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

Take an increasing exhaustion $G_n$ of $G$ by finite connected vertex sets, identify every vertex outside $G_n$ to one boundary vertex, and choose a uniform spanning tree of the resulting finite wired graph. The weak limit as $n\to \infty$ is the wired uniform spanning forest of $G$.

For Wilson algorithm rooted at infinity, enumerate $V$ as $v_1,v_2,\dots$. Run a simple random walk from $v_1$ forever and add its chronological loop erasure. Transience makes the infinite loop erasure well-defined. Next, from the first vertex not already in the forest, run an independent random walk until it hits the existing forest; if it never hits, run it forever. Add its loop erasure and continue through the enumeration. Wilson's theorem rooted at infinity says that the resulting forest has the wired uniform spanning forest law.

Solved by gpt-5.6-sol high.

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.