Nonmonotonic confining potential and eigenvalue density transition for generalized random matrix model

Abstract

We consider a limiting case of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent γ (called the γ-ensembles), which is equivalent to the probability distribution of Laguerre β-ensembles. The effective potential, which is essentially the confining potential for an equivalent ensemble with γ = 1 (called the Muttalib-Borodin ensemble), is a crucial quantity defined in solution to the Riemann-Hilbert problem associated with the γ-ensembles. It enables us to numerically compute the eigenvalue density of Laguerre β-ensembles for all β > 1. In addition, the effective potential shows a non-monotonic behavior for γ < 1, suggestive of its possible role in developing a transition from a diverging to a non-diverging density near the hard-edge of the eigenvalue spectrum. Such a transition in eigenvalue density is of critical importance for the random matrix theory describing disordered mesoscopic systems. For γ-ensembles with γ > 1, the effective potential also shows transition from non-monotonic to monotonic behavior near the origin when a parameter of the additional interaction term is varied.