Nonmonotonic confining potential and eigenvalue density transition for generalized random matrix model
Abstract
We consider several limiting cases of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent γ (called the γ-ensembles). The effective potential, which is essentially the single-particle 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 γ-ensembles for all γ > 0. We show that one important effect of the two-particle interaction parameter γ is to generate or enhance the non-monotonicity in the effective single-particle potential. For suitable choices of the initial single-particle potentials, reducing γ can lead to a large non-monotonicity in the effective potential, which in turn leads to significant changes in the density of eigenvalues. For a disordered conductor, this corresponds to a systematic decrease in the conductance with increasing disorder. This suggests that appropriate models of γ-ensembles can be used as a possible framework to study the effects of disorder on the distribution of conductances.