Limit laws of transient excited random walks on integers

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.Comment: 27 page

Excited Random Walks with Non-nearest Neighbor Steps

“…While self-interacting random walks are typically very difficult to study, much is known about one-dimensional nearest neighbor excited random walks. Under mild assumptions there are explicit criteria for recurrence/transience, ballisticity, and a characterization of the limiting distributions of the excited random walks [Zer05,BS08a,BS08b,KZ08,KM11]. In the current paper, we study a model for excited random walks that allows for jumps that can even be unbounded.…”

“…In particular, multi-dimensional excited random walks with local drifts contained in a fixed half-plane are known to be transient with a non-zero limiting speed and CLT type limiting distributions [BR07,MPRV12]. In contrast, for one-dimensional excited random walks it is known that the walks can be transient with sublinear speed and with non-Gaussian limiting distributions [BS08a,BS08b,KM11].…”

“…That is, the excited random walk is recurrent if δ ∈ [−1, 1], transient to the right if δ > 1 and transient to the left if δ < −1. Moreover, in the case of finitely many cookies per site, the parameter δ gives a criterion for when the limiting speed of the excited random walk is non-zero [BS08a,KZ08] and also determines the limiting distribution for the excited random walk [BS08b,KM11,KZ08,DK12].…”

Extreme slowdowns for one-dimensional excited random walks

“…If δ > 1 so that the excited random walk is transient to the right, then the following limiting distributions are known [2,11,12].…”

“…The limiting distributions in the borderline cases δ ∈ {2, 4} [11,13] and for recurrent ERW [6,7] are also known. In these cases as well the parameter δ determines both the scaling needed and the type of the limiting distribution.…”