  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 letter, we
show that the orthogonal polynomials, and their Cauchy transforms, evolve
according to a viscid Burgers equation with an effective "spectral viscosity"
$\nu_s=1/2N$, 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 obtaining a new
perspective on universality.
