Related papers: Random matrices, free probability and the invarian…
The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The…
As more of topology's tools become popular in analyzing high dimensional data sets, the goal of understanding the underlying probabilistic properties of these tools becomes even more important. While much attention has been given to…
We characterize the limiting second order distributions of certain independent complex Wigner and deterministic matrices using Voiculescu's notions of freeness over the diagonal. If the Wigner matrices are Gaussian, Mingo and Speicher's…
We give a new Banach module characterization of $W^*$-modules, also known as selfdual Hilbert $C^*$-modules over a von Neumann algebra. This leads to a generalization of the notion, and the theory, of W*-modules, to the setting where the…
The curious connection between the spacings of the eigenvalues of random matrices and the corresponding spacings of the non trivial zeros of the Riemann zeta function is analyzed on the basis of the geometric dynamical global program of…
Examples of operator algebras with involution include the operator $*$-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and others, several classical function algebras, triangular matrix…
In a seminal 2005 paper, Haagerup and Thorbj{\o}rnsen discovered that the norm of any noncommutative polynomial of independent complex Gaussian random matrices converges to that of a limiting family of operators that arises from…
Let $R$ be a rational function. The iterations $(R^n)_n$ of $R$ gives a complex dynamical system on the Riemann sphere. We associate a $C^*$-algebra and study a relation between the $C^*$-algebra and the original complex dynamical system.…
Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or…
This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of…
We study the relationship between operator algebras, $C^*$ and von Neumann, acting on a Hilbert space and unitary representations of topological groups on the same space. We obtain certain correspondences between both these families of…
We review recent work connected with the invariant subspace problem for operators, in particular new developments in the last 15 years. In particular, we include discussions of almost-invariant subspaces, universal operators, specific…
Wigner-type randomizations of the tangent spaces of classical symmetric spaces can be thought of as ordinary Wigner matrices on which additional symmetries have been imposed. In particular, they fall within the scope of a framework, due to…
Matrices are said to behave as free non-commuting random variables if the action which governs their dynamics constrains only their eigenvalues, i.e. depends on traces of powers of individual matrices. The authors use recently developed…
We review the state of the art of the theory of Euclidean random matrices, focusing on the density of their eigenvalues. Both Hermitian and non-Hermitian matrices are considered and links with simpler, standard random matrix ensembles are…
In this set of five lectures the authors have presented techniques to analyze open classical and quantum systems using correlation matrices. For diverse reasons we shall see that random matrices play an important role to describe a null…
We review recent progress in the analytic study of random matrix models suggested by noncommutative geometry. One considers fuzzy spectral triples where the space of possible Dirac operators is assigned a probability distribution. These…
This paper explores the Invariant Subspace Problem in operator theory and functional analysis, examining its applications in various branches of mathematics and physics. The problem addresses the existence of invariant subspaces for bounded…
Neural network models are one of the most successful approaches to machine learning, enjoying an enormous amount of development and research over recent years and finding concrete real-world applications in almost any conceivable area of…
There exists an exact relationship between the quasi-exactly solvable problems of quantum mechanics and models of square and rectangular random complex matrices. This relationship enables one to reduce the problem of constructing…