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 $\operatorname{LE}(x_1\to a)$. Rooting the same uniform law at $x_1$ shows that the same undirected path with its orientation reversed has the law of $\operatorname{LE}(a\to x_1)$. Consequently
$$
(L(\tau),L(\tau-1),\ldots,L(0))
\overset d=\operatorname{LE}(a\to x_1).
$$
This is <reversibility of loop-erased random walk>.

Solution

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

On a finite undirected graph, the reversal of a loop-erased random walk from $x$ to $y$ has the law of a loop-erased random walk from $y$ to $x$. The uniform-spanning-tree path representation proves the identity.


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 Incoming links: Reversibility of loop-erased random walk