reference request: voltage in a resistor network is a unique harmonic function
If I am reading correctly, this appears as Theorem 1.15 of Geoffrey Grimmett's book Probability on Graphs.
It is available for free on Grimmett's web site and has also been published by Cambridge University Press.
My prefered reference for the subject is David's Wagner lecture notes:
https://www.math.uwaterloo.ca/~dgwagner/Networks.pdf