For large beta the TW density takes the form exp(-beta Phi(a)) with Phi(a) obtained as the solution of a Painleve II equation via saddle-point analysis of the stochastic Airy operator.
Wishart,The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A, 32 (1928)
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The Tracy-Widom distribution at large Dyson index
For large beta the TW density takes the form exp(-beta Phi(a)) with Phi(a) obtained as the solution of a Painleve II equation via saddle-point analysis of the stochastic Airy operator.