Abstract
We consider the double scaling limit in the random matrix ensemble with an external source
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.
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Communicated by H. Spohn.
The first author was supported in part by the National Science Foundation (NSF) Grant DMS-0354962.
The second author was supported by FWO-Flanders project G.0455.04, by K.U. Leuven research grant OT/04/24, by INTAS Research Network 03-51-6637, by a grant from the Ministry of Education and Science of Spain, project code MTM2005-08648-C02-01, and by the European Science Foundation Program MISGAM.
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Bleher, P.M., Kuijlaars, A.B.J. Large n Limit of Gaussian Random Matrices with External Source, Part III: Double Scaling Limit. Commun. Math. Phys. 270, 481–517 (2007). https://doi.org/10.1007/s00220-006-0159-1
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DOI: https://doi.org/10.1007/s00220-006-0159-1