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

Harnack inequality and one-endedness of UST on reversible random graphs