Trajectories and Curvatures of Hermitian and Non-Hermitian Random Matrix Eigenvalues| Chapter 2 | New Frontiers in Physical Science Research Vol. 1

 Study is done on the characteristics of the evolution of the eigenvalues of Hermitian and non-Hermitian matrices as a function of a parameter that simulates time. The so-called Olshanetsky-Perelomov Projection Method, which is based on the Lax equation, is applied in the Hermitian case. As a result, the Dyson index, which in the Gaussian ensemble has values of 1, 2, and 3, can take on any real value in the loggas Hamiltonian. As a result, one has a 1D loggas dynamics for the ensemble's tridiagonal matrices' eigenvalues. On the other hand, in the non-Hermitian scenario, a model is built to investigate the statistical characteristics of erroneous log-gas trajectories whose positions correspond to the complex eigenvalues of the unitary Ginibre ensembre.  It is shown that statistical analysis of the trajectories creates a shell structure that makes the eigenvalue departure sites visible. The eigenvalue curvatures are also demonstrated to be the universal distribution for 1D trajectories and a Cauchy distribution for trajectories in the complex plane as a function of the parameter.

Author(s) Details:

M. P. Pato,
Inst´ıtuto de F´ısica, Universidade de S˜ao Paulo CEP 05508-090 S˜ao Paulo, S.P., Brazil.

Please see the link here: https://stm.bookpi.org/NFPSR-V1/article/view/8118