Topics in random matrix theory Terence Tao Department of Mathematics, UCLA, Los Angeles, CA 90095 E-mail address: [email protected] To Garth Gaudry, who set me on the road; To my family, for their constant support; And to the readers of my blog, for their feedback and contributions. Contents Preface ix Acknowledgments x Chapter 1. Preparatory material 1 x1.1. A review of probability theory 2 x1.2. Stirling's formula 41 x1.3. Eigenvalues and sums of Hermitian matrices 45 Chapter 2. Random matrices 65 x2.1. Concentration of measure 66 x2.2. The central limit theorem 93 x2.3. The operator norm of random matrices 124 x2.4. The semicircular law 159 x2.5. Free probability 183 x2.6. Gaussian ensembles 217 x2.7. The least singular value 246 x2.8. The circular law 263 Chapter 3. Related articles 277 x3.1. Brownian motion and Dyson Brownian motion 278 x3.2. The Golden-Thompson inequality 297 vii viii Contents x3.3. The Dyson and Airy kernels of GUE via semiclassical analysis 305 x3.4. The mesoscopic structure of GUE eigenvalues 313 Bibliography 321 Index 329 Preface In the spring of 2010, I taught a topics graduate course on random matrix theory, the lecture notes of which then formed the basis for this text. This course was inspired by recent developments in the subject, particularly with regard to the rigorous demonstration of universal laws for eigenvalue spacing distributions of Wigner matri- ces (see the recent survey [Gu2009b]). This course does not directly discuss these laws, but instead focuses on more foundational topics in random matrix theory upon which the most recent work has been based. For instance, the first part of the course is devoted to basic probabilistic tools such as concentration of measure and the central limit theorem, which are then used to establish basic results in ran- dom matrix theory, such as the Wigner semicircle law on the bulk distribution of eigenvalues of a Wigner random matrix, or the cir- cular law on the distribution of eigenvalues of an iid matrix. Other fundamental methods, such as free probability, the theory of deter- minantal processes, and the method of resolvents, are also covered in the course. This text begins in Chapter 1 with a review of the aspects of prob- ability theory and linear algebra needed for the topics of discussion, but assumes some existing familiarity with both topics, as will as a first-year graduate-level understanding of measure theory (as covered for instance in my books [Ta2011, Ta2010]). If this text is used ix x Preface to give a graduate course, then Chapter 1 can largely be assigned as reading material (or reviewed as necessary), with the lectures then beginning with Section 2.1. The core of the book is Chapter 2. While the focus of this chapter is ostensibly on random matrices, the first two sections of this chap- ter focus more on random scalar variables, in particular discussing extensively the concentration of measure phenomenon and the cen- tral limit theorem in this setting. These facts will be used repeatedly when we then turn our attention to random matrices, and also many of the proof techniques used in the scalar setting (such as the moment method) can be adapted to the matrix context. Several of the key results in this chapter are developed through the exercises, and the book is designed for a student who is willing to work through these exercises as an integral part of understanding the topics covered here. The material in Chapter 3 is related to the main topics of this text, but is optional reading (although the material on Dyson Brow- nian motion from Section 3.1 is referenced several times in the main text). Acknowledgments I am greatly indebted to my students of the course on which this text was based, as well as many further commenters on my blog, including Ahmet Arivan, Joshua Batson, Alex Bloemendal, Andres Caicedo, Emmanuel Cand´es, J´er^omeChauvet, Brian Davies, Ben Golub, Stephen Heilman, John Jiang, Li Jing, Sungjin Kim, Allen Knutson, Greg Kuperberg, Choongbum Lee, George Lowther, Rafe Mazzeo, Mark Meckes, William Meyerson, Samuel Monnier, Andreas Naive, Giovanni Peccati, Leonid Petrov, Anand Rajagopalan, James Smith, Mads Sørensen, David Speyer, Luca Trevisan, Qiaochu Yuan, and several anonymous contributors, for comments and corrections. These comments, as well as the original lecture notes for this course, can be viewed online at terrytao.wordpress.com/category/teaching/254a-random-matrices The author is supported by a grant from the MacArthur Founda- tion, by NSF grant DMS-0649473, and by the NSF Waterman award. Chapter 1 Preparatory material 1 2 1. Preparatory material 1.1. A review of probability theory Random matrix theory is the study of matrices whose entries are ran- dom variables (or equivalently, the study of random variables which take values in spaces of matrices). As such, probability theory is an obvious prerequisite for this subject. As such, we will begin by quickly reviewing some basic aspects of probability theory that we will need in the sequel. We will certainly not attempt to cover all aspects of probability theory in this review. Aside from the utter foundations, we will be focusing primarily on those probabilistic concepts and operations that are useful for bounding the distribution of random variables, and on ensuring convergence of such variables as one sends a parameter n off to infinity. We will assume familiarity with the foundations of measure the- ory, which can be found in any text book (including my own text [Ta2011]). This is also not intended to be a first introduction to probability theory, but is instead a revisiting of these topics from a graduate-level perspective (and in particular, after one has under- stood the foundations of measure theory). Indeed, it will be almost impossible to follow this text without already having a firm grasp of undergraduate probability theory. 1.1.1. Foundations. At a purely formal level, one could call prob- ability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate. At a practical level, the opposite is true: just as number theorists study concepts (e.g. primality) that have the same meaning in every numeral system that models the natural numbers, we shall see that probability theorists study concepts (e.g. independence) that have the same meaning in every measure space that models a family of events or random variables. And indeed, just as the natural numbers can be defined abstractly without reference to any numeral system (e.g. by the Peano axioms), core concepts of probability theory, such as random variables, can also be defined ab- stractly, without explicit mention of a measure space; we will return to this point when we discuss free probability in Section 2.5. 1.1. A review of probability theory 3 For now, though, we shall stick to the standard measure-theoretic approach to probability theory. In this approach, we assume the pres- ence of an ambient sample space Ω, which intuitively is supposed to describe all the possible outcomes of all the sources of randomness that one is studying. Mathematically, this sample space is a proba- bility space Ω = (Ω; B; P) - a set Ω, together with a σ-algebra B of subsets of Ω (the elements of which we will identify with the proba- bilistic concept of an event), and a probability measure P on the space of events, i.e. an assignment E 7! P(E) of a real number in [0; 1] to every event E (known as the probability of that event), such that the whole space Ω has probability 1, and such that P is countably additive. Elements of the sample space Ω will be denoted !. However, for reasons that will be explained shortly, we will try to avoid actually referring to such elements unless absolutely required to. If we were studying just a single random process, e.g. rolling a single die, then one could choose a very simple sample space - in this case, one could choose the finite set f1;:::; 6g, with the dis- crete σ-algebra 2f1;:::;6g := fA : A ⊂ f1;:::; 6gg and the uniform probability measure. But if one later wanted to also study addi- tional random processes (e.g. supposing one later wanted to roll a second die, and then add the two resulting rolls), one would have to change the sample space (e.g. to change it now to the product space f1;:::; 6g × f1;:::; 6g). If one was particularly well organised, one could in principle work out in advance all of the random variables one would ever want or need, and then specify the sample space accord- ingly, before doing any actual probability theory. In practice, though, it is far more convenient to add new sources of randomness on the fly, if and when they are needed, and extend the sample space as nec- essary. This point is often glossed over in introductory probability texts, so let us spend a little time on it. We say that one probability space (Ω0; B0; P0) extends1 another (Ω; B; P) if there is a surjective map π :Ω0 ! Ω which is measurable (i.e. π−1(E) 2 B0 for every E 2 B) and probability preserving (i.e. P0(π−1(E)) = P(E) for every 1Strictly speaking, it is the pair ((Ω0; B0; P0); π) which is the extension of (Ω; B; P), not just the space (Ω0; B0; P0), but let us abuse notation slightly here.