# Capacity of infinite graphs over non-Archimedean ordered fields

Florian Fischer, Matthias Keller, Anna Muranova, Noema Nicolussi  Florian
Fischer, Matthias Keller: Institut für Mathematik, Universität Potsdam 14476
Potsdam, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>Anna
Muranova: Faculty of Mathematics and Computer Science, University of Warmia
and Mazury in Olsztyn, ul. Sloneczna 54, 10-710 Olsztyn, Poland
<EMAIL_ADDRESS>Noema Nicolussi: Faculty of Mathematics,
University of Vienna, Oskar-Morgenster-Platz 1, 1090 Vienna, Austria
<EMAIL_ADDRESS>
###### 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.

MSC2020: 31C20, 47S10, 05C50, 05C22, 12J15

Keywords: capacity, locally finite graphs, weighted graphs, non-Archimedean
field, ordered field

11footnotetext: The first author acknowledges support by the Heinrich Böll
foundation, grant nr. P139140. The second author is grateful for the financial
support of the DFG and is supported in part at the Technion by a fellowship
from the Swiss Foundation. The third author acknowledges financial support by
the NCN (National Science Center, Poland), grant nr. 2022/06/X/ST1/00910. The
fourth author acknowledges financial support by the FWF (Austrian Science
Fund), grant nr. J4497N.

##  1\. Introduction and fundamental notions

The physical quantity capacitance describes the capacity of a capacitor to
store an electric charge. This fundamental notion in the physics of
electrostatics found its way into various fields of pure mathematics and has
proven to be a most fruitful tool ever since. It is widely used in potential
theory, analysis, probability theory and metric geometry, see e.g. [2, 3, 8,
20].

Recently graphs with weights in non-Archimedean ordered fields have been
introduced in [13, 14]. Such graphs are the object of study of the present
work. While this allows for the flexibility to study families of graphs with,
for example, rational functions as weights, one still has a notion of
positivity which allows for the use of a maximum principle. This maximum
principle is a powerful tool to study the Dirichlet problem and to introduce
the capacity of a vertex for finite graphs with boundaries [13, 14]. There the
capacity is studied under the name admittance. For infinite graphs, however,
the situation becomes substantially more complicated. The capacity at a vertex
on an infinite graph can be defined via three equivalent approaches:

  * (1)

… as the limit of the charge at said vertex for the solution of the Dirichlet
problem along an exhaustion of the space (for example by distance balls).

  * (2)

… as the infimum of the energy over functions normalized at said vertex.

  * (3)

… as the reciprocal of the Green function at said vertex.

For graphs with real weights, the existence of the capacity is in each of
these approaches guaranteed by the monotonicity of the approximating quantity,
cf. eg. [12, 20]. However, in a non-Archimedean ordered field monotone limits
do not need to exist any longer. Thus, it is only natural to expect that next
to the case of a vertex having positive or null capacity there is also the
case where the limit does not exist. We show that, if the limit exists for any
of these approaches, then all of them lead to the same quantity in our
situation. Additionally, we show that positive, null and divergent capacity is
independent of the choice of the vertex for connected graphs. As in the
context of real graphs this allows us to define the capacity type of the
graph. In the real case the capacity type is a crucial property and relates to
different aspects of analysis on graphs, such as transience and recurrence of
random walks, superharmonic functions, Hardy inequalities and many more. Here
we investigate this notion for non-Archimedean graphs.

As mentioned above, studying more general weights allows for the study of edge
weights of functions which may take non-positive or even non-real values, cf.
[13, 14]. However, there is another advantage to study graphs over general
ordered fields. Such weights make it more apparent which phenomena have their
origin in the combinatorial geometry of the graph and which phenomena are a
result of the geometry induced by the weights. This shall give insights into
the understanding of weighted graphs on a structural level.

The paper is organized as follows. In the following two subsections we
introduce the Laplacian over non-Archimedean ordered fields together with its
energy form. In Section 2 we present the notion of capacity in terms of the
charge of a vertex via approximating the space with distance balls, cf. (1)
above. Before we investigate the relation of this notion to minimizing the
energy, i.e., (2) or to a Green’s function, i.e., (3) in Section 4, we
illustrate our notion by examples in Section 3. Here we discuss weakly
spherically symmetric graphs, path graphs and the relationship between
capacity types of graphs over the real field and the Levi–Civita field. In
Section 5, we give a comparison principle and a sufficient criteria for null
capacity in terms of a Nash-Williams test. The relation to Hardy inequalities
is explored in Section 6. The existence of positive non-constant superharmonic
functions is studied in Section 7. The dichotomy of non-existence of such
functions for null capacity and existence for positive capacity is in line
with real graphs. However, in the case of divergent capacity the question is
open and we present examples of existence of positive superharmonic functions
for divergent capacity graphs. Finally, we study the relation of capacity and
convergence of powers of a transition operator in Section 8. In case these
powers converge to zero the corresponding power series can be shown to
converge as well. This power series is then an inverse of the corresponding
Laplacian. Furthermore, this can be understood as a first step towards a
probabilistic investigation of such graphs.

###  1.1. Weighted graphs over non-Archimedean ordered field

An ordered field $(\mathbb{K},\succ)$ is called non-Archimedean if there
exists an infinitesimal, i.e., a $\tau\succ 0$ such that

${\tau}\prec\dfrac{1}{n}=\dfrac{1}{\underbrace{1+\dots+1}_{n\;\mbox{\scriptsize
times}}}$

for any $n\in\mathbb{N}\subset\mathbb{K}$. Otherwise, the field is called
Archimedean. Any element $\mathcal{N}$ with $\mathcal{N}\succ N$ for all
$N\in\mathbb{N}$ is called infinitely large. Obviously, the ordered field is
non-Archimedean if and only if there exists an infinitely large element. In
contrast, all Archimedean ordered fields are order isomorphic to subfields of
$\mathbb{R}$, cf. [19].

We denote the positive elements of $\mathbb{K}$ by

$\mathbb{K}^{+}=\\{k\in\mathbb{K}\;\mid\;k\succ 0\\}.$

By the _absolute value of $k\in\mathbb{K}$_, we mean the element $|k|$ in
$\mathbb{K}^{+}\cup\\{0\\}$ defined as

$|k|=\begin{cases}k,&\mbox{if }k\succeq 0,\\\
-k,&\mbox{otherwise.}\end{cases}$

The convergence of a sequence $(k_{n})\subset\mathbb{K}$ to $k\in\mathbb{K}$
in order topology of the ordered field means by definition, that for any
element $c\in\mathbb{K}^{+}$ there exists $N_{0}\in\mathbb{N}$ such that for
any $n\geq N_{0}$ we have

$|k_{n}-k|\preceq c.$

Further, the convergence of a sequence $(k_{n})\subset\mathbb{K}$ to $\infty$
means, that for any element $c\in\mathbb{K}$ there exists $N_{0}\in\mathbb{N}$
such that for any $n\geq N_{0}$ we have

$k_{n}\succ c.$

An ordered field $\mathbb{K}$ is said to have a countable cofinality if there
exists a countable subset $S\subset\mathbb{K}$ such that for any
$k\in\mathbb{K}$ there exists $s\in S$ with $k\preceq s$. The following are
equivalent, see [1, Theorem 6]:

  1. (1)

$\mathbb{K}$ has countable cofinality;

  2. (2)

$\mathbb{K}$ admits a convergence sequence which is not eventually a constant.

An ordered field $\mathbb{K}$ is real-closed if every positive element in
$\mathbb{K}$ has a square root and every polynomial over $\mathbb{K}$ of odd
degree has a root in $\mathbb{K}$.

######  Assumption 1.1.

In this paper we assume that all non-Archimedean ordered fields $\mathbb{K}$
are Cauchy complete (in some literature the term sequentially complete is used
[1], [6]) and have countable cofinality.

Next, we introduce graphs over ordered fields.

######  Definition 1.2.

A _graph_ is a triple $(V,b,m)$, where $V$ is an at most countable set,
$m:V\to\mathbb{K}^{+}$ and $b:V\times V\rightarrow\mathbb{K}^{+}\cup\\{0\\}$
satisfies the following properties:

  1. (1)

$b(x,y)=b(y,x)$ for any $x,y\in V$,

  2. (2)

$b(x,x)=0$ for any $x\in V$.

We call the elements of $V$ _vertices_ and also speak of $b$ being a graph
over $(V,m)$ or $V$.

Property (1) means that our graphs are symmetric and property (2) means that
we consider graphs without loops. We write $x\sim y$ whenever $b(x,y)\neq 0$
and say that there is an edge between $x$ and $y$. A path between vertices
$x,y\in V$ is a finite sequence $(x_{0},\ldots,x_{n}),n\in\mathbb{N}$, of
vertices such that

$x=x_{0}\sim x_{1}\sim x_{2}\sim\dots\sim x_{n}=y.$

We call the minimal value of $n$ such that there exists a path connecting $x$
and $y$ the (combinatorial) _graph distance_ $d\left(x,y\right)$. Furthermore,
we denote the distance balls of radius $n\in\mathbb{N}$ about a vertex $a\in
V$ by

$\displaystyle B_{n}(a)=\\{x\in V\mid d(a,x)<n\\}.$

A set $K\subseteq V$ is called connected, if there exists a path between any
two vertices of it, and so we call the graph _connected_ if $V$ is connected.
Furthermore, a graph is called locally finite, if for all $x\in V$ we have
$\\#\\{y\in V\;\mid\;y\sim x\\}<\infty$.

######  Assumption 1.3.

In this paper we assume that all graphs $(V,b,m)$ over $\mathbb{K}$ are
locally finite and connected.

We denote the _edge boundary_ of a set $W\subseteq V$ by

$\displaystyle\partial W=W\times(V\setminus W).$

For $x\in V$ and any finite subset $W\subset V$, we denote

$b(x)=\sum_{y\in V}b(x,y)\qquad\mbox{ and }\qquad b(W)=\sum_{y\in W}b(y).$

Furthermore, for the boundary $\partial W$ of a finite set $W\subset V$, we
denote

$b(\partial W)=\sum_{{x\in W,\,y\in V\setminus W}}b(x,y).$

###  1.2. Laplace operator and energy form

For a graph $(V,b,m)$, the Laplace operator is defined on the set of all
functions

$C(V)=\\{f\mid f:V\to\mathbb{K}\\}$

as follows

$\Delta f(x)=\dfrac{1}{m(x)}\sum_{y\in V}(f(x)-f(y))b(x,y).$

The support of a function $f\in C(V)$ is defined by
$\operatorname{supp}(f)=\\{x\in V|\,f(x)\neq 0\\}$. We introduce the set of
compactly supported functions as

$C_{c}(V)=\\{\phi\mid\phi:V\to\mathbb{K},\phi\mbox{ has finite support}\\}.$

Then we define a quadratic form

$Q(f,g)=\dfrac{1}{2}\sum_{x,y\in V}(f(x)-f(y))(g(x)-g(y))b(x,y)$

whenever $f\in C(V)$ or $g\in C(V)$ is compactly supported. The energy of a
function $\phi\in C_{c}(V)$ is defined then as

$Q(\phi)=\dfrac{1}{2}\sum_{x,y\in
V}(\phi(x)-\phi(y))^{2}b(x,y)=\dfrac{1}{2}\sum_{x,y\in
V}(\nabla_{xy}\phi)^{2}b(x,y),$

where $\nabla_{xy}\phi=\phi(y)-\phi(x)$ is the _difference operator_.

The analogue of a dual pairing can be defined for $f,g\in C(V)$ by

$\langle f,g\rangle=\sum_{x\in V}f(x)g(x)m(x),$

again whenever $f$ or $g$ is compactly supported. Since the following Green’s
formula [13, Theorem 22] holds when $f$ or $g$ is compactly supported

(GF) $\langle\Delta f,g\rangle=Q(f,g)=\frac{1}{2}\sum_{x,y\in
V}(\nabla_{xy}f)(\nabla_{xy}g)b(x,y),$

we get the following equality for the energy of a compactly supported function
$\phi\in C_{c}(V)$

$Q(\phi)=\langle\Delta\phi,\phi\rangle=\sum_{x\in V}\Delta\phi(x)\phi(x)m(x).$

For a finite set $K\subseteq V$, we let $C_{c}(K)$ be the functions with
support in $K$ and we denote by $C(K)$ the space of functions
$K\to\mathbb{K}$. We consider the Dirichlet restriction $\Delta_{K}$ of the
operator $\Delta$ to the finite subset $K$ which is the restriction of
$\Delta$ to $C_{c}(K)\to C_{c}(V)$ (defined on a subspace of $C(V)$). By
identifying functions in $C(K)$ with functions in $C_{c}(K)$ we also write
$\Delta_{K}$ for the operator $C(K)\to C(K)$ (defined on the finite
dimensional space $C(K)$) with slight abuse of notation. Clearly, on infinite
graphs the operator $C_{c}(K)\to C_{c}(V)$ is not invertible, however, on
connected graphs the operator $C(K)\to C(K)$ is invertible as we will see
below, cf. Lemma 2.5. In this case we denote by $\Delta_{K}^{-1}$ its inverse
and again by slight abuse of notation we extend the functions in the image of
$\Delta_{K}^{-1}$ in $C(K)$ by $0$ to $V$.

##  2\. The notion of capacity

In this section we introduce the effective capacity of a vertex. It turns out
that in contrast to the classical situation of graphs over the real field
there is a third option next to positive and null capacity. We will refer to
the latter as divergent capacity. We first define the capacity on a finite
subset via the the charge of the solution of the Dirichlet problem on this
vertex. This capacity will always be a positive element of $\mathbb{K}$. The
capacity of a vertex on the infinite graph is then obtained by approximation
provided the limit exists. In what follows let $b$ be an infinite locally
finite connected graph over $(V,m)$ with respect to a non-Archimedean field
$\mathbb{K}$ which satisfy Assumption 1.1 and 1.3.

###  2.1. Dirichlet problem on finite subsets

We formulate a Dirichlet problem, which is known for real weighted graphs and
naturally arises for the field of rational functions in electrical engineering
(see [13, 14]).

Let $\emptyset\neq K\subseteq V$ be a finite connected subset of $V$ and $a\in
K$. The Dirichlet problem for $K$ and $a$ then looks for a solution
$v:V\to\mathbb{K}$ of the equation

(DP) $\begin{cases}\Delta v=0&\mbox{ on }K\setminus\\{a\\},\\\ v=1&\mbox{ on
}\\{a\\},\\\ v=0&\mbox{ on }V\setminus K.\end{cases}$

This problem was studied in [13] for graphs over non-Archimedean fields.
Existence and uniqueness were answered affirmatively there. Moreover,
positivity is concluded with the help of the following maximum-minimum
principle from [13]. We denote by $S_{1}(K)=\left(\bigcup_{x\in
K}B_{2}(x)\right)\setminus K$ for a finite set $K\subseteq V$.

######  Proposition 2.1 (Maximum principle, Lemma 18 in [13]).

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

$\displaystyle\max_{K}v\preceq\max_{S_{1}(K)}v$

and if $\Delta v\succeq 0$ on $K$, then

$\displaystyle\min_{K}v\succeq\min_{S_{1}(K)}v.$

With this maximum-minimum principle at hand we are ready to address the
fundamental result about solving the Dirichlet problem.

######  Proposition 2.2 (Dirichlet problem [13]).

Let $K\subseteq V$ be finite and connected and $a\in K$. The Dirichlet problem
(DP) has a unique solution $v$ which satisfies

$0\prec v\preceq 1\qquad\mbox{on }K$

and

$Q(v)=\Delta v(a)m(a)=-\sum_{x\in V\setminus K}\Delta v(x)m(x).$

Moreover, for any function $f\in C_{c}(V)$ with $f(a)=1$ and $f=0$ on
$V\setminus K$,

$Q(v)\preceq Q(f),$

i.e., the solution of the Dirichlet problem minimizes the energy.

###### Proof.

All statements except for $0\prec v$ follow from Theorem 17, 25, 26 and
Corollary 20 in [13] (where only $0\preceq v$ was established). To see $0\prec
v$ on $K$, assume $v(x)=0$ for some $x\in K$. By the equation $\Delta v(x)=0$
and $v\succeq 0$ we see that $v$ vanishes in all neighbors of $x$. By
iteration and connectivity of $K$ we infer $v(a)=0$ which is a contradiction.
∎

The effective capacity for a finite graph with boundary was introduced in
[13], where it is called the admittance of the graph. Here we formulate it for
finite subsets.

######  Definition 2.3 (Effective capacity).

The _effective capacity_ for a finite and connected $K\subseteq V$ and $a\in
K$ is defined as

$\mathrm{cap}_{K}(a)=\Delta v(a)m(a),$

where $v$ is the solution of the Dirichlet problem (DP).

######  Remark 2.4.

Note that although $m$ appears in the definition above, the effective capacity
indeed does not depend on $m$ since it cancels with the $m$ appearing in the
definition of the Laplacian.

We also need a renormalized version of the Dirichlet problem, where instead of
normalizing the potential we normalize the charge in a vertex $a$ of a finite
connected set $K\subseteq V$

($\widetilde{\mathrm{DP}}$) $\begin{cases}\Delta\widetilde{v}=0&\mbox{ on
}K\setminus\\{a\\},\\\ \Delta\widetilde{v}=1&\mbox{ on }\\{a\\},\\\
\widetilde{v}=0&\mbox{ on }V\setminus K.\end{cases}$

Existence and uniqueness of solutions of the renormalized Dirichlet problem
follow from the corresponding consideration of the Dirichlet problem above.
For a vertex $a\in V$, we denote by $1_{a}$ the function taking the value $1$
at $a$ and $0$ otherwise.

######  Lemma 2.5 (Renormalized Dirichlet problem).

Let $K\subseteq V$ be finite and connected and $a\in K$. The Dirichlet problem
($\widetilde{\mathrm{DP}}$) has a unique solution $\widetilde{v}$ which
satisfies

$\widetilde{v}=\dfrac{v}{\Delta v(a)}\qquad\mbox{and }\qquad
v=\dfrac{\widetilde{v}}{\widetilde{v}(a)},$

where $v$ is the solution of the Dirichlet problem (DP). Furthermore, the
Dirichlet Laplacian $\Delta_{K}$ is invertible as an operator $C(K)\to C(K)$
and

$\widetilde{v}=\Delta^{-1}_{K}1_{a}$

and

$\mathrm{cap}_{K}(a)=\dfrac{m(a)}{\widetilde{v}(a)}.$

###### Proof.

For existence, we observe first that $v\neq const$ as
$\Delta\widetilde{v}(a)=1$. Hence, by Proposition 2.2 that the denominator
satisfies $\Delta v(a)=Q(v)/m(a)\neq 0$. This gives a solution
$\widetilde{v}=v/\Delta v(a)$ by direct calculation.

To see uniqueness, assume $\widetilde{v}$ is a solution of
($\widetilde{\mathrm{DP}}$). To show $\widetilde{v}(a)\neq 0$, we assume
$\widetilde{v}(a)=0$. Then, $\Delta\widetilde{v}=1_{a}$ on $K$ gives by
Green’s formula

$\displaystyle
Q(\widetilde{v})=\langle\Delta\widetilde{v},\widetilde{v}\rangle=\widetilde{v}(a)=0$

which yields $\widetilde{v}=\mathrm{const}=0$ which contradicts
$\Delta\widetilde{v}(a)=1$. Thus, $\widetilde{v}(a)\neq 0$ and
$v=\widetilde{v}/\widetilde{v}(a)$ is a solution of (DP) which is unique by
Proposition 2.2. Thus, $\widetilde{v}$ is unique.

Furthermore, ($\widetilde{\mathrm{DP}}$) can be rewritten as the problem

$\displaystyle\Delta_{K}\widetilde{v}=1_{a}\qquad\mbox{ on }C(K).$

So existence and uniqueness shows that $\Delta_{K}$ is invertible on $C(K)$
which allows us to write $\widetilde{v}=\Delta_{K}^{-1}1_{a}$ for the solution
of ($\widetilde{\mathrm{DP}}$). The last formula follows now directly from
Proposition 2.2. ∎

The following lemma shows how solutions of the Dirichlet problem (DP),
normalized in different vertices are related.

######  Lemma 2.6.

Let $K\subseteq V$ be finite and connected and $x,y\in K$. Let $v^{x}$,
respectively $v^{y}$, be solutions of (DP) for $K$ and $x$, respectively $y$,
as well as $\widetilde{v}^{x}$, $\widetilde{v}^{y}$ corresponding solutions to
($\widetilde{\mathrm{DP}}$). Then,

$\dfrac{v^{x}(y)}{\mathrm{cap}_{K}(x)}=\dfrac{v^{y}(x)}{\mathrm{cap}_{K}(y)}$

and

$\widetilde{v}^{x}(y)m(y)=\widetilde{v}^{y}(x){m(x)}.$

###### Proof.

By Lemma 2.5 above we have $\widetilde{v}^{z}=\Delta^{-1}_{K}1_{z}$ for
$z=x,y$. Now, by Green’s formula (GF) for
$Q(\Delta^{-1}_{K}1_{y},\Delta^{-1}_{K}1_{x})$, we get

$\sum_{z\in
K}{1_{y}(z)}\Delta^{-1}_{K}1_{x}(z)m(z)=Q(\Delta^{-1}_{K}1_{y},\Delta^{-1}_{K}1_{x})=Q(\Delta^{-1}_{K}1_{x},\Delta^{-1}_{K}1_{y})=\sum_{z\in
K}{1_{x}(z)}\Delta^{-1}_{K}1_{y}(z)m(z)$

from which the second statement follows. The first statement now follows from
Lemma 2.5 as

$\displaystyle\dfrac{v^{x}(y)}{\mathrm{cap}_{K}(x)}=\frac{\widetilde{v}^{x}(y)}{m(x)}=\frac{\widetilde{v}^{y}(x)}{m(y)}=\dfrac{v^{y}(x)}{\mathrm{cap}_{K}(y)}.$

This finishes the proof. ∎

###  2.2. Capacity as a charge

To define the capacity of an infinite graph we start with finite
approximations, being distance balls of some fixed vertex, and follow the
approach from [4] and [14] for networks.

Let $a\in V$ be fixed. Recall the distance balls $B_{n}(a)=\\{x\in
V\mid\operatorname{dist}(a,x)<n\\}$, $n\in\mathbb{N}$, about $a$ and denote

$\mathrm{cap}_{n}(a):=\mathrm{cap}_{B_{n}(a)}(a).$

Similar to the case of graphs over the real field $\mathbb{R}$ it is known
that effective capacity is monotonically decreasing in $n$.

######  Proposition 2.7 (Theorem 15 in [14]).

For all $a\in V$ and $n\in\mathbb{N}$

$\mathrm{cap}_{n+1}(a)\preceq\mathrm{cap}_{n}(a).$

Beware that even in a Cauchy complete non-Archimedean ordered field the
inequalities above do not imply that the sequence $(\mathrm{cap}_{n}(a))$
converges.

######  Definition 2.8 (Definition 17 in [14]).

If for a fixed vertex $a\in V$ the limit of the sequence
$(\mathrm{cap}_{n}(a))$ exists in the order topology of $\mathbb{K}$, we call
it _effective capacity of $a$_ and denote it by

$\mathrm{cap}(a):=\lim_{n\to\infty}\mathrm{cap}_{n}(a).$

The following theorem shows that existence of the limit is independent of the
base vertex $a$, which can be considered as one of the main results of this
work. Its proof is given in Section 4.

######  Theorem 2.9 (Independence on the vertex).

If $\mathrm{cap}(a)$ exists for some $a\in V$, then it exists for all $x\in
V$. Moreover, if $\mathrm{cap}(a)=0$ for some $a\in V$, then
$\mathrm{cap}(x)=0$ for all $x\in V$.

The previous theorem gives rise to the following definition of the capacity
type of a graph.

######  Definition 2.10 (Capacity of an infinite graph).

The graph is said to have

  1. (1)

null capacity, if $\mathrm{cap}(a)=0$ for some (all) $a\in V$,

  2. (2)

positive capacity, if $\mathrm{cap}(a)$ exists and $\mathrm{cap}(a)\neq 0$ for
some (all) $a\in V$,

  3. (3)

divergent capacity, if $\mathrm{cap}(a)$ does not exist for some (all) $a\in
V$.

######  Remark 2.11.

(a) Note that $\mathrm{cap}_{n}(a)$ does not depend on $m$, confer Remark 2.4,
so $\mathrm{cap}(a)$ does not depend on $m$.

(b) When the limit does not exist, the sequence $\mathrm{cap}_{n}(a)$ is still
bounded as it is positive and decaying by Theorem 2.7.

######  Remark 2.12.

For real weighted graphs the capacity is intimately related to the notion of
recurrence and transience. Specifically, recurrent graphs are characterized by
null capacity and transient graphs by positive capacity which holds as a
dichotomy for connected graphs. At this point, we refrain from elaborating on
this relation with respect to non-Archimedean fields. The reason is that a
probabilistic interpretation is more subtle in this setting. Specifically, in
the non-Archimedean context, a property is said to hold almost surely if it
holds outside of an event $N$ which has probability less than every positive
non-infinitesimal quantity [16, p. 25]. Thus, one should for example be
hesitant to call a graph with positive but infinitesimal small capacity
transient. The probabilistic interpretation of these notions will be addressed
in a later investigation.

##  3\. Examples

Before going deeper into the general theory, we first give some examples to
establish a basic intuition. We start with a general class of examples, so
called weakly spherically symmetric graphs which allow for an explicit formula
to calculate the capacity. Secondly, we will discuss specific examples over
the Levi-Civita field. Finally, in this section we explore the relation
between capacity of graphs over the Levi–Civita field and the real field.

###  3.1. Weakly spherically symmetric graphs

In this subsection, we consider weakly spherically graphs over a non-
Archimedean ordered field $\mathbb{K}$ as they were considered over the real
field in [8] (see also [7, Chapter 9]).

Fix a reference vertex $o\in V$ which we refer to as a root. For $x\in V$, we
define

$\displaystyle b_{\pm}(x)=\sum_{y\in S_{|x|\pm 1}}b(x,y),$

where $|x|=d(x,o)$ is the combinatorial graph distance from $x$ to $o$ and the
sphere of radius $k\geq 0$ about $o$ is given by $S_{k}=\\{x\in
V\mid|x|=k\\}=B_{k+1}(o)\setminus B_{k}(o)$.

A graph is called _weakly spherically symmetric_ with respect to $o\in V$ if
$b_{\pm}$ are functions of the combinatorial graph distance, i.e., there are
$\widetilde{b}_{\pm}:\mathbb{N}_{0}\to\mathbb{K}^{+}\cup\\{0\\}$ such that
$b_{\pm}(x)=\widetilde{b}_{\pm}(|x|)$, $x\in V$. With slight abuse of notation
we do not distinguish between $b_{\pm}$ and $\widetilde{b}_{\pm}$ in notation.

Firstly, we formulate several useful results on convergence in non-Archimedean
fields. We refer to the following simple fact as “a student’s dream”, a term
we picked up from a talk of Khodr Shamseddine111confer
https://youtu.be/T8yw7yn8zII?t=745 at minute 12:25.

######  Proposition 3.1 (“A student’s dream” [1]).

In a Cauchy complete non-Archimedean field $\mathbb{K}$, a sequence $(a_{n})$
converges to an element in $\mathbb{K}$ if and only if $(a_{n+1}-a_{n})$
converges to zero. In particular, the series $\sum_{n\geq 0}a_{n}$ converges
if and only if $(a_{n})$ converges to zero.

The following lemma concerns the convergence to $\infty$ and complements “a
student’s dream”.

######  Lemma 3.2.

Let $a_{n}\succ 0,n\in\mathbb{N}_{0}$, in a non-Archimedean field with
countable cofinality. Then,

$\sum_{n=0}^{\infty}{a_{n}}=\infty$

if and only if there exists a subsequence $(a_{n_{k}})$ with
$a_{n_{k}}\to\infty$ as $k\to\infty$.

###### Proof.

The “if” direction is clear. Conversely if $(a_{n})$ is bounded, i.e.
$a_{n}\preceq M$, we have $\sum_{n=0}^{N}{a_{n}}\preceq\mathcal{N}\cdot M$,
where $\mathcal{N}$ is any infinitely large element. Thus, the partial sums
cannot exceed any element and thus cannot diverge to infinity. In case that
$(a_{n})$ is not bounded, by countable cofinality of the field $\mathbb{K}$ we
can extract a subsequence $(a_{n_{k}})$ which tends to $\infty$. ∎

We now come to the fundamental result for weakly spherical symmetric graphs.

######  Theorem 3.3 (Weakly spherically symmetric graphs).

For a weakly spherically symmetric graph, we have

$\mathrm{cap}(o)=\left(\sum_{k=0}^{\infty}\dfrac{1}{b(\partial
B_{k}(o))}\right)^{-1}$

whenever either side exists where $1/\infty=0$. In particular, we have the
following equivalences:

  1. (1)

The graph has null capacity if and only if there exists a subsequence
$(b_{+}(n_{k}))$ of $(b_{+}(n))$ with

$\lim_{n_{k}\to\infty}b_{+}(n_{k})=0.$

  2. (2)

The graph has positive capacity if and only if

$\lim_{n\to\infty}b_{+}(n)=\infty.$

  3. (3)

The graph has divergent capacity if and only if there exist
$c,C\in\mathbb{K}^{+}$ such that

$c\preceq b_{+}(n)$

for all $n\in\mathbb{N}_{0}$ and

$b_{+}(n)\preceq C$

for infinitely many $n\in\mathbb{N}_{0}$.

###### Proof.

Observe that on the ball $B_{n}(o)$ the function

$\displaystyle\widetilde{v}(x)=m(o)\sum_{k=|x|}^{n-1}\dfrac{1}{b_{+}(k)\\#S_{k}}=m(o)\sum_{k=|x|}^{n-1}\dfrac{1}{b(\partial
B_{k+1}(o))}$

solves the Dirichlet problem ($\widetilde{\mathrm{DP}}$) since
$\\#S_{k}b_{+}(k)=\\#S_{k+1}b_{-}(k+1)$ by weak spherical symmetry. Hence, by
Lemma 2.5 we infer

$\displaystyle\mathrm{cap}_{n}(o)=\left(\sum_{k=1}^{n-1}\dfrac{1}{b(\partial
B_{k}(o))}\right)^{-1}.$

Taking the limit $n\to\infty$ yields the first statement. The equivalences now
follow immediately due to Proposition 3.1, Lemma 3.2, and the estimate
$b_{+}(k)\preceq b(\partial B_{k+1}(o))\preceq\mathcal{N}b_{+}(k)$ valid for
any $k\in\mathbb{N}$ and infinitely large element
$\mathcal{N}\in\mathbb{K}^{+}$. ∎

######  Remark 3.4.

Observe that the theorem shows that the combinatorial structure of a weakly
spherically graph is completely irrelevant for the capacity type. Indeed, the
capacity type is determined by the weights alone.

###  3.2. Graphs over the Levi-Civita field

In the examples below we discuss the Levi-Civita field $\mathcal{R}$ which
satisfies all our requirements (it is ordered, non-Archimedean, Cauchy
complete in order topology and has countable cofinality). Moreover, it is
real-closed. It was introduced by Tullio Levi-Civita in 1862 [11], see e. g.
[17], [18] for more details.

######  Definition 3.5 (Levi-Civita field [11]).

The _Levi-Civita field_ $\mathcal{R}$ is defined as the field of formal power
series

$a=\sum_{i=0}^{\infty}a_{i}\epsilon^{q_{i}}$

of the variable $\epsilon$ with coefficients $a_{i}\in\mathbb{R}$, $a_{0}\neq
0$, and a strictly increasing unbounded sequence $q_{i}\in\mathbb{Q}$,
$i\in\mathbb{N}_{0}$.

Addition and multiplication are defined naturally as for formal power series.
The order is defined as follows: $a\succ 0$ if $a_{0}>0$. The Levi-Civita
field contains a subfield isomorphic to field of rational functions of one
variable $\mathbb{R}(r)$, which arises naturally in theory of electrical
networks see [13], [14].

Let us consider _path graphs_ over the Levi-Civita field $\mathcal{R}$, i.e.,
$V=\mathbb{N}_{0}$ and $b(i,j)\neq 0$ if and only if $|i-j|=1$, and
$m:V\to\mathbb{K}^{+}$ is arbitrary, see Figure 1. These are the simplest
examples of weakly spherically symmetric graphs.

$0$$1$$2$$k$$k+1$ Figure 1. Path graph

######  Example 3.6 (Null capacity).

If $b(k,k+1)=\epsilon^{k}\in\mathcal{R}$ for any $k\in\mathbb{N}_{0}$, then
due to Theorem 3.3 above or the series law, cf. [14], we have

$\lim_{n\to\infty}\mathrm{cap}_{n}(0)=\lim_{n\to\infty}\left(\sum_{k=0}^{n-1}\epsilon^{-k}\right)^{-1}=\lim_{n\to\infty}\epsilon^{n-1}\left(\sum_{k=0}^{n-1}\epsilon^{k}\right)^{-1}=0.$

Therefore, the graph has null capacity.

######  Example 3.7 (Positive capacity).

If $b(k,k+1)=\epsilon^{-k}\in\mathcal{R}$ for any $k\in\mathbb{N}_{0}$, then

$\lim_{n\to\infty}\mathrm{cap}_{n}(0)=\lim_{n\to\infty}\left(\sum_{k=0}^{n-1}\epsilon^{k}\right)^{-1}=\left(\sum_{k=0}^{\infty}\epsilon^{k}\right)^{-1}=1-\epsilon.$

Therefore, the graph has positive capacity.

######  Example 3.8 (Divergent capacity).

If $b(k,k+1)=1$ for any $k\in\mathbb{N}_{0}$, then the limit
$\mathrm{cap}_{n}(0)=1/n$ does not exists for $n\to\infty$ over $\mathcal{R}$
and the graph has divergent capacity.

The last example is indeed symptomatic for graphs with rational weights. It
turns out that such graphs over a non-Archimedean field always have divergent
capacity, cf. Corollary 5.2 below.

###  3.3. Relation to real weighted graphs

In this section, we study the relationship between capacity types of graphs
over the Levi–Civita field $\mathcal{R}$ and sequences of graphs over the real
field $\mathbb{R}$. More precisely, consider a graph $b$ over $\mathcal{R}$
which is given by power series which simultaneously converge for small $r>0$.
Replacing the formal parameter $\varepsilon$ in Definition 3.5 by small $r>0$,
we obtain a family of weights $b_{r}$ over $\mathbb{R}$ which are all finite
by the assumption that the power series converge for small $r>0$. Moreover,
for each pair of vertices $x,y\in V$ there is also $r_{x,y}>0$ such that
$b_{r}(x,y)\geq 0$ for $r<r_{x,y}$ by $b(x,y)\succeq 0$. If there is an
$r_{0}>0$ such that $r_{x,y}\geq r_{0}$ for all $x,y$, then $b_{r}$ is a graph
over $\mathbb{R}$ for all small $r>0$. In this case, we denote by
$\mathrm{cap}^{\mathbb{R}}_{r}(a)$ the corresponding classical notion of
capacity of a vertex $a\in V$ for the graph $b_{r}$ over $\mathbb{R}$ .

A natural question is in which sense the capacity type of the graphs $b_{r}$
over $\mathbb{R}$ relates to the capacity type of the graph $b$ over the
Levi–Civita field $\mathcal{R}$. The next two examples show that the capacity
can behave somewhat differently.

######  Example 3.9.

The path graph $\mathbb{N}_{0}$ with $b(k,k+1)=k!\epsilon^{k}$ has null
capacity over the Levi-Civita field $\mathcal{R}$ by Theorem 3.3. However, for
the graph over reals with the weights $b_{r}(k,k+1)=k!r^{k}$, $r>0$, we obtain

$\operatorname{cap}^{\mathbb{R}}_{r}(0)=\left(\sum_{k=0}^{\infty}\dfrac{1}{k!r^{k}}\right)^{-1}=e^{-\frac{1}{r}}.$

That is, the graph over $\mathbb{R}$ has positive capacity for all $r>0$. A
handwaving justification of this phenomena is that $\epsilon^{-k}$ in
$\mathcal{R}$ is “significantly larger” than $r^{-k}$ in $\mathbb{R}$. Thus,
the series with parameter $\epsilon$ diverges in $\mathcal{R}$ but converges
in $\mathbb{R}$ with the parameter $r$.

######  Example 3.10.

The path graph $\mathbb{N}_{0}$ with $b(k,k+1)={\epsilon^{-k}}/{k!}$ has
positive capacity over the Levi-Civita field $\mathcal{R}$ by Theorem 3.3. But
if we consider the graph over reals with the weights
$b_{r}(k,k+1)=1/\left(k!r^{k}\right)$, $r>0$, we get the capacity

$\operatorname{cap}^{\mathbb{R}}_{r}(0)=\left(\sum_{k=0}^{\infty}{k!r^{k}}\right)^{-1}=0.$

That is, the graph has null capacity for all $r>0$. The phenomena is similar
to the one above. While the power series $\sum_{k}k!r^{k}$ diverges in
$\mathbb{R}$ the corresponding power series $\sum_{k}k!\epsilon^{k}$ converges
in $\mathcal{R}$.

In Theorem 3.11 below we prove that the limiting behavior of the capacity with
respect to a variable $r\in\mathbb{R}$ can be related to capacity over the
Levi-Civita field $\mathcal{R}$, as long as the edge weights are rational
functions of $r$. More precisely, null capacity over $\mathcal{R}$ leads to
$\operatorname{cap}^{\mathbb{R}}_{r}(a)\to 0$ as $r\to 0$ for $a\in V$. Note,
that the field of rational functions $\mathbb{R}(r)$ is isomorphic to a
subfield of the Levi-Civita field $\mathcal{R}$ with $\epsilon:=r$. The
corresponding order in $\mathbb{R}(r)$ is the following: for any
$g(r)\in\mathbb{R}(r)$ we can write

$g(r)=\dfrac{a_{1}r^{n_{1}}+\dots+a_{k}r^{n_{k}}}{b_{1}r^{m_{1}}+\dots+b_{s}r^{m_{s}}},$

where powers of $r$ are ordered from smaller to larger in both numerator and
denominator and $a_{1}\neq 0,b_{1}\neq 0$. Then

$g(r)\succ 0\mbox{ if and only if }a_{1}/b_{1}>0.$

######  Theorem 3.11.

Let $b$ be a graph over the field of rational functions
$\mathbb{R}(r)\subset\mathcal{R}$ with null capacity over $\mathcal{R}$ such
that the corresponding $b_{r}$ give rise to a graph with real weights for
small $r>0$. Then, for any $a\in V$ and for any $n\in\mathbb{N}\cup\\{0\\}$ we
have

$\lim_{r\to+0}r^{-n}\operatorname{cap}^{\mathbb{R}}_{r}(a)=0,$

where the capacity $\mathrm{cap}^{\mathbb{R}}_{r}$ is taken in the usual sense
in $\mathbb{R}$ and depends on $r\in\mathbb{R}$.

###### Proof.

Let $n\in\mathbb{N}$ be fixed. By null capacity in the Levi-Civita field we
know that there exists a finite set $K\subset V$ with $a\in K$ such that for
the capacity $\mathrm{cap}^{\mathcal{R}}_{K}$ within $\mathcal{R}$

$\mathrm{cap}_{K}^{\mathcal{R}}(a)\prec\epsilon^{n+1}.$

Since by definition in this case $\mathrm{cap}_{K}^{\mathcal{R}}(a)$ is a
rational function on $\epsilon$, we can write

$\epsilon^{n+1}-\mathrm{cap}_{K}^{\mathcal{R}}(a)=\dfrac{a_{1}\epsilon^{n_{1}}+\dots+a_{k}\epsilon^{n_{k}}}{b_{1}\epsilon^{m_{1}}+\dots+b_{s}\epsilon^{m_{s}}},$

where the powers are increasing, i.e., $n_{1}<\dots<n_{k}$ and
$m_{1}<\dots<m_{s}$. Furthermore, from
$\mathrm{cap}_{K}^{\mathcal{R}}(a)\prec\epsilon^{n+1}$ we conclude that
$a_{1}/b_{1}>0$ and without loss of generality $a_{1},b_{1}>0$. Therefore,
there exists $r_{0}\in\mathbb{R}^{+}$ such that for any $0<r<r_{0}$

$\dfrac{a_{1}r^{n_{1}}+\dots+a_{k}r^{n_{k}}}{b_{1}r^{m_{1}}+\dots+b_{s}r^{m_{s}}}>0,$

i.e., $\operatorname{cap}_{K}(a)<r^{n+1},$ which gives as the capacity is
decreasing in $K$

$r^{-n}\mathrm{cap}^{\mathbb{R}}_{r}(a)\leq
r^{-n}\operatorname{cap}^{\mathbb{R}}_{K,r}(a)<r.$

Taking the limit $r\to 0$ yields the statement. ∎

######  Remark 3.12.

If we have a closer look on the above proof, we observe that for fixed
$n\in\mathbb{N}$ we get by
$\mathrm{cap}_{K}^{\mathcal{R}}(a)\prec\epsilon^{n+1}$ the approximation $K$
such that $\operatorname{cap}^{\mathbb{R}}_{K,r}(a)<r^{n+1}$ for small
$r\in\mathbb{R}$. That is we can construct finite graphs with a very small
energy, keeping control over the number of vertices. This could be useful for
numerical simulations on real recurrent graphs.

##  4\. Characterizations of the capacity

In Section 2 we defined the capacity of an infinite graph as the charge of a
vertex. In this section we show that the capacity can be also obtained as an
infimum of energies of functions normalized at a vertex and as the reciprocal
of the Green function. Moreover, we prove Theorem 2.9 in Section 4.4 which
shows that null, positive and divergent capacity does not depend on the vertex
and is therefore a global property.

###  4.1. Capacity as an infimum of energies

The first theorem shows that the effective capacity can be obtained as the
infimum of energies. Recall that, given a subset $A\subset\mathbb{K}$ of an
ordered field $(\mathbb{K},\succ)$, an infimum is a lower bound $a$ for $A$
such that $a\succeq a^{\prime}$ for every other lower bound $a^{\prime}$. In a
general ordered field $(\mathbb{K},\succ)$, a lower bounded set $A$ does not
necessarily have an infimum. However, if the infimum exists, it is unique and
we denote it by $\inf A$.

Recall that we consider graphs $(V,b,m)$ over non-Archimedean ordered fields
$(\mathbb{K},\succ)$ which satisfy Assumption 1.1 and Assumption 1.3.
Moreover, recall that $\mathrm{cap}_{n}$ denotes the capacity with respect to
distance balls of radius $n$.

######  Theorem 4.1.

For $a\in V$, we have

$\mathrm{cap}(a)=\inf\\{Q(\phi)\;\mid\;\phi\in C_{c}(V),\phi(a)=1\\}$

whenever either side exists.

###### Proof.

By Proposition 2.2 we have

$\displaystyle\mathrm{cap}_{n}(a)=\min\\{Q(\phi)\;\mid\;\phi\in
C_{c}(B_{n}(a)),\phi(a)=1\\}.$

We denote the set $E:=\\{Q(\phi)\;\mid\;\phi\in C_{c}(V),\phi(a)=1\\}$.

Assume the infimum $c=\inf E$ exists. For $\tau\succ 0$, let $\phi_{\tau}$ in
$C_{c}(V)$ and $\phi_{\tau}(a)=1$ be such that

$\displaystyle|Q(\phi_{\tau})-c|\preceq\tau.$

Then there exists $n(\tau)$ such that $\phi_{\tau}\in C_{c}(B_{n(\tau)}(a))$.
Hence, for all $n\geq n(\tau)$,

$\displaystyle c\preceq\mathrm{cap}_{n}(a)\preceq Q(\phi_{\tau})\preceq
c+\tau.$

This shows that $\mathrm{cap}_{n}(a)$ converges to $c$ as $n\to\infty$.

On the other hand, assume $c=\lim_{n\to\infty}\mathrm{cap}_{n}(a)$ exists.
Then, $c$ is a lower bound on $E$ since every $\phi\in C_{c}(V)$ is supported
in some ball. Assume now that there exists a lower bound $d\succ c$ on $E$.
Then there exists $n\in\mathbb{N}$ such that $d\succ\mathrm{cap}_{n}(a)\succeq
c$ which means there exists $\phi_{n}\in C_{c}(B_{n}(a))$ such that
$d\succ\mathrm{cap}_{n}(a)=Q(\phi_{n})$, which is a contradiction. ∎

###  4.2. Independence of null capacity on the vertex

In this subsection, we show that if the capacity is zero for one vertex, then
it is zero for all vertices. We start with the following lemma.

######  Lemma 4.2.

Let $x,y\in V$. Then, there is a constant $C_{x,y}$ such that for all $\phi\in
C_{c}(V)$

$\displaystyle|\phi(x)-\phi(y)|^{2}$ $\displaystyle\preceq C_{x,y}Q(\phi).$

###### Proof.

Let $x=x_{0}\sim x_{1}\sim x_{2}\sim\cdots\sim x_{n-1}\sim x_{n}=y$ be a path
connecting between $x$ and $y$. Then by triangle inequality we have for all
compactly supported functions $\phi\in C_{c}(V)$

$\displaystyle|\phi(x)-\phi(y)|^{2}$
$\displaystyle\preceq\left(\sum_{i=1}^{n}|\phi(x_{i-1})-\phi(x_{i})|\right)^{2}$
$\displaystyle\preceq\frac{n}{\min\limits_{i=1,\ldots,n}b(x_{i},x_{i-1})}\sum_{i=1}^{n}|\phi(x_{i})-\phi(x_{i-1})|^{2}b(x_{i},x_{i-1})$
$\displaystyle\preceq
2\left(\frac{n}{\min\limits_{i=1,\ldots,n}b(x_{i},x_{i-1})}\right)Q(\phi).$

This finishes the proof. ∎

######  Corollary 4.3 (Null capacity does not depend on the vertex).

If $\mathrm{cap}(a_{0})=0$ for some $a_{0}\in V$, then $\mathrm{cap}(a)=0$ for
all $a\in V$.

###### Proof.

By Theorem 4.1 we have $\mathrm{cap}(a_{0})=0$ if and only if there is a
sequence of functions $\phi_{k}\in C_{c}(V),k\in\mathbb{N}$ with
$\phi_{k}(a_{0})=1$ and $Q(\phi_{k})\to 0$. If $\mathrm{cap}(a_{0})=0$ for
some $a_{0}\in V$, then $\phi_{k}(a)\to 1$ for all $a\in V$ by the lemma
above. In turn, the sequence $\psi_{k}:=\dfrac{\phi_{k}}{\phi_{k}(a)}$ belongs
to $C_{c}(V)$, $\psi_{k}(a)=1$ and

$0\preceq Q\left(\psi_{k}\right)=\dfrac{Q(\phi_{k})}{\phi_{k}^{2}(a)}\to 0,$

which finishes the proof. ∎

###  4.3. Convergence of the solution of the Dirichlet problem

Next, we investigate convergence of solutions of the Dirichlet problems on
finite approximations of graph. It will allow us to prove that positive
capacity is independent of the vertex.

######  Definition 4.4 (Exhaustion).

A sequence $K_{n}$, $n\in\mathbb{N}_{0}$, of finite subsets $K_{n}\subset V$
is called an exhaustion of the graph, if for all $n\in\mathbb{N}_{0}$,

$K_{n}\subset K_{n+1}\qquad\mbox{and}\qquad V=\bigcup_{n=0}^{\infty}K_{n}.$

Next, we show that solutions of the Dirichlet problem pointwise increase along
an exhaustion. This is also referred to as domain monotonicity.

######  Lemma 4.5 (Domain monotonicity).

Let $a\in V$ be a fixed vertex and $(K_{n})$ be an exhaustion with $a\in
K_{0}$. Let $v_{n}$, $\widetilde{v}_{n}$ be the solution of the Dirichlet
problem (DP) and ($\widetilde{\mathrm{DP}}$) for $a$ and $K_{n}$. Then, for
any $x\in V,n\in\mathbb{N}$,

$\displaystyle 0\preceq v_{n}(x)$ $\displaystyle\preceq v_{n+1}(x)\preceq 1,$
$\displaystyle 0\preceq\widetilde{v}_{n}(x)$
$\displaystyle\preceq\widetilde{v}_{n+1}(x).$

Moreover, for any $x\in V$ there exists $n_{0}\in\mathbb{N}$ such that
$v_{n}(x),\widetilde{v}_{n}(x)\succ 0$ for all $n\geq n_{0}$.

###### Proof.

Firstly we note that $1\succeq v_{n}\succeq 0$ due to Proposition 2.2. We
observe that $u_{n}:=v_{n+1}-v_{n}$ satisfies $\Delta u_{n}=0$ on
$K_{n}\setminus\\{a\\}$. Hence, by the maximum principle, Proposition 2.1,

$\displaystyle\min_{K_{n}\setminus\\{a\\}}u_{n}\succeq\min_{S_{1}(K_{n})\cup\\{a\\}}u_{n}\succeq
0$

since $v_{n+1}\succeq 0$ and $v_{n}=0$ on $S_{1}(K)$. Hence, $v_{n+1}\succeq
v_{n}$ on $K_{n}\setminus\\{a\\}$ and, furthermore, $v_{n}(a)=1=v_{n+1}(a)$.
Finally, the eventual strict positivity in $x$ follows from Proposition 2.2 by
choosing $n_{0}$ such that $x\in K_{n_{0}}$.

The argument for $\widetilde{v}_{n}$ is analogous by setting
$\widetilde{u}_{n}:=\widetilde{v}_{n+1}-\widetilde{v}_{n}$ and observing
$\Delta\widetilde{u}_{n}=0$ on $K_{n}$ and $\widetilde{u}_{n}\succeq 0$
outside of $K_{n}$. Application of the maximum principle yields the
corresponding result. ∎

######  Theorem 4.6 (Convergence of solutions to (DP)).

Let $a\in V$ be a fixed vertex and $(K_{n})$ be an exhaustion with $a\in
K_{0}$. Let $v_{n}$ be the solution of the Dirichlet problem for $K_{n}$ and
$a$. Then,

$\displaystyle\mathrm{cap}(a)=m(a)\lim_{n\to\infty}\Delta
v_{n}(a)=b(a)-\lim_{n\to\infty}\sum_{y\in V}v_{n}(y)b(a,y)$

in case either limit exists. Moreover,

  1. (1)

$\mathrm{cap}(a)=0$ if and only if ${\lim_{n\to\infty}v_{n}(x)}=1$ for all
$x\in V$.

  2. (2)

$\mathrm{cap}(a)\succ 0$ if and only if ${\lim_{n\to\infty}v_{n}(x)}\prec 1$
for some $x\in V$ and the limit exists for all $x$.

  3. (3)

$\mathrm{cap}(a)$ does not exist if and only if ${\lim_{n\to\infty}v_{n}(x)}$
does not exist for some $x\in V$.

###### Proof.

For the formula, we use the definition of effective capacity for finite graphs

$\mathrm{cap}_{n}(a)=\Delta v_{n}(a)m(a)=\sum_{y\in V}(1-v_{n}(y))b(a,y).$

From this immediately follows that convergence of $v_{n}$ for all vertices
implies existence of capacity as $0\preceq v_{n}(y)\preceq 1$. Moreover,
”$\Leftarrow$” in (1) holds.

For “$\Rightarrow$” in (1) assume the graph has null capacity, i.e.,
$Q(v_{n})\to 0$. Then the statement follows from Lemma 4.2.

For “$\Leftarrow$” in (2) assume, that there is some $x\in V$ with
$\lim_{n\to\infty}v_{n}(x)\prec 1$ and the limit exists for all vertices. Then
$\mathrm{cap}(a)$ exists, and is not zero by “$\Rightarrow$” in (1).
Therefore, $\mathrm{cap}(a)\succ 0$.

To see “$\Rightarrow$ ” in (2) observe that in a non-Archimedean field, the
convergence of the sequence $v_{n}(x)$, $n\in\mathbb{N}$, is equivalent to the
convergence of $u_{n}(x):=v_{n+1}(x)-v_{n}(x)$ to $0$ by Proposition 3.1.
First, by linearity, Green’s formula and the fact that $v_{n},v_{n+1}$ are the
solutions of the Dirichlet problems and Proposition 2.2

$\displaystyle Q(u_{n})$ $\displaystyle=Q(v_{n+1})+Q(v_{n})-2\sum_{x\in
V}\Delta v_{n+1}(x)v_{n}(x)m(x)$
$\displaystyle=\mathrm{cap}_{n+1}(a)+\mathrm{cap}_{n}(a)-2\Delta
v_{n+1}(a)m(a)=\mathrm{cap}_{n}(a)-\mathrm{cap}_{n+1}(a)\to 0.$

Hence, we deduce by $u_{n}(a)=0$ and Lemma 4.2

$|v_{n+1}(x)-v_{n}(x)|=|u_{n}(x)-u_{n}(a)|\to 0.$

Thus, $(v_{n}(x))$ converges to a limit and this limit is strictly less than
$1$ for at least one vertex since otherwise we would be in the null capacity
case.

(3) “$\Leftrightarrow$”: This follows from all the above. ∎

######  Remark 4.7.

In (2) and (3) of Theorem 4.6, some can not be changed to all: Consider a
vertex, which is trivially connected to $a$ by one edge and is not connected
to any other vertex.

The theorem above gives a characterization of null capacity via null
sequences.

######  Corollary 4.8 (Null sequences).

The graph has null capacity if and only if there is a sequence of compactly
supported functions, converging pointwise to $1$, whose energies converge to
$0$.

###### Proof.

If the graph has null capacity, then for the sequence of solutions $(v_{n})$
of the Dirichlet problem on balls $B_{n}(a)$ we have $Q(v_{n})=\Delta
v_{n}(a)m(a)\to 0$ by Proposition 2.2. By Theorem 4.6 these solutions converge
to $1$ pointwise. On the other hand, if such a sequence exists, then infimum
of the energies in Theorem 4.1 is $0$, therefore, the graph has null capacity.
∎

###  4.4. Independence of positive capacity on the vertex

To see the independence of positive capacity on the vertex, we also need the
following lemma.

######  Lemma 4.9.

Let $(a_{n})$ and $(b_{n})$ be sequences in $\mathbb{K}$ satisfying
$a_{n+1}\succeq a_{n}\succ 0$ and $b_{n+1}\succeq b_{n}\succ 0$ for all
$n\in\mathbb{N}$. If the limit $\lim_{n\to\infty}a_{n}b_{n}$ exists, then the
limits $\lim_{n\to\infty}a_{n}$ and $\lim_{n\to\infty}b_{n}$ exist.

###### Proof.

We have

$\displaystyle
a_{n+1}b_{n+1}-a_{n}b_{n}=a_{n+1}b_{n+1}-a_{n+1}b_{n}+a_{n+1}b_{n}-a_{n}b_{n}=a_{n+1}(b_{n+1}-b_{n})+b_{n}(a_{n+1}-a_{n}).$

By the assumption the left hand side converges to $0$. Since

$a_{n+1}(b_{n+1}-b_{n})\succeq a_{1}(b_{n+1}-b_{n})\succeq 0$

we have $b_{n+1}-b_{n}\to 0$, which by “a student’s dream”, Proposition 3.1,
means that $(b_{n})$ converges. The same argument applies to $(a_{n})$. ∎

######  Corollary 4.10 (Positive capacity does not depend on the vertex).

If $\mathrm{cap}(a_{0})\succ 0$ for some $a_{0}\in V$, then $\mathrm{cap}(a)$
exists and $\mathrm{cap}(a)\succ 0$ for all $a\in V$.

###### Proof.

Assume $\mathrm{cap}(a_{0})\succ 0$. Let $v^{0}_{n}$ and $v_{n}$ be the
solutions of the Dirichlet problem (DP) on $K_{n}$ with respect to $a_{0}$ and
$a$. Then $Q(v^{0}_{n})\to\mathrm{cap}(a_{0})$ by Proposition 2.2 and the
definition of effective capacity. By Theorem 4.6, $v^{0}_{n}(x)$ converges for
all $x\in V$, in particular, $v^{0}_{n}(a)$ converges. Therefore, by Lemma 2.6
we have

$\dfrac{v^{0}_{n}(a)}{Q(v^{0}_{n})}=\dfrac{v_{n}(a_{0})}{Q(v_{n})}$

and, we infer that the right hand side converges as well. Moreover,
$v_{n+1}(a_{0})\succeq v_{n}(a_{0})\succ 0$ for $n$ large enough by Lemma 4.5
and $Q(v_{n})\succeq Q(v_{n+1})$ by Proposition 2.7 and Proposition 2.2.
Therefore, by Lemma 4.9 we get that $1/Q(v_{n})$ converges, i.e.
$\mathrm{cap}(a)$ exists. Finally, $\mathrm{cap}(a)\neq 0$ due to Lemma 4.3. ∎

Putting these results together we conclude Theorem 2.9, which states that
null, positive and divergent capacity is independent of the vertex.

###### Proof of Theorem 2.9.

The statements follow from Corollary 4.3 and Corollary 4.10. ∎

###  4.5. Capacity in terms of a Green function

In the next theorem we compute the capacity of a graph in terms of a Green
function.

######  Theorem 4.11 (Green function).

Let $(V,b)$ be an infinite graph, $(K_{n})$ an exhaustion and $a\in V$ a fixed
vertex. Then, for $G_{n}(x,y)=\Delta_{K_{n}}^{-1}1_{y}(x)$, $x,y\in V$, we
have

$\mathrm{cap}(a)=\lim_{n\to\infty}\dfrac{m(a)}{G_{n}(a,a)},$

when either side exists. Furthermore,

  1. (1)

the graph has null capacity if and only if
$\lim_{n\to\infty}G_{n}(x,y)=\infty$ for some (all) $x,y\in V$,

  2. (2)

the graph has positive capacity if and only if
$\lim_{n\to\infty}G_{n}(x,y)\in\mathbb{K}$ for some (all) $x,y\in V$,

  3. (3)

the graph has divergent capacity if and only if $\lim_{n\to\infty}G_{n}(x,y)$
does not exist for some (all) $x,y\in V$.

Moreover, in the positive capacity case $G(\cdot,a)$ is the smallest positive
solution $u$ of

$\displaystyle\Delta u=1_{a}.$

###### Proof.

Let $v_{n}$, $n\in\mathbb{N}$, be the solutions of the (DP) for $K_{n}$ and
$y$. We recall that by Lemma 2.5

$G_{n}(x,y)=\Delta_{K_{n}}^{-1}1_{y}(x)=\dfrac{v_{n}(y)m(x)}{Q(v_{n})}.$

This formula yields the results (1)-(3) via Theorem 4.6. In the positive
capacity case it is immediate to see by local finiteness

$\Delta G(x,a)=\lim_{n\to\infty}\Delta
G_{n}(x,a)=\lim_{n\to\infty}\Delta\Delta_{K_{n}}^{-1}1_{a}(x)=1_{a}(x).$

Furthermore, $G$ is positive by Proposition 2.2. For any other positive
solution $u$ we have that $w=u-G_{n}(\cdot,a)$ is harmonic on $K_{n}$ and
$w=u\succeq 0$ outside of $K_{n}$. Hence, $w\succeq 0$ by the maximum
principle, Proposition 2.1 and, therefore $u\succeq G(\cdot,a)$. ∎

######  Definition 4.12 (Green function).

For a positive capacity graph $(V,b,m)$, we call the function $G:V\times
V\to\mathbb{K}$ from the theorem above the _Green function_.

##  5\. Monotonicity law and Nash-Williams test

In this section we explore sufficient conditions on edge weights to determine
the type. We prove a Nash-Williams test which may come in a surprising form to
the reader who is not well acquainted with non-Archimedean fields.

######  Theorem 5.1 (Monotonicity law).

Let $b$ and $b^{\prime}$ be two graphs on $V$ with $b\preceq b^{\prime}$. Then

$\displaystyle\mathrm{cap}_{n}\preceq\mathrm{cap}_{n}^{\prime}$

and, in particular,

$\mathrm{cap}\preceq\mathrm{cap}^{\prime}$

whenever both effective capacities exist.

###### Proof.

For any $\phi\in C_{c}(V)$, we have by assumption

$\displaystyle Q(\phi)\preceq Q^{\prime}(\phi).$

Thus, Theorem 4.1 gives the statement. ∎

Next, we give a criterion for a graph to not have null capacity.

######  Corollary 5.2.

If the weights on the edges are bounded from below, i.e., there exists
$\tau\in\mathbb{K}^{+}$ such that $b(x,y)\succeq\tau 1_{b(x,y)\neq 0}$,
$x,y\in V$, then the graph does not have null capacity.

###### Proof.

Let $b^{\prime}=1_{b\neq 0}$. Then, by assumption there exists
$\tau\in\mathbb{K}_{+}$ such that $b\succeq\tau b^{\prime}$. By the theorem
above we infer

$\mathrm{cap}_{n}(a)\succeq\tau\mathrm{cap}^{\prime}_{n}(a).$

Then $\mathrm{cap}^{\prime}_{n}(a)$ is a rational number for any $n$ and can
not converge to zero in $\mathbb{K}$ as $n\to\infty$. ∎

Next, we turn to the non-Archimedean Nash-Williams test.

######  Theorem 5.3 (Nash-Williams test).

Let $a\in V$ be a vertex. Then, the following statements are equivalent:

  * (i)

One has

$\sum_{r=0}^{\infty}\dfrac{1}{b(\partial B_{r})}=\infty.$

  * (ii)

There is a subsequence of balls $(B_{n_{k}}(a))$ such that

$\lim_{{k}\to\infty}b(\partial B_{n_{k}}(a))=0.$

  * (iii)

There is a subsequence of balls $(B_{n_{k}}(a))$ such that

$\lim_{{k}\to\infty}\max_{(x,y)\in\partial B_{n_{k}(a)}}b(x,y)=0.$

If one of the above conditions is satisfied, then the graph has null capacity.

###### Proof.

The equivalence “(i) $\Leftrightarrow$ (ii)” is due to Lemma 3.2 in non-
Archimedean fields with countable cofinality.

The equivalence “(ii) $\Leftrightarrow$ (iii)” follows from the estimate

$0\preceq b(\partial
B_{n}(a))\preceq\mathcal{N}\max_{\begin{subarray}{c}d(x,a)=n\\\
d(y,a)=n-1\end{subarray}}b(x,y),$

where $\mathcal{N}$ is any infinitely large element in $\mathbb{K}$.

For the final implication, consider the test functions
$\phi_{k}:=1_{B_{n_{k}}(a)}$, $k\in\mathbb{N}$, which are in $C_{c}(V)$ due to
local finiteness. Then,

$\displaystyle
Q(\phi_{k})=\frac{1}{2}\sum_{x,y}b(x,y)(\phi_{k}(x)-\phi_{k}(y))^{2}=b(\partial
B_{n_{k}}(a))\to 0$

as $k\to\infty$. Hence, the statement follows from Theorem 4.1. ∎

######  Remark 5.4.

The classical Nash-Williams test [15] states that a graph over the real field
$\mathbb{R}$ has null capacity if condition (i) above is fulfilled. In the
real setting one only has the implications (i) $\Rightarrow$ (ii)
$\Rightarrow$ (iii). Moreover, the capacity type of graphs over $\mathbb{R}$
with weights $b(x,y)\in\\{0,1\\}$ corresponds to recurrence/transience of
simple random walks on graphs, which is a highly non-trivial question [20].
This is in sharp contrast to Corollary 5.2. The criteria in Corollary 5.2 and
Theorem (i) thus indicate that graphs over $\mathbb{R}$ and non-Archimedean
fields have rather different properties.

The Nash-Williams test, Theorem 5.3, has the following immediate implication.

######  Corollary 5.5.

Suppose that the edge weights $b(x,y)$ tend to zero towards infinity, i.e.,
for every $\tau\succ 0$ the set $\\{(x,y)\mid b(x,y)\succ\tau\\}$ is finite,
then the graph has null capacity.

###### Proof.

This follows by applying the condition (iii) from Theorem 5.3 above. ∎

Next, we study the Nash-Williams test in the special case of weakly
spherically symmetric graphs. Observe that in this special case the criterion
becomes indeed a characterization.

######  Corollary 5.6 (Nash-Williams test is a characterization for weakly
spherically symmetric graphs).

On a weakly spherically symmetric graph the following statements are
equivalent:

  1. (i)

The graph has null capacity.

  2. (ii)

There exists a subsequence of balls $(B_{n_{k}}(o))$ such that

$\lim_{{n_{k}}\to\infty}b(\partial B_{n_{k}}(o))=0.$

###### Proof.

The implication (ii)$\Rightarrow$(i) is immediate from Theorem 5.3 above. On
the other hand, if the graph has null capacity, i.e., (i) then
$b_{+}(n_{k})\to 0$ for some subsequence $(n_{k})$ by Theorem 3.3. However,
this is equivalent to (ii) since $b(\partial
B_{n_{k}+1}(o))=\\#S_{n_{k}}b_{+}(n_{k})$. ∎

##  6\. Hardy inequality

In the classical setting it is well known that a Hardy inequality on the graph
fails to hold if and only if the graph has null capacity, see e.g. [7, Chapter
6]. On non-Archimedean fields it is possible to deduce a similar statement,
see Theorem 6.2 below. As an important application of the Hardy inequality we
discuss an uncertainty-type principle afterwards.

######  Definition 6.1 (Hardy weight).

A Hardy weight is a non-trivial function
$\omega:V\to\mathbb{K}^{+}\cup\\{0\\}$ such that the corresponding Hardy
inequality

$Q(\phi)\succeq\sum_{x\in V}\phi^{2}(x)\omega(x),\qquad\phi\in C_{c}(V),$

holds true.

The following is the main result of this section.

######  Theorem 6.2 (Hardy inequality).

The following statements are equivalent:

  1. (i)

The graph is not of null capacity.

  2. (ii)

The Hardy inequality holds for some strictly positive Hardy weight.

  3. (iii)

The Hardy inequality holds for some Hardy weight.

  4. (iv)

There exists a non-trivial weight $\omega:V\to\mathbb{K}^{+}\cup\\{0\\}$ such
that

$Q(\phi)\succeq\left(\sum_{x\in V}\phi(x)\omega(x)\right)^{2},\qquad\phi\in
C_{c}(V).$

###### Proof.

(i) $\implies$ (ii): Assume that the graph is not of null capacity (i.e.,
positive or divergent capacity). Then for each $x\in V$ exists a positive
lower bound $m_{x}\succ 0$ such that

$Q(v_{n})=\mathrm{cap}_{n}(x)\succeq m_{x},$

for all $n\in\mathbb{N}$, where $v_{n}$ is the solution of the (DP) for
$B_{n}(x)$ and the first equality holds according to Proposition 2.2. Now, the
vertex set $V$ is countable. Since countable cofinality is equivalent to the
existence of a function $c:V\to\mathbb{K}^{+}$ with $\sum_{x\in V}c_{x}=1$
(see [1, Theorem 6]), we can define a strictly positive function
$\omega:V\to\mathbb{K}^{+}$ by

$\omega(x)=m_{x}c_{x}.$

For $\phi\in C_{c}(V)$, by Proposition 2.2 we have for all sufficiently large
$n\in\mathbb{N}$ that

$\sum_{x\in V}\omega(x)\phi^{2}(x)=\sum_{x\in
V}c_{x}m_{x}\phi^{2}(x)\preceq\sum_{x\in
V}c_{x}\mathrm{cap}_{n}(x)\phi^{2}(x)\preceq\sum_{x,\phi(x)\neq
0}c_{x}Q\left(\frac{\phi}{\phi(x)}\right)\phi^{2}(x)\preceq Q(\phi).$

(ii) $\implies$ (iii): This is clear.

(iii) $\implies$ (iv): Assume there is a Hardy weight $\omega$. Since any
smaller function is also a Hardy weight we can assume without loss of
generality that $\sum_{x}\omega(x)\preceq 1$ and that $\omega$ is the square
of a non-trivial positive function. Thus, the statement follows directly by
the Cauchy–Schwarz inequality

$\left(\sum_{x\in V}\phi(x)\omega(x)\right)^{2}\preceq\left(\sum_{x\in
V}\phi^{2}(x)\omega(x)\right)\left(\sum_{x\in V}\omega(x)\right)\preceq
Q(\phi).$

(iv) $\implies$ (i): Let $x\in V$ be such that $\omega(x)\succ 0$. For
$n\in\mathbb{N}$, let $v_{n}$ be the solution of the Dirichlet problem (DP) on
$B_{n}(x)$. Then, since $v_{n}\succ 0$ on $B_{n}(x)$ and $v_{n}(x)=1$ we have

$\displaystyle 0\prec\omega(x)^{2}\preceq\left(\sum_{y\in
B_{n}(x)}\omega(y)v_{n}(y)\right)^{2}\preceq Q(v_{n})=\mathrm{cap}_{n}(x)$

for all $n\in\mathbb{N}$. As there is a lower bound on $\mathrm{cap}_{n}(x)$,
the quantity cannot converge to zero as $n\to\infty$. Therefore, the graph is
not of null capacity . ∎

######  Remark 6.3.

By going in the proof directly from (ii) to (iv) it is not hard to see that
the weight $\omega$ in (iv) can also be chosen strictly postive.

######  Remark 6.4 (Uncertainty-type principle).

Assume that the graph is not of null capacity with a strictly positive Hardy
weight $\omega$ and the underlying field is real-closed. (That means in
particular that one can take square roots of positive elements in
$\mathbb{K}$.) Then by the Cauchy–Schwarz and Hardy inequality, we derive for
all $\phi\in C_{c}(V)$ that

$\displaystyle\sum_{x\in V}\phi^{2}(x)$ $\displaystyle=\sum_{x\in
V}\bigl{(}\omega^{-1/2}(x)\phi(x)\bigr{)}\bigl{(}\omega^{1/2}(x)\phi(x)\bigr{)}$
$\displaystyle\preceq\left(\sum_{x\in
V}\frac{\phi^{2}(x)}{\omega(x)}\right)^{1/2}\left(\sum_{x\in
V}\omega(x)\phi^{2}(x)\right)^{1/2}$ $\displaystyle\preceq\left(\sum_{x\in
V}\frac{\phi^{2}(x)}{\omega(x)}\right)^{1/2}\cdot Q^{1/2}(\phi).$

This is a version of the Heisenberg–Pauli–Weyl inequality on graphs which is
also known as uncertainty principle. An interpretation is the following: the
position and momentum of a particle can not be determined simultaneously, if
the graph is not of null capacity.

######  Remark 6.5 (Largest Hardy weights).

Theorem 6.2 says that existence of a Hardy weight is equivalent to having
either divergent or positive capacity. We mention briefly that divergent and
positive capacity graphs can be further distinguished by existence of Hardy
weights with a certain maximality property.

Let $a\in V$. If the graph has positive capacity, then
$\omega_{a}=\mathrm{cap}(a)\cdot 1_{a}$ is a Hardy weight. Moreover,
$\omega_{a}$ is maximal at $a$ in the sense that
$\omega(a)\preceq\omega_{a}(a)=\mathrm{cap}(a)$ for all Hardy weights
$\omega$. On the other hand, if the graph has divergent capacity, then there
exists no Hardy weight $\omega$ with this maximality property. More precisely,
for every Hardy weight $\omega$, there exists a Hardy weight $\omega^{\prime}$
with $\omega(a)\prec\omega^{\prime}(a)$. These claims are straightforward
consequences of Theorem 4.1.

##  7\. Positive superharmonic functions

In this section we discuss existence of positive superharmonic functions. We
consider a graph $b$ over $(V,m)$ and a non-Archimedean ordered field
$\mathbb{K}$. A function $u:V\to\mathbb{K}$ is called superharmonic if $\Delta
u\succeq 0$.

###  7.1. Ground state transform and local Harnack inequality

In this subsection we discuss two standard tools to study superharmonic
functions. The first is a ground state transform and the second one is a local
Harnack inequality. For analogous results on graphs over the real field see
e.g. [5, 7, 10] and references therein. While the proofs carry over verbatim
from this case we still give them here for the sake of being self-contained.

######  Lemma 7.1.

For all $\phi\in C_{c}(V)$ and $u:V\to\mathbb{K}$

$\Delta\left(u\cdot\phi(x)\right)={\phi(x)}\Delta{u(x)}+\frac{1}{m(x)}\sum_{y\in
V}b(x,y)u(y)(\phi(x)-\phi(y)),$

and if $u\succ 0$, then

$\Delta\phi(x)=\dfrac{\phi(x)}{u(x)}\Delta{u(x)}+\dfrac{1}{u(x)m(x)}\sum_{y\in
V}b(x,y)u(y)u(x)\left(\dfrac{\phi(x)}{u(x)}-\dfrac{\phi(y)}{u(y)}\right).$

###### Proof.

We compute straightforwardly

$\displaystyle\Delta\left(u\cdot\phi(x)\right)$
$\displaystyle=\frac{1}{m(x)}\sum_{y\in
V}\left(u(x)\phi(x)-u(y)\phi(x)+u(y)\phi(x)-u(y)\phi(y)\right)b(x,y)$
$\displaystyle=\frac{1}{m(x)}\sum_{y\in
V}\left(u(x)-u(y)\right)\phi(x)b(x,y)+\frac{1}{m(x)}\sum_{y\in
V}u(y)\left(\phi(x)-\phi(y)\right)b(x,y)$

and the statement follows. For the second statement consider the function
${\phi}/{u}$ instead of $\phi$. ∎

######  Theorem 7.2 (Ground state transform).

For all $\phi\in C_{c}(V)$ and $u:V\to\mathbb{K}$ with $u\succ 0$

$Q(\phi)-\langle\frac{\Delta
u}{u}\phi,\phi\rangle=Q_{u}\left(\dfrac{\phi}{u}\right),$

where

$Q_{u}(\psi):=\dfrac{1}{2}\sum_{x,y\in
V}b(x,y)u(x)u(y)\left(\psi(x)-\psi(y)\right)^{2},\qquad\psi\in C_{c}(V).$

In fact, $Q_{u}(\psi)$ is the energy of $\psi$ with respect to the graph
$b_{u}(x,y)=b(x,y)u(x)u(y)$, $x,y\in V$.

###### Proof.

Applying Green’s formula (GF) first to the graph $b$, then using Lemma 7.1 and
then applying (GF) to the graph $b_{u}$ yields

$\displaystyle Q(\phi)-\langle\frac{\Delta u}{u}\phi,\phi\rangle$
$\displaystyle=\langle(\Delta-\frac{\Delta u}{u})\phi,\phi\rangle=\sum_{x\in
V}\dfrac{\phi(x)}{u(x)}\sum_{y\in
V}b(x,y)u(y)u(x)\left(\dfrac{\phi(x)}{u(x)}-\dfrac{\phi(y)}{u(y)}\right)=Q_{u}\left(\frac{\phi}{u}\right).$

This finishes the proof. ∎

Next, we show a local Harnack inequality.

######  Theorem 7.3 (Local Harnack inequality).

Let $W\subset V$ be finite and connected. Then there exists
$C_{W}\in\mathbb{K}^{+}$ such that for every $u:V\to\mathbb{K}^{+}\cup\\{0\\}$
which is superharmonic on $W$ we have

$\max_{x\in W}u(x)\preceq C_{W}\cdot\min_{x\in W}u(x).$

In particular, $u\succ 0$ on $W$ whenever $u\not\equiv 0$ on $W$.

###### Proof.

Rewriting $\Delta u\succeq 0$, we get

$b(x)u(x)\succeq\sum_{y\in V}b(x,y)u(y)\succeq b(x,z)u(z)$

for all $x\in W,z\sim x$. Let $x,y\in W$ and $x=x_{0}\sim x_{1}\sim\dots\sim
x_{n}=y$ be a path in $W$. Then by the above

$u(x)=u(x_{0})\preceq\dfrac{b(x_{1})}{b(x_{0},x_{1})}u(x_{1})\preceq\prod_{i=1}^{n}\dfrac{b(x_{i})}{b(x_{i-1},x_{i})}u(x_{n})=C_{x,y}u(y).$

The maximum $C_{W}=\max_{x,y\ W}C_{x,y}$ exists due to finiteness of $W$ and
the statement follows. The “in particular” is obvious. ∎

###  7.2. Necessary and sufficient conditions for existence

In this subsection we prove the existence of positive superharmonic functions
for positive capacity graphs and non-existence for null capacity graphs.
Afterwards, we consider the divergent capacity case and give sufficient
criteria for existence.

######  Theorem 7.4.

(a) If there exists a positive superharmonic function which is not harmonic,
then the graph does not have null capacity.
(b) If the non-Archimedean field is real closed, then existence of a non-
constant positive superharmonic function implies that the graph does not have
null capacity.

###### Proof.

(a) By the local Harnack inequality, Theorem 7.3, any non-constant positive
superharmonic function $u$ is strictly positive. Therefore, by the ground
state transform, Theorem 7.2, $\omega=m\cdot\Delta u/u$ is a Hardy weight, as
$Q_{u}\succeq 0$. Thus, the graph does not have null capacity by Theorem 6.2.
(b) In a real closed field one can take square roots and therefore can
consider the function $u^{1/2}$. A direct calculation, cf. [9, Lemma 2.2.],
yields

$\displaystyle\Delta u^{1/2}(x)=\frac{1}{2u^{1/2}(x)}\Delta
u(x)+\frac{1}{2u^{1/2}(x)m(x)}\sum_{y\sim x}(\nabla_{xy}u^{1/2})^{2}b(x,y).$

Hence $u^{1/2}$ is non-harmonic whenever $u$ is non-constant and the same
argument as above while replacing $u$ with $u^{1/2}$ yields the result. ∎

######  Theorem 7.5.

If the graph has positive capacity, then there exists a strictly positive non-
constant superharmonic function.

###### Proof.

For positive capacity graphs the Green function $G(\cdot,y)$ from Theorem 4.11
is a positive superharmonic function. By the Harnack inequality, Theorem 7.3
it is also strictly positive. ∎

In the case of divergent capacity graphs the question of existence of positive
superharmonic function remains open. In the following, we consider some
examples and present a sufficient condition on existence of positive
superharmonic functions on divergent capacity graphs.

######  Example 7.6 (Existence and non-uniqueness).

In Example 3.8, i.e. the path graph over the Levi-Civita field $\mathcal{R}$
with $b(k,k+1)=1$ and arbitrary $m:\mathbb{N}_{0}\to\mathbb{K}^{+}$, we can
easily see that the function

$u(k)=1-k\epsilon$

is positive and superharmonic. Specifically, $\Delta u(0)=\epsilon/m(0)$ and
for any $k\in\mathbb{N}$ we have $\Delta u(k)=0.$ Therefore, $u$ is not just
superharmonic, but also a solution of the Dirichlet problem on the infinite
graph $V$ with $u(0)=1$. The solution is obviously not unique, since one can
take any arbitrary infinitesimal $\tau\in\mathcal{R}$ instead of $\epsilon$.

Next, we present a theorem which gives a sufficient condition for having a
positive superharmonic function on a graph. Recall the notation of $S_{n}$ and
$b_{\pm}$ which were defined with respect to a root vertex $o$ from Section
3.1. The following theorem is a generalization of Example 7.6.

######  Theorem 7.7.

Let $o\in V$ be given. Suppose that there exists $c\in\mathbb{K}^{+}$ such
that

$\dfrac{b_{-}(x)}{b_{+}(x)}\preceq c$

for all $x\in V$ and $c^{n}\prec\tau^{-1}$ for all $n\in\mathbb{N}$ and some
infinitesimal $\tau\in\mathbb{K}^{+}$.

  * (a)

If $c\succ 1$, then the following function is positive and superharmonic

$u(x)=1-{c^{|x|}}\tau,\qquad x\in V.$

  * (b)

If $c\preceq 1$, then the following function is positive and superharmonic

$v(x)=1-|x|\tau\qquad x\in V.$

In particular, in either case there exists a positive superharmonic function
on $V$.

###### Proof.

(a) Since $c^{-n}\succ\tau$, we have $u\succ 0$. As $c\succ 1$, we have that
$u$ is decaying in $|x|$ and $\Delta u(o)\succ 0$. Moreover,

$\displaystyle\frac{m(x)}{\tau b_{+}(x)}\Delta
u(x)=(c^{|x|+1}-c^{|x|})+\frac{b_{-}(x)}{b_{+}(x)}(c^{|x|-1}-c^{|x|})\succeq
0.$

(b) Clearly, $v\succ 0$ and $v$ is decaying which gives $\Delta v(o)\succeq
0$. Moreover,

$\displaystyle\frac{m(x)}{\tau b_{+}(x)}\Delta
v(x)=1-\frac{b_{-}(x)}{b_{+}(x)}\succeq 0.$

This shows the existence of positive superharmonic functions in either case. ∎

######  Remark 7.8.

The conditions of Theorem 7.7 are satisfied if there exists a rational
$c\in\mathbb{Q}$ with ${b_{-}}/{b_{+}}\preceq c$.

The following example shows that the condition ${b_{-}}/{b_{+}}\preceq c$ in
Theorem 7.7 is not necessary.

######  Example 7.9.

Let a path graph with $m(k)=1$, $b(2k,2k+1)=1$ and $b(2k-1,2k)=\epsilon$ for
$k\in\mathbb{N}$ over the Levi-Civita field $\mathcal{R}$ be given. Then the
function

$\displaystyle u(2k)=1-k\epsilon-k\epsilon^{2},\qquad
u(2k-1)=1-(k-1)\epsilon-k\epsilon^{2}$

is positive and superharmonic. Indeed,

$\displaystyle\Delta u(0)=\epsilon^{2}$ and $\displaystyle\Delta
u(n)=0,\;n>0.$

One can easily see that this graph does not allow for a positive superharmonic
function in the form $1-c_{n}\tau$, where $\tau$ is a fixed infinitesimal,
while $c_{n}\in\mathbb{Q}$, i.e. we need two different infinitesimals to
compensate $b_{-}(1)/b_{+}(1)=\epsilon^{-1}$. Furthermore, if we compare this
example with the null capacity graph of Example 3.6, we see that
$\max_{S_{n}}b_{-}/b_{+}=\epsilon^{-1}$ for both graphs. Therefore, the
necessary condition on having a positive superharmonic function can not be
formulated just in terms of an upper bound for $b_{-}/b_{+}$.

##  8\. The transition operator

In this section we study the transition operator $P=I-\Delta$ arising from a
graph, which is a well known analytical tool in the study of random walks.
Specifically, throughout this section we choose the function
$m:V\to\mathbb{K}^{+}$ to be given by the vertex degree

$m(x)=b(x)=\sum_{z\in V}b(x,z),\qquad x\in V.$

While this section can be seen as a first step towards a probabilistic
interpretation, we refrain from doing so at this point. The reason is that
such an interpretation is somewhat more involved in the non-Archimedean
setting, as already discussed in Remark 2.12. Here we rather study the
operator theoretic properties of $P$. Specifically, we show how this operator
can be used to obtain an inverse of the Laplacian under certain circumstances.
For a background on the probability theory of random walks on real graphs we
refer the reader to [20].

###  8.1. Basic properties

For a graph $b$ over $V$, the _transition probabilities_ are defined by

$p(x,y)=\dfrac{b(x,y)}{b(x)},\qquad x,y\in V.$

Then, we have

$\displaystyle\sum_{y\in V}p(x,y)=1,\qquad x\in V.$

These transition probabilities give rise to the transition operator $P:C(V)\to
C(V)$ acting on functions $f:V\to\mathbb{K}$ as

$\displaystyle Pf(x)=\sum_{y\in V}p(x,y)f(y),\qquad x\in V.$

With the choice of the function $m(x)=b(x)$, $x\in V$, we have

$\Delta=I-P.$

In analogy with real valued probability we can define powers $P^{n}$,
$n\in\mathbb{N}_{0}$, of $P$ where $P^{0}=I$ by definition. This is possible
as by local finiteness of the graph, each matrix element of $P^{n}$ is a
finite sum of products of finitely many matrix elements of $P$, i.e.,

$P^{n}(x,y)=P^{n}1_{y}(x)=\sum_{\begin{subarray}{c}\text{paths $(x_{i})$ of
length $n$}\\\ \text{from $x=x_{0}$ to
$y=x_{n}$}\end{subarray}}p(x,x_{1})p(x_{1},x_{2})\dots p(x_{n-1},y),\qquad
x,y\in V.$

Let us further denote

$\Pi^{n}(x,y)=\max_{\begin{subarray}{c}\text{paths $(x_{i})$ of length $n$}\\\
\text{from $x=x_{0}$ to $y=x_{n}$}\end{subarray}}p(x,x_{1})p(x_{1},x_{2})\dots
p(x_{n-1},y),\qquad x,y\in V.$

For a finite subset $K\subseteq V$, we denote by $P_{K}$ the restriction of
$P$ to $C_{c}(K)$ or $C(K)$, respectively. We also denote the corresponding
restriction to paths within $K$ by $\Pi^{n}_{K}$.

The next lemma shows that the behavior of the matrix elements $P^{n}(x,y)$ of
$P^{n}$ for non-Archimedean fields is completely determined by the most
probable paths.

######  Lemma 8.1 (Basic properties).

  * (a)

For $x,y\in V$, we have $\lim_{n\to\infty}P^{n}(x,y)=0$ if and only if
$\lim_{n\to\infty}\Pi^{n}(x,y)=0.$

  * (b)

If $\lim_{n\to\infty}P^{n}(x,y)=0$ for some $x,y\in V$, then
$\lim_{n\to\infty}P^{n}(x,y)=0$ for all $x,y\in V$.

  * (c)

For $x,y\in V$, we have $\lim_{n\to\infty}P^{n}(x,y)=0$ if and only if
$\sum_{n=0}^{\infty}P^{n}(x,y)$ exists.

Corresponding statements hold if $P^{n}$ is replaced by $P^{n}_{K}$ for some
finite $K\subseteq V$.

###### Proof.

(a) Let $\mathcal{N}$ be an infinitely large element of $\mathbb{K}$. Then
since the number of paths from $x$ to $y$ of given length $n\in\mathbb{N}$ is
a natural number, we have

$\Pi^{n}(x,y)\preceq P^{n}(x,y)\preceq\mathcal{N}\Pi^{n}(x,y),$

and the statement follows.

(b) Let $x_{1},y_{1},x_{2},y_{2}\in V$ and
$d_{x}=\operatorname{dist}(x_{1},x_{2})$ and
$d_{y}=\operatorname{dist}(y_{1},y_{2})$. For $n$ large enough, we have

$\Pi^{n}(x_{1},y_{1})\succeq\Pi^{d_{x}}(x_{1},x_{2})\Pi^{n-d_{x}-d_{y}}(x_{2},y_{2})\Pi^{d_{y}}(y_{2},y_{1}).$

Since $\Pi^{d_{x}}(x_{1},x_{2})$ and $\Pi^{d_{y}}(y_{2},y_{1})$ does not
depend on $n$, the statement follows.

(c) This follows from the “students dream” Proposition 3.1. ∎

###  8.2. Inverting the Laplacian

Let $K\subseteq V$ be finite and connected. Recall that the operator
$\Delta^{-1}_{K}$ is defined via the solution of the renormalized Dirichlet
problem ($\widetilde{\mathrm{DP}}$) on $K$, confer Lemma 2.5.

######  Theorem 8.2.

If $\lim_{n\to\infty}P^{n}(x,y)=0$ for some (all) $x,y\in V$, then for all
$\phi\in C_{c}(V)$

$\Delta\sum_{n=0}^{\infty}P^{n}\phi=\sum_{n=0}^{\infty}P^{n}\Delta\phi=\phi.$

A corresponding statement holds for the restrictions $P_{K}$ and $\Delta_{K}$
to finite subsets $K\subseteq V$. In particular, if
$\lim_{n\to\infty}P^{n}_{K}(x,y)=0$ for some (all) $x,y\in K$, then for all
$\phi\in C_{c}(K)$

$\displaystyle\Delta^{-1}_{K}\phi=\sum_{n=0}^{\infty}P^{n}_{K}\phi.$

###### Proof.

Since $\Delta=I-P$, the Laplacian and the partial sums clearly commute.
Moreover, we can explicitly compute for the matrix elements

$\displaystyle\left((I-P)\sum_{n=0}^{N}P^{n}\right)(x,y)=\sum_{n=0}^{N}P^{n}(x,y)-\sum_{n=0}^{N}P^{n+1}(x,y)=I(x,y)-P^{N+1}(x,y)\to
I(x,y)$

as $N\to\infty$. Thus, we can conclude the statements for finitely supported
function. The argument for the restrictions is the same verbatim. Furthermore,
since inverses are unique the equality follows. ∎

######  Remark 8.3.

Over a non-Archimedean field the equality
$\Delta^{-1}_{K}=\sum_{n=0}^{\infty}P_{K}^{n}$ is _not_ necessarily true in
general, since the right hand side may not converge. For example, consider any
graph with real/rational weights over a non-Archimedean ordered field.

######  Lemma 8.4.

Let $(K_{j})$ be an exhaustion. Then for any $j,n\in\mathbb{N}$ and for any
$x,y\in V$,

$P^{n}(x,y)\succeq P^{n}_{K_{j+1}}(x,y)\succeq P^{n}_{K_{j}}(x,y).$

Moreover, for any $x,y\in V$ and $n\in\mathbb{N}$ there exists
$j_{0}\in\mathbb{N}$ such that for any $j\geq j_{0}$

$P^{n}_{K_{j}}(x,y)=P^{n}(x,y).$

###### Proof.

The first statement is obvious. For the equality take ${j_{0}}$ such that
$B_{n}(x),B_{n}(y)\subset K_{j_{0}}$. ∎

Recall that in the positive capacity case, we have the existence of a Green’s
function $G$ by Theorem 4.11.

######  Theorem 8.5.

If $\lim_{n\to\infty}P^{n}(x,y)=0$ for some $x,y\in V$, then the graph has
positive capacity. In this case, we have for any exhaustion $(K_{j})$ and all
$\phi\in C_{c}(V)$

$\sum_{n=0}^{\infty}P^{n}\phi=\lim_{j\to\infty}\Delta^{-1}_{K_{j}}\phi.$

In particular, for all $x,y\in V$

$\sum_{n=0}^{\infty}P^{n}(x,y)=G(x,y).$

###### Proof.

Let $x,y\in V$. From Lemma 8.4 it follows, that if $P^{n}(x,y)\to 0$, then
$P^{n}_{K}(x,y)\to 0$ for any finite $K\subset V$. Theorem 8.2, therefore,
yields together with Lemma 8.4

$\sum_{n=0}^{\infty}P^{n}(x,y)-\Delta^{-1}_{K_{j}}1_{x}(y)=\sum_{n=0}^{\infty}P^{n}(x,y)-\sum_{n=0}^{\infty}P^{n}_{K_{j}}(x,y)\succeq
0.$

Moreover, let $\tau\in\mathbb{K}^{+}$ and choose $k$ such that
$\sum_{n=k}^{\infty}P^{n}(x,y)\preceq\tau.$ By Lemma 8.4 we can choose $j_{0}$
such that $P_{K_{j}}^{n}(x,y)=P^{n}(x,y)$ for $n\leq k$ and $j\geq j_{0}$.
Thus,

$\sum_{n=0}^{\infty}P^{n}(x,y)-\Delta^{-1}_{K_{j}}1_{x}(y)=\sum_{n=k}^{\infty}(P^{n}(x,y)-P^{n}_{K_{j}}(x,y))\preceq\sum_{n=k}^{\infty}P^{n}(x,y)\preceq\tau.$

Thus, the desired convergence
$G_{j}(x,y)=\Delta^{-1}_{K_{j}}1_{y}(x)\to\sum_{n=0}^{\infty}P^{n}(x,y)$
follows and the graph has positive capacity by Theorem 4.11 which also gives
the “in particular” statement. ∎

######  Theorem 8.6.

Let a positive capacity graph be given. If for some $x,y\in V$ and some
exhaustion $(K_{j})$ we have $\lim_{n\to\infty}P_{K_{j}}^{n}(x,y)=0$ for all
$j$, then $P^{n}(x,y)\to 0$ for all $x,y\in V$.

###### Proof.

If $P_{K_{j}}^{n}(x,y)\to 0,$ $n\to\infty$, for some $x,y\in V$, then this
holds for all $x,y\in V$ by Lemma 8.1. So fix $x,y\in V$. By Theorem 4.11 and
the second part of Theorem 8.2 we have

$G(x,y)=\lim_{j\to\infty}\Delta^{-1}_{K_{j}}1_{x}(y)=\lim_{j\to\infty}\sum_{n=0}^{\infty}P_{K_{j}}^{n}(x,y).$

We now show that this is equal to $\sum_{n=0}^{\infty}P^{n}(x,y)$. To this end
observe that for all $k$ and $j$ large enough by Lemma 8.4

$\displaystyle
G(x,y)-\sum_{n=0}^{k}P^{n}(x,y)=G(x,y)-\sum_{n=0}^{k}P_{K_{j}}^{n}(x,y)\succeq
G(x,y)-\Delta^{-1}_{K_{j}}1_{x}(y)\succeq 0,$

where the last inequality follows from domain monotonicity of solutions
$\widetilde{v}_{j}=\Delta^{-1}_{K_{j}}1_{x}$ of the renormalized Dirichlet
problem ($\widetilde{\mathrm{DP}}$), Lemma 4.5. On the other hand, by the
equality in the beginning of the proof, for given $\tau\in\mathbb{K}^{+}$ we
obtain the existence of $j_{0}\in\mathbb{N}$ such that for $j\geq j_{0}$

$\displaystyle G(x,y)\preceq\sum_{n=0}^{\infty}P_{K_{j}}^{n}(x,y)+\tau.$

Moreover, by convergence of the series $\sum_{n}P^{n}_{K_{j}}$ for every $j$
there is $k_{j}$ such that for all $k\geq k_{j}$

$\displaystyle\sum_{n=k+1}^{\infty}P_{K_{j}}^{n}(x,y)\preceq\tau.$

Thus, we obtain for all $k\geq k_{j_{0}}$

$\displaystyle 2\tau\succeq
G(x,y)-\sum_{n=0}^{\infty}P_{K_{j_{0}}}^{n}(x,y)+\sum_{n=k+1}^{\infty}P_{K_{j_{0}}}^{n}(x,y)\succeq
G(x,y)-\sum_{n=0}^{k}P^{n}(x,y).$

Hence, we infer, that $\sum_{n=0}^{\infty}P^{n}(x,y)$ converges. Thus, the
$P^{n}(x,y)$ converge to zero by the “student’s dream”, Proposition 3.1. ∎

The following example shows that the condition of positive capacity is
necessary in Theorem 8.6, namely, a graph with
$\lim_{n\to\infty}P_{K_{j}}^{n}(x,y)=0$ for some exhaustion $(K_{j})$ can have
divergent capacity. Then $P^{n}(x,y)$ does not converge to zero due to Theorem
8.5. Below in Theorem 8.8 we will show that it can be not of null capacity
either.

######  Example 8.7 ($P_{K_{j}}^{n}(x,y)\to 0$ but $P^{n}(x,y)\not\to 0$).

We consider the path graph on $V=\mathbb{N}_{0}$, see Figure 1, over the Levi-
Civita field $\mathcal{R}$ with

$b(k,k+1)=\epsilon^{\frac{1}{2^{k}}}.$

Then, we can estimate
$b(k)=b(k,k+1)+b(k,k-1)=\epsilon^{\frac{1}{2^{k}}}(1+\epsilon^{\frac{1}{2^{k}}})$
by

$\epsilon^{\frac{1}{2^{k}}}\prec b(k)\prec 2\epsilon^{\frac{1}{2^{k}}}$

which readily gives

$\dfrac{1}{2}\prec P(k,k+1)\preceq 1\qquad\mbox{ and
}\qquad\dfrac{1}{2}\epsilon^{\frac{1}{2^{k}}}\prec
P(k,k-1)\prec\epsilon^{\frac{1}{2^{k}}}.$

Then, for the finite approximation $L=\\{0,1,\dots,l\\}$ of length
$l\in\mathbb{N}$, we have

$\Pi^{2n}_{L}(0,0)\preceq\epsilon^{\frac{n}{2^{l}}}\to 0,\qquad n\to\infty.$

Hence, $P_{L}^{n}(0,0)\to 0$ for $n\to\infty$.

On the other hand, for the infinite graph

$\Pi^{2n}(0,0)\succeq\dfrac{1}{2^{2n}}\prod_{k=1}^{n}\epsilon^{\frac{1}{2^{k}}}=\dfrac{1}{2^{2n}}\epsilon^{\sum_{k=1}^{n}\frac{1}{2^{k}}}\succ\dfrac{1}{2^{2n}}\epsilon\succ\epsilon^{2},$

so $P^{n}(0,0)\not\to 0$.

Moreover, the effective capacity of finite approximations is given by Theorem
3.3 as

$\mathrm{cap}_{L}(0)=\left(\sum_{k=0}^{l}\dfrac{1}{b(k,k+1)}\right)^{-1}=\left(\sum_{k=0}^{l}\epsilon^{-\frac{1}{2^{k}}}\right)^{-1}.$

Since $\epsilon^{-\frac{1}{2^{k}}}\succ 1$ the series does not converge to an
element of $\mathbb{K}^{+}$ and since it is bounded above by $\epsilon^{-2}$
it does not converges to $\infty$. Thus, the graph has divergent capacity.

######  Theorem 8.8.

If for some $x,y\in V$ and some exhaustion $(K_{j})$, we have
$P_{K_{j}}^{n}(x,y)\to 0$, $n\to\infty$ for all $j$, then the graph does not
have null capacity.

###### Proof.

Observe that $P^{n}_{K_{j}}(x,y)\preceq 1$. We have by Theorem 8.2 and for any
infinitely large element $\mathcal{N}\in\mathbb{K}$ by assumption

$G_{j}(x,y)=\Delta^{-1}_{K_{j}}(x,y)=\sum_{n=0}^{\infty}P^{n}_{K_{j}}(x,y)\prec\mathcal{N}.$

Hence, $G_{j}(x,y)$ does not converge to infinity and therefore the graph does
not have null capacity by Theorem 4.11. ∎

######  Theorem 8.9 (Sufficient condition for $P^{n}(x,y)\not\to 0$ ).

If the graph contains a vertex $x_{0}\in V$ such that
$P^{k}(x_{0},x_{0})\succeq c$ for some $c\in\mathbb{Q}$ and some $k\geq 2$,
then $P^{n}(x,y)\not\to 0$ as $n\to\infty$ for all $x,y\in V$.

###### Proof.

By assumption there is rational number $c\succ 0$ such that for any $n$

$\displaystyle P^{kn}(x_{0},x_{0})\succeq c^{n}\succ\tau$

for an infinitesimal $\tau\in\mathbb{K}$. Hence, $P^{kn}(x_{0},x_{0})\not\to
0$ and by Lemma 8.1 this holds for all vertices. ∎

We finish our considerations with giving an example of a positive capacity
graph for which $P^{n}(x,y)\not\to 0$.

######  Example 8.10 (Positive capacity but $P^{n}(x,y)\not\to 0$).

Again we consider a path graph $V=\mathbb{N}_{0}$, see Figure 1, over the
Levi-Civita field $\mathcal{R}$. Let $b(0,1)=b(1,2)=1$ and
$b(k,k+1)=\epsilon^{-k+1}$ for $k>2$. Then by Theorem 3.3 the graph has
positive capacity. On the other hand, the graph satisfies the condition of the
lemma above since we have $P^{2}(0,0)=P(0,1)P(1,0)={1}/2.$ Therefore,
$P^{n}(x,y)\not\to 0$.

## References

  * [1] Pete L. Clark and Niels J. Diepeveen.  Absolute convergence in ordered fields.  The American Mathematical Monthly, 121(10):pp. 909–916, 2014.
  * [2] M. Fukushima, Y. Oshima, and M. Takeda.  Dirichlet Forms and Symmetric Markov Processes.  De Gruyter studies in mathematics. De Gruyter, 2011.
  * [3] Alexander Grigorian.  Analytic and geometric background of recurrence and non-explosion of the brownian motion on riemannian manifolds.  Bulletin of the American Mathematical Society, 36:135–249, 04 1999\.
  * [4] Geoffrey Grimmett.  Probability on Graphs: Random Processes on Graphs and Lattices.  Institute of Mathematical Statistics Textbooks. Cambridge University Press, 2 edition, 2018.
  * [5] Sebastian Haeseler and Matthias Keller.  Generalized solutions and spectrum for dirichlet forms on graphs.  In Daniel Lenz, Florian Sobieczky, and Wolfgang Woess, editors, Random Walks, Boundaries and Spectra, pages 181–199, Basel, 2011. Springer Basel.
  * [6] J. F. Hall and T. D Todorov.  Ordered fields, the purge of infinitesimals from mathematics and the rigorousness of infinitesimal calculus.  Bulgarian Journal of Physics, 42(2):99–127, 2015.
  * [7] Matthias Keller, Daniel Lenz, and Radosław K. Wojciechowski.  Graphs and discrete Dirichlet spaces, volume 358 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer, Cham, [2021] ©2021.
  * [8] Matthias Keller, Daniel Lenz, and Radosław K. Wojciechowski.  Volume growth, spectrum and stochastic completeness of infinite graphs.  Mathematische Zeitschrift, 274(3):905–932, Aug 2013.
  * [9] Matthias Keller, Yehuda Pinchover, and Felix Pogorzelski.  Optimal hardy inequalities for schrödinger operators on graphs.  Communications in Mathematical Physics, 358:767–790, 2016.
  * [10] Matthias Keller, Yehuda Pinchover, and Felix Pogorzelski.  Criticality theory for schrödinger operators on graphs.  Journal of Spectral Theory, 2017.
  * [11] Tullio Levi-Civita.  Sugli infiniti ed infinitesimi attuali quali elementi analitici.  Atti Ist. Veneto di Sc., Lett. ed Art., IV(7a), 1892-1893.
  * [12] Russell Lyons and Yuval Peres.  Probability on Trees and Networks, volume 42 of Cambridge Series in Statistical and Probabilistic Mathematics.  Cambridge University Press, New York, 2016.  Available at https://rdlyons.pages.iu.edu/.
  * [13] Anna Muranova.  On the notion of effective impedance.  Oper. Matrices, 14(3):723–741, 2020.
  * [14] Anna Muranova.  Effective impedance over ordered fields.  Journal of Mathematical Physics, 62(3):033502, 03 2021.
  * [15] C. St J. A. Nash-Williams.  Random walk and electric currents in networks.  Mathematical Proceedings of the Cambridge Philosophical Society, 55(2):181–194, 1959.
  * [16] Edward Nelson.  Radically Elementary Probability Theory. (AM-117), Volume 117.  Princeton University Press, Princeton, 1988.
  * [17] Khodr Shamseddine and Martin Berz.  Analysis on the Levi-Civita field, a brief overview.  In Advances in $p$-adic and non-Archimedean analysis, volume 508 of Contemp. Math., pages 215–237. Amer. Math. Soc., Providence, RI, 2010.
  * [18] K.M. Shamseddine.  New Elements of Analysis on the Levi-Civita Field.  Michigan State University. Department of Mathematics and Department of Physics and Astronomy, 1999.
  * [19] B.L. van der Waerden.  Algebra.  Number Vol. 1. Springer New York, NY, 2003.
  * [20] Wolfgang Woess.  Random Walks on Infinite Graphs and Groups.  Cambridge Tracts in Mathematics. Cambridge University Press, 2000.