Lemma 9.6 -- the Green's function and effective resistance

Let $c$ be a network on a finite vertex set: a symmetric nonnegative conductance function with total conductance $c(x)=\sum_yc(x,y)>0$ at every vertex, carrying the walk $P(x,y)=c(x,y)/c(x)$, assumed irreducible. Fix distinct vertices $a\ne z$. The Green's function of the walk stopped at $z$ is the expected number of visits to a vertex before first hitting $z$:

the walk starting at $a$ and $\tau_z$ being the hitting time of $z$. The effective resistance $R(a\leftrightarrow z)$ is defined through the voltage: with $W(x)=\mathbb P_x\{\tau_a<\tau_z\}$ (the harmonic function with boundary values $W(a)=1$, $W(z)=0$), the current flowing out of $a$ is $\|I\|=\sum_yc(a,y)\bigl[W(a)-W(y)\bigr]$, and $R(a\leftrightarrow z)=\|I\|^{-1}$.

The theorem (Lemma 9.6 of Levin–Peres–Wilmer) asserts:

The expected number of returns to the starting point before reaching $z$ is exactly the vertex conductance times the effective resistance — the identity through which escape probabilities and resistances translate into each other.