Extension of Foster's averaging formula to infinite networks with moderate growth (Q1895777)

We show that in a network whose underlying graph has moderate growth, the average of the effective resistances across the edges of suitable subnetworks tends to 1 as the subnetworks expand to the whole network. As a consequence we show that a graph with moderate growth such that the automorphism group acts with finitely many orbits has no nonconstant Dirichlet finite functions.