In these proceedings, we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of \(k\)-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on \(k\geq 2\) complex eigenvalues. As a new result, we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.

Random matrices are nowadays classical tools for modeling multiantenna wireless channels. Scattering phenomena typical of cellular frequencies and channel reciprocity features led to the adoption of matrices sampled either from the Gaussian Unitary Ensemble (GUE) or from more general Polynomial Ensembles (PE). Such matrices can be used to model the random impairments of the radio channel on the transmitted signal over a wireless link whose transmitter and receiver are both equipped with antenna arrays. The exploitation of the millimeter-wave (mmWave) frequency band, planned for 5G and beyond mobile networks, prevents the use of GUE and PE elements as candidate models for channel matrices. This is mainly due to the lack of scattering richness compared to microwave-based transmissions. In this work, we propose to model mmWave Multi-Input–Multi-Output (MIMO) systems via products of random Vandermonde matrices. We illustrate the physical motivation of our model selection, discuss the meaning of the parameters and their impact on the spectral properties of the random matrix at hand, and provide both a list of results of immediate use for performance analysis of mmWave MIMO systems, as well as a list of open problems in the field.

We discuss the concept of width-to-spacing ratio which plays the central role in the description of local spectral statistics of evolution operators in multiplicative and additive stochastic processes for random matrices. We show that the local spectral properties are highly universal and depend on a single parameter being the width-to-spacing ratio. We discuss duality between the kernel for Dysonian Brownian motion and the kernel for the Lyapunov matrix for the product of Ginibre matrices.

Two methods of eigeninference, using one-point and two-point Green’s functions, are compared and tested on large samples of random matrices. We note that the first method is mostly robust and very fast, whereas the second method is based on a flawed assumption (the matrix appearing in an optimization problem being in many cases not positive-definite) and computationally complex, which limits its usability.

We use the Kac–Rice method to analyze statistical features of an “optimization landscape” of the loss function in a random version of the Oblique Procrustes Problem, one of the simplest optimization problems of the least-square-type on a sphere.

We use freeness assumptions of random matrix theory to analyze the dynamical behavior of inference algorithms for probabilistic models with dense coupling matrices in the limit of large systems. For a toy Ising model, we are able to recover previous results such as the property of vanishing effective memories and the analytical convergence rate of the algorithm.

The study of the statistical distribution of the eigenvalues of Wishart matrices finds application in many fields of physics and engineering. Here, we consider a special case of finite dimensions correlated complex central Wishart matrices, characterized by the fact that the covariance matrix has all eigenvalues equal, except for one which is the largest. Starting from the knowledge of the joint probability distribution function (p.d.f.) of this kind of Wishart matrices, we focus on the evaluation of a tractable form for the distribution of each individual eigenvalue. In particular, we derive an expression for the p.d.f. of the \(\ell ^{\underline {\mathrm {th}}}\) largest eigenvalue as a sum of terms of the type \(x^{\beta } \mathrm {e}^{-x \delta }\), which allows us to write a large class of statistical averages involving functions of eigenvalues in closed form.