By Russell Lyons, Yuval Peres

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A set Π of edges separates A and Z if every path with one endpoint in A and the other endpoint in Z must include an edge in Π; we also call Π a cutset. The Nash-Williams Inequality. If a and z are distinct vertices in a finite network that are separated by pairwise disjoint cutsets Π1 , . . , Πn , then ( )−1 n ∑ ∑ R(a ↔ z) ≥ c(e) . 13) e∈Πk k=1 Proof. 7), it suffices to show that the unit current flow i from a to z has energy at least the right-hand side. Now given a finite cutset Π that separates a from z, let Z be the set of endpoints of Π that are separated by Π from a.

With this network in hand, the Markov chain may be described as a random walk on G: when the walk is at a vertex x, it chooses randomly among the vertices adjacent to x with transition probabilities proportional to the weights of the edges incident to x. Conversely, every connected graph with (positive) weights on the edges such that the sum of the weights incident to every vertex is finite gives rise to a random walk with transition probabilities proportional to the weights. * The most well-known example is gambler’s ruin.

Proof. We have [ E[Sxy ] = E = ∞ ∑ ] 1[Xk =x,Xk+1 =y] = k=0 ∞ ∑ ∞ ∑ P[Xk = x, Xk+1 = y] k=0 ∞ ∑ [ P[Xk = x]p(x, y) = E k=0 ] 1[Xk =x] p(x, y) = GZ (a, x)p(x, y) . 1, we have ∀x, y E[Sxy − Syx ] = GZ (a, x)p(x, y) − GZ (a, y)p(y, x) = v(x)π(x)p(x, y) − v(y)π(y)p(y, x) = [v(x) − v(y)]c(x, y) = i(x, y) . ◀ Effective conductance is a key quantity because of the following relationship to the question of transience and recurrence when G is infinite. 6) x∼y so that the associated random walk is well defined.

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Probability on Trees and Networks by Russell Lyons, Yuval Peres

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