Transience, recurrence and the speed of a random walk in a site-based feedback environment
成果类型：
Article
署名作者：
Pinsky, Ross G.; Travers, Nicholas F.
署名单位：
Technion Israel Institute of Technology; Indiana University System; Indiana University Bloomington
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-016-0695-3
发表日期：
2017
页码：
917-978
关键词：
摘要：
We study a random walk on 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 , while from a site in the q-mode these probabilities are q and . We prove a sharp cutoff for right/left transience of the random walk in terms of an explicit function of the parameters . For the walk is transient to for any initial environment, whereas for the walk is transient to for any initial environment. In the critical case, , 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 or and when . 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.
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