A Vector Equilibrium Problem for Muttalib–Borodin Biorthogonal Ensembles

The Muttalib–Borodin biorthogonal ensemble is a joint density function for $n$ particles on the positive real line that depends on a parameter $\theta$. There is an equilibrium problem that describes the large $n$ behavior. We show that for rational values of $\theta$ there is an equivalent vector equilibrium problem.