An Overview of the Balanced Excited Random Walk
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Date
2021
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Volume Title
Publisher
Birkhauser
Abstract
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in 2011, is defined as a discrete time stochastic process in ℤd, depending on two integer parameters 1 ≤ d1, d2 ≤ d, which whenever it is at a site x∈ ℤd at time n, it jumps to x ± ei with uniform probability, where e1, …, ed are the canonical vectors, for 1 ≤ i ≤ d1, if the site x was visited for the first time at time n, while it jumps to x ± ei with uniform probability, for 1 + d − d2 ≤ i ≤ d, if the site x was already visited before time n. Here we give an overview of this model when d1 + d2 = d and introduce and study the cases when d1 + d2 > d. In particular, we prove that for all the cases d ≥ 5 and most cases d = 4, the balanced excited random walk is transient.
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Keywords
Excited random walk, Transience