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.
翻译:Gaussian ensensles 软边缘缩放底限最大值的美元分布值最大,高斯星团的分布是随机矩阵理论中最重要的分布,其数字评价是一个具有极大实际重要性的主题。一个用于评价发行量的数字方法使用一个事实,即它们可以作为Fredholm 决定因素来表示,涉及所谓的Airy整体操作员。当集成操作员的频谱通过直接分解来计算时,就会以最精确的绝对精确度来计算。值得注意的是,Airy 整体操作员是一个所谓的非双光谱操作员的范例,它承认了一个共享相同电子元元功能的最佳计算器。在这个手稿中,我们开发了一个高效的数字算法,用来评价航空整体操作员的egendomcom定位,使用通缩放差操作员的eigendecommation。这使我们能够快速评价美元最高值的分布到完全相对精确度,除非在左尾部,它们被计算为绝对精确度。此外,我们将一个最精确度的计算为最精确度操作员。我们在空气操作员的精确度中,我们将一个内置的内置的内值和空操作员的内置的内置的内置的内定值描述一个内置的内置的内置的内置的内置和内置的内置的内置的内置。我们正的内置的内置的内置的内置的内,我们的内置的内置的内置的内,我们正的内置的内置的内置的内置的内置的内置的内置的内置是一个内置。我们的内置的内置的内置的内置的内置的内置的内置的内置的内置的内置的内置的内置的内置的内置的内置。