On the Numerical Evaluation of Fredholm Determinants

TL;DR

This paper introduces a simple, efficient numerical method based on the Nystrom approach for evaluating Fredholm determinants, crucial in mathematics and physics, with proven exponential convergence for analytic kernels and applications to random matrix theory.

Contribution

It develops a general, easily implementable numerical method for Fredholm determinants, filling a gap in existing literature and enabling new computations in random matrix theory.

Findings

The method achieves exponential convergence for analytic kernels.

It successfully evaluates distribution functions in random matrix theory.

First numerical evaluation of two-point correlation functions for Airy processes.

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 Painleve 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 Nystrom 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 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 Airy1 processes for the first time.