Diffusivity of a random walk on random walks

Abstract

We consider a random walk $(Z_n^{(1)},...,Z_n^{(K+1)})\in {\mathbb{Z}}^{K+1}$ 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.