The shape of a memorised random walk

TL;DR

This paper models how animals' memory of visited locations affects their foraging paths, showing that longer-lasting memories result in less elongated mental maps, with exact formulas and simulations supporting these findings.

Contribution

It introduces an analytic framework linking memory decay functions to the shape of memorised random walks, providing exact expressions and simulation validation.

Findings

Slower memory decay leads to less elongated walks.

Exact analytic expressions relate memory kernel to walk shape.

Simulations confirm theoretical predictions.

Abstract

We view random walks as the paths of foraging animals, perhaps searching for food or avoiding predators while forming a mental map of their surroundings. The formation of such maps requires them to memorise the locations they have visited. We model memory using a kernel, proportional to the number of locations recalled as a function of the time since they were first observed. We give exact analytic expressions relating the elongation of the memorised walk to the structure of the memory kernel, and confirm these by simulation. We find that more slowly decaying memories lead to less elongated mental maps.