A vague memory can affect first-return time

Abstract

First-return time is an important property for the return of particles or walkers to a start point. Recursive walks, which may be related to first-return time, are found in both random walk models and memory-based walk models. Achieving a balance between recursive walks and diffusive movements is a crucial but difficult modeling problem. Here, starting with a simple Brownian-walk model, I investigated how vague memorized information influences the first-return times of a walker. In the proposed model, the walker memorizes recently visited positions and recalls the direction in which it previously moved when returning to those positions. Using the recalled information, the walker then moves in the opposite direction to that previously traveled. In addition, the walker considers its recent experience and modifies its directional rules, i.e., memorized information, when the rule disturbs the recent flow of its movement. Thus, the proposed model effectively produces recursive walks in which a walker returns to a start point while demonstrating diffusive movements.