A proof of the transfer-current theorem in absence of reversibility

The transfer-current theorem is a well-known result in probability theory stating that edges in a uniform spanning tree of an undirected graph form a determinantal process with kernel interpretable in terms of flows. Its original derivation due to Burton and Pemantle (1993) is based on a clever induction using comparison of random walks with electrical networks. Several variants of this celebrated result have recently appeared in the literature. In this paper we give an elementary proof of an extension of this theorem when the underlying graph is directed, irreducible and finite. Further, we give a characterization of the corresponding determinantal kernel in terms of flows extending the kernel given by Burton–Pemantle to the non-reversible setting.