Free Probability Theory and Random Matrices

2007 ◽  
pp. 53-73 ◽  
Author(s):  
R. Speicher
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
Free Probability ◽  
Free Probability Theory

scholarly journals On the Outlying Eigenvalues of a Polynomial in Large Independent Random Matrices

2019 ◽  
Author(s):  
Serban T Belinschi ◽  
Hari Bercovici ◽  
Mireille Capitaine
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
Empirical Distribution ◽  
Free Probability ◽  
Probability Measures ◽  
Unitary Operators ◽  
Free Probability Theory ◽  
Wigner Matrix ◽  
Asymptotic Eigenvalue ◽  
Special Case

Abstract Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A_N,B_N)$, where $A_N$ and $B_N$ are independent Hermitian random matrices and the distribution of $B_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A_N$ and $B_N$ converge almost surely to deterministic probability measures $\mu$ and $\nu$, respectively. In addition, the eigenvalues of $A_N$ and $B_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu ,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A_N,B_N)$ converges to a certain deterministic probability measure $\eta \ (\textrm{sometimes denoted}\ P^\square(\mu,\nu))$ and, when there are no spikes, the eigenvalues of $P(A_N,B_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A_N,B_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu ,\nu ,P$, and the spikes of $A_N$. We establish a similar result when $B_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

scholarly journals Identifiability of parametric random matrix models

2019 ◽  
Vol 22 (03) ◽  
pp. 1950018
Author(s):  
Tomohiro Hayase
Keyword(s):  
Probability Theory ◽  
Matrix Models ◽  
Random Matrices ◽  
Random Matrix ◽  
Free Probability ◽  
Parameter Identifiability ◽  
Free Probability Theory ◽  
Noise Type ◽  
Random Matrix Models ◽  
Spectral Distributions

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

scholarly journals Limiting spectral distributions of sums of products of non-Hermitian random matrices

2018 ◽  
Vol 38 (2) ◽  
pp. 359-384
Author(s):  
Holger Kösters ◽  
Alexander Tikhomirov
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
General Framework ◽  
Free Probability ◽  
Singular Value ◽  
Eigenvalue Distribution ◽  
Limiting Distributions ◽  
Free Probability Theory ◽  
Spectral Distributions ◽  
Sums Of Products

For fixed l≥0 and m≥1, let Xn0, Xn1,..., Xnl be independent random n × n matrices with independent entries, let Fn0 := Xn0, Xn1-1,..., Xnl-1, and let Fn1,..., Fnm be independent random matrices of the same form as Fn0 . We show that as n → ∞, the matrices Fn0 and m−l+1/2Fn1 +...+ Fnm have the same limiting eigenvalue distribution. To obtain our results, we apply the general framework recently introduced in Götze, Kösters, and Tikhomirov 2015 to sums of products of independent random matrices and their inverses.We establish the universality of the limiting singular value and eigenvalue distributions, and we provide a closer description of the limiting distributions in terms of free probability theory.

Free probability theory

2018 ◽  
Author(s):  
Gerard Ben Arous ◽  
Alice Guionnet
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
Random Matrix ◽  
Matrix Theory ◽  
Free Probability ◽  
Combinatorial Theory ◽  
Free Probability Theory ◽  
Asymptotic Eigenvalue ◽  
Basic Ideas ◽  
The Moment

This article focuses on free probability theory, which is useful for dealing with asymptotic eigenvalue distributions in situations involving several matrices. In particular, it considers some of the basic ideas and results of free probability theory, mostly from the random matrix perspective. After providing a brief background on free probability theory, the article discusses the moment method for several random matrices and the concept of freeness. It then gives some of the main probabilistic notions used in free probability and introduces the combinatorial theory of freeness. In this theory, freeness is described in terms of free cumulants in relation to the planar approximations in random matrix theory (RMT). The article also examines free harmonic analysis, second-order freeness, operator-valued free probability theory, further free-probabilistic aspects of random matrices, and operator algebraic aspects of free probability.

Free Probability Theory

2019 ◽  
Vol 15 (4) ◽  
pp. 3147-3215
Author(s):  
Alice Guionnet ◽  
Roland Speicher ◽  
Dan-Virgil Voiculescu
Keyword(s):  
Probability Theory ◽  
Free Probability ◽  
Free Probability Theory

Free Probability Theory

2005 ◽  
pp. 827-880
Author(s):  
Philippe Biane ◽  
Roland Speicher ◽  
Dan-Virgil Voiculescu
Keyword(s):  
Probability Theory ◽  
Free Probability ◽  
Free Probability Theory
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