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 exponent $\gamma$ (called $\gamma$-ensembles). The effective potential, which is essentially single-particle confining potential equivalent $\gamma=1$ Muttalib-Borodin ensemble), crucial quantity defined in solution to Riemann-Hilbert problem associated $\gamma$-ensembles. It enables us numerically compute eigenvalue density $\gamma$-ensembles all $\gamma > 0$. show that one important effect two-particle parameter generate or enhance non-monotonicity potential. For suitable choices initial potentials, reducing can lead large turn leads significant changes eigenvalues. disordered conductor, this corresponds systematic decrease conductance increasing disorder. This suggests appropriate models be used as possible framework study effects disorder on conductances.