Consider the graph with $\mathbb{Z}^2$ as a vertex set, and with an edge between two vertices iff their $L^{\infty}$-distance is $1$. Imagine we perform a random walk on this graph, starting at the origin.
Equivalently, imagine a chess king moving randomly on an infinite chessboard.
Does anyone know whether this process has been studied? I am particuarly curious as to whether there is some sort of asymptotical rotational invariance.