ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS

Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment...

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Bibliographic Details
Published in:Mathematics of computation 2010-04, Vol.79 (270), p.871-915
Main Author: BORNEMANN, FOLKMAR
Format: Article
Language:English
Subjects:
Approximation
Determinants
Differential equations
Eigenvalues
Exact sciences and technology
Gaussian quadratures
Integral equations
Mathematical analysis
Mathematical integrals
Mathematics
Matrices
Methods of scientific computing (including symbolic computation, algebraic computation)
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Numerical quadratures
Operator theory
Polynomials
Sciences and techniques of general use
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Summary:Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implement able, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss-Legendre or Clenshaw-Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk scaling limit and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the two-point correlation functions of the more recently studied Airy and Airy 1 processes.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-09-02280-7