Tipsy cop and tipsy robber: collisions of biased random walks on graphs

Abstract

Introduced by Harris, Insko, Prieto Langarica, Stoisavljevic, and Sullivan, the \emph{tipsy cop and drunken robber} is a variant of the cop and robber game on graphs in which the robber simply moves randomly along the graph, while the cop moves directed towards the robber some fixed proportion of the time and randomly the remainder. In this article, we adopt a slightly different interpretation of tipsiness of the cop and robber where we assume that in any round of the game there are four possible outcomes: a sober cop move, a sober robber move, a tipsy (uniformly random) move by the cop, and a tipsy (uniformly random) move by the robber. We study this tipsy cop and tipsy robber game on the infinite grid graph and on certain families of infinite trees including $\delta$-regular trees %infinite binary trees with an infinite path rooted at every vertex, and $\delta$-regular trees rooted to a $\Delta$-regular tree, where $\Delta \geq \delta$. Our main results analyze strategies for the cop and robber on these graphs. We conclude with some directions for further study.