= Uniform-spanning-tree path representation of loop-erased random walk
On a finite connected unweighted <graph>, the <graph path> between $x$ and $y$ in a <uniform spanning tree> has the law of <loop-erased random walk> from $x$ to $y$. Root <Wilson's algorithm> at $y$ and process $x$ first: the first attached <graph path> is exactly that <loop erasure>, and later attachments cannot change it. Changing the root to $x$ proves <reversibility of loop-erased random walk>, even though chronological <loop erasure> does not commute with reversing each individual random-walk trajectory.
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