In this thesis, we discuss some topics in random matrix theory which have applications to probability, statistics and quantum information theory. In Chapter 2, by relying on the spectral properties of an associated adjacency matrix, we find the distribution of the maximum of a Dyck path and show that it has the same distribution function as the unsigned Brownian excursion which was first derived in 1976 by Kennedy. We obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung, Kennedy and Khorunzhiy and Marckert. In Chapter 3, we discuss a method of sampling called the Gibbs-slice sampler. This method is based on Neal's slice sampling combined with Gibbs sampling. In Chapter 4, we discuss several examples which have applications in physics and quantum information theory.

This thesis is about high–dimensional problems considered under the so{called Kolmogorov condition. Hence, we consider research questions related to random matrices with p rows (corresponding to the parameters) and n columns (corresponding to the sample size), where p &gt; n, assuming that the ratio <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%5Cfrac%7Bp%7D%7Bn%7D" /> converges when the number of parameters and the sample size increase. We focus on the eigenvalue distribution of the considered matrices, since it is a well–known information–carrying object. The spectral distribution with compact support is fully characterized by its moments, i.e., by the normalized expectation of the trace of powers of the matrices. Moreover, such an expectation can be seen as a free moment in the non–commutative space of random matrices of size p x p equipped with the functional <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20%5Cfrac%7B1%7D%7Bp%7DE%5BTr%5C%7B%5Ccdot%5C%7D%5D" />. Here, the connections with free probability theory arise. In the relation to that eld we investigate the closed form of the asymptotic spectral distribution for the sum of the quadratic forms. Moreover, we put a free cumulant–moment relation formula that is based on the summation over partitions of the number. This formula is an alternative to the free cumulant{moment relation given through non{crossing partitions ofthe set. Furthermore, we investigate the normalized <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20E%5B%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D%5D" /> and derive, using the dierentiation with respect to some symmetric matrix, a recursive formula for that expectation. That allows us to re–establish moments of the Marcenko–Pastur distribution, and hence the recursive relation for the Catalan numbers. In this thesis we also prove that the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D" />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20W%5Csim%5Cmathcal%7BW%7D_p(I_p,n)" />, is a consistent estimator of the <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20E%5B%5Cprod_%7Bi=1%7D%5Ek%20Tr%5C%7BW%5E%7Bm_i%7D%5C%7D%5D" />. We consider <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20Y_t=%5Csqrt%7Bnp%7D%5Cbig(%5Cfrac%7B1%7D%7Bp%7DTr%5Cbig%5C%7B%5Cbig(%5Cfrac%7B1%7D%7Bn%7DW%5Cbig)%5Et%5Cbig%5C%7D-m%5E%7B(t)%7D_1%20(n,p)%5Cbig)," />, where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csmall%20m%5E%7B(t)%7D_1%20(n,p)=E%5Cbig%5B%5Cfrac%7B1%7D%7Bp%7DTr%5Cbig%5C%7B%5Cbig(%5Cfrac%7B1%7D%7Bn%7DW%5Cbig)%5Et%5Cbig%5C%7D%5Cbig%5D" />, which is proven to be normally distributed. Moreover, we propose, based on these random variables, a test for the identity of the covariance matrix using a goodness{of{t approach. The test performs very well regarding the power of the test compared to some presented alternatives for both the high–dimensional data (p &gt; n) and the multivariate data (p ≤ n).

It was conjectured in the early 1950¡¯s that the empirical spectral distribution (ESD) of an $n \times n$ matrix whose entries are independent and identically distributed with mean zero and variance one, normalized by a factor of $\frac{1}{\sqrt{n}}$, converges to the uniform distribution over the unit disk on the complex plane, which is called the circular law. The goal of this thesis is to prove the so called Replacement Principle introduced by Tao and Vu which is a crucial step in their recent proof of the circular law in full generality. It gives a general criterion for the difference of the ESDs of two normalised random matrices $\frac{1}{\sqrt{n}}A_n$, $\frac{1}{\sqrt{n}}B_n$ to converge to 0. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2009-07-11 14:57:44.225

In this thesis, we discuss some topics in random matrix theory which have applications to probability, statistics and quantum information theory. In Chapter 2, by relying on the spectral properties of an associated adjacency matrix, we find the distribution of the maximum of a Dyck path and show that it has the same distribution function as the unsigned Brownian excursion which was first derived in 1976 by Kennedy. We obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung, Kennedy and Khorunzhiy and Marckert. In Chapter 3, we discuss a method of sampling called the Gibbs-slice sampler. This method is based on Neal's slice sampling combined with Gibbs sampling. In Chapter 4, we discuss several examples which have applications in physics and quantum information theory.

Thesis (Ph. D.)--Brandeis University, 2008. / "UMI:3306459." MICROFILM COPY ALSO AVAILABLE IN THE UNIVERSITY ARCHIVES. Includes bibliographical references.

Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This talk is a survey specifically oriented towards the needs and interests of a computationally inclined audience. We include the important mathematics (free probability) that permit the characterization of a large class of random matrices. We discuss how computational software is transforming this theory into practice by highlighting its use in the context of a stochastic eigen-inference application. / Singapore-MIT Alliance (SMA)

Cette thèse s'inscrit dans le domaine des matrices aléatoires et des techniques de grandes déviations. On s'attachera dans un premier temps à donner des inégalités de déviations pour différentes fonctionnelles du spectre qui reflètent leurs comportement de grandes déviations, pour des matrices de Wigner vérifiant une propriété de concentration indexée par un paramètre alpha ∈ (0,2]. Nous présenterons ensuite le principe de grandes déviations obtenu pour la plus grande valeur propre des matrices de Wigner sans queues Gaussiennes, dans la lignée du travail de Bordenave et Caputo, puis l'étude des grandes déviations des traces de matrices aléatoires que l'on aborde dans trois cas : le cas des beta-ensembles, celui des matrices de Wigner Gaussiennes, et enfin des matrices de Wigner sans queues Gaussiennes. Le cas Gaussien a été l'occasion de revisiter la preuve de Borell et Ledoux des grandes déviations des chaos de Wiener, que l'on prolonge en proposant un énoncé général de grandes déviations qui nous permet donner une autre preuve des principes de grandes déviations des matrices de Wigner sans queues Gaussiennes. Enfin, nous donnons une nouvelle preuve des grandes déviations de la mesure spectrale empirique des beta-ensembles associés à un potentiel quadratique, qui ne repose que sur leur représentation tridiagonale. / This thesis falls within the theory of random matrices and large deviations techniques. We mainly consider large deviations problems which involve a heavy-tail phenomenon. In a first phase, we will focus on finding concentration inequalities for different spectral functionals which reflect their large deviations behavior, for random Hermitian matrices satisfying a concentration property indexed by some alpha ∈ (0,2]. Then we will present the large deviations principle we obtained for the largest eigenvalue of Wigner matrices without Gaussian tails, in line with the work of Bordenave and Caputo. Another example of heavy-tail phenomenon is given by the large deviations of traces of random matrices which we investigate in three cases: the case of beta-ensembles, of Gaussian Wigner matrices, and the case of Wigner matrices without Gaussian tails. The Gaussian case was the opportunity to revisit Borell and Ledoux's proof of the large deviations of Wiener chaoses, which we investigate further by proposing a general large deviations statement, allowing us to give another proof of the large deviations principles known for the Wigner matrices without Gaussian tail. Finally, we give a new proof of the large deviations principles for the beta-ensembles with a quadratic potential, which relies only on the tridiagonal representation of these models. In particular, this result gives a proof of the large deviations of the GUE and GOE which does not rely on the knowledge of the law of the spectrum.

In the first part of this thesis, we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables. In the second part we deal with the spectrum of products of Ginibre matrices. Exact eigenvalue density is known for a very few matrix ensembles. For the known ones they often lead to determinantal point process. Let X1, X2,..., Xk be i.i.d Ginibre matrices of size n ×n whose entries are standard complex Gaussian random variables. We derive eigenvalue density for matrices of the form X1 ε1 X2 ε2 ... Xk εk , where εi = ±1 for i =1,2,..., k. We show that the eigenvalues form a determinantal point process. The case where k =2, ε1 +ε2 =0 was derived earlier by Krishnapur. In the case where εi =1 for i =1,2,...,n was derived by Akemann and Burda. These two known cases can be obtained as special cases of our result.

Orientador: Dimitar Kolev Dimitrov / Banca: Carlos Tomei / Banca: José  Alberto Cuminato / Resumo: Neste trabalho estudaremos a relação existente entre polinômios ortogonais e matrizes aleatórias. Exibiremos uma caracterização de polinômios ortogonais via problemas de Riemann-Hilbert, a qual tem se mostrado uma ferramenta única para obtenção de assintóticas de polinômios ortogonais. Posteriormente, estudaremos a teoria básica dos ensembles unitários de matrizes aleatórias. Por fim, mostraremos como a teoria de assintóticas de polinômios ortogonais pode ser usada na análise assintótica de estatísticas de matrizes aleatórias, nos levando a resultados de universalidade para os ensembles unitários / Abstract: We will exhibit a characterization of orthogonal p olynomials via Riemann-Hilbert problems, which has been shown a powerful to ol for studying asymptotics of orthogonal polynomials. Posteriorly we will review the basic theory of unitary ensembles of random matrices. At the end, we will show how asymptotics of orthogonal polynomials can be used to study asymptotics of several statistics in random matrix theory, obtaining universality results for the unitary ensembles / Mestre

In this thesis, we discuss some topics in random matrix theory which have applications to probability, statistics and quantum information theory. In Chapter 2, by relying on the spectral properties of an associated adjacency matrix, we find the distribution of the maximum of a Dyck path and show that it has the same distribution function as the unsigned Brownian excursion which was first derived in 1976 by Kennedy. We obtain a large and moderate deviation principle for the law of the maximum of a random Dyck path. Our result extends the results of Chung, Kennedy and Khorunzhiy and Marckert. In Chapter 3, we discuss a method of sampling called the Gibbs-slice sampler. This method is based on Neal's slice sampling combined with Gibbs sampling. In Chapter 4, we discuss several examples which have applications in physics and quantum information theory.