Recurrence and transience for non‐Archimedean and directed graphs

Matthias Keller and Anna Muranova

Mathematische Nachrichten, 2025, vol. 298, issue 10, 3307-3330

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.

Date: 2025
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