Universality of a family of random matrix ensembles with logarithmic soft-confinement potentials

Recently we introduced a family of U N invariant random matrix ensembles which is characterized by a parameter describing logarithmic soft-confinement potentials V H ln H 1+ 0 . We showed that we can study eigenvalue correlations of these “ ensembles” based on the numerical construction of the corresponding orthogonal polynomials with respect to the weight function exp − ln x 1+ . In this work, we expand our previous work and show that: i the eigenvalue density is given by a power law of the form x ln x −1 /x and ii the two-level kernel has an anomalous structure, which is characteristic of the critical ensembles. We further show that the anomalous part, or the so-called “ghost-correlation peak,” is controlled by the parameter ; decreasing increases the anomaly. We also identify the two-level kernel of the ensembles in the semiclassical regime, which can be written in a sinh-kernel form with more general argument that reduces to that of the critical ensembles for =1. Finally, we discuss the universality of the ensembles, which includes Wigner-Dyson universality → limit , the uncorrelated Poisson-type behavior →0 limit , and a critical behavior for all the intermediate 0 in the semiclassical regime. We also comment on the implications of our results in the context of the localization-delocalization problems as well as the N dependence of the two-level kernel of the fat-tail random matrices.