Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 209 / 4 / 2 / Solution

Sample a uniform spanning tree $T$. Rooting Wilson's algorithm at $a$ shows that the path in $T$ oriented from $x_1$ to $a$ has the law of $\mathrm{LE}(x_1\to a)$. Rooting the same uniform law at $x_1$ shows that the same undirected path with its orientation reversed has the law of $\mathrm{LE}(a\to x_1)$. ConsequentlyThis is reversibility of loop-erased random walk.