Related papers: Free Probability Theory
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can…
In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D.…
The purpose of this article is to put forward the claim that Hurwitz's paper "Uber die Erzeugung der Invarianten durch Integration." [Gott. Nachrichten (1897), 71-90] should be regarded as the origin of random matrix theory in mathematics.…
This paper describes the expected characteristic polynomial of the commutator of randomly rotated matrices, in the context of the finite free probability theory initiated by Marcus, Spielman, and Srivastava. The key technical features are…
One can build an operatorial model for freeness by considering either the right-handed or the left-handed representation of algebras of operators acting on the free product of the underlying pointed Hilbert spaces. Considering both at the…
The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important…
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…
The amalgamated $T$-transform of a non-commutative distribution was introduced by K.~Dykema. It provides a fundamental tool for computing distributions of random variables in Voiculescu's free probability theory. The $T$-transform…
We introduce a new class of large structured random matrices characterized by four fundamental properties which we discuss. We prove that this class is stable under matrix-valued and pointwise non-linear operations. We then formulate an…
In this work, which was inspired by the article [2] by M. V. Velasco and A. R. Villena, we obtain a characterization for probably continuous operators and show that the probability of a linear random operator being continuous coincides with…
These are notes from a three-lecture mini-course on free probability given at MSRI in the Fall of 2010 and repeated a year later at Harvard. The lectures were aimed at mathematicians and mathematical physicists working in combinatorics,…
Around 1950, Wigner introduced the idea of modelling physical reality with an ensemble of random matrices while studying the energy levels of heavy atomic nuclei. Since then, the field of random-matrix theory has grown tremendously, with…
It has been shown by Voiculescu that important classes of square independent random matrices are asymptotically free, where freeness is a noncommutative analog of classical independence. Recently, we introduced the concept of matricial…
In classical probability the law of large numbers for the multiplicative convolution follows directly from the law for the additive convolution. In free probability this is not the case. The free additive law was proved by D. Voiculescu in…
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…
We characterize semicircular distribution by the freeness of linear and quadratic forms in noncommutative random variables from a tracial $W^*$-probability space with relaxed moment conditions.
This expository paper advocates an approach to physics in which ``typicality" is identified with a suitable form of algorithmic randomness. To this end various theorems from mathematics and physics are reviewed. Their original versions…
Free probability provides a framework for describing correlations between non-commuting observables in complex quantum systems whose Hilbert-space states follow maximum-entropy distributions. We examine the robustness of this framework…
We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of "second order freeness" and derive the global fluctuations of Gaussian and…
We prove that independent rectangular random matrices, when embedded in a space of larger square matrices, are asymptotically free with amalgamation over a commutative finite dimensional subalgebra $D$ (under an hypothesis of unitary…