A Random Walk Proof of Matrix Tree Theorem

Start a simple random walk at x.Application: Cayley’s FormulaSuppose that ∆ ⊂ V\{x}, where ∆ ≠ ∅, |∆| = m.Recall.r ∆ (x) is the probability that simple random walk starting at x returns to x beforeentering ∆.Let r ∆ (x;k) be the probability that simple random walk starting at x returns to xin exactly k steps without entering ∆ so thatr ∆ (x) =∞∑r ∆ (x;k).k=2Note that a SRW cannot return to its starting point in only 1 step.25