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

Abstract

We continue the study of the Hermitian random matrix ensemble with external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ where $A$ has two distinct eigenvalues $\pm a$ of equal multiplicity. This model exhibits a phase transition for the value $a=1$, since the eigenvalues of $M$ accumulate on two intervals for $a>1$, and on one interval for $0<a<1$. The case $a>1$ was treated in part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case $0<a<1$. As in part I we apply the Deift/Zhou steepest descent analysis to a $3\times 3$-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems.

24 pages, 7 figures

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