Limit of Gaussian Random Matrices with External Source, Part III: Double Scaling Limit

We consider the double scaling limit in the random matrix ensemble with an external source 1 Zn e−nTr( 1 2 M −AM)d M defined on n×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 → ∞ 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 Brézin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a 3×3-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.