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Averages of characteristic polynomials in random matrix theory
Author(s) -
Borodin A.,
Strahov E.
Publication year - 2006
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20092
Subject(s) - pfaffian , mathematics , random matrix , orthogonal polynomials , circular ensemble , symplectic geometry , matrix (chemical analysis) , mehler–heine formula , scaling , limit (mathematics) , scaling limit , gaussian , pure mathematics , mathematical analysis , classical orthogonal polynomials , gegenbauer polynomials , eigenvalues and eigenvectors , geometry , quantum mechanics , physics , materials science , composite material
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. © 2005 Wiley Periodicals, Inc.