Computing the Tracy-Widom Distribution for Arbitrary 𝛽 > 0
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-di...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212118 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Computing the Tracy-Widom Distribution for Arbitrary 𝛽 > 0. Thomas Trogdon and Yiting Zhang. SIGMA 20 (2024), 005, 26 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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 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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| ISSN: | 1815-0659 |