Fluctuation moments induced by asymptotically liberating random matrix ensembles

Abstract

The topics covered in this thesis belong to the theories of free probability and random matrices. Random matrix ensembles composed of independent Haar-unitary random matrices, or independent Haar-orthogonal random matrices, are known to be asymptotically liberating, they give rise to asymptotic free independence when used for conjugation of constant matrices. G. Anderson and B. Farrel showed that certain family of discrete random unitary matrices can actually be used to same end. We investigate fluctuation moments and higher order moments induced on constant matrices by conjugation with asymptotically liberating random matrix ensembles. We show for the first time that the fluctuation moments associated to second order free independence can be obtained from conjugation with an ensemble consisting of signed permutation matrices and the Discrete Fourier Transform matrix. We also determine fluctuation moments induced by various related ensembles where we do not get known expressions but others related to traffic free independence.