Generalized random matrix model with additional interactions

We introduce a log-gas model that is a generalization of a random matrix ensemble with an additional interaction, whose strength depends on a parameter $\gamma$. The equilibrium density is computed by numerically solving the Riemann-Hilbert problem associated with the ensemble. The effect of the additional parameter $\gamma$ associated with the two-body interaction can be understood in terms of an effective $\gamma$-dependent single-particle confining potential.