Excited Random Walk

A random walk on Z is excited if the first time it visits a vertex there is a bias in one direction, but on subsequent visits to that vertex the walker picks a neighbor uniformly at random. We show that excited random walk on Z is transient iff d > 1. 1. Excited Random Walk A random walk on Z is excited (with bias ε/d) if the first time it visits a vertex it steps right with probability (1 + ε)/(2d) (ε > 0), left with probability (1 − ε)/(2d), and in other directions with probability 1/(2d), while on subsequent visits to that vertex the walker picks a neighbor uniformly at random. This model was studied heavily in the framework of perturbing 1-dimensional Brownian motion, see for instance [5, 14] and reference therein. Excited random walk falls into the notorious wide category of self-interacting random walks, such as reinforced random walk, or self-avoiding walks. These models are difficult to analyze in general. The reader should consult [4, 11, 16, 15, 1], and especially the survey paper [13] for examples. Simple coupling and an additional neat observation allow us to prove that excited random walk is recurrent only in dimension 1. The proof uses and studies a special set of points (“tan points”) for the simple random walk. 2. Recurrence in Z It is already known that excited random walk in Z is recurrent, indeed, a great deal more is known about it [6]. But for the reader’s convenience we provide a short proof. On the first visit to a vertex there is probability p > 1/2 of going right and 1− p of going left, while on subsequent visits the probabilities are 1/2. Suppose that the walker is at x > 0 for the first time, and that all vertices between 0 and x have been visited. The probability that the walker goes to x+ 1 before going to 0 is p+ (1− p)(1− 2/(x+ 1)) = 1− 2(1− p)/(x+ 1). Multiplying over the x’s, we see that the random walk returns to 0 with probability 1. 86 Excited random walk 87 3. Transience in Z The simple random walk (SRW) in Z visits about n/ log n points by time n, and if the excited random walk (ERW) gets pushed to the right n/ log n times, it would very quickly depart its start location and never return. But it is not clear what effect that the perturbations have on the number of visited points, and it is not obvious that the excited random walk will visit n/ log n distinct points by time n. To lower bound the number of points that the excited random walk visits, we couple it with the SRW in the straightforward way, and count the number of “tan points” visited by the SRW. We define the coupling as follows: if the SRW moves up, down, or right, then so does the ERW. If the SRW moves left, then the ERW moves left if it is at a previously visited point, and if the ERW is at a new point, it moves either left or right with suitable probabilities. At all times, the y-coordinates of the SRW and ERW are identical. To explain the concept of a “tan point”, we imagine that the simple random walker leaves behind an opaque trail, and that the sun is shining from infinitely far away in the positive x-direction. If the SRW visits a point (x, y) such that no point (x, y) with x > x has been visited, then the sun shines upon (x, y), and this point becomes tanned. Formally, we define a tan point for the SRW to be a vertex (x, y) that is visited by the SRW before any point of the form (x, y) with x > x. If the sun shines upon the simple random walker the first time it is at (x, y), it is straightforward to check that ERW is at a new point. We will show that with high probability there are many tan points (so the ERW visits many new points), and that this implies that the ERW is transient. The probability that a point (x, y) will be tan follows from some enumerative work of BousquetMélou and Schaeffer on random walks in the slit plane [3]. Lemma 1. Let r and θ be the polar coordinates of the point (x, y), i.e. r ≥ 0, 0 ≤ θ < 2π, x = r cos θ, and y = r sin θ. Then Pr[(x, y) is tan] = (1 + o(1)) √ 1 + √ 2 2π sin(θ/2) √ r , (1) where the o(1) term goes to 0 as r tends to ∞. This equation does not explicitly appear in [3], but all the real work that goes into proving it is in [3]. In the interest of completeness, we explain how this equation follows from explicit results in [3]: Proof. Let an be the number of walks of length n that start from (0, 0), and avoid the nonnegative real axis at all subsequent times, and let px,y,n denote the probability that a random such walk ends at the point (x, y). By reversibility of the random walks,