  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.
