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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 [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 1 It is already known that excited random walk in Z 1 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 8

Topics:
Perturbed random walk, transience

Year: 2003

OAI identifier:
oai:CiteSeerX.psu:10.1.1.172.8444

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