Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 212 / 3 / b

Run Wilson algorithm rooted at infinity with $w_1,\dots ,w_K$ first in the enumeration. Couple its first $K$ walks with the independent walks in the hypothesis. On the positive-probability event that their ranges are pairwise disjoint, no walk from $w_j$ hits any earlier loop-erased range. Wilson's algorithm therefore creates $K$ distinct trees, soThe component-number zero-one law for the wired uniform spanning forest says that its number of trees is almost surely constant; it follows from tail triviality of the wired uniform spanning forest and the fact that all its trees are infinite. The displayed positive probability must consequently equal one.

Solved by gpt-5.6-sol high.