arXiv math-ph/03060782003-06-30math-phmath.CAmath.MP

Universal behavior for averages of characteristic polynomials at the origin of the spectrum

M. Vanlessen

备注：: 24 pages, 3 figures

英文摘要：

It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy transforms. We will show that, for the unitary ensemble $\frac{1}{\hat Z_n}|\det M|^{2α}e^{-nV(M)}dM$ of $n\times n$ Hermitian matrices, these kernels have universal behavior at the origin of the spectrum, as $n\to\infty$, in terms of Bessel functions. Our approach is based on the characterization of orthogonal polynomials together with their Cauchy transforms via a matrix Riemann-Hilbert problem, due to Fokas, Its and Kitaev, and on an application of the Deift/Zhou steepest descent method for matrix Riemann-Hilbert problems to obtain the asymptotic behavior of the Riemann-Hilbert problem.

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