Free probability for purely discrete eigenvalues of random matrices

We study the eigenvalues of polynomials of large random matrices which have only discrete spectra. Our model is closely related to and motivated by spiked random matrices, and in particular to a recent result of Shlyakhtenko in which asymptotic infinitesimal freeness is proved for rotationally invariant random matrices and finite rank matrices. We show the almost sure convergence of Shlyakhtenko's result. Then we show the almost sure convergence of eigenvalues of our model when it has a purely discrete spectrum. We define a framework for analyzing purely discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of purely discrete eigenvalues of our model. This talk is based on a joint work with B. Collins and T. Hasebe.Non UBCUnreviewedAuthor affiliation: Aichi University of EducationFacult