Averages of characteristic polynomials in random matrix theory

A. Borodin*, E. Strahov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

80 Scopus citations

Abstract

We compute averages of products and ratios of characteristic polynomials associated with orthogonal, unitary, and symplectic ensembles of random matrix theory. The Pfaffian/determinantal formulae for these averages are obtained, and the bulk scaling asymptotic limits are found for ensembles with Gaussian weights. Classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation. We employ a discrete approximation method: the problem is solved for discrete analogues of random matrix ensembles originating from representation theory, and then a limit transition is performed. Exact Pfaffian/determinantal formulae for the discrete averages are proven using standard tools of linear algebra; no application of orthogonal or skew-orthogonal polynomials is needed.

Original languageAmerican English
Pages (from-to)161-253
Number of pages93
JournalCommunications on Pure and Applied Mathematics
Volume59
Issue number2
DOIs
StatePublished - Feb 2006
Externally publishedYes

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