Nonmonotonic Confining Potential And Eigenvalue Density Transition For Generalized Random Matrix Model

We consider several limiting cases of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent gamma (called the gamma ensembles). The effective potential, which is essentially the single-particle confining potential for an equivalent ensemble with gamma = 1 (called the Muttalib-Borodin ensemble), is a crucial quantity defined in solution to the Riemann-Hilbert problem associated with the gamma ensembles. It enables us to numerically compute the eigenvalue density of gamma ensembles for all gamma > 0. We show that one important effect of the two-particle interaction parameter gamma is to generate or enhance the nonmonotonicity in the effective single-particle potential. For suitable choices of the initial single-particle potentials, reducing gamma can lead to a large nonmonotonicity 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 gamma ensembles can be used as a possible framework to study the effects of disorder on the distribution of conductances.