Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula
Copyright policy )Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula
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.
Quantum optics, Condensed Matter - Mesoscale and Nanoscale Physics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Mathematical Physics (math-ph), 510, Random matrices (probabilistic aspects), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [PHYS.COND.CM-GEN]Physics [physics]/Condensed Matter [cond-mat]/Other [cond-mat.other], Numerical interpolation, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Many-body theory; quantum Hall effect, Mesoscale and Nanoscale Physics (cond-mat.mes-hall), FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Quantum chaos, Mathematical Physics
Quantum optics, Condensed Matter - Mesoscale and Nanoscale Physics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Mathematical Physics (math-ph), 510, Random matrices (probabilistic aspects), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [PHYS.COND.CM-GEN]Physics [physics]/Condensed Matter [cond-mat]/Other [cond-mat.other], Numerical interpolation, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Many-body theory; quantum Hall effect, Mesoscale and Nanoscale Physics (cond-mat.mes-hall), FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Quantum chaos, Mathematical Physics
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