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Free Probability and Random Matrices

Authors: James A. Mingo, Roland Speicher

Publisher: Springer New York

Book Series : Fields Institute Monographs

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume opens the world of free probability to a wide variety of readers. From its roots in the theory of operator algebras, free probability has intertwined with non-crossing partitions, random matrices, applications in wireless communications, representation theory of large groups, quantum groups, the invariant subspace problem, large deviations, subfactors, and beyond. This book puts a special emphasis on the relation of free probability to random matrices, but also touches upon the operator algebraic, combinatorial, and analytic aspects of the theory.

The book serves as a combination textbook/research monograph, with self-contained chapters, exercises scattered throughout the text, and coverage of important ongoing progress of the theory. It will appeal to graduate students and all mathematicians interested in random matrices and free probability from the point of view of operator algebras, combinatorics, analytic functions, or applications in engineering and statistical physics.

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Table of Contents

Frontmatter

Chapter 1. Asymptotic Freeness of Gaussian Random Matrices

Abstract

In this chapter we shall introduce a principal object of study: Gaussian random matrices. This is one of the few ensembles of random matrices for which one can do explicit calculations of the eigenvalue distribution. For this reason the Gaussian ensemble is one of the best understood. Information about the distribution of the eigenvalues is carried by it moments: {E(tr(X ^k))}_k where E is the expectation, tr denotes the normalized trace (i.e. tr(I _N) = 1), and X is an N × N random matrix.

James A. Mingo, Roland Speicher

Chapter 2. The Free Central Limit Theorem and Free Cumulants

Abstract

Recall from Chapter 1 that if \((\mathcal{A},\varphi )\) is a non-commutative probability space and \(\mathcal{A}_{1},\ldots,\mathcal{A}_{s}\) are subalgebras of \(\mathcal{A}\) which are free with respect to φ, then freeness gives us in principle a rule by which we can evaluate φ(a ₁ a ₂⋯a _k) for any alternating word in random variables a ₁, a ₂, …, a _k. Thus we can in principle calculate all mixed moments for a system of free random variables. However, we do not yet have any concrete idea of the structure of this factorization rule. This situation will be greatly clarified by the introduction of free cumulants. Classical cumulants appeared in Chapter 1, where we saw that they are intimately connected with the combinatorial notion of set partitions. Our free cumulants will be linked in a similar way to the lattice of non-crossing set partitions; the latter were introduced in combinatorics by Kreweras [113]. We will motivate the appearance of free cumulants and non-crossing partition lattices in free probability theory by examining in detail a proof of the central limit theorem by the method of moments.

James A. Mingo, Roland Speicher

Chapter 3. Free Harmonic Analysis

Abstract

In this chapter we shall present an approach to free probability based on analytic functions. At the end of the previous chapter, we defined the Cauchy transform of a random variable a in an algebra \(\mathcal{A}\) with a state φ to be the formal power series \(G(z) = \frac{1} {z}M(\frac{1} {z})\) where M(z) = 1 + ∑ _{n ≥ 1} α _n z ⁿ and α _n = φ(a ⁿ) are the moments of a. Then R(z), the R-transform of a, was defined to be the formal power series R(z) = ∑ _{n ≥ 1} κ _n z ⁿ⁻¹ determined by the moment-cumulant relation which we have shown to be equivalent to the equations

$$\displaystyle{G\big(R(z) + 1/z\big) = z = 1/G(z) + R(G(z)).}$$

James A. Mingo, Roland Speicher

Chapter 4. Asymptotic Freeness for Gaussian, Wigner, and Unitary Random Matrices

Abstract

After having developed the basic theory of freeness, we are now ready to have a more systematic look into the relation between freeness and random matrices. In Chapter 1, we showed the asymptotic freeness between independent Gaussian random matrices. This is only the tip of an iceberg. There are many more classes of random matrices which show asymptotic freeness. In particular, we will present such results for Wigner matrices, Haar unitary random matrices and treat also the relation between such ensembles and deterministic matrices. Furthermore, we will strengthen the considered form of freeness from the averaged version (which we considered in Chapter 1) to an almost sure one.

James A. Mingo, Roland Speicher

Chapter 5. Fluctuations and Second Order Freeness

Abstract

Given an N × N random matrix ensemble, we often want to know, in addition to its limiting eigenvalue distribution, how the eigenvalues fluctuate around the limit. This is important in random matrix theory because in many ensembles, the eigenvalues exhibit repulsion, and this feature is often important in applications (see, e.g. [112]). If we take a diagonal random matrix ensemble with independent entries, then the eigenvalues are just the diagonal entries of the matrix and by independence do not exhibit any repulsion. If we take a self-adjoint ensemble with independent entries, i.e. the Wigner ensemble, the eigenvalues are not independent and appear to spread evenly, i.e. there are few bald spots and there is much less clumping; see Fig. 5.1. For some simple ensembles, one can obtain exact formulas measuring this repulsion, i.e. the two-point correlation functions; unfortunately these exact expressions are usually rather complicated. However, just as in the case of the eigenvalue distributions themselves, the large N limit of these distributions is much simpler and can be analysed.

James A. Mingo, Roland Speicher

Chapter 6. Free Group Factors and Freeness

Abstract

The concept of freeness was actually introduced by Voiculescu in the context of operator algebras, more precisely, during his quest to understand the structure of special von Neumann algebras, related to free groups. We wish to recall here the relevant context and show how freeness shows up there very naturally and how it can provide some information about the structure of those von Neumann algebras.

James A. Mingo, Roland Speicher

Chapter 7. Free Entropy χ: The Microstates Approach via Large Deviations

Abstract

An important concept in classical probability theory is Shannon’s notion of entropy. Having developed the analogy between free and classical probability theory, one hopes to find that a notion of free entropy exists in counterpart to the Shannon entropy. In fact there is a useful notion of free entropy. However, the development of this new concept is at present far from complete. The current state of affairs is that there are two distinct approaches to free entropy. These should give isomorphic theories, but at present we only know that they coincide in a limited number of situations.

James A. Mingo, Roland Speicher

Chapter 8. Free Entropy χ ∗: The Non-microstates Approach via Free Fisher Information

Abstract

In classical probability theory, there exist two important concepts which measure the amount of “information” of a given distribution. These are the Fisher information and the entropy. There exist various relations between these quantities, and they form a cornerstone of classical probability theory and statistics. Voiculescu introduced free probability analogues of these quantities, called free Fisher information and free entropy, denoted by Φ and χ, respectively. However, there remain some gaps in our present understanding of these quantities. In particular, there exist two different approaches, each of them yielding a notion of entropy and Fisher information. One hopes that finally one will be able to prove that both approaches give the same result, but at the moment this is not clear. Thus, for the time being, we have to distinguish the entropy χ and the free Fisher information Φ coming from the first approach (via microstates) and the free entropy χ ^∗ and the free Fisher information Φ ^∗ coming from the second non-microstates approach (via conjugate variables).

James A. Mingo, Roland Speicher

Chapter 9. Operator-Valued Free Probability Theory and Block Random Matrices

Abstract

Gaussian random matrices fit quite well into the framework of free probability theory, asymptotically they are semi-circular elements, and they have also nice freeness properties with other (e.g. non-random) matrices. Gaussian random matrices are used as input in many basic models in many different mathematical, physical, or engineering areas. Free probability theory provides then useful tools for the calculation of the asymptotic eigenvalue distribution for such models. However, in many situations, Gaussian random matrices are only the first approximation to the considered phenomena, and one would also like to consider more general kinds of such random matrices. Such generalizations often do not fit into the framework of our usual free probability theory. However, there exists an extension, operator-valued free probability theory, which still shares the basic properties of free probability but is much more powerful because of its wider domain of applicability. In this chapter, we will first motivate the operator-valued version of a semi-circular element and then present the general operator-valued theory. Here we will mainly work on a formal level; the analytic description of the theory, as well as its powerful consequences, will be dealt with in the following chapter.

James A. Mingo, Roland Speicher

Chapter 10. Deterministic Equivalents, Polynomials in Free Variables, and Analytic Theory of Operator-Valued Convolution

Abstract

The notion of a “deterministic equivalent” for random matrices, which can be found in the engineering literature, is a non-rigorous concept which amounts to replacing a random matrix model of finite size (which is usually unsolvable) by another problem which is solvable, in such a way that, for large N, the distributions of both problems are close to each other. Motivated by our example in the last chapter, we will in this chapter propose a rigorous definition for this concept, which relies on asymptotic freeness results. This “free deterministic equivalent” was introduced by Speicher and Vargas in [166].

James A. Mingo, Roland Speicher

Chapter 11. Brown Measure

Abstract

The Brown measure is a generalization of the eigenvalue distribution for a general (not necessarily normal) operator in a finite von Neumann algebra (i.e. a von Neumann algebra which possesses a trace). It was introduced by Larry Brown in [46], but fell into obscurity soon after. It was revived by Haagerup and Larsen [85] and played an important role in Haagerup’s investigations around the invariant subspace problem [87].

James A. Mingo, Roland Speicher

Chapter 12. Solutions to Exercises

Abstract

Let ν be a probability measure on \(\mathbb{R}\) such that \(\int _{\mathbb{R}}\vert t\vert ^{n}\,d\nu (t) <\infty\). For m ≤ n,

$$\displaystyle\begin{array}{rcl} \int _{\mathbb{R}}\vert t\vert ^{m}\,d\nu (t)& =& \int _{ \vert t\vert \leq 1}\vert t\vert ^{m}\,d\nu (t) +\int _{ \vert t\vert>1}\vert t\vert ^{m}\,d\nu (t) {}\\ & \leq & \int _{\vert t\vert \leq 1}1\,d\nu (t) +\int _{\vert t\vert>1}\vert t\vert ^{n}\,d\nu (t) {}\\ & \leq & \nu (\mathbb{R}) +\int _{\mathbb{R}}\vert t\vert ^{n}\,d\nu (t) {}\\ & <& \infty. {}\\ \end{array}$$

James A. Mingo, Roland Speicher

Backmatter

Title: Free Probability and Random Matrices
Authors: James A. Mingo
Roland Speicher
Copyright Year: 2017
Publisher: Springer New York
Electronic ISBN: 978-1-4939-6942-5
Print ISBN: 978-1-4939-6941-8
DOI: https://doi.org/10.1007/978-1-4939-6942-5