Capacity of infinite graphs over non-Archimedean ordered fields

TL;DR

This paper explores the concept of vertex capacity in infinite graphs over non-Archimedean fields, revealing new phenomena like divergent capacity and establishing its connections to energy minimization, Dirichlet problems, and Green's functions.

Contribution

It introduces the notion of capacity in non-Archimedean graphs, analyzes its properties, and links it to potential theory and operator analysis, extending classical graph theory concepts.

Findings

Capacity can be positive, null, or divergent, and these are global properties independent of vertex choice.

Capacity relates to energy minimization, Dirichlet problem solutions, and Green's function existence.

Provides criteria for capacity using Nash-Williams test and discusses connections to Hardy inequalities.

Abstract

In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null capacity there is a third case of divergent capacity. However, we show that either of these cases is independent of the choice of the vertex and is therefore a global property for connected graphs. The capacity is shown to connect the minimization of the energy, solutions of the Dirichlet problem and existence of a Green's function. We furthermore give sufficient criteria in form of a Nash-Williams test, study the relation to Hardy inequalities and discuss the existence of positive superharmonic functions. Finally, we investigate the analytic features of the transition operator in relation to the inverse of the Laplace operator.