Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Capacity of infinite graphs over non-Archimedean ordered fields

Florian Fischer, Matthias Keller, Anna Muranova, Noema Nicolussi

TL;DR

The paper develops a comprehensive capacity theory for infinite graphs over non-Archimedean ordered fields, linking vertex capacity to Dirichlet problems, energy minimization, and Green functions. Unlike the real-field setting, monotone limits can fail, creating three capacity types—positive, null, and divergent—and, on connected graphs, these types are global and vertex-independent. It establishes multiple equivalent characterizations of capacity, including infima of energy, convergence of Dirichlet solutions, and a Green-function formulation, and it connects capacity with Nash-Williams criteria, Hardy inequalities, and the existence of positive superharmonic functions. The transition operator is analyzed to relate Laplacian inversion to sums of powers, with explicit behavior in positive-capacity cases and nuanced convergence issues in the non-Archimedean context, supported by concrete Levi-Civita-field examples and comparisons to real-weighted graphs.

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.

Capacity of infinite graphs over non-Archimedean ordered fields

TL;DR

The paper develops a comprehensive capacity theory for infinite graphs over non-Archimedean ordered fields, linking vertex capacity to Dirichlet problems, energy minimization, and Green functions. Unlike the real-field setting, monotone limits can fail, creating three capacity types—positive, null, and divergent—and, on connected graphs, these types are global and vertex-independent. It establishes multiple equivalent characterizations of capacity, including infima of energy, convergence of Dirichlet solutions, and a Green-function formulation, and it connects capacity with Nash-Williams criteria, Hardy inequalities, and the existence of positive superharmonic functions. The transition operator is analyzed to relate Laplacian inversion to sums of powers, with explicit behavior in positive-capacity cases and nuanced convergence issues in the non-Archimedean context, supported by concrete Levi-Civita-field examples and comparisons to real-weighted graphs.

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.
Paper Structure (24 sections, 38 theorems, 136 equations, 1 figure)

This paper contains 24 sections, 38 theorems, 136 equations, 1 figure.

Table of Contents

  1. Introduction and fundamental notions
  2. Weighted graphs over non-Archimedean ordered field
  3. Laplace operator and energy form
  4. The notion of capacity
  5. Dirichlet problem on finite subsets
  6. Capacity as a charge
  7. Examples
  8. Weakly spherically symmetric graphs
  9. Graphs over the Levi-Civita field
  10. Relation to real weighted graphs
  11. Characterizations of the capacity
  12. Capacity as an infimum of energies
  13. Independence of null capacity on the vertex
  14. Convergence of the solution of the Dirichlet problem
  15. Independence of positive capacity on the vertex
  16. ...and 9 more sections

Key Result

Proposition 2.1

Let $K\subseteq V$ be finite and $v$ a function on $V$. If $\Delta v\preceq 0$ on $K$, then and if $\Delta v\succeq 0$ on $K$, then

Figures (1)

  • Figure 1: Path graph

Theorems & Definitions (102)

  • Definition 1.2
  • Proposition 2.1: Maximum principle, Lemma 18 in Muranova1
  • Proposition 2.2: Dirichlet problem Muranova1
  • proof
  • Definition 2.3: Effective capacity
  • Remark 2.4
  • Lemma 2.5: Renormalized Dirichlet problem
  • proof
  • Lemma 2.6
  • proof
  • ...and 92 more