The bold/timorous walker on the trek from home

TL;DR

This paper introduces a one-dimensional random walk model with memory, where the walker's behavior at maximum distance from home varies between bold and timorous, affecting its diffusion properties.

Contribution

It presents a novel mathematical analysis of a memory-dependent random walk, revealing continuous scaling behavior from subdiffusive to superdiffusive regimes.

Findings

Scaling behavior varies continuously from subdiffusive to superdiffusive.

New mathematical approach determines the scaling exponents.

Behavior depends on the probability of moving forward at maximum distance.

Abstract

We study a one-dimensional random walk with memory. The behavior of the walker is modified with respect to the simple symmetric random walk (SSRW) only when he is at the maximum distance ever reached from his starting point (home). In this case, having the choice to move farther or to move closer, he decides with different probabilities. If the probability of a forward step is higher then the probability of a backward step, the walker is bold, otherwise he is timorous. We investigate the asymptotic properties of this bold/timorous random walk (BTRW) showing that the scaling behavior vary continuously from subdiffusive (timorous) to superdiffusive (bold). The scaling exponents are fully determined with a new mathematical approach.