Log-periodicity can appear in a non-Markovian random walk even if there is perfect memory of its history

Abstract

The exactly solvable Elephant Random Walk (ERW) model introduced by Schütz and Trimper 15 years ago stimulated research that led to many new insights and advances in understanding anomalous diffusion. Such models have two distinct ingredients: i) long-range —possibly complete— memory of the past behavior and ii) a decision-making rule that makes use of the memory. These models are memory-neutral: the decision-making rule does not distinguish between short-term (or recent) memories and long-term (or old) memories. Here we relax the condition of memory neutrality, so that memory and decision-making become interconnected. We investigate the diffusive properties of random walks that evolve according to memory-biased decision processes and find remarkably rich phase diagrams, including a phase of log-periodic superdiffusion that may be associated with old memory and negative feedback regulating mechanisms. Our results overturn the conventional wisdom concerning the origin of log-periodicity in non-Markovian models. All previously known non-Markovian random walk models that exhibit log-periodicities in their behavior have incomplete (or damaged) memory of their history. Here we show that log-periodicity can appear even if the memory is complete, so long as there is a memory bias.