Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

Abstract

We investigate the asymptotic behavior of the Selberg-like integral \documentclass[12pt]{minimal}\begin{document}$\frac{1}{N!}\int _{[0,1]^N}x_1^p$\break $\prod _{i<j}(x_i-x_j)^2\prod _ix_i^{a-1}(1-x_i)^{b-1}dx_i,$\end{document}1N!∫[0,1]Nx1p∏i<j(xi−xj)2∏ixia−1(1−xi)b−1dxi, as N → ∞ for different scalings of the parameters a and b with N. Integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting N channels in the two attached leads. Making use of Newton's interpolation formula, we show that an asymptotic limit exists and we compute it explicitly.