Universal shocks in random matrix theory

Understand

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.