The full edge scaling limit of singular values of GL_N(C) Brownian motion solves an infinite log-interacting SDE system, and the limiting reverse characteristic polynomial satisfies a nonlinear multiplicative-noise SPDE.
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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.
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$\mathsf{GL}_N(\mathbb{C})$ Brownian motion and stochastic PDE on entire functions
The full edge scaling limit of singular values of GL_N(C) Brownian motion solves an infinite log-interacting SDE system, and the limiting reverse characteristic polynomial satisfies a nonlinear multiplicative-noise SPDE.
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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.