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 non-monotonicity in the effective single-particle potential. For suitable choices of the initial single-particle potentials, reducing $\gamma$ 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 $\gamma$-ensembles can be used as a possible framework to study the effects of disorder on the distribution of conductances.