Positively and negatively excited random walks on integers, with branching processes

TL;DR

This paper studies excited random walks on integers with bounded cookies, extending recurrence, transience, and speed criteria, and proves an annealed central limit theorem using branching process techniques and renewal structures.

Contribution

It generalizes existing criteria for recurrence, transience, and positive speed to cases with both positive and negative drifts, and establishes a central limit theorem for such walks.

Findings

Extended criteria for recurrence and transience.

Criteria for positivity of speed.

Proved annealed central limit theorem.

Abstract

We consider excited random walks on the integers with a bounded number of i.i.d. cookies per site which may induce drifts both to the left and to the right. We extend the criteria for recurrence and transience by M. Zerner and for positivity of speed by A.-L. Basdevant and A. Singh to this case and also prove an annealed central limit theorem. The proofs are based on results from the literature concerning branching processes with migration and make use of a certain renewal structure.