Scaling limits of recurrent excited random walks on integers

Dmitry Dolgopyat,Elena Kosygina

doi:10.1214/ecp.v17-2213

Abstract

We describe scaling limits of recurrent excited random walks (ERWs) on $\mathbb{Z}$ in i.i.d. cookie environments with a bounded number of cookies per site. We allow both positive and negative excitations. It is known that ERW is recurrent if and only if the expected total drift per site, $\delta$, belongs to the interval $[-1,1]$. We show that if $|\delta|<1$ then the diffusively scaled ERW under the averaged measure converges to a $(\delta,-\delta)$-perturbed Brownian motion. In the boundary case, $|\delta|=1$, the space scaling has to be adjusted by an extra logarithmic term, and the weak limit of ERW happens to be a constant multiple of the running maximum of the standard Brownian motion, a transient process.

Highlights

Introduction and main resultsGiven an arbitrary positive integer M let ΩM := ((ωz(i))i∈N)z∈Z | ωz(i) ∈ [0, 1], for i ∈ {1, 2, . . . , M } and ωz(i) = 1/2, for i > M, z ∈ Z .An element of ΩM is called a cookie environment
excited random walks (ERWs) on Z in a non-negative cookie environment and its natural extension to Zd were considered previously by many authors
In this paper we study scaling limits of recurrent ERW under P0

Summary

Introduction and main results

ERW on Z in a non-negative cookie environment and its natural extension to Zd (when there is a direction in Rd such that the projection of a drift induced by every cookie on that direction is non-negative) were considered previously by many authors (see, for example, [4], [22], [23], [2], [3], [17] [5], [9], [16], and references therein) Our model allows both positive and negative cookies but restricts their number per site to M. The functional limit theorem for recurrent ERW in stationary ergodic non-negative cookie environments on strips Z × (Z/LZ), L ∈ N, under the quenched measure was proven in [9]. The stated result comes from the fact that with an overwhelming prob√ability the maximum amount of “ba√cktracking” of Xj from Sj for j ≤ [T n] is of order n, which is negligible on the scale n log n (see Lemma 10)

Notation and preliminaries

Non-boundary case: two useful lemmas

Non-boundary case

Boundary case