  We prove that for recurrent, reversible graphs, the following conditions are
equivalent: (a) existence and uniqueness of the potential kernel, (b) existence
and uniqueness of harmonic measure from infinity, (c) a new anchored Harnack
inequality, and (d) one-endedness of the wired Uniform Spanning Tree. In
particular this gives a proof of the anchored (and in fact also elliptic)
Harnack inequality on the UIPT. This also complements and strengthens some
results of Benjamini, Lyons, Peres and Schramm. Furthermore, we make progress
towards a conjecture of Aldous and Lyons by proving that these conditions are
fulfilled for strictly subdiffusive recurrent unimodular graphs. Finally, we
discuss the behaviour of the random walk conditioned to never return to the
origin, which is well defined as a consequence of our results
