2006/02/28 by Pavel M. Bleher, Arno B. J. Kuijlaars
Computer Science · Mathematics · Physics and Astronomy · #Mathematical functions and polynomials #Matrix Theory and Algorithms #Random Matrices and Applications #math-ph #math.MP #math.PR
paper · pdf · doi:10.1007/s00220-006-0159-1
published as Communications in Mathematical Physics 270 (2007), 481--517 · 36 pages, 9 figures
arxiv created 2006/02/28 · openalex publication_date 2006/12/08 · arxiv updated 2010/07/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We consider the double scaling limit in the random matrix ensemble with an external source (1)/(Zn) e^-n \Tr(1/2M2 -AM) dM 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.