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Large \(n\) limit of Gaussian random matrices with external source. I. (English) Zbl 1124.82309

Summary: We consider the random matrix ensemble with an external source \[ \frac 1{Z_n} e^{-n\text{Tr}(\frac12 M^2 - AM)} \,dM \] defined on \(n \times n\) Hermitian matrices, where A is a diagonal matrix with only two eigenvalues \(\pm a\) of equal multiplicity. For the case \(a >1\), we establish the universal behavior of local eigenvalue correlations in the limit \(n \rightarrow \infty\), 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 \(3\times3\)-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large \(n\) limit.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
15A42 Inequalities involving eigenvalues and eigenvectors
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
60F99 Limit theorems in probability theory
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