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• W3105334966 abstract "We consider the double scaling limit in the random matrix ensemble with an external source $${1over{Z_n}} e^{-n hbox{Tr}({1over 2}M^2 -AM)} dM$$ 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." @default.
• W3105334966 created "2020-11-23" @default.
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• W3105334966 date "2006-12-09" @default.
• W3105334966 modified "2023-09-25" @default.
• W3105334966 title "Large n Limit of Gaussian Random Matrices with External Source, Part III: Double Scaling Limit" @default.
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• W3105334966 doi "https://doi.org/10.1007/s00220-006-0159-1" @default.
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