  We theoretically study propagating correlation fronts in non-interacting
fermions on a one-dimensional lattice starting from an alternating state, where
the fermions occupy every other site. We find that, in the long-time asymptotic
regime, all the moments of dynamical fluctuations around the correlation fronts
are described by the universal correlation functions of Gaussian orthogonal and
symplectic random matrices at the soft edge. Our finding here sheds light on a
hitherto unknown connection between random matrix theory and correlation
propagation in quantum dynamics.
