| 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. |
