Transience, recurrence and the speed of a random walk in a site-based feedback environment

Ross G. Pinsky, Nicholas F. Travers

Research output: Contribution to journalArticlepeer-review

Abstract

We study a random walk on Z which evolves in a dynamic environment determined by its own trajectory. Sites flip back and forth between two modes, p and q. R consecutive right jumps from a site in the q-mode are required to switch it to the p-mode, and L consecutive left jumps from a site in the p-mode are required to switch it to the q-mode. From a site in the p-mode the walk jumps right with probability p and left with probability 1 - p, while from a site in the q-mode these probabilities are q and 1 - q. We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters α= α(p, q, R, L). For α> 1 / 2 the walk is transient to + ∞ for any initial environment, whereas for α< 1 / 2 the walk is transient to - ∞ for any initial environment. In the critical case, α= 1 / 2 , the situation is more complicated and the behavior of the walk depends on the initial environment. Nevertheless, we are able to give a characterization of transience/recurrence in many instances, including when either R= 1 or L= 1 and when R= L= 2. In the noncritical case, we also show that the walk has positive speed, and in some situations are able to give an explicit formula for this speed.

Original languageEnglish
Pages (from-to)917-978
Number of pages62
JournalProbability Theory and Related Fields
Volume167
Issue number3-4
DOIs
StatePublished - 1 Apr 2017

Keywords

  • Ballistic
  • Recurrence
  • Self-interacting random walks
  • Transience

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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