On the Evaluation of the Eigendecomposition of the Airy Integral Operator

The distributions of random matrix theory have seen an explosion of interest in recent years, and have been found in various applied fields including physics, high-dimensional statistics, wireless communications, finance, etc. The Tracy-Widom distribution is one of the most important distributions in random matrix theory, and its numerical evaluation is a subject of great practical importance. One numerical method for evaluating the Tracy-Widom distribution uses the fact that the distribution can be represented as a Fredholm determinant of a certain integral operator. However, when the spectrum of the integral operator is computed by discretizing it directly, the eigenvalues are known to at most absolute precision. Remarkably, the integral operator is an example of a so-called bispectral operator, which admits a commuting differential operator that shares the same eigenfunctions. In this manuscript, we develop an efficient numerical algorithm for evaluating the eigendecomposition of the integral operator to full relative precision, using the eigendecomposition of the differential operator. With our algorithm, the Tracy-Widom distribution can be evaluated to full absolute precision everywhere rapidly, and, furthermore, its right tail can be computed to full relative precision.