A Random Walk Proof of Matrix Tree Theorem

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Some History• Kirchhoff - 1800sGustav Kirchhoff was motivated to study spanning trees by problems arisingfrom his work on electrical networks.• Wilson’s Algorithm - 1996David Wilson used “cycle-popping” to prove his algorithm generated a uniformspanning tree. His original proof is of a very different flavour. The Matrix TreeTheorem does not follow directly from the cycle-popping proof.• Greg Lawler - 1999Lawler discovered a new, computational proof of Wilson’s Algorithm viaGreen’s functions. The Matrix Tree Theorem follows immediately as a corollaryto his proof.Our original goal was to give an expository account of Lawler’s proof. However, inaddition to simplifying his proof, we discovered that these ideas could be applied todeduce results for Markov processes.15
Suppose ∆ ⊂ V, ∆ ≠ ∅.Computing a Loop-Erased Walk ProbabilityLet x 1 ,··· ,x K be distinct elements of a connected subset of V\∆ labelled in sucha way that x j ∼ x j+1 for j = 1,··· ,K. Note that x K+1 ∈ ∆.Consider simple random walk on Γ starting at x 1 . Set ξ ∆ = inf{j ≥ 0 : S j ∈ ∆}.LetP ∆ (x 1 ,··· ,x K ,x K+1 ) := P{L({S j , j = 0,··· ,ξ ∆ }) = [x 1 ,··· ,x K ,x K+1 ]}denote the probability that loop-erasure of {S j , j = 0,··· ,ξ ∆ } is exactly[x 1 ,··· ,x K+1 ].16
Page 1 and 2: A Random Walk Proof of Matrix Tree
Page 3 and 4: Set-upSuppose that Γ = (V,E) is a
Page 5 and 6: Kirchhoff’s Matrix Tree TheoremSu
Page 7 and 8: Example (cont.)This graph has 29 sp
Page 9 and 10: Random Walk on a GraphFormally, a s
Page 11 and 12: The Key Random Walk FactsLet ζ ∆
Page 13 and 14: Example: Wilson’s Algorithm on Γ
Page 15: Example: Wilson’s Algorithm on Γ
Page 19 and 20: Computing a Loop-Erased Walk Probab
Page 21 and 22: Proof of Wilson’s AlgorithmRecall
Page 23 and 24: Proof of Wilson’s AlgorithmRecall
Page 25 and 26: Application: Cayley’s FormulaIf
Page 27 and 28: Application: Cayley’s FormulaSinc
Page 29: Thank you.28

A Random Walk Proof of Matrix Tree Theorem

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