Random normal matrices ensembles: projection, asymptotics behavior and universality of ugenvalues

Author(s)

Alexei Magalhães Veneziani

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2008-03-12

Examining board members

Domingos Humberto Urbano Marchetti; Nestor Felipe Caticha Alfonso; Joao Carlos Alves Barata; Luiz Renato Goncalves Fontes; Maria Eulalia Vares

• Advisor: Domingos Humberto Urbano Marchetti

Abstract

A matrix `A IND.N´ of order N is normal if and only if it commutes with its adjoint. In the present thesis we investigate the eigenvalues statistics (in the complex plane) of ensembles of normal random matrices when their order N tends to infinite. The probability distribution function in the space of normal matrices attributes, as in statistical mechanics, a Boltzmann weight `e POT.-NF(`A IND.N´)´ at each matrix realization `A IND.N´, where F is a real-valued function invariant by unitary transformations. By performing a change of variables (from entry variables to spectral variables) we write the marginal joint distribution of eigenvalues {`z IND.i´} POT.N´ `IND.i=1´, as well as the n-points functions corresponding to several ensembles, as the determinant of an associated integral kernel. From this formalism well-established in the literature, we shall present in this thesis two types of results: Firstly, exploiting the similarity of joint distribution of eigenvalues to a variational problem on electrostatic equilibrium measures of charges subjected to an external potential V : C - > R (by choosing F(`A IND.N´) = ```sigma´ POT.N´ IND.i´=1 V (`z IND.i´)), we can apply the theory of logarithmic potentials to obtain the unique equilibrium measure coinciding with the 1-point function of these ensembles. Based on this theory, we propose in this thesis a method of analytical interpolation capable of projecting the equilibrium measure of normal ensembles in equilibrium measures of corresponding Hermitian and unitary ensembles. We give several applications of this procedure. The second type of results utilizes the saddle point method applied to integral kernel of a family of normal matrix ensembles with potentials `V IND.`alfa´´ (z) = `|z| POT.`alfa´´ , z `PERTENCE A´ C e `alfa´ `PERTENCE A´ ]0,`INFINITO´[. Similarly to what has been shown in hermitian ensembles by Deift, we established by mean of this expansion a similar concept of universality for this family, making use of conformal maps and theory of Segal-Bargmann space. Concerning the universality defined by G. Oas, we show that the universality claimed by this author is incorrect when the tail of this probability is taking into account. (AU)