Impatient random walk

TL;DR

This paper introduces a novel type of random walk called 'impatient' random walk, which features a unique edge reinforcement mechanism, and explores its fundamental properties like recurrence, transience, and speed.

Contribution

It proposes a new class of random walks with distinct edge reinforcement rules and analyzes their basic probabilistic behaviors.

Findings

Impatient random walks exhibit different recurrence properties.

The speed of impatient random walks varies with reinforcement rules.

Transience and null/positive recurrence depend on specific parameters.

Abstract

We introduce a new type of random walk where the definition of edge reinforcement is very different from the one in the reinforced random walk models studied so far, and investigate its basic properties, such as null/positive recurrence, transience, and speed. Two basic cases will be dubbed "impatient" and"ageing" random walks.