Every recurrent network has a potential tending to infinity

A rooted network consists of a connected, locally finite graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , equipped with edge conductances and a distinguished vertex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>o</mml:mi> </mml:math> . A nonnegative function on the vertices of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> which vanishes at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>o</mml:mi> </mml:math> , has Laplacian <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>o</mml:mi> </mml:math> , and is harmonic at all other vertices is called a potential. We prove that every infinite recurrent rooted network admits a potential tending to infinity. This is an analogue of classical theorems due to Evans and Nakai in the settings of Euclidean domains and Riemannian surfaces.