Transience, recurrence and the speed of a random walk in a site-based feedback environment

We study a random walk on Z which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, p and q. R consecutive right jumps from a site in the q-mode are required to switch it to the p-mode, and L consecutive left jumps from a site in the p-mode are required to switch it to the q-mode. From a site in the p-mode the walk jumps right with probability p and left with probability 1− p, while from a site in the q-mode these probabilities are q and 1 − q. We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters α = α(p, q, R, L). For α > 1/2 the walk is transient to +∞ for any initial environment, whereas for α < 1/2 the walk is transient to−∞ for any initial environment. In the critical case, α = 1/2, the situation is more complicated and the behavior of the walk depends on the initial environment. Nevertheless, we are able to give a characterization of transience/recurrence in many instances, including when either R = 1 or L = 1 and when R = L = 2. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed.