Recurrence and transience for non-Archimedean and directed graphs

TL;DR

This paper extends the concepts of recurrence and transience to graphs over non-Archimedean fields, relating them to real directed graphs and providing capacity-based characterizations.

Contribution

It introduces recurrence and transience for non-Archimedean graphs and characterizes which real directed graphs can be represented in this framework.

Findings

Characterization of real directed graphs from non-Archimedean graphs

Capacity-related criteria for recurrence and transience

Link between non-Archimedean and real graph random walks

Abstract

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.