Integrals of rational symmetric functions, two matrix models and biorthogonal polynomials

2006/03/14 by J. Harnad, Harnad, J., A. Yu. Orlov +1
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #hep-th #math-ph #math.MP #nlin.SI

paper · pdf · doi:10.48550/arxiv.math-ph/0603040

21 pages

arxiv created 2006/03/20 · arxiv updated 2009/12/01

Abstract

We give a new method for the evaluation of a class of integrals of rational symmetric functions in N pairs of variables x_a, y_{aa=1,... N} arising in coupled matrix models, valid for a broad class of two-variable measures. The result is expressed as the determinant of a matrix whose entries consist of the associated biorthogonal polynomials, their Hilbert transforms, evaluated at the zeros and poles of the integrand, and bilinear expressions in these. The method is elementary and direct, using only standard determinantal identities, partial fraction expansions and the property of biorthogonality. The corresponding result for one-matrix models and integrals of rational symmetric functions in N variables x_{aa=1,... N} is also rederived in a simple way using this method

Related

• Matrix Integrals, Symmetric Functions theory and matrix integrals
• Matrix biorthogonal polynomials, associated polynomials and functions of the second kind
• The 2-matrix model, biorthogonal polynomials, Riemann-Hilbert problem, and algebraic geometry
• Zeros of some bi-orthogonal polynomials
• Universality in the two matrix model: a Riemann-Hilbert steepest descent analysis