Pseudo-Hermitian β-Laguerre Ensemble with Nonpositive Metric

A β-Laguerre ensemble with nonpositive metric is introduced considering first the case of 2 × 2 matrices. Extending then the procedure to the case of large matrix, it is shown that when the eigenvalues evaporated from the real axis, they fill up a hyperbola in the complex plane. Moreover, by performing the numerical simulations, it is found that the parameters of the hyperbola are determined by the parameters of the density of eigenvalues of the Hermitian β-Laguerre ensemble.