On the non-equivalence of two standard random walks

TL;DR

This paper compares two common models of random walks on lattices, showing they behave similarly under symmetry but differ significantly when external bias is applied.

Contribution

It demonstrates that the discrete-time Polya walk and the continuous-time Montroll-Weiss walk are not equivalent under biased conditions, highlighting the importance of model choice.

Findings

Symmetric walks exhibit identical long-term behavior in both models.

Biased walks show markedly different behaviors between the two models.

The difference impacts the modeling of real-world processes with external forces.

Abstract

We focus on two models of nearest-neighbour random walks on d-dimensional regular hyper-cubic lattices that are usually assumed to be identical - the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll-Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed random variables with mean 1. We show that while for symmetric random walks both models indeed lead to identical behaviour in the long time limit, when there is an external bias they lead to markedly different behaviour.