Related papers: A Simple Introduction to Free Probability Theory a…
Large-dimensional random matrix theory, RMT for short, which originates from the research field of quantum physics, has shown tremendous capability in providing deep insights into large dimensional systems. With the fact that we have…
We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…
These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition…
Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of…
The theory of large deviations deals with the probabilities of rare events (or fluctuations) that are exponentially small as a function of some parameter, e.g., the number of random components of a system, the time over which a stochastic…
We show how the combination of new "linearization" ideas in free probability theory with the powerful "realization" machinery -- developed over the last 50 years in fields including systems engineering and automata theory -- allows solving…
The goal of these expository notes is to give an introduction to random matrices for non-specialist of this topic focusing on the link between random matrices and systems of particles in interaction. We first recall some general results…
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they…
Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of inter-dependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier…
This is an introduction to some of the most probabilistic aspects of free probability theory.
We investigate the implications of free probability for random matrices. From rules for calculating all possible joint moments of two free random matrices, we develop a notion of partial freeness which is quantified by the breakdown of…
Random matrix theory allows for the deduction of stability criteria for complex systems using only a summary knowledge of the statistics of the interactions between components. As such, results like the well-known elliptical law are…
Random matrices have their roots in multivariate analysis in statistics, and since Wigner's pioneering work in 1955, they have been a very important tool in mathematical physics. In functional analysis, random matrices and random structures…
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…
This paper is a brief review of recent developments in random matrix theory. Two aspects are emphasized: the underlying role of integrable systems and the occurrence of the distribution functions of random matrix theory in diverse areas of…
We introduce the concept of matrix liberation process, a random matrix counterpart of the liberation process in free probability, and prove a large deviation upper bound for its empirical distribution with several properties on its rate…
A combinatorial approach to free probability theory has been developped by Roland Speicher, based on the notion of noncrossing cumulants, a free analogue of the classical theory of cumulants in probability theory. We review this theory, and…
Some tools and ideas are interchanged between random matrix theory and multivariate statistics. In the context of the random matrix theory, classes of spherical and generalised Wishart random matrix ensemble, containing as particular cases…
This paper covers two topics: first an introduction to Algorithmic Complexity Theory: how it defines probability, some of its characteristic properties and past successful applications. Second, we apply it to problems in A.I. - where it…
We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that…