  We study the return probability for the Anderson model on the random regular
graph and give evidence of the existence of two distinct phases: a fully
ergodic and nonergodic one. In the ergodic phase, the return probability decays
polynomially with time with oscillations, being the attribute of the
Wigner-Dyson-like behavior, while in the nonergodic phase the decay follows a
stretched exponential decay.We give a phenomenological interpretation of the
stretched exponential decay in terms of a classical random walker. Furthermore,
comparing typical and mean values of the return probability, we show how to
differentiate an ergodic phase from a nonergodic one. We benchmark this method
first in two random matrix models, the power-law random banded matrices, and
the Rosenzweig-Porter matrices, which host both phases. Second, we apply this
method to the Anderson model on the random regular graph to give further
evidence of the existence of the two phases.
