Computing the Tracy–Widom Distribution for Arbitrary $\beta>0$

Thomas Trogdon, Yiting Zhang

Department of Applied Mathematics, University of Washington, Seattle, Washington, USA

Abstract: 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 in his Ph.D. Thesis (2011). 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.

Keywords: numerical differential equation, Tracy–Widom distribution, Fourier transformation.

MSC: 65M06, 60B20, 60H25

Received: April 19, 2023; in final form January 3, 2024; Published online January 13, 2024

Language: English

DOI: 10.3842/SIGMA.2024.005

ArXiv: 2304.04951