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 tailcan be computed to full relative precision .
The distributions of the k-th largest level at the soft edge scaling limit of Gaussian ensembles are some of the most important distributions in random matrix theory, and their numerical evaluation is a subject of great practical importance. One numerical method for evaluating the distributions uses the fact that they can be represented as Fredholm determinants involving the so-called Airy integral operator. When the spectrum of the integral operator is computed by discretizing it directly, the eigenvalues are known to at most absolute precision. Remarkably, the Airy 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 Airy integral operator to full relative precision, using the eigendecomposition of the commuting differential operator. This allows us to rapidly evaluate the distributions of the k-th largest level to full relative precision rapidly everywhere except in the left tail, where they are computed to absolute precision . In addition, we characterize the eigenfunctions of the Airy integral operator, and describe their extremal properties in relation to an uncertainty principle involving the Airy transform. We observe that the Airy integral operator is fairly universal, and we describe a separate application to Airy beams in optics. Using the eigenfunctions, we compute a finite-energy Airy beam that is optimal, in the sense that the beam is both maximally concentrated, and maximally non-diffracting and self-accelerating .
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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
the k-th largest level at the soft edge scaling limit of Gaussian ensembles are some
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With our algorithm, the Tracy-Widom distribution can be evaluated to full
This allows us to rapidly evaluate the distributions of the k-th largest level to full relative precision rapidly everywhere except in the left tail, where they are computed to
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everywhere rapidly , and, furthermore, its right tailcan be computed to full relative precision
. In addition, we characterize the eigenfunctions of the Airy integral operator, and describe their extremal properties in relation to an uncertainty principle involving the Airy transform. We observe that the Airy integral operator is fairly universal, and we describe a separate application to Airy beams in optics. Using the eigenfunctions, we compute a finite-energy Airy beam that is optimal, in the sense that the beam is both maximally concentrated, and maximally non-diffracting and self-accelerating