Limit laws of transient excited random walks on integers

TL;DR

This paper investigates the limiting behavior of excited random walks on integers with varying drift conditions, revealing stable law limits for certain drift ranges and extending previous results to more general cookie configurations.

Contribution

It extends the understanding of ERWs by characterizing their limit laws for delta in (2,4], including cases with both positive and negative cookies, beyond prior non-negativity assumptions.

Findings

For delta in (2,4], ERWs follow a stable law with parameter delta/2.

When delta > 4, ERWs obey the Central Limit Theorem.

The results generalize previous work to settings with both positive and negative cookies.

Abstract

We consider excited random walks (ERWs) on integers with a bounded number of i.i.d. cookies per site without the non-negativity assumption on the drifts induced by the cookies. Kosygina and Zerner [KZ08] have shown that when the total expected drift per site, delta, is larger than 1 then ERW is transient to the right and, moreover, for delta>4 under the averaged measure it obeys the Central Limit Theorem. We show that when delta in (2,4] the limiting behavior of an appropriately centered and scaled excited random walk under the averaged measure is described by a strictly stable law with parameter delta/2. Our method also extends the results obtained by Basdevant and Singh [BS08b] for delta in (1,2] under the non-negativity assumption to the setting which allows both positive and negative cookies.