Processos de punts determinantals generats per ensembles de matrius aleatòries

Abstract

[en] Random matrices and the distribution of their eigenvalues, is a probability problem that arises from mathematical physics. The eigenvalues are point processes. In some special cases, they are a determinantal point process. This study has been very important over the last century, because it has been used for the distribution of electrons in heavy atoms. In this work, we will consider the distribution of the eigenvalues of a specific type of random matrix, the complex Gaussian matrix, also known as Ginibre ensemble. The distribution of the eigenvalues of Ginibre ensemble has a determinantal form. Therefore, we will first study the point processes in general, the distribution function of the Ginibre ensemble and the correlation functions. Secondly, we will also study the relationship between the distribution of point processes and the correlation functions. Thirdly, we will study determinantal point process and their properties in a general way. Finally, we will see the same types of ensembles matrices that will fulfill this same type of relationship with the determinantal point process. These examples of matrices will develop the processes known as Gaussian Unitary Ensemble (GUE) and the Circular Unitary Ensemble (CUE). We will also perform experiments to see the distribution of the eigenvalues of Ginibre ensemble, GUE and CUE in the complex plane.