A random walk on Z d 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 d is transient iff d> 1. 1. Excited Random Walk A random walk on Z d 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  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 1 It is already known that excited random walk in Z 1 is recurrent, indeed, a great deal more is known about it . 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 8
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