FREE ENTROPY

2002 ◽  
Vol 34 (3) ◽  
pp. 257-278 ◽  
Author(s):  
DAN VOICULESCU
Keyword(s):  
Probability Theory ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Von Neumann Algebras ◽  
Matrix Theory ◽  
Free Probability ◽  
Open Problems ◽  
Von Neumann ◽  
Free Probability Theory ◽  
Free Entropy

Free entropy is the analogue of entropy in free probability theory. The paper is a survey of free entropy, its applications to von Neumann algebras and its connections to random matrix theory, as well as a discussion of open problems and of a basic variational problem, connected to random multimatrix models.

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.

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 A Random Matrix Approach to Absorption in Free Products

2020 ◽  
Author(s):  
Ben Hayes ◽  
David Jekel ◽  
Brent Nelson ◽  
Thomas Sinclair
Keyword(s):  
Random Matrix ◽  
Von Neumann Algebras ◽  
Concentration Of Measure ◽  
Unified Approach ◽  
Free Products ◽  
Von Neumann ◽  
Free Entropy ◽  
Famous Result ◽  
Random Matrix Models ◽  
Entropy Zero

Abstract This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if ${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$, then ${\mathcal{P}}$ absorbs any subalgebra of ${\mathcal{M}}$ that intersects it diffusely and that has $1$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra ${\mathcal{P}} \leq{\mathcal{M}}$ to have this absorption property, it suffices for ${\mathcal{M}}$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto ${\mathcal{P}}$.

SSA, Random Matrix Theory, and Noise-Reduced Correlations

2016 ◽  
Author(s):  
Jan W Dash ◽  
Xipei Yang ◽  
Mario Bondioli ◽  
Harvey J. Stein
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory
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