Barriers, trapping times and overlaps between local minima in the dynamics of the disordered Ising $p$-spin Model

We study the low temperature out of equilibrium Monte Carlo dynamics of the disordered Ising $p$-spin Model with $p=3$ and a small number of spin variables. We focus on sequences of configurations that are stable against single spin flips obtained by instantaneous gradient descent from persistent ones. We analyze the statistics of energy gaps, energy barriers and trapping times on sub-sequences such that the overlap between consecutive configurations does not overcome a threshold. We compare our results to the predictions of various trap models finding the best agreement with the step model when the $p$-spin configurations are constrained to be uncorrelated.