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Fredholm determinants, differential equations and matrix models

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 01 Jun 1994Embargo end date: 01 Jan 1993 United States English Publisher:Springer Science and Business Media LLCJournal:Communications in Mathematical Physics, volume 163, pages 33-72 (issn: 0010-3616, eissn: 1432-0916,

Authors: Tracy, Craig A.; Widom, Harold;

doi: 10.1007/bf02101734 , 10.48550/arxiv.hep-th/9306042

arXiv: http://arxiv.org/abs/hep-th/9306042

Fredholm determinants, differential equations and matrix models

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Abstract

Orthogonal polynomial random matrix models of NxN hermitian matrices lead to Fredholm determinants of integral operators with kernel of the form (phi(x) psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm determinants of integral operators having kernel of this form and where the underlying set is a union of open intervals. The emphasis is on the determinants thought of as functions of the end-points of these intervals. We show that these Fredholm determinants with kernels of the general form described above are expressible in terms of solutions of systems of PDE's as long as phi and psi satisfy a certain type of differentiation formula. There is also an exponential variant of this analysis which includes the circular ensembles of NxN unitary matrices.

34 pages, LaTeX using RevTeX 3.0 macros; last version changes only the abstract and decreases length of typeset version

Country

United States

Related Organizations

University of California System
United States
University of California, Davis
United States
University of California, Santa Cruz
United States

Keywords

High Energy Physics - Theory, Hermite ensembles, 33C90, math-ph, 47N55, FOS: Physical sciences, Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.), Painlevé transcendents, math.MP, Jacobi ensembles, nlin.SI, solv-int, Mathematical Physics, sine, Airy and Bessel kernels, Nonlinear Sciences - Exactly Solvable and Integrable Systems, Laguerre ensembles, 82B10, hep-th, Mathematical Physics (math-ph), KdV equations (Korteweg-de Vries equations), High Energy Physics - Theory (hep-th), 47A75, Bessel and Airy functions, cylinder functions, \({}_0F_1\), Exactly Solvable and Integrable Systems (nlin.SI), 82B05, 47G10

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