A Random Walk Proof of Matrix Tree Theorem

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Random Walk on a GraphRecall. The graph Laplacian matrix L is defined by L = D −A.We can rewrite it asL = D(I−D −1 A) = D(I−P).Let ∆ ⊂ V, ∆ ≠ ∅. ThenL ∆ = D ∆ (I ∆ −P ∆ )for the matrices obtained by deleting the rows and columns associated to theentries in ∆.Note that P ∆ is strictly substochastic; that is, non-negative entries and rows sumto at most 1 with at least one row sum less than 1.Thus (I ∆ −P ∆ ) −1 exists.9
The Key Random Walk FactsLet ζ ∆ = inf{j ≥ 0 : S j ∈ ∆} be the first time the random walk visits ∆ ⊂ V.For x,y /∈ ∆, let[ ∞]∑G ∆ (x,y) = E x 1{S k = y, k < ζ ∆ }k=0be the expected number of visits to y by simple random walk on Γ starting at xbefore entering ∆.This is often called the random walk Green’s function.• If G ∆ = [G ∆ (x,y)] x,y∈V \∆ , thenG ∆ = (I ∆ −P ∆ ) −1 .• If r ∆ (x) denotes the probability that simple random walk starting at x returnsto x before entering ∆, thenG ∆ (x,x) =∞∑r ∆ (x) k =k=011−r ∆ (x) .10
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: Random Walk on a GraphFormally, a s
Page 13 and 14: Example: Wilson’s Algorithm on Γ
Page 15 and 16: Example: Wilson’s Algorithm on Γ
Page 17 and 18: Suppose ∆ ⊂ V, ∆ ≠ ∅.Comp
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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