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

Nathanaël Berestycki; Diederik van Engelenburg

doi:10.1007/s00440-023-01239-z

Titel

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

Autor*in

Nathanaël Berestycki

Universität Wien

Diederik van Engelenburg

Universität Wien

Abstract

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 et al. (Ann Probab 29(1):1–65, 2001). 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.

Stichwort

Uniform spanning treeReversible random graphsHarnack inequalityPotential theory

Objekt-Typ

journal article

Sprache

Englisch [eng]

Persistent identifier

https://phaidra.univie.ac.at/o:2044800

DOI

10.1007/s00440-023-01239-z

Erschienen in

Titel