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

2010/03/31 by Christophe Carré, Matthieu Deneufchâtel, Matthieu Deneufchatel +2
Mathematics · Physics and Astronomy · #Classical mechanics #Combinatorics #Interpolation (computer graphics) #Law #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Motion (physics) #Physics #Pure mathematics #Quantum chaos and dynamical systems #Random Matrices and Applications #Riemann hypothesis #Selberg trace formula #Theoretical and Computational Physics #Unitary matrix #Unitary state #cond-mat.mes-hall #math-ph #math.CO #math.MP

paper · pdf · doi:10.1063/1.3514535

arxiv created 2010/03/31 · openalex publication_date 2010/12/01 · arxiv updated 2015/05/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We investigate the asymptotic behavior of the Selberg-like integral \documentclass[12pt]minimal\begindocument(1)/(N!)∫ _[0,1]^Nx₁^p\break ∏ _i<j(x_i-x_j)²∏ _ix_i^a-1(1-x_i)^b-1dx_i,\enddocument1N!∫[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.

Citations

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