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?WHAT IS... a Random ?

When I became a professor at Harvard, David Kazh- by applying the Gram-Schmidt dan was running a “basic notions” seminar—things algorithm; make the first row have norm one, take every graduate student (and perhaps also faculty the first row out of the second row and then make member) should know. He asked me to give a talk this have norm one, and so on. The resulting ma- on “what is a .” Since a random ma- trix X is random (because it was based on the ran-

trix is a random variable taking values in the space dom Yij) and orthogonal (because we forced it to of matrices, perhaps it’s good to start with random be). Using the orthogonal invariance of the normal variables. distribution it is not hard to prove that X has the I was surprised by Kazhdan’s request since invariant Haar measure “everybody knows” that a random variable is just probability(X ∈ A) = µ(A). a measurable function Let us now translate the algorithmic description X X(ω) from Ω to . of a random orthogonal matrix into random vari- 2 He answered “yes, but that’s not what it means to able language. Let Ω = lR n . Let X be the orthogonal people working in probability” and of course he was group. The Gram-Schmidt algorithm gives a map right. Let us consider the phrase X(ω) from almost all of Ω onto X. This X(ω) is our random variable. To prescribe its probability Pick a random matrix from Haar measure −x2/2 distribution, put product measure of e√ dx on lR n2. on the orthogonal group On 2π The push forward of this measure under the map X Here O is the group of n × n real matrices X with n is Haar measure µ. XXT = id. Most of us learn that O has an invari- n Often, one is interested in the eigenvalues of the ant probability measure µ, that is, a measure on the matrix. For orthogonal matrices, these are n points Borel sets A of O such that for every set A and n on the unit circle. Figure 1a shows the eigenvalues matrix M of a random 100 × 100 orthogonal matrix. While µ(A) = µ(MA),µ(On) = 1. there is some local variation, the eigenvalues are Let me tell you how to “pick X from µ.” To begin very neatly distributed. For contrast, Figure 1b shows 100 points put down independently at with, you will need a sample Yij of picks from the standard normal density (bell-shaped curve). This random on the unit circle. There are holes and −x2/2 clusters that do not appear in Figure 1a. For details, is the measure on the real line with density e√ . 2π applications and a lot of theory supplementing Even if you don’t know what it means to “pick Yij these observations, see Diaconis (2003). from the normal density,” your computer knows. So far, I have answered the question “what is a You can just push a button and get a stream of in- random orthogonal matrix?” For a random unitary dependent normal picks. matrix replace the on lR with × Now things are easy. Fill up an empty n n the normal distribution on C. This has density ≤ ≤ 2 array with Yij, 1 i,j n . Turn this into an e−|z| π dz. We choose a random complex normal Persi Diaconis is the Mary V. Sunseri Professor of Statis- variable Z on a computer by choosing real tics and Mathematics at Stanford University. independent normals Y1 and Y2 and setting

1348 NOTICES OF THE AMS VOLUME 52, NUMBER 11 Figure 1a. Eigenvalues of a random orthogonal Figure 1b. Random points on the circle. matrix.

= 1 + 1 applied areas from psychology to oceanography Z 2 Y1 i 2 Y2 . For a random , 1 and is a crucial ingredient of search engines such use the and Z = (Y1 + iY2+ 4 as Google. See Diaconis (2003) for more details. jY3 + kY4). Random matrices in orthogonal, uni- Physicists began to study random matrix theory tary, and symplectic groups are called the classical in the 1950s as a useful description of energy dif- circular ensembles in the physics literature. ferences in things like slow neutron scattering. There are also noncompact ensembles; to choose This has grown into an enormous literature which a random , fill out an n × n array has been developed to study new materials (quan- by putting picks from the standard complex nor- tum dots) and parts of string theory. There have mal above the diagonal, picks from the standard also been wonderful applications of random ma- real normal on the diagonal, and finally filling trix theory in combinatorics and . below the diagonal by using complex conjugates One can find the literature on all of these topics of what is above the diagonal. This is called GUE by browsing in Forrester et al (2003). (the Gaussian unitary ensemble) in the physics Returning finally to random matrices as math- literature because the random matrices have dis- ematical objects, the reader will see that we have tribution invariant under multiplication by the been treating them as “real” rather than as ab- unitary group. There are other useful ensembles stract mathematical objects. From a matrix one considered in the classical book by Mehta (2004). passes to the eigenvalues and then perhaps to a One of the interesting claims argued there is that spectrum renormalized to have average spacing one only three universal families (orthogonal, unitary, and then to the histogram of spacings. All of this and symplectic) need be considered; many large n is mechanically translatable to the language of problems have answers the same as for these measurable functions. However, this is a bit like families, no matter what working in assembly language instead of just governs the matrices involved. naturally programming your Mac. Random matrix Historically, random matrix theory was started theory has evolved as the high order descriptive by statisticians studying correlations between language of this rich body of results. different features of a population (height, weight, income...). This led to correlation matrices with References (i,j) entry the correlation between the ith and jth P. DIACONIS, Patterns in Eigenvalues: The 70th Josiah features. If the data was based on a random sam- Willard Gibbs Lecture, Bull. Amer. Math. Soc. 40, ple from a larger population, these correlation 155–178 (2003). matrices are random; the study of how the eigen- P. FORRESTER, N. SNAITH, and V. VERBAARSCHOT, Introduction Review to Special Issue on Random Matrix Theory, values of such samples fluctuate was one of the first Jour. Physics A: Mathematical and General 36, R1–R10 great accomplishments of random matrix theory. (2003). These values and the associated eigenvectors are M. MEHTA, Random Matrices, 3rd Edition, Elsevier, a mainstay of a topic called “principle components Amsterdam. (2004). analysis”. This seeks low-dimensional descriptions of high-dimensional data; it is widely used across

DECEMBER 2005 NOTICES OF THE AMS 1349