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Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 005, 26 pages      arXiv:2304.04951      https://doi.org/10.3842/SIGMA.2024.005
Contribution to the Special Issue on Evolution Equations, Exactly Solvable Models and Random Matrices in honor of Alexander Its' 70th birthday

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

Thomas Trogdon and Yiting Zhang
Department of Applied Mathematics, University of Washington, Seattle, Washington, USA

Received April 19, 2023, in final form January 03, 2024; Published online January 13, 2024

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.

Key words: numerical differential equation; Tracy-Widom distribution; Fourier transformation.

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