One class of random walks with infinite memory, so-called elephant random walks, are simple models describing anomalous diffusion. We present a surprising connection between these models and bond percolation on random recursive trees. We use a coupling between the two models to translate results from elephant random walks to the percolation process. We calculate, besides other quantities, exact expressions for the first and the second moment of the root cluster size and of the number of nodes in child clusters of the first generation. We further introduce another model, the skew elephant random walk, and calculate the first and second moment of this process.
Dynamical properties of self-propelled particles obeying a bounded confidence rule are investigated by means of kinetic theory and agent-based simulations. While memory effects are observed in disordered systems, we show that they also occur in active matter systems. In particular, we find that the system exhibits a giant Kovacs-like memory effect that is much larger than predicted by a generic linear theory. Based on a separation of time scales we develop a nonlinear theory to explain this effect. We apply this theory to driven granular gases and propose further applications to spin glasses.
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