Suppose we have a network on $\mathcal{Z}^d$ with unit resistors between neighbouring points. Let $X$ be a simple symmetric random walk on $\mathcal{Z}^d$.
I would like to prove that for any two arbitrary neighbouring points $x_0$ and $x_1$ the effective conductance, $C_{\text{eff}}(x_0,x_1)$, is $d$.
Any hint? I tried to work on total currents and similar with no results.
Let me know if more context is needed. Thanks for the help.
This is also part of exercise 19.4.1 of Probability Theory by A. Klenke (3rd version).