Dynamical properties of the Anderson model on the random regular graph

We study the dynamics of system initially prepared in a highly-excited non-stationary state (wave packet) confined in space and containing a fixed fraction of all eigenstates. Our focus is on the probability for finding the initial state later in time, the so-called return probability. We show that comparing typical and mean values of the return probability one can differentiate between ergodic and multifractal dynamical phases in some random matrix model (power-law random banded matrices and Rosenzweig-Porter matrices), and in the Anderson model on a random regular graph. For the latter we show that the decay of the return probability follows a stretched exponential law, whose exponent changes continuously with the disorder.