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

• 成果类型: Article
• 署名作者: Berestycki, Nathanael; van Engelenburg, Diederik
• 署名单位: University of Vienna
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-023-01239-z
• 发表日期: 2024
• 页码: 487-548
• 关键词: incipient infinite cluster; random-walk; interlacements

摘要

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.