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On the numerical evaluation of Fredholm determinants

Author: Folkmar Bornemann
Journal: Math. Comp. 79 (2010), 871-915
MSC (2000): Primary 65R20, 65F40; Secondary 47G10, 15A52
Published electronically: September 24, 2009
MathSciNet review: 2600548
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Abstract: 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 implementable, 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 $\text {Airy}_1$ processes.

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Additional Information

Folkmar Bornemann
Affiliation: Zentrum Mathematik – M3, Technische Universität München, Boltzmannstr. 3, 85747 Garching bei München, Germany

Received by editor(s): June 24, 2008
Received by editor(s) in revised form: March 16, 2009
Published electronically: September 24, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.