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Softcover ISBN:  9781470442989 
Product Code:  MEMO/267/1300 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
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Product Code:  MEMO/267/1300.E 
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MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470442989 
eBook ISBN:  9781470463991 
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MAA Member Price:  $153.00 $114.75 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 267; 2020; 88 ppMSC: Primary 15; 46; Secondary 60; 18
Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability.
The author first establishes this construction for random matrices and then defines the traffic distribution of random matrices, which is richer than the \(^*\)distribution of free probability. The knowledge of the individual traffic distributions of independent permutation invariant families of matrices is sufficient to compute the limiting distribution of the join family. Under a factorization assumption, the author calls traffic independence the asymptotic rule that plays the role of independence with respect to traffic distributions. Wigner matrices, Haar unitary matrices and uniform permutation matrices converge in traffic distributions, a fact which yields new results on the limiting \(^*\)distributions of several matrices the author can construct from them.
Then the author defines the abstract traffic spaces as non commutative probability spaces with more structure. She proves that at an algebraic level, traffic independence in some sense unifies the three canonical notions of tensor, free and Boolean independence. A central limiting theorem is stated in this context, interpolating between the tensor, free and Boolean central limit theorems.

Table of Contents

Chapters

Introduction

1. The Asymptotic Traffic Distributions of Random Matrices

1. Statement of the Main Theorem and Applications

2. Definition of Asymptotic Traffic Independence

3. Examples and Applications for Classical Large Matrices

2. Traffics and their Independence

4. Algebraic Traffic Spaces

5. Traffic Independence and the Three Classical Notions

6. Limit theorems for independent traffics


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Voiculescu's notion of asymptotic free independence is known for a large class of random matrices including independent unitary invariant matrices. This notion is extended for independent random matrices invariant in law by conjugation by permutation matrices. This fact leads naturally to an extension of free probability, formalized under the notions of traffic probability.
The author first establishes this construction for random matrices and then defines the traffic distribution of random matrices, which is richer than the \(^*\)distribution of free probability. The knowledge of the individual traffic distributions of independent permutation invariant families of matrices is sufficient to compute the limiting distribution of the join family. Under a factorization assumption, the author calls traffic independence the asymptotic rule that plays the role of independence with respect to traffic distributions. Wigner matrices, Haar unitary matrices and uniform permutation matrices converge in traffic distributions, a fact which yields new results on the limiting \(^*\)distributions of several matrices the author can construct from them.
Then the author defines the abstract traffic spaces as non commutative probability spaces with more structure. She proves that at an algebraic level, traffic independence in some sense unifies the three canonical notions of tensor, free and Boolean independence. A central limiting theorem is stated in this context, interpolating between the tensor, free and Boolean central limit theorems.

Chapters

Introduction

1. The Asymptotic Traffic Distributions of Random Matrices

1. Statement of the Main Theorem and Applications

2. Definition of Asymptotic Traffic Independence

3. Examples and Applications for Classical Large Matrices

2. Traffics and their Independence

4. Algebraic Traffic Spaces

5. Traffic Independence and the Three Classical Notions

6. Limit theorems for independent traffics