1. Methods of free probability
  2. Preface

Preface

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Probability theory, and the probabilistic way of thinking, have seen a considerable surge in the last years, with virtually every single branch of mathematics being affected. It goes without saying that everything mathematics coming from quantum mechanics, which actually accounts for a big part of pure mathematics as we know it, has some probability behind, and this has become more and more visible during recent years. The same goes of course for statistical mechanics, once again somehow by definition. As in what regards classical mechanics, randomness of the initial data is certainly a very fruitful idea too. Finally, old branches of pure mathematics, such as number theory, are increasingly becoming more analytic, and more probabilistic too.


At the technical level, probability theory comes in many flavors. However, if there is one thing to be known, having interesting mathematics and physics behind, this is the fact that classical probability theory has a “twin sister”, namely free probability.


Free probability was introduced by Voiculescu in the mid 1980s, with motivation coming from general quantum mechanics, and more specifically with a number of operator algebra questions in mind. Among the main discoveries of Voiculescu was the fact that Wigner's semicircle law, coming from advanced quantum physics and random matrices, appears as the “free analogue” of the normal law. This has led to a lot of interest in free probability, with the subject having now deep ties to operator algebras, random matrices, quantum groups, noncommutative geometry, and virtually any other branch of mathematics coming from quantum mechanics, or statistical mechanics.


This book is an introduction to free probability, with the aim of keeping things as simple and concrete as possible, while still being relatively complete. Our goals will be on one hand that of explaining the definition and main properties of free probability, in analogy with the definition and main properties of classical probability, and by keeping the presentation as elementary as possible, and on the other hand to go, at least a little bit, into each of the above-mentioned classes of examples and applications, namely operator algebras, random matrices, quantum groups and noncommutative geometry.


The first half of the book contains basic material, all beautiful and useful things, leading to free probability. Part I is concerned with classical probability, or rather with selected topics from classical probability, which extend well to the free case. These include the standard classical limiting theorems (CLT, CCLT, PLT, CPLT), all done via the moment method and combinatorics, and then a discussion regarding Lie groups, and Weingarten calculus. Part II is an introduction to the random matrices, benefiting from the probability theory learned in Part I, and making a transition towards the free probability theory from Parts III-IV. The main results here are the classical limiting theorems of Wigner and Marchenko-Pastur, both done via the moment method and combinatorics, and with a look into the block-modified random matrices too.


The second half of the book is concerned with free probability itself, and applications. Part III deals with the definition and main properties of free probability, central here being, besides the foundations, the free analogues of the classical limiting theorems (CLT, CCLT, PLT, CPLT), following Voiculescu. Our approach is based on standard calculus and basic operator algebra theory, a bit in the spirit of the original book by Voiculescu, Dykema and Nica [1], but by attempting to make things a bit simpler, with the whole presentation meant to be as accessible to everyone as possible. Also, we will explain here the Bercovici-Pata bijection, and the block-modified random matrix models for the corresponding main free laws. As for Part IV, this deals with applications to quantum groups, noncommutative geometry, operator algebras and subfactors.


All in all, many things to be discussed. As a complement to what we will be doing here, for advanced combinatorics and operator algebra aspects you have [2], [3], [1], and for advanced random matrix theory you have [4], [5], [6]. So, in the hope that you will like free probability, and end up learning everything, from here and from [4], [5], [2], [6], [3], [1], with the precise order being more a matter of taste.


I learned myself free probability long ago, as a graduate student, from [1], with my first research paper being a 1996 note on the circular variables [7]. Later I started doing quantum groups, and some random matrices too, with free probability always in mind. I am grateful to Mireille Capitaine, Beno\^it Collins, Steve Curran, Ion Nechita, Roland Speicher and the others, for substantial joint work on the subject. Many thanks go as well to my cats. No serious science can be done without advice from a cat or two.


\ Cergy, August 2024

Teo Banica \baselineskip=15.95pt \tableofcontents \baselineskip=14pt

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].

References

  1. 1.0 1.1 1.2 1.3 D.V. Voiculescu, K.J. Dykema and A. Nica, Free random variables, AMS (1992).
  2. 2.0 2.1 F. Hiai and D. Petz, The semicircle law, free random variables and entropy, AMS (2000).
  3. 3.0 3.1 A. Nica and R. Speicher, Lectures on the combinatorics of free probability, Cambridge Univ. Press (2006).
  4. 4.0 4.1 G.W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices, Cambridge Univ. Press (2010).
  5. 5.0 5.1 A. Bose, Random matrices and non-commutative probability, CRC Press (2021).
  6. 6.0 6.1 J.A. Mingo and R. Speicher, Free probability and random matrices, Springer (2017).
  7. T. Banica, On the polar decomposition of circular variables, Integral Equations Operator Theory 24 (1996), 372--377.