Computing the Tracy-Widom distribution for arbitrary β>0
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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 β is simply a parameter in the boundary-value problem, any β> 0 can be used, in principle. The limiting distribution of the nth largest eigenvalue can also be computed. Our methods are available in the Julia package TracyWidomBeta.jl.
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