Trajectories and Curvatures of Hermitian and Non-Hermitian Random Matrix Eigenvalues

Properties of the evolution of the eigenvalues of Hermitian and non-Hermitian matrices as a function of a parameter that plays the role of time are studied. In the Hermitian case, the so-called Olshanetsky-Perelomov Projection Method based on Lax equation it is used. This leads to a loggas Hamiltonian in which the Dyson index that in tha Gaussian ensemble has the values 1, 2 and 3, can assume any real value. One has, therefore, a 1D loggas dynamics for the eigenvalues of the tridiagonal matrices of the ensemble. In the non-Hermitian case, on the other hand, a model is constructed to study statistical properties of irregular trajectories of a log-gas whose positions are those of the complex eigenvalues of the unitary Ginibre ensembre. It is demonstrated that statistical analysis of the trajectories produces a shell structure that exposes the locations of the eigenvalue departures. It is also shown that the eigenvalue curvatures are, as a funnction of the parameter ; the universal distribution for the 1D trajectories and, a Cauchy distribution, for the trajectories in the complex plane.