We compute the Tracy-Widom distribution describing the asymptotic distribution of the largest eigenvalue of a large random matrix by solving a boundary-value problem posed by Bloemendal. The distribution is computed in two ways. The first method is a second-order finite-difference method and the second is a highly accurate Fourier spectral method. Since $\beta$ is simply a parameter in the boundary-value problem, any $\beta> 0$ can be used, in principle. The limiting distribution of the $n$th largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.

翻译：我们通过求解 Bloemendal 提出的边值问题，计算了描述大型随机矩阵最大特征值渐近分布的 Tracy-Widom 分布。该分布采用两种方法进行计算：第一种是二阶有限差分法，第二种是高精度傅里叶谱方法。由于 $β$ 仅是边值问题中的一个参数，原则上可适用于任意 $β>0$ 的情况。此外，第 $n$ 大特征值的极限分布同样可被计算。我们的方法已集成于 Julia 软件包 TracyWidomBeta.jl 中。