1. Introduction
A finite or countably infinite connected graph whose edges carry positive real weights can be considered as an electrical network with resistors, and this is closely related with the intensively studied field of random walks on graphs. See the books [
1,
2,
3,
4,
5], and a great number of papers, among which for example [
6,
7,
8]. In [
9,
10,
11,
12], the wider class of networks with resistors, coils, and capacitors are considered as complex-weighted graphs. In the present note, we use the corresponding model from [
12,
13], i.e., we assume that
is a connected, locally finite graph without loops, where each (non-oriented) edge
is equipped with an
admittance
where
with
, and
. Here,
is the inductance,
the resistance and
the capacitance of the edge, and
is the inverse of the impedance. From a viewpoint of physics,
s is a complex frequency, and the admittance of an edge is the complex-valued analogue of a conductance. Indeed, when
is real,
can be interpreted as a conductance of the underlying edge.
In the present paper we consider exclusively the case
, the right half plane consisting of all complex numbers with
. While this is a technical assumption which is crucial for the present approach, it is also typical in network theory: the admittance (
1) is a positive-real function, that is,
when
; see [
12,
14,
15,
16,
17]. We set
if
is not an edge, so that
is a function on
. We call the couple
a
complex (electrical) network.
We introduce the
admittance operator , which acts on functions
as follows:
When , we see that is a stochastic transition matrix which governs a nearest neighbour random walk. This is also true when all the vectors are collinear (proportional). In particular, if they are same on each edge, then is the transition matrix of the simple random walk on the graph, independently of s. In all these cases, our network can be interpreted as a purely resistive one, where the admittance of an edge is just its conductance. The study of the network properties is then an issue of the discrete potential theory related with and the corresponding discrete Laplacian. This has its natural counterpart in the probabilistic study of the random walk (reversible Markov chain) governed by , confer the references right at the beginning.
The main questions addressed in this note are threefold:
How can the concept of transience (resp. recurrence) be formulated?
In the transient case, how can one construct (the analogue of) the Green function ≡ potential kernel?
To which extent can the latter be computed in terms of power series?
We analyse the analogues of the different Laplace type equations associated with when s is complex, as compared to the well-understood case when it is real.
We first prove, resp. recall some basic estimates of admittances in
Section 2. In
Section 3, we introduce the Green function for finite networks with boundary, a non-empty subset of the vertex set where the network is grounded. We relate the Green function, resp., the analogues of escape probabilities, with the effective impedance defined in [
12,
13]. It is convenient to work with the inverse of effective impedance, that is, the
effective admittance, which corresponds to the total amount of current in the electrical network. In this context, we provide first comparisons of associated power series with analogous ones for reversible Markov chains.
Our main effort concerns infinite networks, and in
Section 4, we study the effective admittance both in presence of a boundary
as well as the effective admittance between a source vertex and infinity. The latter leads to the notion of transience, resp. recurrence, and our main result is that this does not depend on the parameter
, and that in the transient case, one always can construct a Green kernel in extension of the well-understood case when
. In the final
Section 5, we show how that Green kernel can be used when the network is a tree. We construct the Martin kernel and provide a Poisson type integral representation of all harmonic functions over the boundary at infinity of the tree. In the specific case of a free group, we have a closer look at the applicability of our comparison results between the complex network and the ones associated wiith positive real weights.
2. Inequalities for Admittance Operators
Notational convention. In the sequel, we shall compare the complex-weighted admittance operators with non-negative, stochastic transition operators. In order to better visualize these different types, we shall use slightly different fonts: P and will refer to stochastic transition operators—even though when .
Lemma 1. The admittance (1) of any edge is a positive-real function of s. The following estimates hold. Proof. We first reconsider the property of being positive-real. Note that for any complex number
,
if and only if
. We have
Therefore
which proves (
3). Regarding (
4), we use that for
, one has
.
In addition to the operators (matrices)
(resp.
when
) we also introduce the transition operators
and
with matrix entries
where
. From Lemma 1, we get the following comparison.
Proposition 1. For any and all , Proof. Using (
4), we obtain
and the first two of the proposed inequalities follow. Combining the above with (
3) and (
5) yields the third one. □
Recall that when is real, is the transition operator of a random walk. This also holds when all three-dimensional vectors are collinear, in which case is independent of the value of s. We also have the following comparison.
Lemma 2. If (both real) then for all Proof. This is elementary: for real
with
, consider the function
For maximising, resp. minimising g, it suffices to consider , and one finds that in the simplex , the extrema of lie in the corners, whence the maximum is and the minimum is . □
Corollary 1. For and , we have for all (Note the particular case .) This means that we can investigate some properties of our complex-weighted network via comparison with the corresponding random walks with transition probabilities , where , or , respectively.
Notation: in the sequel, we shall write
for the collection of the stochastic matrices that come up in our context, and
3. The Green Function on Finite Networks with Boundaries
Let
be a finite network. We fix a non-empty proper subset
of
V. This is our (generic) boundary, where the network is grounded. (This does not have to be what was introduced as a “natural” boundary of a finite graph in terms of dominating graph distances [
18,
19].) We consider
as the
interior of our graph.
If
is any real or complex matrix indexed by
V, then we let
We write , so that in particular, is the identity matrix over .
Definition 1. Whenever the matrix is invertible, let Its matrix elements are called the Green function or Green kernel of P with respect to the chosen interior .
Since each of the stochastic matrices
is irreducible, it is a quite elementary fact that
exists; see, e.g., [
20] (Lemma 2.4). For the Markov chain with transition matrix
P starting at vertex
x, we have that
is the expected number of visits in
y before leaving the interior
. Furthermore, it follows from [
21,
22] that also for complex weights with positive real part,
is invertible for every
. See in particular the proof of [
22] (Theorem 2).
Definition 2. If , then the associated normalized weighted Laplacian
is the operator acting on functions by A function
is called
harmonic on with respect to if
for any
. Now, choose
and consider the
augmented boundary as well as the
reduced interior. Harmonic functions come up in the following
Dirichlet problem.Our interest is in
and the associated Dirichlet problem with complex weights. By [
21,
22] this problem has a unique solution
whenever
. Indeed, the function
provides the (augmented) boundary data, and the solution can be given in two ways:
where (as usual) functions are to be seen as column vectors. Indeed, one easily checks that both formulas provide a solution of (
7), and by uniqueness, they coincide.
The Dirichlet problem has a physical interpretation. In the electrical network model, the vertex a is the source, where the potential is kept at 1, and is the set of grounded nodes. Then is the complex voltage at the vertex x (for the complex frequency s). This leads to the following definition.
Definition 3 ([
13,
22])
. For the finite network with , the admittance between the source vertex a and the grounded set is defined by where is the solution of the Dirichlet problem (7) with respect to . In [
22], the symbol
is used for the admittance of the network. When
, this is of course classical, and
is the inverse of the
total resistance between
a and
, while the resistance of a single edge is
. The following is immediate from Formula (
9).
Let us have another look at (
8) and (
9). If we replace the complex matrix
by a stochastic matrix
then we have the Markov chain
with transition matrix
P. Given
and
, we can consider the stopping time of the first visit in
a before leaving
:
Note that
and that
for
. It is well-known and easy to prove that
the solution of our Dirichlet problem when
. Furthermore,
The estimates of
Section 2 suggest that we can compare the solution of (
7) and related items concerning the complex network
with the analogous ones for
. For any
(i.e., including complex weights), we introduce the power series
Proposition 2. (i)
for , let , where is the set of eigenvalues of that matrix. Then , it is an eigenvalue of , and for , each of the power series of (11) converges absolutely.(ii)
If for complex , the seriesconverges absolutely for every , then it is the solution of the Dirichlet problem (7). This holds whenever for In these cases, the series (12) is dominated in absolute value by Proof. (i) The subgraph induced by has one or more connected components . Each of the corresponding sub-matrices of P is irreducible and non-negative, and these matrices give rise to a block-decomposition of . By the Perron–Frobenius theorem, the spectral radius of coincides with its largest eigenvalue, which is positive real. It is , since is substochastic, but not stochastic. The maximum of the Perron–Frobenius eigenvalues of all the matrices is , and the Perron–Frobenius theorem also yields absolute convergence of for and all .
(ii) If
converges absolutely for all
then the value of the series is
, so that (
12) is indeed the solution (
8) of the Dirichlet problem. The last part of the proposition follows from Proposition 1, resp. Corollary 1. □
In statement (ii) above, the most natural choices for
t are
or
. The advantage of the comparison lies in the possibility to use combinatorial methods of generating functions and paths for computing the solution of the Dirichlet problem (
7).
Example 1. We consider the graph with vertex set as in Figure 1. We choose where . Along each edge, the label in the figure is its admittance. Also, , since . We consider and , so that . For our comparison, we choose and write . Then . Also, and . The latter is better (smaller) than the former. We see that the comparison of Proposition 2 works whenever . On the other hand, the spectral radius of satisfies One gets that if and only if , and precisely in this case, the series (12) converges absolutely, while the comparison with (resp. or ) is not useful when . Finally, if , the series (12) diverges and cannot be used for solving the Dirichlet problem. We also remark that for comparison with yields no improvement when . 4. Admittance and Green Function on Infinite Networks
The above can also be performed when the network is infinite. Recall that we assume local finiteness (each node has finitely many neighbours). In this case, the set
of grounded states may be finite or infinite. It may also be empty, in which case we are considering a complex-valued flow from
a to
∞. (Indeed, the boundary should rather be thought of as
.) We can again consider the power series (
11). When is
non-empty, we get that
because
is a set of absorbing states for this Markov chain. In addition, we can also consider the unrestricted Green function
and when
and the series converges absolutely, we write once more
. Finiteness of
means that the associated Markov chain with transition matrix
is
transient: with probability 1, each vertex is visited only finitely often by the random process, and finiteness is independent of
x and
y by connectedness of the graph. More generally, consider the
spectral radiusIt is well known that this number is independent of
x and
y. It is indeed the spectral radius (norm) of
P acting as a self-adjoint operator on
, where the weights are
Furthermore, the radius of convergence of the power series
is
, and at
the latter either converges for all
or diverges for all
. In the first of those two cases,
P is called
-transient, in the second case
-recurrent. See, e.g., [
23]. In the case of a finite network, we have of course
, and the respective Green function diverges at
.
Proposition 3. (a) The Markov chains induced by are either all recurrent or all transient.
(b) We either have for all or for all .
Proof. Let
be the space of all finitely supported real or complex functions on
V. With
as in (
17), the
Dirichlet sum associated with
is
By Proposition 1 and Corollary 1, the Dirichlet sums associated with distinct
compare above and below by positive multiplicative constants. Thus, by [
3] (Corollary 2.14) (to cite one among various sources), transience of
P implies transience of
Q and vice versa. This proves (a).
Regarding the spectral radius, by [
3] (Theorem 10.3) (once more to cite one among various sources) we have
if and only if there is
such that
Besides the Dirichlet forms, also the weights m with respect to different compare up to positive multiplicative constants. This yields (b). □
Let us now consider the effective admittance of our infinite network, as defined in [
12,
22]. Let
be the ball of radius
n around a choosen the root vertex
with respect to the integer-valued graph metric, and let
be the set of edges whose endpoints lie in
. Thus,
is the subgraph of
induced by
. We write
for the resulting finite sub-network of
, where more precisely, the admittance function
is the restriction of the given one to
. Given the set of grounded states
in the infinite network, as well as an input node
, we take
n large enough such that
and define
Now let
be the unique solution of the Dirichlet problem
7 on
with respect to
, as given by (
8) and (
9). Its dependence on
s is important here. The associated effective admittance is
Proposition 4 ([
22] (Theorem 22))
. As a function of , the sequence converges locally uniformly to a holomorphic function:The limit is the effective admittance of the infinite network.
We are led to the following.
Definition 4. Given the parameters and the admittances on all edges of , where , the infinite network is called transient, if for some source vertex . Otherwise, it is called recurrent.
The definition is motivated by the case
, in which case we know that
is the transition matrix of a reversible Markov chain, or equivalently, a resistive network, were the edge resistances are
. This Markov chain is transient (i.e., it tends to
∞ almost surely) if and only if the effective conductance from any vertex
a to infinity is positive (equivalently, the effective resistance is finite). Based on the previous results of [
22], the following is now quite easy to prove, but striking.
Theorem 1. (a) Transience (resp., recurrence) is independent of the source vertex a as well as of the parameter .
(b) If the (finite) set of grounded nodes is non-empty, then we have for all and .
Proof. By Proposition 4,
is the locally uniform limit of a sequence of real-positive holomorphic functions of the variable
. Hence it is holomorphic on the right half plane. By Hurwitz’ Theorem (see, e.g., [
24] (p. 178)), it is either nowhere zero or constant equal to zero on
.
(a) Suppose that
. We know already from Proposition 3 that for real
, transience of the reversible Markov chain with transition matrix
in independent of
s. In this case it is well known that the effective admittance (or rather conductance in this situation) is
. Transience then means that
. It is also well known that in this case,
for all
. See, e.g., [
2,
23].
Thus, when for some and , then one also has for all and all .
The proof of (b) is analogous: when
, then
for all
, as observed in (
14). Again, in this case, the effective admittance is
, and the extension to complex
works as in (a). □
In the transient case (with
), if
for
, we have by monotone convergence
and
where
, the solution of the corresponding Dirichlet problem with source node
a and grounded set
, see above. The analogous statement is true for
, when
is non-empty.
In the general case of complex
, it is natural to
define the on-diagonal Green kernel by
We shall unify notation, writing
in both cases of (
19), so that the index
can be omitted when
. This also applies to the following consideration of the off-diagonal elements.
Theorem 2. For complex ,exists for all and defines a holomorphic function of s. Proof. (Note that when
, the sequence is constant
.) We set
. Then [
22] (Corollary. 1) shows that for any
n,
We now use the last identity of Definition 3, also proved in [
22]. It yields
Note that each edge appears twice in that sum. For each
, we choose a shortest path
from
a to
x in our graph. If
n is sufficently large then it is contained in
. Recall that
. We now use the Cauchy–Schwarz inequality:
Combinig (
4), (
5) and Lemma 2, we get that for any edge
,
Therefore
which of course depends on the chosen shortest path from
a to
x. We conclude that
We can now proceed as in the proof of [
22] (Theorem. 6a). For any fixed
x and
a in
, the sequence of holomorphic (rational) functions
is bounded in any domain
where
. By Montel’s theorem [
25] (p. 153), this sequence of functions is precompact in
with respect to uniform convergence on compact sets. The limit of any convergent subsequence must be holomorphic in
. Now, if
is real, then
by monotone convergence. But a holomorphic function on
is determined by its values on the positive real half-axis. Therefore, we have convergence on all of
. □
Corollary 2. In the recurrent case (with ),for all and all . This holds once more by Montel’s theorem, since for all (stochastic case).
We now can define the Green kernel of the transient infinite network by
with
given by (
19). (Recall that
.) Then, in matrix notation,
where
is the identity matrix over
V. In precisely the same way, we also get the Green kernel
when
, and it satisfies
Question 1. Is it true that in the transient case, the analogue of the last identity of Definition 3 holds for the infinite network ? That is, setting , then is it true that For , this is well-known to hold; see, e.g., [3] (Exercise 2.13). Let us take up the notation of (
10), for arbitrary
:
Once more,
when
. By local finiteness, the sum is finite. We use the analogous notation with respect to
and
, where
,
, are the vertex sets of our increasing family of finite subnetworks. We also consider the generating power series
For
, we extend the definition of (
16) by
Observe that deletion of the non-empty set
leaves at most countably many connected components
of our graph. Then each
is an irreducible, substochastic matrix, so that by old and well-known results on infinite, non-negative matrices (see [
26]),
is independent of
, while
when
x and
y do not belong to the same component. This means that
. Recalling that
, we thus get the following also in the infinite case.
Lemma 4. If then the power series defining and thus also , as well as , converge absolutely whenever .
We now have the following comparison result for convergence of the respective power series.
Theorem 3. Let , and . Ifthen the power series of (22) converge absolutely for . Furthermore This also holds when i.e., and .
Proof. A
walk in
is a sequence
of vertices such that
for all
i. Its length is
k, and for
, its
z-weight with respect to
is
We also admit
, in which case the walk consists of a single vertex, and its weight is defined as 1. If
is a set of walks, then
When
is infinite, we require absolute convergence. For any subset
U of
V, and
, let
be the set of all walks within
U which start at
x and end at
y, and
the set of those walks which meet
y only at their endpoint. Finally, the superscript
refers to the respective walks of length
k. Note that
is finite. Then, referring to (
21), for
we have
The analogous identities hold when we replace
V by
. If
is sufficiently small to yield absolute convergence, we get
and with the same
z, we may again replace
V by
.
Now suppose that
. Then “sufficiently small” just means that
. We can apply this to
with
, and to
or
or
with
. Then
with
or
, respectively. Then
for all
and all
. When, as assumed,
, we get that both power series
and
are dominated in element-wise absolute value by
The use of weights of walks serves in particular to verify the second statement of the theorem: for
, absolute convergence allows us to estimate
By monotone convergence, the last difference tends to 0. □
We would like to apply the last theorem in particular to the unsrestricted transient case (
) with
. When
s is non-real, this requires that the stochastic comparison matrix
satisfies
. This is independent of the specific choice of
by Proposition 3, and then Theorem 3 applies when
is sufficiently small. Compare this with Example 1. For general
, let
Then
is the radius of convergence of the power series
, defined as in (
15). However, contrary to the stochastic case, we do not see a general argument that this should be independent of
x and
y. Let us call
the
spectral radius of
. In the stochastic case, this is indeed the spectral radius (norm) as a self-adjoint operator, see (
17). For general
, we are not aware of an analogous interpretation.
Another question is the following. For stochastic
, the function
is the
-matrix element of the resolvent operator
, so that it extends analytically to
, where
is the spectrum of
P as an operator as described via (
17). Since
is real, we get that
extends as a holomorphic function from the disk
to all
with
. Is there a similar property for general
?
These observations and questions are also valid when . The same is true for the next identities which we state only for empty boundary. Recall that we have for every .
Lemma 5. For every , the following holds.
- a
is recurrent if and only if for some and all . In this case, for all .
- b
In the transient case, for every , - c
For all with (not necessarily neighbours) - d
If y is a cut vertex between x and a (i.e., every path from x to a passes through y) then
All these identities hold for
; see, e.g., [
23] (Section 1.D), and extend to complex
by analytic extension, compare with the proof of Theorem 2. They also hold for the generating functions
and
with the adaptations
as long as
, but it is not clear to us how to bridge the gap between these values of
z and the value 1 corresponding to the statements of Lemma 5.
5. Trees and Free Groups
In this section, we concentrate on the infinite, transient case in absence of a finite set of grounded vertices. For , we shall write and for the associated kernels. The fact that we have these kernels and that their matrix elements are holomorphic functions of allows us to transport a variety of methods and results from the stochastic case to this complex-weighted one. Here, we present some example classes of this kind.
- A.
Trees and harmonic functions
We assume that
is (the vertex set of) an infinite, locally finite tree, i.e., a connected graph wihout closed walks whose vertices are all distinct. We assume that each vertex has at least two neighbours. We also assume that our complex weights
are such that
is transient for some (
all)
. Taking up the definition of
Section 3, a function
is called
harmonic on
T if for all
In this sub-section, we shall explain that every harmonic function has a Poisson-type boundary integral representation.
We start by recalling the
boundary at infinity of the tree. First of all, for any pair of vertices
there is a unique geodesic path
in
T from
x to
y. A
geodesic ray is a sequence
of distinct vertices such that
for all
k. Two rays are called equivalent if (as sets) their symmetric difference is finite, that is, they differ at most for finitely many initial vertices. An equivalence class of rays is an
end of
T. It represents a way (direction) of going to infinity in
T. The set of all ends
is the boundary at infinity of
T. For every
and
, there is a unique ray
with initial vertex
x which represents
. We get the compact metric space
as follows. We fix a “root” vertex
o. The length
of a vertex
is its graph distance from
o, that is, the number of edges of
. For distinct
, their
confluent is the last common vertex on the geodesics
and
. Then
defines an (ultra)metric on
, and
T becomes a discrete, dense subset of the compact space
. A basis of the toppology on
is given by all
boundary arcsEach boundary arc is open-compact. A
successor of a vertex
is a neighbour
y of
x such that
, and then we call
the
predecessor of
y. We have
a disjoint union.
Definition 5. A distribution
on is a finitely additive complex measure ν on , that is, for all .
Remark 1. In [27], the following is proved. A distribution ν on extends to a complex Borel measure on the compact space if an only if for any family of mutually disjoint boundary arcs , , one has If
is a locally constant function on
then it can be written as a linear combination of indicator functions of boundary arcs,
and in this case, the arcs can be forced to be pairwise disjoint. For a distribution
as in Definition 5, we then set
As a matter of fact, via this definition, the linear space of all distributions is the dual of the space of all locally constant functions on
, compare with [
27].
In addition to Lemma 5, we now shall need the following, which is specific to trees.
Lemma 6. Suppose that is transient. Then for every and every pair of neighbours , Proof. For real
, the identity is derived in [
23] (9.35). Once more, it must hold for all
by analytic extension. □
Note that for arbitrary
, if
is the geodesic path connecting the two, then the tree structure and Lemma 5(c) imply that
Corollary 3. In the transient case, for all .
At this point, we can define the
Martin kernel as in the stochastic case:
The second identity follows from (
26). Note that for any fixed
, the function
is locally constant on
. We now get the following extension of a result which is well-known in the stochastic case.
Theorem 4. Harmonic functions are in one-to-one correspondence with distributions on : for every harmonic function h on T with respect to (), there is a unique distribution on such that The distribution is given byand . The proof is exactly as in [
23] (Theorem 9.36). It goes back to [
28].
More generally, for
, a function
is called
λ-harmonic with respect to
if
. For suitable values of
, the above extends to
-harmonic functions. Namely, if for
then we can use comparison and work with
and
. The associated Martin kernel is then
In this case, the arguments of [
23] (9.35) that lead to Lemma 6 can be applied directly via “path composition” as in that reference, and one gets
Thereafter, everything works as in [
29] (with a little care concerning the slightly different notation), and one gets the analogue of Theorem 4 with
as in that theorem, replacing the appearing terms
with
. Following the methods of [
29], one also gets boundary integral representations of
λ-polyharmonic functions for complex
in the range of (
27).
However, in general
does not belong to that range, unless the stochastic operators have spectral radius strictly
and
is sufficiently close to 1. One of the future issues is to understand if and how the gap between
and
in the range of (
27) can be bridged. The (finite) Example 1 indicates that this will not always be possible.
- B.
Free groups
We consider the case when
is a finitely generated group and
A is a finite, symmetric set of generators of
which does not contain the group identity
e. The Cayley graph of
has vertex set
, and two vertices
are neighbours if and only if
. Then it is natural to require that the edge admittances (
1) satisfy
so that
is a non-zero, symmetric function
. We then have
for the total admittance at any group element (vertex)
x. We get that
is a symmetric, complex measure supported by
A with total sum 1. The transition operator
is then the right convolution operator by
, and in the subsequent notation, we shall always refer to
in the place of
. It is natural to consider the action on
, the Hilbert space of all square summable complex functions on
. The operator is symmetric, but not self-adjoint unless
. We are interested in its norm
and its operator spectral radius
where
is the
convolution power of
. For the “spectral radius”
defined in (
24), we have
When
, the three numbers coincide, with
being the associated Markov chain spectral radius (
16), and
is of course a probability measure on
.
For any , when , we get convergence of the power series . This holds in particular, when .
We now consider the case when
is a
free group with free generators
(
). We set
and
for our symmetric generating set. Recall that
consists of all
reduced wordsWhen , this is the empty word, which stands for the group identity e. The group operation is concatenation of words followed by reduction, i.e., cancellation of successive “letter” pairs , .
The Cayley graph of
with respect to
A is the regular tree where each vertex has
neighbours. It is very well known since [
30] that
in the stochastic case
, and we have transience. In particular, the results of the preceding sub-section apply here. The following important result is due to [
31]; for a simple “random walk” proof, see [
32].
In particular, the norm is the same as for
, where
. The latter is in general not a probability measure, its total mass is
. Thus, we may have
when
s is complex. We have by Proposition 1
and since
is a probability measure, its operator norm (= spectral radius) is
. The stochastic transition operator induced by this probability measure is
. Again, if
is sufficiently close to 1, we can use the comparison method described in the previous sections, including the Green kernel at
. As a matter of fact, this applies to any non-amenable group, but here we have a specific formula for the norm.
Example 2. We suppose that and that with .
Note that then admittance means , admittance means , and admittance means . Let Then is equidistribution on A, and it is very well known that the norm of the associated convolution operator, i.e., the spectral radius of simple random walk is . Consequently, If thenand if is fixed, then this will be for k sufficiently large, so that the Green kernel is defined via the corresponding power series for complex z in an open disk around the origin that contains . The same is true when k is small and the angle α is sufficiently close to 0. For example, when and then which is when .
Of course, the general estimate (28) yields a smaller range of angles α for which one obtains , like in the finite network of Example 1. In all those cases, the power series representation of the Green kernel in a neighbourhood of allows to derive a variety of further results, such as the study of polyharmonic functions as in [29].