I conducted experiments on the following random walk:
(i) You start with $n=1$ and $k=0$.
(ii) For each $k\in\{0,\ldots,n-1\}$ :
you walk $\omega$ steps where $\omega$ is the order of $k$ in the additive group $\mathbb Z/n\mathbb Z$,
you turn $90^\circ$ to the left.
(iii) When you have done all $k\in\{0,\ldots,n-1\}$, you increase $n$ by $1$.
It gives the following drawings, respectively until $n=4$, $n=10$, $n=30$ and $n=60$.
(0) Can you help me visualise this walk better? And for $n$ greater thatn $60$? (I didn't succeed in finishing the computation.)
(1) Can we prove that we will walk as far off $0$ as we want?
(2) Will we cover all $\mathbb Z^2$?
(3) Will we go back to $0$ infinitely many times?