English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We study a random walk on \({\mathbb {Z}}\) 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 \(\alpha = \alpha (p,q,R,L)\). For \(\alpha > 1/2\) the walk is transient to \(+\infty \) for any initial environment, whereas for \(\alpha < 1/2\) the walk is transient to \(-\infty \) for any initial environment. In the critical case, \(\alpha = 1/2\), 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.

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