Sixty years after the works of Wigner and Dyson, Random Matrix Theory still remains a very active and challenging area of research, with countless applications in mathematical physics, statistical mechanics and beyond. In this thesis, we focus on rotationally invariant models where the requirement of independence of matrix elements is dropped. Some classical examples are the Jacobi and Wishart-Laguerre (or chiral) ensembles, which constitute the core of the present work. The Wishart-Laguerre ensemble contains covariance matrices of random data, and represents a very important tool in multivariate data analysis, with recent applications to finance and telecommunications. We will first consider large deviations of the maximum eigenvalue, providing new analytical results for its large N behavior, and then a power-law deformation of the classical Wishart-Laguerre ensemble, with possible applications to covariance matrices of financial data. For the Jacobi matrices, which arise naturally in the quantum conductance problem, we provide analytical formulas for quantities of interest for the experiments.

Random numbers are important for gaming, simulation and cryptography. Random numbers have been generated using analog circuitry. Two problems exist with using analog circuits in a digital design: (1) analog components require an analog circuit designer to insure proper structure and functionality and (2) analog components are not easily transmigrated into a different fabrication technology. This paper proposes a class of random number generators that are constructed using only digital components and typical digital design methodology. The proposed classification is called divergent path since the path of generated numbers through the range of possible values diverges at every sampling. One integrated circuit was fabricated and several models were synthesized into a FPGA. Test results are given.

Scattering is a fundamental phenomenon in physics, e.g. large parts of the knowledge about quantum systems stem from scattering experiments. A scattering process can be completely characterized by its K-matrix, also known as the \Wigner reaction matrix" in nuclear scattering or \impedance matrix" in the electromagnetic wave scattering. For chaotic quantum systems it can be modelled within the framework of Random Matrix Theory (RMT), where either the K-matrix itself or its underlying Hamiltonian is taken as a random matrix. These two approaches are believed to lead to the same results due to a universality conjecture by P. Brouwer, which is equivalent to the claim that the probability distribution of K, for a broad class of invariant ensembles of random Hermitian matrices H, converges to a matrix Cauchy distribution in the limit of large matrix dimension of H. For unitarily invariant ensembles, this conjecture will be proved in the thesis by explicit calculation, utilising results about ensemble averages of characteristic polynomials. This thesis furthermore analyses various characteristics of the K-matrix such as the distribution of a diagonal element at the spectral edge or the distribution of an off-diagonal element in the bulk of the spectrum. For the latter it is necessary to know correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices for the Gaussian Orthogonal Ensemble (GOE), which is an interesting and important topic in itself, as they frequently arise in various other applications of RMT to physics of quantum chaotic systems, and beyond. A larger part of the thesis is dedicated to provide an explicit evaluation of the large-N limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method.

Random triangulated surfaces are created by taking an even number, n, of triangles and arbitrarily ”gluing” together pairs of edges until every edge has been paired. The resulting surface can be described in terms of its number of boundary cycles, a random variable denoted by h. Building upon the work of Nicholas Pippenger and Kristin Schleich, and using a recent result from Alex Gamburd, we establish an improved approximation for the expectation of h for certain values of n. We use a computer simulation to exactly determine the distribution of h for small values of n, and present a method for calculating these probabilities. We also conduct an investigation into the related problem of creating one connected component out of n triangles.

The objective of this study is to answer the following research question: How large is a diversified portfolio? Although previous work is abundant, very little progress has been made in answering this question since the seminal work of Evans and Archer (1968). This study proposes two approaches to address the research question. The first approach is to measure the rate of risk reduction as diversification increases. For the first approach, I identify two kinds of risks: (1) risk that portfolio returns vary across time (Evans and Archer (1968), and Campbell et al. (2001)); and (2) risk that returns vary across portfolios of the same size (Elton and Gruber (1977), and O'Neil (1997)). I show that the times series risk reaches an asymptote as portfolio size increases. Cross sectional risk, on the other hand, does not appears to reach an asymptote as portfolio size increases. The second approach consists of comparing portfolios' performance to a benchmark portfolio that is assumed to be diversified (Statman (1987)). I develop a performance index. The performance index is calculated, for any given test portfolio, as the ratio of the Sharpe-like measure of the test portfolio to the Sharpe-like measure of the benchmark portfolio that is assumed to be diversified. The index is based on the intuition that an increase in portfolio size reduces times series risk and cross sectional risk, and increases transaction costs. Portfolio size is worth increasing as long as the marginal increase in the performance index from a decrease in risk is greater than the marginal decrease of the performance index from an increase in transaction costs. Diversification is attained when the value of the index reaches one. The results of my simulations indicate that the size of a well diversified portfolio is at the very least 30. This number can be substantially higher if, for example, the investment horizon length, the benchmark portfolio, and/or the cost of investing in the benchmark portfolio are changed. The active diversification strategy considered in this study, which consists of optimizing randomly selected portfolios, does not seem to produce smaller diversified portfolios. This result supports the market efficiency hypothesis.

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Includes bibliographical references (p. 131-134).

Thesis (Ph. D.)--University of Chicago, School of Business, 2001. / Includes bibliographical references. Also available on the Internet.