Séminaire exceptionnel : Maciej Nowak


11:00 - 12:15

LPTMS, salle 201, 2ème étage, Bât 100, Campus d'Orsay
15 Rue Georges Clemenceau, Orsay, 91405
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« Burgers dynamics  in hermitian and nonhermitian RMM ».

Maciej Nowak, Jagellonian University, Cracovie.


We obtain several classes of non-linear partial differential equations for various random matrix ensembles undergoing Brownian type of random walk. These equations for spectral flow of eigenvalues as a function of dynamical parameter (« time ») are exact for any finite size N of the random matrix ensemble and resemble viscid Burgers-like equations known in the theory of turbulence. In the limit of infinite size of the matrix, these equations reduce complex inviscid Burgers equations, proposed originally by Voiculescu in the context of free processes. We identify spectral shock waves for these equations in the limit of the infinite size of the matrix, and then we solve exact, finite N nonlinear equations in the vicinity of the shocks, obtaining in this way universal, microscopic scalings equivalent to Airy, Bessel and cuspoid kernels. Finally, we show that similar but  hidden Burgers-like structures appear (surprisingly)  also in nonhermitian random matrix models, e.g. in the Ginibre-Girko ensemble, where they govern the concurent  evolution of both eigenvalues and eigenvectors.

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