Pseudo-Hermitian Random Matrices

Mauricio Porto Pato is a Senior Professor at the University of São Paulo with a large experience in the field of random matrices theory and applications. In the early 90’s, in a collaboration with the nuclear physicist M. S. Hussein, he began a study of random matrices that resulted in the construction of an ensemble to be applied to a situation of partial conservation of a quantum number. The model was then, successfully, applied to the description of isospin data. In a collaboration with O. Bohigas, another important contribution of him to be highlighted, was the formalism to deal with missing levels in correlated spectra, a study that evolved from his work with the experimentalist G. E. Mitchell. About ten years ago, his interest moved from Hermitian to non-Hermitian operators and this led to his involvement with the studies of the class of pseudo-Hermitian matrices associated to PT-symmetric systems, that is, systems invariant under parity and time-reversal transformations. This investigation started with the introduction of the pseudo-Hermiticity condition in the sparse tridiagonal matrices of the so-called beta-ensembles of the random matrix theory. Next, the pseudo-Hermiticity condition was extended to the standard Gaussian matrices with the creation of the pseudo-Hermitian Gaussian ensembles. All this effort, along a decade, comprises about one dozen of works among articles and thesis.