Diffusivity of a random walk on random walks

TL;DR

This paper analyzes a constrained random walk on a lattice, demonstrating that its diffusivity is reduced by a specific variance factor compared to an unconstrained simple random walk.

Contribution

It introduces a new model of a constrained random walk and derives an explicit formula for its variance reduction factor.

Findings

The constrained walk has a variance factor of 2/(K+2).

The walk's diffusivity is slower than the classical simple random walk.

Explicit variance reduction formula derived for the constrained walk.

Abstract

We consider a random walk $(Z^{(1)}_n, ..., Z^{(K+1)}_n) \in \mathbb{Z}^{K+1}$ with the constraint that each coordinate of the walk is at distance one from the following one. In this paper, we show that this random walk is slowed down by a variance factor $\sigma_K^2 = \frac{2}{K+2}$ with respect to the case of the classical simple random walk without constraint.