Generalized Excited Random Walks under Bernoulli excitations

Abstract

We study a variant of the Generalized Excited Random Walk (GERW) on $\mathbb{Z}^d$ introduced by Menshikov, Popov, Ram\'irez and Vachkovskaia in [Ann. Probab. 40 (5), 2012]. It consists of a particular version of the model studied in [arXiv preprint arXiv:2211.05715, 2022] where excitation may or may not occur according to a time-dependent probability. Specifically, given $\{p_n\}_{n \ge 1}$, $p_n \in (0, 1]$ for all $n \ge 1$, whenever the process visits a site at time $n$ for the first time, with probability $p_n$ it gains a drift in a fixed direction. Otherwise, it behaves as a $d$-martingale with zero-mean vector. We refer to the model as $p_n$-GERW. Assuming bounded jumps and $p_n \approx n^{-\beta}$, we show a series of results for the $p_n$-\Name{} depending on the value of $\beta$ and on the dimension $d$. Specifically, for every $\beta\in(0,1]$ and $d=2$ or $d>h(\beta)$, with $h$ a decreasing function of $\beta$, we prove a SLLN for the range, while for $\beta<1/2$ we prove a sub-ballistic SLLN for the process whenever the SLLN for the range holds. We also study the $p_n$-\Name{} under diffusive scaling, and we obtain a Functional Central Limit Theorem for $\beta > 1/2$ and $d\geq 2$, or $\beta=1/2$ and $d=2$. Finally, for $\beta=1/2$ and $d \ge 11$ we show that the diffusively rescaled $p_n$-\Name{} converges in distribution to a Brownian Motion plus a multiple of the square root of time.