A Random Walk Proof of Matrix Tree Theorem

Computing a Loop-Erased Walk ProbabilityQuestion: How can we computeP{L({S j , j = 0,··· ,ξ ∆ }) = [x 1 ,··· ,x K ,x K+1 ]}?For the loop-erasure to be exactly [x 1 ,··· ,x K+1 ], we need that:• the SRW started at x 1 , then• made a number of loops back to x 1 without entering ∆, then• took a step from x 1 to x 2 , then• made a number of loops back to x 2 without entering ∆∪{x 1 }, then• took a step from x 2 to x 3 , then• made a number of loops back to x 3 without entering ∆∪{x 1 ,x 2 }, then• ···• made a number of loops back to x K without entering∆∪{x 1 ,x 2 ,··· ,x K−1 }, then• took a step from x K to x K+1 ∈ ∆.17