Related papers: Asymptotic cyclic-conditional freeness of random m…
Non-asymptotic theory of random matrices strives to investigate the spectral properties of random matrices, which are valid with high probability for matrices of a large fixed size. Results obtained in this framework find their applications…
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 show that finite rank perturbations of certain random matrices fit in the framework of infinitesimal (type B) asymptotic freeness. This can be used to explain the appearance of free harmonic analysis (such as subordination functions…
We prove a number of results concerning the large $N$ asymptotics of the free energy of a random matrix model with a polynomial potential $V(z)$. Our approach is based on a deformation $\tau_tV(z)$ of $V(z)$ to $z^2$, $0\le t<\infty$ and on…
Unitary randomness underpins both fundamental tasks in quantum information and the modern theory of quantum chaos. On one side, a central concept is that of approximate unitary designs: circuits that look random according to small moments…
We characterize asymptotic collective behaviour of rectangular random matrices, the sizes of which tend to infinity at different rates: when embedded in a space of larger square matrices, independent rectangular random matrices are…
We present an alternative proof of asymptotic freeness of independent sample covariance matrices, when the dimension and the sample size grow at the same rate, by embedding these matrices into Wigner matrices of a larger order and using…
We extend the relation between random matrices and free probability theory from the level of expectations to the level of all correlation functions (which are classical cumulants of traces of products of the matrices). We introduce the…
We introduce a new kind of free independence, called real infinitesimal freeness. We show that independent orthogonally invariant with infinitesimal laws are asymptotically real infinitesimally free. We introduce new cumulants, called real…
Random matrix theory has played a major role in several areas of pure and applied mathematics, as well as statistics, physics, and computer science. This lecture aims to describe the intrinsic freeness phenomenon and how it provides new…
We establish a large deviation principle for the empirical spectral measure of a sample covariance matrix with sub-Gaussian entries, which extends Bordenave and Caputo's result for Wigner matrices having the same type of entries [7]. To…
Let A_1,A_2,...,A_s be a finite sequence of (not necessarily disjoint, or even distinct) non-empty sets of positive integers satisfying a certain condition. It is shown that an independent family U_1,U_2,...,U_s of random NxN permutation…
We study the asymptotic behavior of the free cumulants (in the sense of free probability theory of Voiculescu) of Jucys--Murphy elements--or equivalently--of the transition measure associated with a Young diagram. We express these cumulants…
A central tool in the study of nonhomogeneous random matrices, the noncommutative Khintchine inequality, yields a nonasymptotic bound on the spectral norm of general Gaussian random matrices $X=\sum_i g_i A_i$ where $g_i$ are independent…
We consider the algebra of square matrices of bounded non-commutative (NC) functions over NC operator unit balls (unit balls corresponding to finite-dimensional operator spaces) and characterize cyclic matrix free polynomials with respect…
We derive a multiplication law for free non-hermitian random matrices allowing for an easy reconstruction of the two-dimensional eigenvalue distribution of the product ensemble from the characteristics of the individual ensembles. We define…
Let $\Sigma_{g}$ be a closed surface of genus $g\geq 2$ and $\Gamma_{g}$ denote the fundamental group of $\Sigma_{g}$. We establish a generalization of Voiculescu's theorem on the asymptotic $*$-freeness of Haar unitary matrices from free…
Using the standard concepts of free random variables, we show that for a large class of nonhermitean random matrix models, the support of the eigenvalue distribution follows from their hermitean analogs using a conformal transformation. We…
Recent work of Belinschi, Mai and Speicher resulted in a general algorithm to calculate the distribution of any selfadjoint polynomial in free variables. Since many classes of independent random matrices become asymptotically free if the…
Asymptotic freedom is a feature of quantum chromodynamics that guarantees its well-posedeness. We derive an analog of asymptotic freedom enabling unconditional stability of lattice Boltzmann simulation of hydrodynamics. For the lattice…