  If $\mathcal{W}$ is the simple random walk on the square lattice
$\mathbb{Z}^2$, then $\mathcal{W}$ induces a random walk $\mathcal{W}_G$ on any
spanning subgraph $G\subset \mathbb{Z}^2$ of the lattice as follows: viewing
$\mathcal{W}$ as a uniformly random infinite word on the alphabet
$\{\mathbf{x}, -\mathbf{x}, \mathbf{y}, -\mathbf{y} \}$, the walk
$\mathcal{W}_G$ starts at the origin and follows the directions specified by
$\mathcal{W}$, only accepting steps of $\mathcal{W}$ along which the walk
$\mathcal{W}_G$ does not exit $G$. For any fixed subgraph $G \subset
\mathbb{Z}^2$, the walk $\mathcal{W}_G$ is distributed as the simple random
walk on $G$, and hence $\mathcal{W}_G$ is almost surely recurrent in the sense
that $\mathcal{W}_G$ visits every site reachable from the origin in $G$
infinitely often. This fact naturally leads us to ask the following: does
$\mathcal{W}$ almost surely have the property that $\mathcal{W}_G$ is recurrent
for \emph{every} subgraph $G \subset \mathbb{Z}^2$? We answer this question
negatively, demonstrating that exceptional subgraphs exist almost surely. In
fact, we show more to be true: exceptional subgraphs continue to exist almost
surely for a countable collection of independent simple random walks, but on
the other hand, there are almost surely no exceptional subgraphs for a
branching random walk.
