I discuss an equilibrium problem for a vector of three measures that is associated with the two matrix model with a quartic potential. The first component of the minimizer describes the limiting mean eigenvalue density of one of the random matrices in the two matrix model. There is an external field acting on the first measure, an upper constraint on the second measure, and, in the case of a double well quartic potential, an external field on the third measure as well. The latter fact allows for new critical phenomena that are not observed in the usual one matrix model.
This is joint work with Maurice Duits (CalTech) and Man Yue Mo (Bristol).