DOCSLIB.ORG

The Bi-Free Extension of Free Probability

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Bi-Free Extension of Free Probability The Bi-Free Extension of Free Probability Dan-Virgil Voiculescu Abstract. Free probability is a non-commutative probability the- ory adapted to variables with the highest degree of non-commutativity. The theory has connections with random matrices, combinatorics and operator algebras. Recently we realized that the theory has an extension to systems with left and right variables, based on a notion of bi-freeness. We provide a look at the development of this new direction. The paper is an expanded version of the plenary lecture at the 10th ISAAC Congress in Macau. 1. Introduction Free probability is now in its early thirties. After such a long time I became aware that the theory has a natural extension to a theory with two kinds of variables: left and right. This is not the same as passing from a theory of modules to a theory of bi-modules, since our left and right variables will not commute in general, a non-commutation which will appear already when we shall take a look at what the Gaussian variables of the theory are. We call the theory with left and right variables bi-free probabil- ity and the independence relation that underlies it is called bi-freeness. This new type of independence does not contradict the theorems of Mu- raki [15] and Speicher [20] about the possible types of independence in non-commutative probability with all the nice properties (\classi- cal", free, Boolean and if we give up symmetry also monotonic and anti-monotonic), the reason being that we play a new game here, by replacing the usual sets of random variables by sets with two types of variables. 2000 Mathematics Subject Classification. Primary: 46L54; Secondary: 46L53. Key words and phrases. two-faced pair, bi-free probability, bi-free convolution. Research supported in part by NSF Grant DMS-1301727. 1 2 DAN-VIRGIL VOICULESCU The observation about the possibility of left and right variables could have been made at the very beginning of free probability. At present it becomes necessary to look back and think about the prob- lems which would have appeared earlier had we been aware of the possibility. Several advances on this road have been made. Develop- ments are happening faster since the lines along which free probability developed are serving often as a guide. 2. Free probability background Free probability is a non-commutative probability framework adapted for variables with the highest degree of non-commutativity. By \high- est" non-commutativity we mean the kind of non-commutativity one encounters for instance in free groups, free semigroups or in the creation and destruction operators on a full Fock space. At this heuristic level, Bosonic and Fermionic creation and destruction operators are \less non-commutative" because the commutation or anti-commutation re- lations they satisfy represent restrictions on the non-commutativity. These are the reasons for the adjective \free" in the name of free probability. It should also be noticed that heuristically, when non- commutativity is at the highest, a certain homogeneity appears which simplifies matters. The distinguishing feature of free probability among non-commutative probability theories is the independence relation, called free indepen- dence or freeness, on which it is based. Thus the notions of random variables and of expectation values are the usual ones in non-commutative probability that is quantum mechanical observables, and their expectation values or some purely algebraic version of these. So our random variables will be operators T on some complex Hilbert space H and there will be a unit vector ξ 2 H so that the expectation of T is hT ξ; ξi (we use the mathematician's scalar product which is linear in the first and conjugate linear in the second variable). The purely algebraic version is a \non-commutative probability space" which is a unital algebra A over C with a linear expectation functional ' : A! C, '(1) = 1 and the elements a 2 A are the non-commutative random variables (in the Hilbert space setting A is L(H) the linear operators on H and '(a) = haξ; ξi). The distribution µα of a family α = (ai)i2I of non-commutative random variables in a non-commutative probability space (A;') is the information provided by the collection of non-commutative moments '(ai1 : : : ain ) when n 2 N and i1; : : : ; in 2 I. This information can be structured in better ways. For instance in the case of just one THE BI-FREE EXTENSION OF FREE PROBABILITY 3 variable a, which is a bounded hermitian operator on a Hilbert space H then '(an) = hanξ j ξi are precisely the moments of a probabil- ity measure, which we shall also denote by µa and which is given by µa(!) = hE(a; !)ξ; ξi where E(a; !) is the spectral project of a for the Borel set ! ⊂ R. The definition of freeness, which is the notion of independence, for a family of subalgebras (Ai)i2I which contain the unit 1 2 Ai (i 2 I) of (A;') is that: '(a1 : : : an) = 0 whenever '(aj) = 0, aj 2 Aij 1 ≤ j ≤ n and ij 6= ij+1 if 1 ≤ j < n. A family of sets αi ⊂ (A;')i2I is free if the algebras Ai generated by f1g [ αi are freely independent. When compared with the usual notion of independence (often called \classical" or \tensor product" independence) in quantum mechanics, a key difference is the non- commutation of independent variables in the free case. In the usual definition, at least in distribution, the independent variables commute. So if a; b 2 (A;') are classically independent and '(a) = '(b) = 0 then '(abab) = '(a2)'(b2) which may be 6= 0 while '(abab) = 0 when a; b are freely independent. Once random variables, distributions and independence of random variables in a non-commutative probability theory have been defined, one can imitate classical probability theory and look at the limit pro- cesses which give rise to basic types of variables: Gaussian, Poisson, etc. For instance, a free central limit process will consider a sequence ak, k 2 N of freely independent variables in some (A;') which are p identically distributed, i.e., '(ak) = mp for all k 2 N, and assuming the variables are centered '(ak) = m1 = 0 one looks at the limits as −1=2 n ! +1 of the distributions of n (a1 + ··· + an). One finds that the limit distribution if m2 > 0 is a semi-circle law, that is if we nor- malize m2 = 1 the moments are thosep of a probability measure on R 1 2 with support [−2; 2] and density 2π 4 − t with respect to Lebesgue measure. Similarly, to find a free Poisson distribution one takes for each n (n) freely independent variables ak , 1 ≤ k ≤ n which have identical distri- α α butions corresponding to Bernoulli measures 1 − n δ0 + n δ1 and then (n) (n) looks at the limit distribution of a1 +···+an as n ! 1. The distribu- tion one finds (in the case of α > 0) looks like a tilted shifted semicircle with the possibility of an additional atom at 0, that is the probability measure (1 − a)δ + ν if 0 ≤ a ≤ 1 and only ν if a ≥ 1 where ν has p 0 p support [(1− a)2; (1+ a)2] and density (2πt)−1p4a − (t − (1 + a))2. 4 DAN-VIRGIL VOICULESCU This law is quite different from the classical Poisson law which is X an e−a δ n! n n≥0 a measure concentrated on the natural numbers. Actually the free Gaussian and free Poisson laws are the same as well-known limit eigenvalue distribution laws in random matrix the- ory: the Wigner semicircle law (the limit distribution for eigenvalues of a suitably normalized hermitian Gaussian random matrix with i.i.d. entries) and the Marchenko{Pastur law. Clearly the fact that the free Gauss law and the free Poisson law appear in random matrix theory provided a strong indication of a connection between free probability and random matrices. The explanation for the connection between random matrices and free probability I found in [25] is that free independence appears asymp- totically among large random matrices under suitable conditions. The algebra of N × N random matrices with entries p-integrable for all 1 ≤ p < 1 over a probability space (Ω; Σ; µ) can be endowed with −1 an expectation functional 'N , where 'N (Y ) = N E(TrN Y ). Thus random matrices become non-commutative random variables and by doing this their entries are forgotten and only non-commutative mo- ments are remembered. The simple occurrence of asymptotic freeness (N) (N) is that an m-tuple (Y1 ;:::;Ym ) of Hermitian Gaussian random ma- trices with i.i.d. suitably normalized entries is asymptotically free as N ! 1. This generalizes to sets of independent random matrices with distributions invariant under unitary conjugation and even, using a combination of concentration and operator algebra techniques to a kind of generic asymptotic freeness result. The fact that free probabil- ity can be modeled by random matrices in the limit N ! 1, was the source of the applications of free probability to the operator algebras of free groups (see [26], [28]), a subject we will not discuss here. The computations of non-commutative distributions in free proba- bility has evolved in two directions. On one hand my initial analytic approach using complex analysis and a bit of operator algebras evolved toward an analytic approach and connections with non-commutative analysis. On the other hand the streamlining of the computation of moments led to Roland Speicher's combinatorial approach to free prob- ability. In essence free probability from the point of view of moments and commulants can be viewed as replacing the lattice of partitions of f1; : : : ; ng which underlies classical probability by the lattice of non- crossing partitions.
Load more

Recommended publications
  • How Non-Crossing Partitions Occur in Free Probability Theory
    How non-crossing partitions occur in free probability theory Alexandru Nica (talk at A.I.M. on January 10, 2005) This is one of the survey talks at the workshop on \Braid groups, clusters, and free probability" at the American Institute of Mathematics, January 10-14, 2005. The goal of the talk is to give a quick and elementary introduction to some combinatorial aspects of free probability, addressed to an audience who is not familiar with free probability but is acquainted to lattices of non-crossing partitions. The (attempted) line of approach is to select a cross-section of relevant material, and increase its rate of absorption by working through a set of exercises (the exercises are embedded in the lecture, and are straightforward applications of the material presented). While this is hopefully an efficient approach, it is quite far from being comprehensive; for a more detailed treatment of the subject, one can check the references [3, 5, 8, 9] listed at the end of the lecture. (Among these references, the one which is closest to the spirit of the present lecture is [3] { but, of course, there is a lot of material from [3] that could not be even mentioned here.) The plan of the lecture is as follows. The first section reviews a minimal amount of general terminology, and then the Sections 2 and 3 present some basic tools used in the combinatorics of free probability { the non-crossing cumulants and the R-transform. The main Section 4 is devoted to explaining how convolution in lattices of non-crossing partitions is intimately related to the multiplication of free random variables.
    [Show full text]
  • Free Probability Theory: Deterministic Equivalents and Combinatorics
    Free Probability Theory: Deterministic Equivalents and Combinatorics Dissertation zur Erlangung des Grades des Doktors des Naturwissenschaften der Naturwissenschanftlich-Technischen Fakult¨aten der Universit¨atdes Saarlandes von Carlos Vargas Obieta Saarbr¨ucken 2015 Tag des Kolloquiums: 04.05.2015 Dekan: Prof. Dr. Markus Bläser Prüfungsausschuss: Vorsitzender Prof. Dr. Hannah Markwig Berichterstatter Prof. Dr. Roland Speicher Prof. Dr. Jörg Eschmeier Prof. Dr. Florent Benaych-Georges Akademischer Mitarbeiter Dr. Guillaume Cébron 1 This work is dedicated to all those stubbornly trying to bring education to people who live in remote and vulnerable places. In solidarity with the families of the students of the \Ra´ulIsidro Burgos" Rural Teachers' College of Ayotzinapa: Abel Garc´ıaHern´andez Abelardo V´azquezPeniten Ad´anAbrajan de la Cruz Antonio Santana Maestro Benjam´ınAscencio Bautista Bernardo Flores Alcaraz Carlos Iv´anRam´ırezVillarreal Carlos Lorenzo Hern´andezMu~noz C´esarManuel Gonz´alezHern´andez Christian Alfonso Rodr´ıguezTelumbre Christian Tomas Colon Garnica Cutberto Ortiz Ramos Dorian Gonz´alezParral Emiliano Alen Gaspar de la Cruz. Everardo Rodr´ıguez Bello Felipe Arnulfo Rosas Giovanni Galindes Guerrero Israel Caballero S´anchez Israel Jacinto Lugardo Jes´usJovany Rodr´ıguezTlatempa Jonas Trujillo Gonz´alez Jorge Alvarez´ Nava Jorge An´ıbalCruz Mendoza Jorge Antonio Tizapa Legide~no Jorge Luis Gonz´alezParral Jos´e Angel´ Campos Cantor Jos´e Angel´ Navarrete Gonz´alez Jos´eEduardo Bartolo Tlatempa Jos´eLuis Luna Torres Jhosivani Guerrero de la Cruz Julio C´esarL´opez Patolzin Leonel Castro Abarca Luis Angel´ Abarca Carrillo Luis Angel´ Francisco Arzola Magdaleno Rub´enLauro Villegas Marcial Pablo Baranda Marco Antonio G´omezMolina Mart´ınGetsemany S´anchez Garc´ıa Mauricio Ortega Valerio Miguel Angel´ Hern´andezMart´ınez Miguel Angel´ Mendoza Zacar´ıas Sa´ulBruno Garc´ıa, kidnapped on Sept.
    [Show full text]
  • Free Probability Theory
    Chapter 22 Free Probability Theory Roland Speicher Department of Mathematics and Statistics Queen's University Kingston, Ontario K7L 3N6 Canada Abstract Free probability theory was created by Dan Voiculescu around 1985, motivated by his e®orts to understand special classes of von Neumann algebras. His dis- covery in 1991 that also random matrices satisfy asymptotically the freeness relation transformed the theory dramatically. Not only did this yield spectac- ular results about the structure of operator algebras, but it also brought new concepts and tools into the realm of random matrix theory. In the following we will give, mostly from the random matrix point of view, a survey on some of the basic ideas and results of free probability theory. 22.1 Introduction Free probability theory allows one to deal with asymptotic eigenvalue distribu- tions in situations involving several matrices. Let us consider two sequences AN and BN of selfadjoint N £N matrices such that both sequences have an asymp- totic eigenvalue distribution for N ! 1. We are interested in the asymptotic eigenvalue distribution of the sequence f(AN ;BN ) for some non-trivial self- adjoint function f. In general, this will depend on the relation between the eigenspaces of AN and of BN . However, by the concentration of measure phe- nomenon, we expect that for large N this relation between the eigenspaces 2 CHAPTER 22. concentrates on typical or generic positions, and that then the asymptotic eigenvalue distribution of f(AN ;BN ) depends in a deterministic way only on the asymptotic eigenvalue distribution of AN and on the asymptotic eigen- value distribution of BN .
    [Show full text]
  • De Finetti Theorems for Easy Quantum Groups
    The Annals of Probability 2012, Vol. 40, No. 1, 401–435 DOI: 10.1214/10-AOP619 c Institute of Mathematical Statistics, 2012 DE FINETTI THEOREMS FOR EASY QUANTUM GROUPS By Teodor Banica1, Stephen Curran and Roland Speicher2 Cergy-Pontoise University, UCLA and Queen’s University We study sequences of noncommutative random variables which are invariant under “quantum transformations” coming from an or- thogonal quantum group satisfying the “easiness” condition axioma- tized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups Sn, On, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of K¨ostler and Speicher, and the characterization of operator-valued free semi- circular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman. Introduction. In the study of probabilistic symmetries, the classical groups Sn and On play central roles. De Finetti’s fundamental theorem states that an infinite sequence of random variables whose joint distribu- tion is invariant under finite permutations must be conditionally indepen- dent and identically distributed. In [20], Freedman considered sequences of real-valued random variables whose joint distribution is invariant under or- thogonal transformations, and proved that any infinite sequence with this property must form a conditionally independent Gaussian family with mean zero and common variance. Although these results fail for finite sequences, approximation results may still be obtained (see [17, 18]).
    [Show full text]
  • Combinatorics of Free Probability Theory Roland Speicher
    Combinatorics of free probability theory Roland Speicher Queen’s University, Kingston, Ontario Abstract. The following manuscript was written for my lec- tures at the special semester ’Free probability theory and operator spaces’, IHP, Paris, 1999. A complementary series of lectures was given by A. Nica. It is planned to combine both manuscripts into a book A. Nica and R. Speicher: Lectures on the Combinatorics of Free Probability, which will be published by Cambridge University Press Contents Part 1. Basic concepts 5 Chapter 1. Basic concepts of non-commutative probability theory 7 1.1. Non-commutative probability spaces and distributions 7 1.2. Haar unitaries and semicircular elements 10 Chapter 2. Free random variables 17 2.1. Definition and basic properties of freeness 17 2.2. The group algebra of the free product of groups 21 2.3. The full Fock space 23 2.4. Construction of free products 27 Part 2. Cumulants 35 Chapter 3. Free cumulants 37 3.1. Motivation: Free central limit theorem 37 3.2. Non-crossing partitions 46 3.3. Posets and M¨obius inversion 50 3.4. Free cumulants 52 Chapter 4. Fundamental properties of free cumulants 59 4.1. Cumulants with products as entries 59 4.2. Freeness and vanishing of mixed cumulants 66 Chapter 5. Sums and products of free variables 75 5.1. Additive free convolution 75 5.2. Description of the product of free variables 84 5.3. Compression by a free projection 87 5.4. Compression by a free family of matrix units 91 Chapter 6. R-diagonal elements 95 6.1.
    [Show full text]
  • Free Probability, Random Matrices and the Enumeration of Maps
    Free probability, random matrices and the enumeration of maps Alice Guionnet MIT 30 novembre 2012 Abstract These are the lecture notes for the Colloquium lectures to be held in San Diego, january 2012. These lectures will discuss the relations between free probability, random matrices, and combinatorics. They will also show how these relations can be used to solve problems in each of these domains. Even if related, we will try to make each lecture independent of the others. Introduction Free probability is a theory initiated by D. Voiculescu in the eighties that studies non-commutative random variables. It is equipped with a notion of freeness, which is related with free products, and which plays the same role as independence in standard probability. These two ingredients allows to translate many problems from operator algebra into words more familiar to probabilists, and eventually import tools form probability theory to try to solve them. It turns out that free probability is not only analogue to classical probability but that there is a natural bridge between both ; it is gi- 1 2 ven by random matrices with size going to infinity. Indeed, matrices with size going to infinity provide a reach class of non-commutative random variables, in fact it is still an open question whether all non- commutative laws can be described in this way. Moreover, the notion of freeness of two variables is related with the notion of independence of the basis of the eigenvectors of the two underlying random matrices, in the sense that one is uniformly distributed with respect to the other.
    [Show full text]
  • Le´Vy Laws in Free Probability
    Le´vy laws in free probability Ole E. Barndorff-Nielsen† and Steen Thorbjørnsen‡§ †Department of Mathematical Sciences, University of Aarhus, DK-8000 Aarhus C, Denmark; and ‡Department of Mathematics and Computer Science, University of Southern Denmark, 5230 Odense M, Denmark Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved October 7, 2002 (received for review January 29, 2002) This article and its sequel outline recent developments in the theory of infinite divisibility and Le´vy processes in free probability, a subject area belonging to noncommutative (or quantum) probability. The present paper discusses the classes of infinitely divisible probability measures in classical and free probability, respectively, via a study of the Bercovici–Pata bijection between these classes. n this article and its follow-up (1) we outline certain aspects of the theory of infinite divisibility and Le´vy processes in the Iframework of Voiculescu’s free probability. Many researchers in the area have felt that an analogue of the highly developed theory of Le´vy processes might be constructed in the framework of free probability, but it has not been clear how to do this. (Recall that one major difficulty in noncommutative probability is the lack of a notion of the joint distribution of two noncommuting operators (as a probability measure.) This fact means that many arguments from classical probability cannot be carried over directly to the noncommutative case). This article and its sequel address some central issues of this research program. While the follow-up article focuses on the study of Le´vy processes in free probability, our study in this paper is concentrated around the bijection, introduced by Bercovici and Pata in ref.
    [Show full text]
  • Free Probability Theory
    Free Probability Theory Alexandru Nica (Waterloo), Roland Speicher (Queen's), Dan Voiculescu (Berkeley) October 9{14, 2004 1 Introduction Free probability theory is a line of research which parallels aspects of classical probability, in a non-commutative context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. It grew out from attempts to solve some longstanding problems about von Neumann algebras of free groups. In the twenty years since its creation, free probability has become a subject in its own right, with connections to several other parts of mathematics: operator algebras, the theory of random matrices, classical probability and the theory of large deviations, algebraic combinatorics. Free probability also has connections with some mathematical models in theoretical physics. The BIRS workshop on free probability brought together a very strong group of mathematicians representing the current directions of development in the area. This continued a sequence of very successful 5-day workshops organized on these lines, like the ones at the Fields Institute in March 1995, at CIRM Luminy in January 1998, and at MSRI in January 2001. In this report we look in more detail at what are the current directions of development in free probability, with an emphasis on how they were represented in the BIRS workshop. 2 Developments related to operator algebras and random matrices Free probability has far-reaching connections both with the field of operator algebras (where the subject is originally coming from) and with the one of random matrices. Since the interactions between free probability and these two fields are closely related to each other, we will discuss them together.
    [Show full text]
  • Free Probability and Random Matrices
    J. A. Mingo and R. Speicher Free Probability and Random Matrices Schwinger-Verlag Saarbrucken¨ Kingston 2017 Dedicated to Dan-Virgil Voiculescu who gave us freeness and Betina and Jill who gave us the liberty to work on it. Contents 1 Asymptotic Freeness of Gaussian Random Matrices . 13 1.1 Moments and cumulants of random variables . 13 1.2 Moments of a Gaussian random variable . 15 1.3 Gaussian vectors . 16 1.4 The moments of a standard complex Gaussian random variable . 17 1.5 Wick’s formula . 17 1.6 Gaussian random matrices . 19 1.7 A genus expansion for the GUE . 19 1.8 Non-crossing partitions and permutations . 21 1.9 Wigner’s semi-circle law . 23 1.10 Asymptotic freeness of independent GUE’s . 23 1.11 Freeness and asymptotic freeness . 26 1.12 Basic properties of freeness . 26 1.13 Classical moment-cumulant formulas . 29 1.14 Additional exercises . 30 2 The Free Central Limit Theorem and Free Cumulants . 33 2.1 The classical and free central limit theorems . 33 2.1.1 Classical central limit theorem . 37 2.1.2 Free central limit theorem . 38 2.2 Non-crossing partitions and free cumulants . 41 2.3 Products of free random variables . 48 2.4 Functional relation between moment series and cumulant series . 49 2.5 Subordination and the non-commutative derivative . 52 3 Free Harmonic Analysis ........................................ 59 3.1 The Cauchy transform . 60 3.2 Moments and asymptotic expansions . 72 3.3 Analyticity of the R-transform: compactly supported measures . 74 3.4 Measures with finite variance .
    [Show full text]
  • (Co) Monads in Free Probability Theory
    (Co)monads in Free Probability Theory Roland Friedrich∗ September 12, 2017 Abstract We discuss free probability theory and free harmonic analysis from a categorical perspective. In order to do so, we extend first the set of analytic convolutions and operations and then show that the comonadic structure governing free probability is isomorphic to several well-known categories of algebras, such as, e.g., Witt vectors, differential algebras, etc. Within this framework moment- cumulant formulae are shown to correspond to natural transformations and not to be exclusive to probability theories. Finally, we start to discuss free probability and in particular free harmonic analysis from the point of view of algebraic theories and operads. Contents 1 Introduction 2 2 Algebraic Aspects of Free Probability3 2.1 The category of algebraic probability spaces and freeness . .3 2.2 The affine group schemes related to free additive and multiplicative convolution . .4 2.3 The natural transformations R and S ...........................7 2.4 Free cumulants and the NS-convolution . .8 2.5 Hopf algebras related to free probability . 10 arXiv:1709.02973v1 [math.PR] 9 Sep 2017 3 Co-rings and comonads 12 3.1 Witt vectors . 12 3.2 The λ-ring of formal power series . 13 3.3 Differential algebras . 14 3.4 Endomorphism algebras . 15 3.5 Free convolution coalgebras . 16 ∗Saarland University, [email protected]. Supported by the ERC advanced grant \Noncommutative distribu- tions in free probability". 1 4 Analytic Aspects of Infinitely Divisible Probability Measures 17 4.1 The Borel functor and free harmonic analysis . 17 4.2 Analytic Witt semi-ring .
    [Show full text]
  • Three Lectures on Free Probability
    Random Matrices MSRI Publications Volume 65, 2014 Three lectures on free probability JONATHAN NOVAK with illustrations by Michael LaCroix These are notes from a three-lecture mini-course on free probability given at MSRI in the Fall of 2010 and repeated a year later at Harvard. The lec- tures were aimed at mathematicians and mathematical physicists working in combinatorics, probability, and random matrix theory. The first lecture was a staged rediscovery of free independence from first principles, the second dealt with the additive calculus of free random variables, and the third focused on random matrix models. Introduction 310 1. Lecture one: discovering the free world 310 1.1. Counting connected graphs 310 1.2. Cumulants and connectedness 312 1.3. Cumulants and independence 315 1.4. Central limit theorem by cumulants 318 1.5. Geometrically connected graphs 319 1.6. Noncrossing cumulants 321 1.7. Noncrossing independence 328 1.8. The medium is the message 331 1.9. A brief history of the free world 333 2. Lecture two: exploring the free world 334 2.1. Random walks on the integers 334 2.2. Pólya’s theorem 336 2.3. Kesten’s problem 338 2.4. The classical algorithm 341 2.5. Voiculescu’s algorithm 345 2.6. Solution of Kesten’s problem 349 2.7. Spectral measures and free convolution 354 2.8. Free Poisson limit theorem 356 2.9. Semicircle flow 358 3. Lecture three: modelling the free world 358 3.1. Algebraic model of a free arcsine pair 358 3.2. Algebraic model of a free semicircular pair 359 309 310 JONATHAN NOVAK 3.3.
    [Show full text]
  • Free Probability Notes
    Free probability and combinatorics Preliminary version Michael Anshelevich c 2012 July 17, 2018 Preface These notes were used in a topics course Free probability and combinatorics taught at Texas A&M Uni- versity in the fall of 2012. The course was based on Nica and Speicher’s textbook [NS06], however there were sufficient differences to warrant separate notes. I tried to minimize the list of references, but many more are included in the body of the text. Thanks to the students in the course, and to March Boedihardjo, for numerous comments and corrections; further corrections or comments are welcome. Eventually I will probably replace hand-drawn figures by typeset ones. Contents 1 Noncommutative probability spaces and distributions. 3 1.1 Non-commutative probability spaces. 3 1.2 Distributions. Weak law of large numbers. 6 2 Functional analysis background. 9 2.1 C∗- and von Neumann algebras. Spectral theorem. 9 2.2 Gelfand-Naimark-Segal construction. 11 2.3 Other topics. 14 3 Free independence. 19 3.1 Independence. Free independence. 19 3.2 Free products. 22 4 Set partition lattices. 29 4.1 All, non-crossing, interval partitions. Enumeration. 29 4.2 Incidence algebra. Multiplicative functions. 33 5 Free cumulant machinery. 38 5.1 Free cumulants. 38 5.2 Free independence and free cumulants. 45 5.3 Free convolution, distributions, and limit theorems. 49 1 6 Lattice paths and Fock space models. 54 6.1 Dyck paths and the full Fock space. 54 6.2 Łukasiewicz paths and Voiculescu’s operator model. 62 6.3 Motzkin paths and Schurmann’s¨ operator model.
    [Show full text]