Recurrence and transience for non‐Archimedean and directed graphs

Abstract We introduce notions of recurrence and transience for graphs over a non‐Archimedean ordered field. To achieve this, we establish a connection between these graphs and random walks on directed graphs over the reals. In particular, we give a characterization of the real directed graphs which can arise in such a way. As a main result, we give characterization for recurrence and transience in terms of a quantity related to the capacity.