Full zero–one law for update-stable events

Determine whether Hutchcroft’s conjectured zero–one law holds for all update-stable events of the free or wired spanning forest, beyond the swap-invariant events covered by the proved theorem.

Background

Hutchcroft conjectured a zero–one law for events that remain unchanged under specified random updates of either the free or wired spanning forest. The updates add a specified absent edge and remove a suitably chosen present edge, with the updated forest having a law absolutely continuous with respect to the original forest law.

The paper proves the result for the strictly stronger condition of invariance under every admissible single-edge swap. It establishes that this condition implies update stability, but explicitly states that the converse—and therefore the full conjecture—has not been established.

References

Hutchcroft Conjecture~7.4 proposed a zero--one law for update-stable events of either forest. The updates considered there add a specified absent edge and remove a suitably chosen present edge, so that the updated forest has a law absolutely continuous with respect to the original forest law. Update stability means that each such random update preserves the event a.s. Our theorem makes no absolute-continuity assumption on an update rule: it requires the event to be unchanged under every admissible swap. This implies stability under Hutchcroft's updates. The converse is not established here, so we do not claim the full conjecture.

— A zero-one law for swap-invariant events in uniform spanning forests  (2610.00938 - Li et al., 1 Oct 2026) in Introduction, paragraph following Theorem 1 (discussion of Hutchcroft’s Conjecture 7.4)