effect of a partial transpose on the limit $*$-distribution of a Haar distributed random unitary matrix. If we fix, $b$, the number of blocks, we show that the partial transpose can be decomposed into a sum of $b$ matrices which are asymptotically free and identically distributed. We then consider the joint effect of different block decompositions and show that under some mild assumptions we also get asymptotic freeness.

math.OA↗
arXiv2021

Asymptotic $\ast$--distribution of permuted Haar unitary matrices

We study Haar unitary random matrices with permuted entries. For a sequence of permutations $\left(σ_N\right)_N$, where $σ_N$ acts on $N\times N$ matrices we identify conditions under which the $\ast$--distribution of permuted Haar unitary matrices $U_N^{σ_N}$ is asymptotically circular and free from the unpermuted sequence $U_N$. We show that this convergence takes place in the almost sure sense. Moreover we show that our conditions on the sequence of permutations are generic in the sense that are almost surely satisfied by a sequence of random permutations.

math.PR↗
arXiv2020

The Partial Transpose and Asymptotic Free Independence for Wishart Random Matrices: Part II

Using new combinatorial techniques, we significantly improve the previous results on asymptotic distributions and asymptotic free independence relations of partial transposes of Wishart random matrices. In particular, we give a necessary and sufficient condition for the asymptotic free independence of partial transposes of Wishart matrices with difference block sizes.

math.OA↗
arXiv2017

An asymptotic property of large matrices with identically distributed Boolean independent entries

Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean independent entries. More precisely, we show that, under some moment conditions, such random matrices are asymptotically $ B $-diagonal and Boolean independent from each other. The paper also gives a combinatorial condition under which such matrices are asymptotically Boolean independent from the matrix obtained by permuting the entries (thus extending a recent result in Boolean probability). In particular, we show that random matrices considered are asymptotically Boolean independent from their partial transposes. The main results of the paper are based on combinatorial techniques.

math.OA↗
arXiv2017

Freeness and The Partial Transposes of Wishart Random Matrices

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives a example where the partial transpose produces freeness at the operator level. Finally we investigate the case of real Wishart matrices.

math.OA↗
arXiv2017

H${}^2$ Spaces of Non-Commutative Functions

We define the Hardy spaces of free noncommutative functions on the noncommutative polydisc and the noncommutative ball and study their basic properties. Our technique combines the general methods of noncommutative function theory and asymptotic formulae for integration over the unitary group. The results are the first step in developing the general theory of free noncommutative bounded symmetric domains on the one hand and in studying the asymptotic free noncommutative analogues of classical spaces of analytic functions on the other.

math.OA↗
arXiv2017

A combinatorial result on asymptotic independence relations for random matrices with non-commutative entries

The paper gives a general condition on permutations, condition under which a semicircular matrix is free independent, or asymptotically free independent from the semicircular matrix obtained by permuting its entries. In particular, it is shown that semicircular matrices are asymptotically free from their transposes, a result similar to the case of Gaussian random matrices. There is also an analysis of asymptotic second order relations between semicircular matrices and their transposes, with results not very similar to the commutative (i.e. Gaussian random matrices) framework. The paper also presents an application of the main results to the study of Gaussian random matrices and furthermore it is shown that the same condition as in the case of semicircular matrices gives Boolean independence, or asymptotic Boolean independence when applied to Bernoulli matrices.

math.OA↗
arXiv2016

On the multiplication of operator-valued c-free random variables

We discuss some results concerning the multiplication of non-commutative random variables that are c-free with respect to a pair $( Φ, φ) $, where $ Φ$ is a linear map with values in some Banach or C$^\ast$-algebra and $ φ$ is scalar-valued. In particular, we construct a suitable analogue of the Voiculescu's $ S $-transform for this framework.

math.OA↗
arXiv2014

Freeness and The Transposes of Unitarily Invariant Random Matrices

We show that real second order freeness appears in the study of Haar unitary and unitarily invariant random matrices when transposes are also considered. In particular we obtain the unexpected result that a unitarily invariant random matrix will be asymptotically free from its transpose.

math.OA↗
arXiv2013

Real Second Order Freeness and Haar Orthogonal Matrices

We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier [R1, R2].

math.OA↗
arXiv2013

On fluctuations of traces of large matrices over a non-commutative algebra

The paper investigates the asymptotic behavior of (non-normalized) traces of certain classes of matrices with non-commutative random variables as entries. We show that, unlike in the commutative framework, the asymptotic behavior of matrices with free circular, respectively with Bernoulli distributed Boolean independent entries is described in terms of free, respectively Boolean cumulants. We also present an exemple of relation of monotone independence arising from the study of Boolean independence.

math.OA↗
arXiv2011

Infinite divisibility and a non-commutative Boolean-to-free Bercovici-Pata bijection

We use the theory of fully matricial, or non-commutative, functions to investigate infinite divisibility and limit theorems in operator-valued non-commutative probability. Our main result is an operator-valued analogue of the Bercovici-Pata bijection. An important tool is Voiculescu's subordination property for operator-valued free convolution.

math.OA↗
arXiv2011

Non-Commutative Functions and Non-Commutative Free Levy-Hincin Formula

The paper is discussing infinite divisibility in the setting of operator-valued boolean, free and, more general, c-free independences. Particularly, using Hilbert bimodules and non-commutative functions techniques, we obtain analogues of the Levy-Hincin integral representation for infinitely divisible real measures.

math.OA↗