Networks with complex weights: Green function and power series

We introduce a Green function and analogues of other related kernels for finite and infinite networks whose edge weights are complex-valued admittances with positive real part. We provide comparison results with the same kernels associated with corresponding reversible Markov chains, i.e., where the edge weights are positive. Under suitable conditions, these lead to comparison of series of matrix powers which express those kernels. We show that the notions of transience and recurrence extend by analytic continuation to the complex-weighted case even when the network is infinite. Thus, a variety of methods known for Markov chains extend to that setting.


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 [12], [24], [22] and [15].In [3], [4], [7] and [16], the wider class of networks with resistors, coils, and capacitors are considered as complexweighted graphs.In the present note, we use the corresponding model from [17] and [16], i.e we assume that pV, Eq is a connected, locally finite graph without loops, where each (non-oriented) edge rx, ys is equipped with an admittance (1) ρ s px, yq " ρ s py, xq " s L xy s 2 `Rxy s `Dxy , x, y P V, where L xy , R xy , D xy ě 0 with L xy `Rxy `Dxy ą 0, and s P C. Here, L xy is the inductance, R xy the resistance and D xy the capacitance of the edge, and ρ s px, yq is the inverse of the impedance.From a viewpoint of Physics, s is a complex frequency, and the admittance of an edge is the complexvalued analogue of a conductance.Indeed, when s ą 0 is real, ρ s px, yq can be interpreted as a conductance of the underlying edge.
In the present paper we consider exclusively the case s P H r , the right half plane consisting of all complex nunbers with Re s ą 0. We set ρ s px, yq " 0 if rx, ys is not an edge, so that ρ s is a function on V 2 .We call the couple pV, ρ s q a complex (electrical) network.
We introduce the admittance operator P s , which acts on functions f : V Ñ C as follows: (2) P s f pxq " ÿ y p s px, yqf pyq , where p s px, yq " ρ s px, yq ρ s pxq with ρ s pxq " ÿ y ρ s px, yq .
The admittance ( 1) is a positive-real function, that is, Re ρ s px, yq ą 0 when Re s ą 0 ; see [5], [16].Therefore P s f pxq is well-defined at any vertex of our graph.When s P R `, we see that P s is a stochastic transition matrix which governs a nearest neighbour random walk.This is also true when all the vectors pL xy , R xy , D xy q are collinear (proportional).In particular, if they are same on each edge, then P s is the transition matrix of the simple random walk on the graph, independently of s.
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 analgoue of) the Green function ?‚ 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 P s 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 [17], [16].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 BV Ĺ V 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 s P H r , and that in the transient case, one always can construct a Green kernel in extension of the well-understood case when s ą 0. In the final Section 5, 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.

Inequalities for admittance operators
Notational convention.In the sequel, we shall compare the complexweighted admittance operators P s with non-negative, stochastic transition operators.In order to better visualize these different types, we shall use slightly different fonts: P and ppx, yq will refer to stochastic transition operators -even though P s " P s when s ą 0.
Lemma 2.1.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 s P C, Re s ą 0 if and only if Re 1{s ą 0. We have In addition to the operators (matrices) P s (resp.P s when s ą 0) we also introduce the transition operators r P s and q P s with matrix entries  3) and ( 5) yields the third one.
Recall that when s ą 0 is real, P s the transition operator of a random walk.This also holds when all three-dimensional vectors pL xy , R xy , D xy q are collinear, in which case P s is independent of the value of s.We also have the following comparison.
Proof.This is elementary: for real L, R, D ě 0 with L `R `D ą 0, consider the function For maximising, resp.minimising g, it suffices to consider L `R `D " 1, and one finds that in the simplex tpL, Dq : L `D ď 1, L ě 0, D ě 0u, the extrema of gpL, 1 ´L ´D, Dq lie in the corners, whence the maximum is t{s and the minimum is s{t.
Corollary 2.4.For s P H r and t ą 0, we have for all x, y P V (Note the particular case t " 1.)This means that we can investigate some properties of our complex-weighted network via comparison with the corresponding random walks with transition probabilities p t px, yq, where t ą 0, or r p s px, yq, respectively.Notation: in the sequel, we shall write Π `" tP t , r P s , q P s : t ą 0 , s P H r u for the collection of the stochastic matrices that come up in our context, and Π " Π `Y tP s : s P H r u.

The Green function on finite networks with boundaries
Let pV, ρq be a finite network.We fix a non-empty proper subset BV of V .We consider V ˝" V zBV as the interior of our graph.
If P " `ppx, yq ˘x,yPV is any real or complex matrix indexed by V , then we let `ppx, yq ˘x,yPV ˝and P V ˝, BV " `ppx, yq ˘xPV ˝, yPBV .
We write P n V ˝" `ppnq V ˝px, yq ˘x,yPV ˝, so that in particular, P 0 V ˝" I V ˝is the identity matrix over V ˝.
Definition 3.1.Whenever the matrix I V ˝´P V ˝is invertible, let `IV ˝´P V ˝˘´1 .Its matrix elements G P V ˝px, yq are called the Green function or Green kernel of P with respect to the chosen interior V o .
Since each of the stochastic matrices P P Π `is irreducible, it is a quite elementary fact that G P V ˝exists; see e.g.[13,Lemma 2.4].For the Markov chain with transition matrix P starting at vertex x, we have that G P V ˝px, yq is the expected number of visits in y before leaving the interior V ˝.Furthermore, it follows from [18] and [11] that also for complex weights with positive real part, I V ˝´P s|V ˝is invertible for every s P H r .See in particular the proof of [18,Theorem 2].Definition 3.2.If P P Π, then the associated normalized weighted Laplacian ∆ P is the operator acting on functions f : V Ñ C by ∆ P f pxq " ÿ y:y"x `f pyq ´f pxq ˘ppx, yq.
A function v : V Ñ C is called harmonic on V ˝with respect to ∆ P if ∆ P vpxq " 0 for any x P V ˝.Now choose a P V ˝and consider the augmented boundary B a V " BV Y tau as well as the reduced interior V a " V ˝ztau.Harmonic functions come up in the following Dirichlet problem.
Our interest is in P " P s and the associated Dirichlet problem with complex weights.By [18] and [11] this problem has a unique solution v " v a whenever Re s ą 0. Indeed, the function v | B a V 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 uniquenss, 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 BV is the set of grounded nodes.Then vpxq is the complex voltage at the vertex x (for the complex frequency s).This leads to the following definition.
Definition 3.3.[17], [18] For the finite network pV, ρ s q with s P H r , the admittance1 between the source vertex a and the grounded set BV is defined by Y s pa Ñ BV q " where vpxq " v a pxq is the solution of the Dirichlet problem (7) with respect to ∆ Ps .
When s ą 0, this is of course classical, and Y s pa Ñ BV q is the inverse of the total resistance between a and BV , while the resistance of a single edge is 1{ρ s px, yq.The following is immediate from formula (9).Y s pa Ñ BV q " ρ s paq G Ps V ˝pa, aq .
Let us have another look at (8) and ( 9).If we replace the complex matrix P s by a stochastic matrix P P Π `then we have the Markov chain pX n q ně0 with transition matrix P. Given BV and V ˝, we can consider the stopping time of the first visit in a before leaving V ˝: t a " inftn ě 0 : X n " a , X k P V ˝for all k ď nu ď 8.
Note that f pnq V ˝pa, aq " δ 0 pnq and that f p0q V ˝px, aq " 0 for x P V a .It is well-known and easy to prove that the solution of our Dirichlet problem when P P Π `.Furthermore, (10) px, wq p pw, aq for x P V a , n ě 1.
The estimates of §2 suggest that we can compare the solution of ( 7) and related items concerning the complex network pV, ρq with the analogous ones for P P Π `.For any P P Π (i.e., including complex weights), we introduce the power series Then λpP V a q ă 1, it is an eigenvalue of P V a , and for |z| ă 1{λpP V a q, each of the power series of (11) converges absolutely.
(ii) If for complex s P H r , the series p pnq s|V a px, yqp s py, aq converges absolutely for every x P V a , then it is the solution of the Dirichlet problem (7).
This holds whenever for t ą 0 λp r P s|V a q , or r s ă 1 λp q P s|V a q .
In these cases, the series (12) is dominated in absolute value by ÿ V ˝px, y|r s q q p s py, aq r s .
Proof.(i) The subgraph induced by V a has one or more connected components C 1 , . . ., C k .Each of the corresponding sub-matrices P C i of P is irreducible and non-negative, and these matrices give rise to a blockdecomposition of P V ˝.By the Perron-Frobenius theorem, the spectral radius of P C i coincides with its largest eigenvalue, which is positive real.It is ă 1, since P C i is substochastic, but not stochastic.The maximum of the Perron-Frobenius eigenvalues of all the matrices P C i is λpP V a q, and the Perron-Frobenius theorem also yields absolute convergence of G P V a px, y|zq for |z| ă 1{λpP V ˝q and all x, y P V ˝.
(ii) If ř n p pnq s|V a px, yq converges absolutely for all x, y P V ˝then the value of the series is G Ps V ˝px, yq, so that ( 12) is indeed the solution (8) of the Dirichlet problem.The last part of the proposition follows from Proposition 2.2, resp.Corollary 2.4.
In statement (ii) above, the most natural choices for t are t " |s| or t " 1.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 3.6.We consider the graph with verex set V " t1, 2, 3, 4u as in Figure 1.We choose s " e iα where |α| ă π{2.Along each edge, the label in the figure is its admittance.We have Also, r P s " q P s " P 1 , since |s| " 1.We consider a " 1 and BV " t4u, so that V a " t2, 3u.For our comparison, we choose t " |s| " 1 and write P " P 1 .Then λpP V a q " 1{ ?6.Also, r s,1 " 1{ cos 2 α and r r s " 1{ cos α.The latter is better (smaller) than the former.We see that the comparison of Proposition 3.5 works whenever cos α ą 1{ ?6.On the other hand, the spectral radius of P s|V a satisfies One gets that |λpP s|V a q| ă 1 if and only if cos α ą 1{ ?8, and precisely in this case, the series (12) converges absolutely, while the comparison with P 1 (resp.r P s or q P s ) is not useful when 1{ ?8 ă cos α ď 1{ ?6.Finally, if cos α ď 1{ ?8, the series ( 12) diverges and cannot be used for solving the Dirichlet problem.We also remark that for t ą 0 comparison with P t yields no improvement when t ‰ 1.

Admittance and Green function on infinite networks
The above can also be done when the network is infinite.Recall that we assume local finiteness (each node has finitely many neighbours).In this case, the set BV Ĺ V 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 8. (Indeed, the boundary should rather be thought of as BV Y t8u.)We can again consider the power series (11).When is BV non-empty, we get that ( 14) Furthermore, the radius of convergence of the power series G P px, y|zq is 1{λpP q, and at z " 1{λpP q the latter either converges for all x, y or diverges for all x, y.In the first of those two cases, P is called λpPq-transient, in the second case λpPq-recurrent.See e.g.[23].In the case of a finite network, we have of course λpP q " 1, and the respective Green function diverges at z " 1. Proof.Let ℓ 0 pV q be the space of all finitely supported real or complex functions on V .With mpxq as in (17), the Dirichlet sum associated with P P Π ìs |f pxq ´f pyq| 2 mpxqppx, yq.
By Proposition 2.2 and Corollary 2.4, the Dirichlet sums associated with distinct P, Q P Π `compare above and below by positive multiplicative constants.Thus, by [22, 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 [22,Theorem 10.3] (once more to cite one among various sources) we have λpP q ă 1 if and only if there is κ ą 0 such that }f } 2 m ď κ D P pf q for all f P ℓ 0 pV q.Besides the Dirichlet forms, also the weights m with respect to different P, Q P Π `compare up to positive multiplicative constants.This yields (b).
Let us now consider the effective admittance of our infinite network, as defined in [18] and [16].Let V n " tx P V : dpx, x 0 q ď nu be the ball of radius n around a choosen the root vertex x 0 with respect to the integervalued graph metric, and let E n be the set of edges whose endpoints lie in V n .Thus, pV n , E n q is the subgraph of pV, Eq induced by V n .We write pV n , ρ s q for the resulting finite sub-network of pV, ρ s q, where more precisely, the admittance function ρ s " ρ s,n is the restriction of the given one to E n .Given the set of grounded states BV in the infinite network, as well as an input node a P V zBV , we take n large enough such that a P V n´1 and define (18) BV n " pV n´1 X BV q Y S n , where S n " V n zV n´1 .Now let v n pxq " v a s,n pxq be the unique solution of the Dirichlet problem 7 on pV n , ρ s q with respect to pa, BV n q, as given by ( 8) and (9).Its dependence on s is important here.The associated effective admittance is Y s pa Ñ BV n q " ÿ x:x"a `1 ´vn pxq ˘ρs pa, xq.Proposition 4.2.[18, Thm.22] As a function of s P H r , the sequence `Ys pa Ñ BV n q ˘n converges locally uniformly to a holomorphic function: The limit is the effective admittance of the infinite network.
We are led to the following.Definition 4.3.Given the parameters pL xy , R xy , D xy q and the admittances ρ psq px, yq on all edges of pV, Eq, where s P H r , the infinite network pV, ρ psq q is called transient, if Y s pa Ñ 8q ‰ 0 for some source vertex a P V .Otherwise, it is called recurrent.
The definition is motivated by the case s ą 0, in which case we know that P s is the transition matrix of a reversible Markov chain, or equivalently, a resistive network, were the edge resistances are L xy s `Rxy `Dxy {s .This Markov chain is transient (i.e., it tends to 8 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 [18], the following is now quite easy to prove, but striking.Proof.By Proposition 4.2, Y s pa Ñ BV Y t8uq is the locally uniform limit of a sequence of real-positive holomorphic functions of the variable s P H r .Hence it is holomorphic on the right half plane.By Hurwitz' Theorem (see e.g.[1, p. 178]), it is either nowhere zero or constant equal to zero on H r .
(a) Suppose that BV " H.We know already from Proposition 4.1 that for real s ą 0, transience of the reversible Markov chain with transition matrix P " P s in independent of s.In this case it is well known that the effective admittance (or rather conductance in this situation) is ρ s paq{G P pa, aq.Transience then means that G P pa, aq ă 8 .It is also well known that in this case, G P px, yq ă 8 for all x, y P V .See e.g.[23] or [15].
Thus, when G P s pa, aq ă 8 for some s ą 0 and a P V , then one also has Y s pa Ñ 8q ‰ 0 for all s P H r and all a P V .
The proof of (b) is analogous: when s ą 0, then G Ps V ˝pa, aq ă 8 for all a P V ˝, as observed in (14).Again, in this case, the effective admittance is ρpaq{G Ps V ˝pa, aq, and the extension to complex s P H r works as in (a).In the transient case (with BV " H), if P " P s for s ą 0, we have by monotone convergence " lim nÑ8 G P V n pa, aq and F P px, aq " lim nÑ8 F P V n px, aq , where F P V n px, aq " v s,n pxq, the solution of the corresponding Dirichlet problem with source node a and grounded set BV n , see above.The analogous statement is true for s ą 0, when BV is non-empty.
In the general case of complex s P H r , it is natural to define the ondiagonal Green kernel by (19) G Ps pa, aq " ρ s paq Y s pa Ñ 8q in the transient case, resp.
when BV ‰ H.
We shall unify notation, writing G Ps V ˝pa, aq in both cases of (19), so that the index V ˝can be omitted when BV " H.This also applies to the following consideration of the off-diagonal elements.
Theorem 4.5.For complex s P H r , ēxists for all a, x P V and defines a holomorphic function of s.
Proof.(Note that when x " a, the sequence is constant " 1.) We set Y n " Y s pa Ñ BV n q.Then [18, Cor.1] shows that for any n, Re ρ 1 paq.
Note that each edge appears twice in that sum.For each x P V , we choose a shortest path ra " x 0 , x 1 , . . ., x k " xs from a to x in our graph.If n is sufficently large then it is contained in pV n , E n q.Recall that v n paq " 1.We now use the Cauchy-Schwarz inequality: |v n px j q ´vn px j´1 q| 2 Re ρ s px j´1 , x j q ¸¨˜k ÿ j"1 Combinig (4), ( 5) and Lemma 2.3, we get that for any edge rx, ys, 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 [18,Thm. 6a].For any fixed x and a in V ˝, the sequence of holomorphic (rational) functions s Þ Ñ v n,s pxq " This holds once more by Montel's theorem, since F Ps px, aq " 1 for all s ą 0 (stochastic case).
We now can define the Green kernel of the transient infinite network by (20) G Ps px, aq " F Ps px, aq G Ps pa, aq , s P H r , with G Ps pa, aq given by (19).(Recall that F Ps px, aq " 1.)Then, in matrix notation, pI V ´Ps qG Ps " I V , where I V is the identity matrix over V .In precisely the same way, we also get the Green kernel G Ps V ˝px, aq when BV ‰ H, and it satisfies pI V ˝´P s|V ˝qG Ps V ˝" I V ˝.
Question 4.7.Is it true that in the transient case, the analogue of the last identity of Definition 3.3 holds for the infinite network ?That is, setting vpxq " F Ps px, aq " lim n v n pxq, then is it true that |vpxq ´vpyq| 2 ρ s px, yq , s P H r ?
Let us take up the notation of (10), for arbitrary P P Π : Once more, V ˝" V when BV " H.By local finiteness, the sum is finite.We use the analogous notation with respect to V n and V a n , where V n , n P N, are the vertex sets of our increasing family of finite subnetworks.We also consider the generating power series ( 22) For P P Π `, we extend the definition of ( 16) by ( 23) Observe that deletion of the non-empty set B a V leaves ad most countably many connected components C i of our graph.Then each P C i is an irreducible, substochastic matrix, so that by old and well-known results on infinite, non-negative matrices (see Seneta [20]), is independent of x, y P C i , while p pnq V a px, yq " 0 when x and y do not belong to the same component.This means that λpP V a q " sup i λpP C i q.Recalling that V a " V ˝ztau, we thus get the following also in the infinite case.Lemma 4.8.If P P Π `then the power series defining G P V a px, a|zq and thus also F P V ˝px, a|zq, as well as F P V n px, a|zq, converge absolutely whenever |z| ă 1{λpP V a q.
We now have the following comparison result for convergence of the respective power series.Theorem 4.9.Let s P H r , t ą 0 and z P C. If |z| ă 1 r s,t λpP t|V a q or |z| ă 1 r s λp r P s|V a q or |z| ă 1 r s λp q P s|V a q then the power series of (22) converge absolutely for P " P s .Furthermore This also holds when BV " H, i.e., V ˝" V and V a " V ztau.
Proof.A walk in pV, Eq is a sequence ω " px 0 , x 1 , . . ., x k q of vertices such that rx i´1 , x i s P E for all i.Its length is k, and for z P C, its z-weight with respect to P P Π is We also admit k " 0, in which case the walk consists of a single vertex, and its weight is defined as 1.If Ω is a set of walks, then When Ω is infinite, we require absolute convergence.For any subset U of V , and x, y P U , let Ω U px, yq be the set of all walks within U which start at x and end at y, and p Ω U px, yq the set of those walks which meet y only at their endpoint.Finally, the superscript pkq refers to the respective walks of length k.Note that Ω pkq U px, yq is finite.Then, referring to (21), for x, y P V a we have The analogous identities hold when whe replace V by V n .If |z| is sufficiently small to yield absolute convergence, we get and with the same z, we may again replace V by V n .Now suppose that P P Π `.Then "sufficiently small" just means that |z| ă 1{λpP V a q.We can apply this to P s with s P H r , and to P " r P s or P " q P s or P " P t with t ą 0. Then |p s px, yq| ď r ppx, yq with r " r s,t or r " r s , respectively.Then ˇˇW Ps `Ωpkq V a px, yq|z ˘ˇď W P `Ωpkq V a px, yq ˇˇr|z| for all x, y P V a Ą V a n and all k ě 0. When, as assumed, r|z| ă 1{λp P V a q, we get that both power series F Ps V ˝px, a|zq and F Ps V n px, a|zq are dominated in element-wise absolute value by x, a ˇˇr|z| ˘ă 8 .The use of weights of walks serves in particular to verify the second statement of the theorem: for r|z| ă 1{λp P V a q, absolute convergence allows us two 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 (BV " H) with z " 1.When s is non-real, this requires that the stochastic comparison matrix P P Π `satisfies λpP q ă 1.This is independent of the specific choice of P by Proposition 4.1, and then Theorem 4.9 applies when | Im s| is sufficiently small.Compare this with Example 3.6.For general s P H r , let λpP s q xy " lim sup nÑ8 |p pnq s px, yq| 1{n .
Then 1{λpP s q xy is the radius of convergence of the power series G Ps px, y|zq, 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 (24) λpP s q " suptλpP s q xy : x, y P V u the spectral radius of P s .In the stochastic case, this is indeed the spectral radius (norm) as a self-adjoint operator, see (17).For general s P H r , we are not aware of an analogous interpretation.Another question is the following.For stochastic P P Π `, the function z Þ Ñ G P px, y|1{zq{z is the px, yq-matrix element of the resolvent operator pz ¨I ´P q ´1, so that it extends analytically to CzspecpP q, where specpP q is the spectrum of P as an operator as described (17).Since specpP q Ă r´λpP q , λpP qs is real, we get that G P px, y|zq extends as a holomorphic function from the disk tz P C : |z| ă 1{λpP qu to all z P CzR with |z| ě 1{λpP q.Is there a similar property for general P s ?
These observations and questions are also valid when BV ‰ H.The same is true for the next identities which we state only for empty boundary.Recall that we have F Ps pa, aq " 1 for every a P V .Lemma 4.10.For every s P H r , the following holds.
(a) pV, ρ s q is recurrent if and only if F Ps px, aq " 1 for some a P V and all x " a.In this case, F Ps px, yq " 1 for all x, y P V .(b) In the transient case, for every a P V , G Ps pa, aq " 1 1 ´U Ps pa, aq , where U Ps pa, aq " ÿ x"a p s pa, xq F Ps px, aq .
(c) For all x, a P V with x ‰ a (not necessarily neighbours) F Ps px, aq " ÿ y p s px, yq F Ps py, aq.
(d) If y is a cut vertex between x and a (i.e., every path from x to a passes through y) then F Ps px, aq " F Ps px, yq F Ps py, aq.
All these identities hold for s ą 0, see e.g.[23, §1.D], and extend to complex s P H r by analytic extension, compare with the proof of Theorem 4.5.They also hold for the generating functions G Ps pa, a|zq and F Ps px, y|zq with the adaptations U Ps pa, a|zq " ÿ x"a p s pa, xqz F Ps px, a|zq and F Ps px, a|zq " ÿ y p s px, yqz F Ps px, a|zq as long as |z| ă 1{λpP s q, 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 4.10.

Trees and free groups
In this section we concentrate on the infinite, transient case in absence of a finite set of grounded vertices.For P s P Π, we shall write G s " G Ps and F s " F Ps for the associated kernels.The fact that we have these kernels and that their matrix elements are holomorphic functions of s P H r 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 V " T 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 ρ s px, yq are such that P s is transient for some (ðñ all) s P H r .Taking up the definition of §3, a function h : T Ñ C is called harmonic on T if for all x P T P s h " h , where P s hpxq " ÿ y:y"x p s px, yqhpyq .
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 x, y there is a unique geodesic path πpx, yq " rx " x 0 , x 1 , . . ., x n " ys in T from x to y.A geodesic ray is a sequence π " rx 0 , x 1 , x 2 , . . .s of distinct vertices such that x k " x k´1 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 B 8 T is the boundary at infinity of T .For every x P T and ξ P B 8 T , there is a unique ray πpx, ξq with initial vertex x which represents ξ.We get the compact metric space p T " T YB 8 T as follows.We fix a "root" vertex o.The length |x| of a vertex x P T is its graph distance from o, that is, the number of edges of πpo, xq.For distinct ξ, η P p T , their confluent ξ ^η is the last common vertex on the geodesics πpo, ξq and πpo, ηq.Then θpξ, ηq " defines an (ultra)metric on p T , and T becomes a discrete, dense subset of the compact space p T .A basis of the toppology on B 8 T is given by all boundary arcs B 8 T x " tξ P B 8 T : x P πpo, ξqu , x P T .
Each boundary arc is open-compact.A successor of a vertex x P T is a neighbour y of x such that |y| " |x| `1, and then we call x " y ´the predecessor of y.We have a disjoint union.
Definition 5.1.A distribution on B 8 T is a finitely additive complex measure ν on F " tB 8 T x : x P T u, that is, νpB 8 T x q " ÿ y ´"x νpB 8 T y q for all x P T .
Remark 5.2.In [9], the following is proved.A distribution ν on B 8 T extends to a complex Borel measure on the compact space B 8 T if an only if for any family of mutually disjoint boundary arcs B 8 T xn , n P N, one has ÿ n ˇˇνpB 8 T xn q ˇˇă 8.
If ϕ is a locally constant function on B 8 T 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 ν as in Definition 5.1, we then set (25) 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 B 8 T , compare with [9].
In addition to Lemma 4.10, we now shall need the following, which is specific to trees.Lemma 5.3.Suppose that pT, ρ s q is transient.Then for every s P H r and every pair of neighbours x, y P T , p s px, yq `1 ´F Ps px, yqF Ps py, xq ˘" F Ps px, yq `1 ´U Ps py, xq ˘.
Proof.For real s ą 0, the identity is derived in [23, (9.35)].Once more, it must hold for all s P H r by analytic extension.
Note that for arbitrary x, y P T , if rx " x 0 , x 1 , . . ., x n " ys is the geodesic path connecting the two, then the tree structure and Lemma 4.10(c) imply that (26) F Ps px, yq " F Ps px, x 1 qF Ps px 1 , x 2 q ¨¨¨F Ps px n´1 , yq ‰ 0.
Corollary 5.4.In the transient case, G Ps px, yq ‰ 0 for all x, y P T .
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 x P T , the function ξ Þ Ñ Kpx, ξq is locally constant on B 8 T .We now get the following extension of a result which is well-known in the stochastic case.
Theorem 5.5.Harmonic functions are in one-to-one correspondence with distributions on B 8 T : for every harmonic function h on T with respect to P s (s P H r ) , there is a unique distribution ν h on B 8 T such that hpxq " ż B 8 T K Ps px, ξq dν h pξq for all x P T .
The distribution is given by ν h pB 8 T x q " F Ps po, xq hpxq ´F Ps px, x ´qhpx ´q 1 ´F Ps px, x ´qF Ps px ´, xq , x P T ztou , and ν h pB 8 T q " hpoq.
The proof is exactly as in [23,Theorem 9.36].It goes back to [6].More generally, for λ P C, a function h : T Ñ C is called λ-harmonic with respect to pT, ρ s q if P s h " λ ¨h.For suitable values of λ, the above extends to λ-harmonic functions.Namely, if for t ą 0 (27) |λ| ą r s,t λpP t|V a q or |λ| ą r s λp r P s|V a q or |λ| ą r s λp q P s|V a q then we can use comparison and work with G Ps px, y|1{λq and F Ps px, y|1{λq.
Thereafter, everything works as in [19] (with a little care concerning the slightly different notation), and one gets the analogue of Theorem 5.5 with ν h as in that theorem, replacing the appearing terms F Ps p¨, ¨q with F Ps p¨, ¨|1{λq.
Following the methods of [19], one also gets boundary integral representations of λ-polyharmonic functions for complex λ in the range of (27).However, in general λ " 1 does not belong to that range, unless the stochastic operators have spectral radius strictly ă 1 and |s|{ Re s is sufficiently close to 1.One of the future issues is to understand if and how the gap between λ " 1 and λ in the range of ( 27) can be bridged.The (finite) Example 3.6 indicates that this will not always be possible.

B. Free groups
We consider the case when V " Γ is a finitely generated group and A is a finite, symmetric set of generators of Γ which does not contain the group identity e.The Cayley graph of Γ has vertex set Γ, and two vertices x, y are neighbours if and only if x ´1y P A. Then it is natural to require that the edge admittances (1) satisfy ρ s px, yq " ρ s pe, x ´1yq , so that a Þ Ñ ρ s pe, aq is a non-zero, symmetric function A Ñ C. We then have ρ s pxq " ρ s peq " ř aPA ρ s pe, aq for the total admittance at any group element (vertex) x.We get that p s px, yq " µ s px ´1yq , where µ s pxq " ρ s pe, xq ρ s peq is a symmetric, complex measure supported by A with total sum 1.The transition operator P s is then the right convolution operator by µ s , and in the subsequent notation, we shall alsways refer to µ s in the place of P s .It is natural to consider the action on ℓ 2 pΓq, the Hilbert space of all square summable complex functions on Γ.The operator is symmetric, but not selfadjoint unless s ą 0. We are interested in its norm }µ s } ℓ 2 and its operator spectral radius where µ pnq s is the n th convolution power of µ s .For the "spectral radius" λpµ s q " λpP s q defined in (24), we have λpµ s q ď λ ℓ 2 pµ s q ď }µ s } ℓ 2 , s P H r .
When s ą 0, the three numbers coincide, with λpµ s q being the associated Markov chain spectral radius ( 16), and µ s is of course a probability measure on Γ.
For any s P H r , when |z| ă 1{λpµ s q, we get convergence of the power series G µs px, y|zq.This holds in particular, when |z| ă 1{}µ s } ℓ 2 .
We now consider the case when Γ is a free group with free generators a 1 , . . ., a k (k ě 2).We set a ´j " a ´1 j and A " ta ˘j : j " 1, . . ., ku for our symmetric generating set.Recall that Γ consists of all reduced words x " a j 1 a j 2 ¨¨¨a jn , n ě 0 , j l P J " t˘1, . . ., ˘ku, j l ‰ ´jl´1 .
When n " 0, 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 a j a ´j , j P J.
The Cayley graph of Γ with respect to A is the regular tree where each vertex has 2k neighbours.It is very well known since [14] that λpµ s q ă 1 in the stochastic case s ą 0, and we have transience.In particular, the results of the preceding sub-section apply here.The following important result is due to [2]; for a simple "random walk" proof, see [21].
In particular, the norm is the same as for |µ| s , where |µ| s pxq " |µ s pxq|.The latter is in general not a probability measure, its total mass is ě 1.Thus, we may have }µ s } ℓ 2 ě 1 when s is complex.We have by Proposition 2.2 (28) |µ| s pΓq " 2 k ÿ j"1 |µ s pa j q| " ř j |ρ s pe, a j q| ˇˇř j ρ s pe, a j q ˇˇď |s| Re s , and since 1 |µ|spΓq |µ| s is a probability measure, its operator norm (= spectral radius) is ă 1.The stochastic transition operator induced by this probability measure is q P s .Again, if Re s{|s| is sufficiently close to 1, we can use the comparison method described in the previous sections, including the Green kernel at z " 1.As a matter of fact, this applies to any non-amenable group, but here we have a specific formula for the norm.
Of course, the general estimate (28) yields a smaller range of angles α for which one obtains }µ s } ℓ 2 ă 1, like in the finite network of Example 3.6.
In all those cases, the power series representation of the Green kernel in a neighbourhood of z " 1 allows to derive a variety of further results, such as the study of polyharmonic functions as in [19].
Proposition 4.1.(a) The Markov chains induced by P P Π `are either all recurrent or all transient.(b)We either have λpPq " 1 for all P P Π `or λpPq ă 1 for all P P Π `.

Theorem 4 . 4 .
(a) Transience (resp., recurrence) is independent of the source vertex a as well as of the parameter s P H r .(b)If the (finite) set BV of grounded nodes is non-empty, then we have Y s pa Ñ BV Y t8uq ‰ 0 for all a P V and s P H r .
Proposition 2.2.For any s P H r and all x, y P V , (6) r p s px, yq " Re ρ s px, yq Re ρ s pxq and q p s px, yq " |ρ s px, yq| |ρ| s pxq , where |ρ| s pxq " ř y |ρ s px, yq|.From Lemma 2.1, we get the following comparison.Proof.Using (4), we obtain |ρ s pxq| ě Re ρ s pxq " ÿ y Re ρ s px, yq ě Re s |s| ÿ y |ρ s px, yq| , and the first two of the proposed inequalities follow.Combining the above with ( because BV is a set of absorbing states for this Markov chain.In addition, we can also consider the unrestricted Green function Finiteness of G P px, yq means that the associated Markov chain with transition matrix P P Π `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 radius p pnq px, yq z n , x, y P V , z P C , and when z " 1 and the series converges absolutely, we write once more G P px, yq " G P px, y|1q.nÑ8p pnq px, yq 1{n .It 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 ℓ 2 pV, mq, aq is bounded in any domain ts P C : Re s ą ε, |s| ă cu where 0 ă ε ă c.By Montel's theorem[10, p. 153], this sequence of functions is precompact in H r with respect to uniform convergence on compact sets.The limit of any convergent subsequence must me holomorphic in H r .Now, if s ą 0 is real, then lim nÑ8F Ps V n px, aq " F Ps V ˝px, aq by monotone convergence.But a holomorphic function on H r is determined by its values on the positive real half-axis.Therefore we have convergence on all of H r .Corollary 4.6.In the recurrent case (with BV " H), F Ps px, aq :" lim nÑ8 F Ps V n px, aq " 1 for all x, a P V and all s P H r .