Random Matrix Theory and Its Applications

The following sections are included:

• CONTENTS

• FOREWORD

• PREFACE

These lectures introduce the concept of the Stieltjes transform of a measure, an analytic function which uniquely chacterizes the measure, and its importance to spectral behavior of random matrices.

In applications of random matrix theory to physics, time reversal symmetry implies one of three exponents β = 1, 2 or 4 for the repulsion s^β between eigenvalues at spacing s, s → 0. However, in the corresponding eigenvalue probability density functions (p.d.f.'s), β is a natural positive real variable. The general β p.d.f.'s have alternative physical interpretations in terms of certain classical log-potential Coulomb systems and quantum many body systems. Each of these topics is reviewed, along with the problem of computing correlation functions for general β. There are also random matrix constructions which give rise to these general β p.d.f.'s. An inductive approach to the latter topic using explicit formulas for the zeros of certain random rational functions is given.

With the rapid development of modern computer science, large dimensional data analysis has become more and more important and has therefore received increasing attention from statisticians. In this note, we shall illustrate the difference between large dimensional data analysis and classical data analysis by means of some examples, and we show the importance of random matrix theory and its applications to large dimensional data analysis.

The landmark contributions to the theory of random matrices of Wishart (1928), Wigner (1955), and Marčenko-Pastur (1967), were motivated to a large extent by their applications. In this paper, we report on two transforms motivated by the application of random matrices to various problems in the information theory of noisy communication channels: η and Shannon transforms. Originally introduced in [1, 2], their applications to random matrix theory and engineering applications have been developed in [3]. In this paper, we give a summary of their main properties and applications in random matrix theory.

This review paper gives a tutorial overview of the usage of the replica method in multiuser communications. It introduces the self averaging principle, the free energy and other physical quantities and gives them a meaning in the context of multiuser communications. The technical issues of the replica methods are explained to a non-physics audience. An isomorphism between receiver metrics and the fundamental laws of physics is drawn. The overview is explained at the example of detection of code-division multiple-access with random signature sequences.