2002/10/05 by Eugene Strahov, Yan V. Fyodorov · 1 citation
Mathematics · Physics and Astronomy · #Cauchy distribution #Discrete orthogonal polynomials #Hahn polynomials #Hermitian matrix #Holomorphic and Operator Theory #Integrable system #Invariant (physics) #Mathematical functions and polynomials #Mehler–Heine formula #Orthogonal polynomials #Random Matrices and Applications #Random matrix #cond-mat #hep-th #math-ph #math.MP #nlin.SI
paper · pdf · doi:10.1007/s00220-003-0938-x
published as Commun.Math.Phys. 241 (2003) 343-382 · 34pages
arxiv created 2002/10/05 · openalex publication_date 2003/09/19 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for arbitrary invariant ensemble of β=2 symmetry class.