Abstract

The purpose of this chapter is to give a brief introduction to the theory of Lie groups and matrix algebras in a style that is suited to random matrix theory. Ensembles are probability measures on spaces of random matrices that are invariant under the action of certain compact groups, and the basic examples are known as the orthogonal, unitary and symplectic ensembles according to the group action. One of the main objectives is the construction of Dyson's circular ensembles in Sections 2.7–2.9, and the generalized ensembles from the affine action of classical compact Lie groups on suitable matrix spaces in Section 2.5. As our main interest is in random matrix theory, our discussion of the classification is patchy and focuses on the examples that are of greatest significance in RMT. We present some computations on connections and curvature, as these are important in the analysis in Chapter 3. The functional calculus of matrices is also significant, and Section 2.2 gives a brief treatment of this topic. The chapter begins with a list of the main examples and some useful results on eigenvalues and determinants.

The classical groups, their eigenvalues and norms

Throughout this chapter,

• R = real numbers;

• C = complex numbers;

• H = quaternions;

• T = unit circle.

By a well-known theorem of Frobenius, R, C and H are the only finitedimensional division algebras over R, and the dimensions are β = 1, 2 and 4 respectively; see [90].

Abstract

The contents of this chapter are introductory and covered in many standard books on probability theory, but perhaps not all conveniently in one place. In Section 1.1 we give a summary of results concerning probability measures on compact metric spaces. Section 1.2 concerns the existence of invariant measure on a compact metric group, which we later use to construct random matrix ensembles. In Section 1.3, we resume the general theory with a discussion of weak convergence of probability measures on (noncompact) Polish spaces; the results here are technical and may be omitted on a first reading. Section 1.4 contains the Brunn–Minkowski inequality, which is our main technical tool for proving isoperimetric and concentration inequalities in subsequent chapters. The fundamental example of Gaussian measure and the Gaussian orthogonal ensemble appear in Section 1.5, then in Section 1.6 Gaussian measure is realised as the limit of surface area measure on the spheres of high dimension. In Section 1.7 we state results from the general theory of metric measure spaces. Some of the proofs are deferred until later chapters, where they emerge as important special cases of general results. A recurrent theme of the chapter is weak convergence, as defined in Sections 1.1 and 1.3, and which is used throughout the book. Section 1.8 shows how weak convergence gives convergence for characteristic functions, cumulative distribution functions and Cauchy transforms.

In this chapter we introduce the basic terminology of probability theory. The notions of independence, distribution, and expected value are studied in more detail later, but it is hard to discuss examples without them, so we introduce them quickly here.

Outcomes, events, and probability

The subject of probability can be traced back to the 17th century when it arose out of the study of gambling games. As we see, the range of applications extends beyond games into business decisions, insurance, law, medical tests, and the social sciences. The stock market, “the largest casino in the world,” cannot do without it. The telephone network, call centers, and airline companies with their randomly fluctuating loads could not have been economically designed without probability theory. To quote Pierre-Simon, marquis de Laplace from several hundred years ago:

In order to address these applications, we need to develop a language for discussing them. Euclidean geometry begins with the notions of point and line. The corresponding basic object of probability is an experiment: an activity or procedure that produces distinct, well-defined possibilities called outcomes.

About 15 years ago, I wrote the book Essentials of Probability, which was designed for a one-semester course in probability that was taken by math majors and students from other departments. This book is, in some sense, the second edition of that book, but there are several important changes:

• Chapter 1 quickly introduces the notions of independence, distribution, and expected value, which previously made their entrance in Chapters 2, 3, and 4. This makes it easier to discuss examples; for example, we can now talk about the expected value of bets.

• For 5 years these notes were used in a course for students who knew only a little calculus and were looking to satisfy their distribution requirement in mathematics, so it is aimed at a wider audience.

• Markov chains are covered, and thanks to a suggestion of Lea Popovic, this topic appears right after the notion of conditional probability is discussed. This material is usually covered in an undergraduate stochastic processes course, if you are fortunate enough to offer one in your department, but in our experience this material is popular with students.

• Continuous distributions are presented as an optional topic. This decision originated to minimize the reliance on calculus, but in time I have grown to enjoy abandoning the boring mechanics of marginal and conditional distributions to spend more time talking about probability.