Benjamini–Lyons–Peres–Schramm and Lyons–Morris–Schramm spanning-forest zero–one questions
Let be a connected, locally finite, infinite network, and let denote its free uniform spanning forest. Is it always true that every tree of has exactly one end?
References
Primary source
Additional references
- A zero-one law for swap-invariant events in uniform spanning forests — arXiv — Xinyi Li, Yao Yu
Progress summary
A new specialist preprint claims to settle the remaining spanning-forest zero–one questions, but the claim has not been independently checked.
Lyons–Morris–Schramm posed whether the event that every uniform-spanning-forest tree has one end has probability or , and whether the total number of excessive ends is almost surely constant. The questions build on work of Benjamini, Lyons, Peres, and Schramm.
Known results
- Wired-forest components are one-ended under broad isoperimetric hypotheses, including transient quasi-transitive networks (Lyons, Morris, Schramm).
- Hutchcroft proved that the number of excessive ends of the wired uniform spanning forest is deterministic, answering Question 7.8.
- Hutchcroft and Nachmias proved indistinguishability for free and wired forests on unimodular random rooted networks and the free-forest , , or infinitely-many-components alternative.
- Timár obtained related component-count and indistinguishability results for Cayley graphs.
October 2026 claimed resolution
Xinyi Li and Yao Yu’s preprint claims a general zero–one law for swap-invariant spanning-forest events, implying deterministic component counts and total excessive ends and settling multiple BLPS and LMS questions. It is unrefereed, and no independent mathematical assessment was retrieved.
Current status (as of October 2026): Several special cases and the wired excessive-end count are established, while Li and Yu claim a general resolution; that claim remains unverified.
Solutions 0
No solutions have been posted yet.