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Global and Local Limit Laws for Eigenvalues of the Gaussian Unitary Ensemble and the Wishart Ensemble

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Global and Local Limit Laws for Eigenvalues of the Gaussian Unitary Ensemble and the Wishart Ensemble Global and Local Limit Laws for Eigenvalues of the Gaussian Unitary Ensemble and the Wishart Ensemble Author: Supervisor: Zhou Fan Prof. Horng-Tzer Yau Submitted to the Harvard University Department of Mathematics in partial fulfillment of the requirements for the degree of A.B. in Mathematics March 22, 2010 Contents 1 Introduction 1 2 Convergence of the Empirical Distribution of Eigenvalues 3 2.1 Convergence in moment of the mean empirical distribution Ln . 4 2.2 Convergence of the empirical distribution Ln .................. 7 2.3 Computation of the limit distribution of Ln for the GUE and Wishart Ensemble 9 2.3.1 The GUE and the semicircle law . 10 2.3.2 The Wishart Ensemble and the law of Mar˘cenko-Pastur . 12 3 Joint Distribution of Eigenvalues 15 3.1 Joint eigenvalue density for the GUE . 18 3.2 Joint eigenvalue density for the Wishart Ensemble . 18 4 Convergence of the Local Eigenvalue Correlation Function 20 4.1 Properties of the k-point eigenvalue correlation functions . 21 4.2 Limiting behavior of the correlation function in the bulk of the spectrum . 24 4.2.1 Hermite polynomials and the GUE . 26 4.2.2 Laguerre polynomials and the Wishart Ensemble for p = 1 . 28 1 Introduction Linear algebra deals with the study of vector spaces and linear transformations between them. These transformations can be represented as matrices, and various properties of a linear transformation are reflected in the properties of its corresponding matrix. When the elements of a matrix are replaced by random variables, tools from probability theory can be used to study the resulting properties of these matrices. This paper presents a survey of several results in random matrix theory, the study of spectral properties of matrices with random elements, and analyzes two families of random matrices in a unified way. To motivate our discussion, we begin with three examples: Example 1.1 (Nuclear Physics [18]). Consider the nucleus of a large atom (e.g. Uranium- 238). We are interested in determining the energy levels of this nucleus. From quantum mechanics, the nuclear energy levels En are given as the eigenvalues of the Hamiltonian operator H of the system, H n = En n. Unfortunately, for large atoms, the Hamiltonian H cannot be explicitly computed and so we cannot explicitly determine its spectrum. However, we may instead model the system by approximating the infinite-dimensional Hilbert space of wave functions with a large finite-dimensional space, and approximating H by a matrix operator on this space with random elements. We may then study local statistics of the energy levels, such as their pairwise joint distributions or distributions concerning the spacings between them, using this model. By imposing conditions on the joint distribution of the matrix elements based on the symmetries of the system, we find that the local statistical properties of the eigenvalue distributions of these random matrix models closely match those of observed nuclear energy levels in large atoms. Example 1.2 (Wireless Communications [27]). Consider the transmission of information over a wireless communication channel. The relation between a vector of transmitted data x and received data y can be modeled by a linear channel y = Hx + n, where H is a random channel matrix and n is a vector of random Gaussian noise. An information theoretic quantity of interest is the channel capacity, an upper bound on the rate of information transmission over the channel. Suppose that x 2 Cm and y 2 Cn, and the matrix HH∗ has 1 1 eigenvalues λ1; : : : ; λn. Let FH (x) = n · #fλi ≤ xg be the cumulative distribution function of the probability mass function of these eigenvalues. Then, under suitable assumptions on the distributions of x and H, the channel capacity is given by the expected value of Z 1 2 nE[kxk ] n log 1 + 2 x dFH (x) 0 kE[knk ] over the distribution of the channel matrix H. In particular, the channel capacity is depen- ∗ dent on the global distribution FH of the eigenvalues of HH . Example 1.3 (Financial Portfolio Optimization [16, 21]). Consider a collection of n stocks. We are interested in understanding the risk associated to a portfolio of these stocks and in constructing portfolios of low risk. If we model the return of each stock as a random variable ri and consider the covariance matrix C with entries Cij = Cov(ri; rj), then the risk associated to a portfolio p = (p1; : : : ; pn) where pi is the amount invested in stock i is given by ptCp. Low risk portfolios can be selected to have large components in the directions of eigenvectors of C with the lowest eigenvalues. However, by estimating C using the sample covariance matrix of observed returns for each stock, the noise in the sample matrix can cause us to misidentify the eigenvectors corresponding to the lowest eigenvalues of C and to underestimate the risk associated with the chosen portfolio when n is large. We may use properties of the global eigenvalue distribution of the random sample covariance matrix to devise more accurate methods of portfolio selection and adjust for the noise factor in the computation of risk. There exists a body of research pertaining to each of these applications of random matrix theory. It is not the goal of this paper to discuss the details of these applications; we refer the interested reader to the listed references. We present these examples as an illustration of the diversity of the range of applications of this theory and as motivation for the specific families of random matrices and the specific properties of their eigenvalue distributions that we will examine. In particular, we will focus on two families of random matrices, defined as follows: (n) (n) Definition 1.4. Let fαij gn2N;1≤i≤j≤n and fβij gn2N;1≤i<j≤n be i.i.d. random variables, normally distributed with mean 0 and variance 1. Let Wn be an n × n matrix for each (n) n, with diagonal entries (Wn)ii = αii for 1 ≤ i ≤ n, above-diagonal entries (Wn)ij = (n) (n) (n) (n) p1 (α + iβ ) for 1 ≤ i < j ≤ n, and below-diagonal entries (W ) = p1 (α − iβ ) 2 ij ij n ij 2 ij ij for 1 ≤ j < i ≤ n. We call fWngn2N the Gaussian Unitary Ensemble (GUE). (n) (n) Definition 1.5. Let p ≥ 1, and let fαij gn2N;1≤i≤n;1≤j≤bpnc and fβij gn2N;1≤i≤n;1≤j≤bpnc be i.i.d. random variables, normally distributed with mean 0 and variance 1. Let Yn be an (n) (n) n × bpnc matrix for each n, with entries (Y ) = p1 (α + iβ ), and let M = Y Y ∗. n ij 2 ij ij n n n We call fMngn2N the Wishart Ensemble with parameter p. We note that both families of matrices are Hermitian, and that the matrices of the Wishart Ensemble are, in addition, positive semi-definite. The GUE is relevant as a matrix model for nuclear energy levels under specific symmetries in Example 1.1, while the Wishart Ensemble is the model of interest in Examples 1.2 and 1.3. All three of the examples above deal with random matrices of large dimensionality. In single-variate statistics, large collections of random variables are analyzed using limit theo- rems such as the Law of Large Numbers and the Central Limit Theorem. The goal of this paper is to develop similar limit theorems for spectral properties of interest for the GUE and Wishart Ensemble, as the matrix size tends to infinity. Our examples motivate the study of the limit theorems for two distinct spectral properties: the global distribution of eigen- values, as relevant to Example 1.2, and the local statistics of the eigenvalue distribution, as relevant to Example 1.1. We will address these properties separately in the subsequent 2 sections. We will see that the assumption of a normal distribution in Definitions 1.4 and 1.5 is not necessary for our study of the global eigenvalue distribution, but we will rely on this assumption when we turn to the examination of the local statistics. A theme of this paper is the unified derivation of our results for the GUE and Wishart Ensemble, the studies of which had historical origins in different fields of application. Each section of the paper takes advantage of a similarity between the two families of matrices to derive a general result, which is then specialized to the GUE and Wishart cases. In Section 2, we prove the existence of a limit law for the global empirical distribution of eigenvalues for a class of general band matrices, using a combinatorial and graph theoretic approach. In Section 3, we use a change of variables formula to derive the form of the joint density function for eigenvalues of matrix distributions invariant under conjugation by unitary matrices. Finally, in Section 4, we compute a local correlation function for matrix eigenvalues in terms of orthogonal polynomials and derive a limit law for this function in the cases of the GUE and Wishart Ensemble with p = 1. 2 Convergence of the Empirical Distribution of Eigen- values We prove in this section the convergence in probability of the empirical distribution of eigenvalues for a class of general band matrices, and we specialize the result to the GUE and Wishart Ensemble by an explicit computation of the limit distribution. The result was first proven for the GUE by Wigner in [28] and [29], and we will follow Wigner's general strategy using the method of moments, with the generalizations provided in [2]. Our presentation draws on [1] and [2].
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