Communications in mathematical physics vol:270 issue:2 pages:481-517
We consider the double scaling limit in the random matrix ensemble with an external source 1/Z(n) e(-nTr(1/2 M2-AM))dM defined on n x n Hermitian matrices, where A is a diagonal matrix with two eigenvalues +/- a of equal multiplicities. The value a = 1 is critical since the eigenvalues of M accumulate as n --> infinity on two intervals for a > 1 and on one interval for 0 < a < 1. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case a = 1 new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Brezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a 3 x 3-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.