Generalized Muttalib-Borodin ensembles, Laguerre β-ensembles and effective potentials

The eigenvalue density for Generalized Muttalib-Borodin ensembles (also called γ-ensembles) can be computed by solving equivalent equilibrium problem for Muttalib-Borodin (MB) ensembles with γ-dependent effective potentials. Ideally, an exponent γ-induced hard-edge to soft-edge transition in eigenvalue density could describe a disorder induced metal to insulator transition in mesoscopic conductors. Such a transition in density has previously been produced only by especially designed non-monotonic confining potentials of MB ensembles. We find that the effective potentials computed for a range of parameter of hard-edge γ–ensembles with standard monotonic potentials show increasing non-monotonic behavior near the origin as γ is decreased (or disorder is increased) systematically. While this non-monotonicity is not sufficient to produce a hard-edge to soft-edge transition in density in this toy model, it suggests that with appropriate combination of the additional interaction and the confining potential, such a transition can indeed occur. As a byproduct of the above calculations we also obtain the eigenvalue densities of Laguerre β-ensembles for any β>1.