Eigenvalues distribution of random matrices.

Author(s)

Roberto da Silva

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2000-05-18

Examining board members

Domingos Humberto Urbano Marchetti; Nestor Felipe Caticha Alfonso; Henrique Von Dreifus

• Advisor: Domingos Humberto Urbano Marchetti

Abstract

In a detailed review we obtain a semi-circle law for the density of states in theWigner’s Gaussian Ensemble. Also we talk about Dyson’s Analogy, seeing the eigenvalues like charges that repulse themselves in the unitary circle, showing that this case the density of states is uniform. In a more general context we obtain the semi-circle law, proving the Glivenko-Cantelli Theorem to strongly correlated variables, using a combinatorial method of Paths' Counting. Thus we are showing the stability of the semi-circle Law. Also, in this dissertation we study the correlation functions in the Gaussian and Circular ensembles showing that using the Gram's Method in the case that eigenvalues are limited in a interval. In these ensembles we computed the density of states. More precisely, in a Chebychev ensemble the results were obtained analytically. In this ensemble, we also obtain graphics of the truncated correlation function. (AU)