Year: 2026
Type: preprint
Abstract: We prove that every event invariant under swaps has probability zero or one for either the free or the wired spanning forest on any connected, locally finite, infinite network. Applications include the constancy of the number of components and of the total number of excessive ends. For the free forest, these results resolve longstanding questions of Benjamini, Lyons, Peres, and Schramm [Ann. Probab. (2001), Question 15.7] on constancy of the component count and of Lyons, Morris, and Schramm [Electron. J. Probab. (2008), Question 7.7] on the zero-one law for one-endedness. In the latter case, we obtain the stronger conclusion that the total number of excessive ends is deterministic.
Source: arXiv (Cornell University)
Authors: Li, Xinyi, Yu, Yao
Institutions: Peking University, University of Chicago
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215737134