English

Large n limit of Gaussian random matrices with external source, part I

Mathematical Physics 2009-11-10 v2 math.MP

Abstract

We consider the random matrix ensemble with an external source 1Znen\Tr(1/2M2AM)dM \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on n×nn\times n Hermitian matrices, where AA is a diagonal matrix with only two eigenvalues ±a\pm a of equal multiplicity. For the case a>1a > 1, we establish the universal behavior of local eigenvalue correlations in the limit nn \to \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 3×33 \times 3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large nn limit.

Keywords

Cite

@article{arxiv.math-ph/0402042,
  title  = {Large n limit of Gaussian random matrices with external source, part I},
  author = {Pavel M. Bleher and Arno B. J. Kuijlaars},
  journal= {arXiv preprint arXiv:math-ph/0402042},
  year   = {2009}
}

Comments

32 pages, 4 figures

R2 v1 2026-07-22T16:24:01.090Z