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.
Highlights
We consider nearest-neighbor random walks on the one-dimensional integer lattice in an i.i.d. cookie environment with a uniformly bounded number of cookies per site
The piles of cookies represent the transition probabilities of the random walker: upon each visit to a site the walker consumes the topmost cookie from the pile at that site and makes a unit step to the right or to the left with probabilities prescribed by that cookie
This can be done by making a time diagram of up and down movements of an ant traversing the tree in preorder: the ant starts at the root, always chooses to go up and to the left whenever possible, never returns to an edge that was already crossed in both directions, and finishes the journey at the root (Figure 2, (III))
Summary
We consider nearest-neighbor random walks on the one-dimensional integer lattice in an i.i.d. cookie environment with a uniformly bounded number of cookies per site. I≥1 i=1 which we shall call the average total drift per site It plays a key role in the classification of the asymptotic behavior of the walk as shown by the three main theorems of this paper. Our first result extends [Zer, Theorem 12] about recurrence and transience for non-negative cookies to i.i.d. environments with a bounded number of positive and negative cookies per site. Throughout the paper we shall denote various constants by ci ∈ (0, ∞), i ≥ 1
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