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Excited Random Walks with Non-nearest Neighbor Steps

Author

Listed:
  • Burgess Davis

    (Purdue University)

  • Jonathon Peterson

    (Purdue University)

Abstract

Let W be an integer-valued random variable satisfying $$E[W] =: \delta \ge 0$$ E [ W ] = : δ ≥ 0 and $$P(W 0$$ P ( W 0 , and consider a self-interacting random walk that behaves like a simple symmetric random walk with the exception that on the first visit to any integer $$x\in \mathbb Z$$ x ∈ Z , the size of the next step is an independent random variable with the same distribution as W. We show that this self-interacting random walk is recurrent if $$\delta \le 1$$ δ ≤ 1 and transient if $$\delta >1$$ δ > 1 . This is a special case of our main result which concerns the recurrence and transience of excited random walks (or cookie random walks) with non-nearest neighbor jumps.

Suggested Citation

  • Burgess Davis & Jonathon Peterson, 2017. "Excited Random Walks with Non-nearest Neighbor Steps," Journal of Theoretical Probability, Springer, vol. 30(4), pages 1255-1284, December.
    • Handle: RePEc:spr:jotpro:v:30:y:2017:i:4:d:10.1007_s10959-016-0697-1
    DOI: 10.1007/s10959-016-0697-1
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