Excited against the tide: A random walk with competing drifts

In this paper we study what might be called an excited random walk with drift (ERWD), where the random walker has a drift β d in the positive direction of the first component each time the walker visits a new site, and a drift μ d in the negative direction of the first component on subsequent visits. Models of this type with drift for a fixed finite number of visits to a site have been studied by Zerner and others, usually in 1 dimension, see for example [1, 2, 3, 17]. They are generalisations of the excited random walk introduced in [4]. It is known that excited random walk has ballistic behaviour when d ≥ 2 in [4, 13, 14], while there is no ballistic behaviour (when β < 1) in one dimension [7]. Laws of large numbers and central limit theorems can be obtained for d ≥ 2 using renewal techniques (see for example [16], [17], [5]). Intuitively, the velocity appearing in the laws of large numbers should be increasing in the excitement parameter, β. This has been proved in dimensions d ≥ 9 [11] using a perturbative expansion developed in [10], but the problem remains open for d < 9. Using the same expansion approach it is possible [12] to prove monotonicity for the first coordinate of the speed of random walk in a partially random i.i.d. environment, in the special case where at each vertex, either the left or the right step is not available. We use the same argument in this paper, together with the law of large numbers provided by [6, Theorem 1.4] to prove that for d ≥ 12, for each fixed μ > 0 the speed in the direction of the positive first coordinate is continuous and strictly increasing in β. We give an easy coupling argument showing that the speed is negative when β is sufficiently small (depending on μ and d) and show that when d ≥ 9 the speed is positive when β is sufficiently large. We conclude that when d ≥ 12, for each μ ≥ 0 there is exactly one value β0(μ, d) of β for which the speed is zero.