Universal shocks in random matrix theory.

We link the appearance of universal kernels in random matrix ensembles to the phenomenon of shock formation in some fluid dynamical equations. Such equations are derived from Dyson's random walks after a proper rescaling of the time. In the case of the gaussian unitary ensemble, on which we focus in this paper, we show that the characteristics polynomials and their inverse evolve according to a viscid Burgers equation with an effective "spectral viscosity" ν(s)=1/2N, where N is the size of the matrices. We relate the edge of the spectrum of eigenvalues to the shock that naturally appears in the Burgers equation for appropriate initial conditions, thereby suggesting a connection between the well-known microscopic universality of random matrix theory and the universal properties of the solution of the Burgers equation in the vicinity of a shock.