Excited random walk with periodic cookies

Abstract

In this paper we consider an excited random walk on $\mathbb{Z}$ in identically piled periodic environment. This is a discrete time process on $\mathbb{Z}$ defined by parameters $(p_1,\dots p_M)\in [0,1{]}^M$ for some positive integer $M$, where the walker upon the $i$-th visit to $z\in \mathbb{Z}$ moves to $z+1$ with probability $p_{i(\mathrm{mod}M)}$, and moves to $z-1$ with probability $1-p_{i(\mathrm{mod}M)}$. We give an explicit formula in terms of the parameters $(p_1,\dots ,p_M)$ which determines whether the walk is recurrent, transient to the left, or transient to the right. In particular, in the case that $\frac{1}{M}{\sum }_{i=1}^M{p}_i=\frac{1}{2}$ all behaviors are possible, and may depend on the order of the $p_i$. Our framework allows us to reprove some known results on ERW with no additional effort.

31 pages, no figures. Incorporating several corrections

References in corpus (2)

• Limit laws of transient excited random walks on integers
• Excited random walks: results, methods, open problems

Cited by in corpus (3)

• Solution manifold and Its Statistical Applications
• Random walks with local memory
• Recurrence of horizontal-vertical walks