Universal Results for Correlations of Characteristic Polynomials: Riemann-Hilbert Approach

Eugene Strahov*, Yan V. Fyodorov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

63 Scopus citations

Abstract

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) those 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 the 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 an arbitrary invariant ensemble of β= 2 symmetry class.

Original languageAmerican English
Pages (from-to)343-382
Number of pages40
JournalCommunications in Mathematical Physics
Volume241
Issue number2-3
DOIs
StatePublished - Oct 2003
Externally publishedYes

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