Tampered Memory Elephant Random Walk on One-Dimensional Integer Lattice

Abstract

One of the outstanding questions in the theory of elephant random walks as observed by Gut and Stadtm\"uller (2023), is to determine how much memory is needed for a phase transition between the diffusive, critical and superdiffusive regimes to persist. To investigate this memory breakpoint, we introduce the tampered memory elephant random walk, in which the memory is partitioned into two disjoint sets $D_n$ and $D_n^c$, which may be deterministic or random. On $D_n^c$ the dynamics is the same as an elephant random walk, while on $D_n$ the increments are replaced by independent innovations. The resulting walk is thus driven by two competing components: elephant random walk and an independent simple random walk corresponding to the innovations. We first establish a law of large numbers when the increasing collections $\{D_n\}_{n \ge 1}$ and $\{D^c_n\}_{n \ge 1}$ have a renewal structure with exponential moments. We then identify a sharp threshold that governs the persistence of the phase transition for deterministic memory partitions. We show that if $\{D_n\}_{n \ge 1}$ is non-random increasing collection with increasing complement $\{D^c_n\}_{n \ge 1}$ such that $\lim_{n \to \infty} \frac{\lvert D^c_n\rvert}{n} >1/2$, then a phase transition into diffusive, critical and superdiffusive regimes persists, whereas for $\lim_{n \to \infty} \frac{\lvert D^c_n \rvert}{n}<1/2$, there is only the diffusive regime with $\mathcal{O}(\sqrt{n})$. The case of $\lim_{n\to \infty}\frac{\lvert D^c_n \rvert}{n}=1/2$ is also characterised. Thus, one-half emerges as the sharp breakpoint for the persistence of anomalous diffusion in this competitive setting. We conjecture that the same threshold governs the case when $\{D_n\}_{n \ge 1}$ is random. Our proofs rely on stochastic approximation applied to the two dependent competing components of the walk, representing the retained memory and the innovations.