Recurrence and transience for non-Archimedean and directed graphs

We introduce the notion of recurrence and transience for graphs over non-Archimedean ordered field. To do so we relate these graphs to random walks of directed graphs over the reals. In particular, we give a characterization of the real directed graphs which can arise is such a way. As a main result, we give characterization for recurrence and transience in terms of a quantity related to the capacity.