An Information-Theoretic Proof of the Shannon-Hagelbarger Theorem

The Shannon-Hagelbarger theorem states that the effective resistance across any pair of nodes in a resistive network is a concave function of the edge resistances. We give an information-theoretic proof of this result, building on the theory of the Gaussian free field. This also allows us to prove an extension of the result to determinants of matrices of cross effective resistances.