Spectral Properties of Beta Hermite and Beta Laguerre Ensembles as Beta to Infinity

In 2002, Dumitriu and Edelman introduced three ensembles of tridiagonal random matrix models for a general parameter beta\u3e0. These ensembles generalized the classical ensembles corresponding to beta=1,2, or 4. The generalization of the behavior of the spectrum for two of these models, the beta-Hermite and beta-Laguerre, in the regime of the largest or smallest eigenvalue, was proved by Ramirez, Rider, and Virag in 2007. This thesis describes the behavior of the spectrum of these two ensembles as beta to infinity. It is found that the eigenvalues become deterministic, fixing themselves at the roots of the Hermite or Laguerre orthogonal polynomials. When beta is large, but not infinite, the eigenvalues have first order Gaussian fluctuations around these roots. Laws of Large Numbers, Central Limit Theorems and the covariance structure for these eigenvalues are derived. Connections between the work of Dumitriu and Edelman and Ramirez, Rider, and Virag are examined. Directions for future research and open problems are also discussed