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Large n Limit of Gaussian Random Matrices with External Source, Part I

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Abstract

We consider the random matrix ensemble with an external source

defined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a>1, we establish the universal behavior of local eigenvalue correlations in the limit n→∞, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3×3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN, 46202, U.S.A

    Pavel Bleher

  2. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001, Leuven, Belgium

    Arno B. J. Kuijlaars

Authors
  1. Pavel Bleher
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  2. Arno B. J. Kuijlaars
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Correspondence to Pavel Bleher.

Additional information

Communicated by J.L. Lebowitz

Dedicated to Freeman Dyson on his eightieth birthday

The first author was supported in part by NSF Grants DMS-9970625 and DMS-0354962.

The second author was supported in part by projects G.0176.02 and G.0455.04 of FWO-Flanders, by K.U.Leuven research grant OT/04/24, and by INTAS Research Network NeCCA 03-51-6637.

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Bleher, P., Kuijlaars, A. Large n Limit of Gaussian Random Matrices with External Source, Part I. Commun. Math. Phys. 252, 43–76 (2004). https://doi.org/10.1007/s00220-004-1196-2

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  • DOI: https://doi.org/10.1007/s00220-004-1196-2

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