Functional limit laws for recurrent excited random walks with periodic cookie stacks

TL;DR

This paper establishes functional limit theorems for one-dimensional recurrent excited random walks with periodic cookie stacks, showing convergence to a Brownian motion perturbed at its extrema, extending previous bounded-cookie results.

Contribution

It extends the functional limit theorems for excited random walks to models with infinitely many cookies per site, covering a broader range of parameters and demonstrating convergence to BMPE.

Findings

Rescaled ERWs converge to BMPE in the recurrent regime.

The results apply to models with infinitely many cookies per site.

The scaling limits encompass a wider parameter range than previous models.

Abstract

We consider one-dimensional excited random walks (ERWs) with periodic cookie stacks in the recurrent regime. We prove functional limit theorems for these walks which extend the previous results of D. Dolgopyat and E. Kosygina for excited random walks with "boundedly many cookies per site." In particular, in the non-boundary recurrent case the rescaled excited random walk converges in the standard Skorokhod topology to a Brownian motion perturbed at its extrema (BMPE). While BMPE is a natural limiting object for excited random walks with boundedly many cookies per site, it is far from obvious why the same should be true for our model which allows for infinitely many "cookies" at each site. Moreover, a BMPE has two parameters $\alpha,\beta<1$ and the scaling limits in this paper cover a larger variety of choices for $\alpha$ and $\beta$ than can be obtained for ERWs with boundedly many cookies per site.