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This is the second part of a study of the limiting distributions of the topeigenvalues of a Hermitian matrix model with spiked external source under ageneral external potential. The case when the external source is of rank onewas analyzed in an earlier paper. In the present paper we extend the analysisto the higher rank case. If all the eigenvalues of the external source are lessthan a critical value, the largest eigenvalue converges to the right end-pointof the support of the equilibrium measure as in the case when there is noexternal source. On the other hand, if an external source eigenvalue is largerthan the critical value, then an eigenvalue is pulled off from the support ofthe equilibrium measure. This transition is continuous, and is universal,including the fluctuation laws, for convex potentials. For non-convexpotentials, two types of discontinuous transitions are possible to occurgenerically. We evaluate the limiting distributions in each case for generalpotentials including those whose equilibrium measure have multiple intervalsfor their support.

On the largest eigenvalue of a Hermitian random matrix model with spiked external source II. Higher rank cases