  In this letter we present an analytic method for calculating the transition
probability between two random Gaussian matrices with given eigenvalue spectra
in the context of Dyson Brownian motion. We show that in the Coulomb gas
language, in large $N$ limit, memory of the initial state is preserved in the
form of a universal linear potential acting on the eigenvalues. We compute the
likelihood of any given transition as a function of time, showing that as
memory of the initial state is lost, transition probabilities converge to those
of the static ensemble.
