Run <Wilson algorithm rooted at infinity> with $w_1,\ldots,w_K$ 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, so
$$
\mathbb P(\text{the wired uniform spanning forest has at least $K$ trees})>0.
$$
The <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.

Solution

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

The number of trees in a wired uniform spanning forest is almost surely constant. This follows from its tail triviality together with the fact that every component is infinite.


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