Mutually excited random walks

TL;DR

This paper studies a model of two mutually influencing random walks on integers, proving positive speed for certain parameters, providing an algorithm to approximate speed, and demonstrating non-monotonic behavior in their velocities.

Contribution

It introduces a novel model of mutually excited random walks, proves positive speed in a specific parameter range, and develops an algorithm to approximate and analyze their non-monotonic velocities.

Findings

Proves positive speed for $1/2<p<1$.

Provides an algorithm to approximate the walks' speed.

Shows the non-monotonicity of the speed with respect to parameter p.

Abstract

Consider two random walks on $\mathbb{Z}$. The transition probabilities of each walk is dependent on trajectory of the other walker i.e. a drift $p>1/2$ is obtained in a position the other walker visited twice or more. This simple model has a speed which is, according to simulations, not monotone in $p$, without apparent "trap" behaviour. In this paper we prove the process has positive speed for $1/2<p<1$, and present a deterministic algorithm to approximate the speed and show the non-monotonicity.