The p -harmonic boundary and D p -massive subsets of a graph of bounded degree
aa r X i v : . [ m a t h . F A ] A p r THE p -HARMONIC BOUNDARY AND D p -MASSIVE SUBSETSOF A GRAPH OF BOUNDED DEGREE MICHAEL J. PULS
Abstract.
Let p be a real number greater than one and let Γ be a graph ofbounded degree. We investigate links between the p -harmonic boundary of Γand the D p -massive subsets of Γ. In particular, if there are n pairwise disjoint D p -massive subsets of Γ, then the p -harmonic boundary of Γ consists of atleast n elements. We also show that the converse of this statement is alsotrue. Introduction
Throughout this paper p will always denote a real number greater than one. Agraph is said to have the p -Liouville property if every bounded p -harmonic functionon the graph is constant. Similarly, a graph is said to have the D p -Liouville property if every bounded p -harmonic function of the graph with finite p -Dirichlet sum isconstant. When a graph has the p -Liouville property ( D p -Liouville property), theset of bounded p -harmonic functions (with finite p -Dirichlet sum) can be identifiedwith R , the real numbers. Now let G be a finitely generated group. Our mainmotivation for studying the p -harmonic boundary of a graph arose from the problemof determining the first reduced ℓ p -cohomology space of G . A locally finite graphwith bounded degree, called the Cayley graph of G , can be associated with G .Thus it makes sense to define the p -harmonic boundary for G , and to say that G has the p -Liouville property ( D p -Liouville property). It turns out that the firstreduced ℓ p -cohomology space of G vanishes if and only if G has the D p -Liouvilleproperty if and only if the p -harmonic boundary of G consists of one point or isempty. A more complete discussion about this characterization can be found in [11]and the references therein. Another reason for studying locally finite graphs withbounded degree is there intimate connection via discrete approximation to completeRiemannian manifolds with bounded geometry. The papers [2, 4, 5] contain a wealthof information concerning this link between graphs and manifolds.Recently, a generalized version of the D p -Liouville property for graphs has beenstudied in [8, 9]. More precisely, under what conditions on a graph can the bounded p -harmonic functions with finite p -Dirichlet sum be identified with R n , n ∈ N . When n ≥
2, this also means that there are nonconstant p -harmonic functions on thegraph. Holopainen and Soardi proved in [2, Lemma 5.7] that there is a nonconstantbounded p -harmonic function with finite p -Dirichlet sum on a graph of bounded Date : March 31, 2013.2010
Mathematics Subject Classification.
Primary: 31C20; Secondary: 05C38, 31C45, 60J50.
Key words and phrases. p -harmonic boundary, D p -massive set, p -harmonic function, asymp-totically constant functions, extreme points of a path.The research for this paper was partially supported by PSC-CUNY grant 63873-00 41. degree if and only if there exists two disjoint D p -massive subsets of vertices of thegraph.The purpose of this paper is to bring into sharper focus this connection between D p -massive subsets and nonconstant p -harmonic functions on a graph. As a con-sequence, we are able to determine exactly when the set of bounded p -harmonicfunctions on a graph with finite p -Dirichlet sum can be identified with R n . Themain tool we use to obtain our results is the p -harmonic boundary of a graph.The p -harmonic boundary is a subset of the p -Royden boundary. When p =2 these sets are respectively known as the harmonic boundary and the Roydenboundary. In [15, Chapter 6] the Royden and harmonic boundaries were studiedfor locally finite graphs of bounded degree. Many of the results in [15, Chapter6] were translated from corresponding results on complete Riemannian surfaces.See [13, Chapter 3] for information about the Royden and harmonic boundariesin the setting of complete Riemannian surfaces. However, there are some majordifferences between these two cases. In [15, Example 6.27] it was shown that theRoyden boundary and the harmonic boundary coincide for a locally finite graphof bounded degree that satisfies a strong isoperimetric inequality. This is in starkcontrast with the complete Riemannian surface case. More precisely, if the harmonicboundary is removed from the Royden boundary of a complete Riemannian surface,then the resulting set is dense in the Royden boundary! See [13, page 157] for thedetails of this fact. Furthermore, if the graph is a k -regular tree, k ≥
3, then thereare no isolated points in the harmonic boundary of the tree, [15, page 145].The problem of explicitly computing the p -harmonic boundary of a locally finitegraph of bounded degree appears to be quite difficult. The only result we can findin this direction is in the paper [16] where it is shown that the Royden boundaryof a 2-regular tree, which can be considered as a Cayley graph for the integers,is a quotient space of β N , the Stone- ˇCech compactification of N . In [11, Chapter7] the author gave some examples of finitely generated groups whose p -harmonicboundary is empty or contains exactly one point by using the fact that the firstreduced ℓ p -cohomology of those particular groups is zero.In Section 2 we define the main concepts used in this paper. We also state ourmain result. Section 3 is devoted to the proof of the main result. We explain inSection 4 how our result extends the main result of [9].I would like to thank the referee for several excellent suggestions that improvedthe exposition of this paper.2. Definitions and statement of main result
Let Γ be a graph with vertex set V Γ and edge set E Γ . We will write V for V Γ and E for E Γ . For x ∈ V, N x will be the set of neighbors of x and deg ( x ) will denote thenumber of neighbors of x . We shall say that Γ is of bounded degree if there exists apositive integer k for which deg ( x ) ≤ k for every x ∈ V . A path γ in Γ is a sequenceof vertices x , x , . . . , x n , . . . where x i +1 ∈ N x i for 1 ≤ i ≤ n − x i = x j if i = j . Note that all paths considered in this paper have no self-intersections. Agraph is connected if any two distinct vertices of the graph are joined by a path. Allgraphs considered in this paper will be connected, of bounded degree with no selfloops and have countably infinite number of vertices. By assigning length one toeach edge of Γ, V becomes a metric space with respect to the shortest path metric.We will denote this metric by d ( x, y ), where x, y ∈ V . Thus d ( x, y ) gives the length -HARMONIC BOUNDARY AND D p -MASSIVE SUBSETS 3 of the shortest path joining the vertices x and y . For S ⊆ V , the outer boundary ∂S of S is the set of vertices in V \ S with at least one neighbor in S , and | S | willdenote the cardinality of S . We use 1 V to represent the function that takes thevalue 1 on all elements of V . Finally, if x ∈ V and n ∈ N , the natural numbers,then B n ( x ) will denote the metric ball that contains all elements of V that havedistance less than n from x .We now proceed to define some function spaces that will be used in this paper.Let S ⊆ V and let f be a real-valued function on S ∪ ∂S . We define the p -th powerof the gradient , the p -Dirichlet sum , and the p -Laplacian of x ∈ S by | Df ( x ) | p = X y ∈ N x | f ( y ) − f ( x ) | p ,I p ( f, S ) = X x ∈ S | Df ( x ) | p , ∆ p f ( x ) = X y ∈ N x | f ( y ) − f ( x ) | p − ( f ( y ) − f ( x )) . In the case 1 < p <
2, we make the convention that | f ( y ) − f ( x ) | p − ( f ( y ) − f ( x )) = 0if f ( y ) = f ( x ). A function f is said to be p -harmonic on S if ∆ p f ( x ) = 0 for all x ∈ S . Observe that a function which is p -harmonic on S is also defined on ∂S .We now give an alternate definition that is commonly used for a function to be p -harmonic on S when S is a finite (compact) subset of V . We begin by settingΞ( f, S ) = 12 I p ( f, S ) + X x ∈ ∂S X y ∈ N x ∩ S | f ( x ) − f ( y ) | p . A function f is said to be p -harmonic on S if it is the minimizer of Ξ among thefunctions in S ∪ ∂S with the same value in ∂S as f , that is , ifΞ( f, S ) ≤ Ξ( u, S )for every function u in S ∪ ∂S with f = u in ∂S . The interested reader can findmore information about p -harmonic functions and harmonic functions on graphs inthe papers [1, 2, 3, 6, 8, 9, 11, 14, 15, 17] and the references therein.We shall say that f is p -Dirichlet finite if I p ( f, V ) < ∞ . The set of all p -Dirichletfinite functions on Γ will be denoted by D p (Γ). With respect to the following norm D p (Γ) is a reflexive Banach space, k f k D p = ( I p ( f, V ) + | f ( o ) | p ) /p , where o is a fixed vertex of Γ and f ∈ D p (Γ). We use HD p (Γ) to represent the setof p -harmonic functions on V that are contained in D p (Γ). Note that the constantfunctions are members of HD p (Γ). Let ℓ ∞ (Γ) denote the set of bounded functionson V and let k f k ∞ = sup V | f | for f ∈ ℓ ∞ (Γ). Set BD p (Γ) = D p (Γ) ∩ ℓ ∞ (Γ). Theset BD p (Γ) is a Banach space under the norm k f k BD p = ( I p ( f, V )) /p + k f k ∞ , where f ∈ BD p (Γ). Let BHD p (Γ) be the set of bounded p -harmonic functionscontained in D p (Γ). The space BD p (Γ) is closed under the usual operations ofscalar multiplication, addition and pointwise multiplication. Furthermore, for f, g ∈ BD p (Γ) we have that k f g k BD p ≤ k f k BD p k g k BD p . Thus BD p (Γ) is a commutativeBanach algebra. Let C c (Γ) be the set of functions on V with finite support. Indicate M. J. PULS the closure of C c (Γ) in D p (Γ) by C c (Γ) D p . Set B ( C c (Γ) D p ) = C c (Γ) D p ∩ ℓ ∞ (Γ).Using the fact that the inequality ( a + b ) /p ≤ a /p + b /p is true when a, b ≥ < p ∈ R , we see immediately that k f k D p ≤ k f k BD p . Consequently, B ( C c (Γ) D p )is closed in BD p (Γ).2.1. The p -harmonic boundary. In this subsection we construct the p -harmonicboundary of a graph Γ. For a more detailed discussion about this construction seeSection 2.1 of [11]. Let Sp ( BD p (Γ)) denote the set of complex-valued characterson BD p (Γ), that is the nonzero ring homomorphisms from BD p (Γ) to C . We willimplicitly use the following property of elements in Sp ( BD p (Γ)) throughout thepaper. Lemma 2.1.
Let χ ∈ Sp ( BD p (Γ)) . If f ∈ BD p (Γ) , then χ ( f ) is a real number.Proof. Suppose there exists an f ∈ BD p (Γ) for which χ ( f ) = a + bi , where b = 0.Set F = ( f − a ) /b and observe that χ ( F ) = i . Since BD p (Γ) is a Banach algebra, F, F and F + 1 V all belong to BD p (Γ). Also, χ ( F + 1 V ) = 0. For x ∈ V and y ∈ N x , (cid:12)(cid:12) F ( y ) + 1 V − F ( x ) + 1 V (cid:12)(cid:12) p ≤ | F ( x ) − F ( y ) | p , because F + 1 V ≥ V . It now follows that ( F + 1 V ) − ∈ BD p (Γ), andso F + 1 V has a multiplicative inverse in BD p (Γ). Hence, χ ( F + 1 V ) = 0, acontradiction. Therefore, χ ( f ) is a real number. (cid:3) With respect to the weak ∗ -topology, Sp ( BD p (Γ)) is a compact Hausdorff space.If A ⊆ Sp ( BD p (Γ)) , A will indicate the closure of A in Sp ( BD p (Γ)). Given atopological space X , let C ( X ) denote the ring of continuous functions on X en-dowed with the sup-norm. The Gelfand transform defined by ˆ f ( χ ) = χ ( f ) yields amonomorphism of Banach algebras from BD p (Γ) into C ( Sp ( BD p (Γ))) with denseimage. Furthermore, the map i : V → Sp ( BD p (Γ)) given by ( i ( x ))( f ) = f ( x ) isan injection, and i ( V ) is an open dense subset of Sp ( BD p (Γ)). For the rest ofthis paper we shall write f for ˆ f , where f ∈ BD p (Γ). The p -Royden boundary ofΓ, which we shall denote by R p (Γ), is the compact set Sp ( BD p (Γ)) \ i ( V ). The p -harmonic boundary of Γ is the following subset of R p (Γ): ∂ p (Γ) : = { χ ∈ R p (Γ) | ˆ f ( χ ) = 0 for all f ∈ B ( C c (Γ) D p ) } . We shall write | ∂ p (Γ) | to indicate the cardinality of ∂ p (Γ).2.2. D p -massive sets. We now define the concept of a D p -massive subset of agraph. An infinite connected subset U of V with ∂U = ∅ is called a D p -massivesubset of V if there exists a nonnegative function u ∈ BD p (Γ) with the followingproperties:(1) ∆ p u ( x ) = 0 for x ∈ U ,(2) u ( x ) = 0 for x ∈ ∂U , and(3) sup x ∈ U u ( x ) = 1.We call any u that satisfies these conditions an inner potential of the D p -massivesubset U . The next result is Proposition 4.11 of [11] and will be needed in thesequel. Proposition 2.2. If U is a D p -massive subset of V , then i ( U ) contains at leastone point of ∂ p (Γ) . -HARMONIC BOUNDARY AND D p -MASSIVE SUBSETS 5 Statement of the main result.
The main result of this paper is:
Theorem 2.3.
Let < p ∈ R and let Γ be a graph of bounded degree. Suppose n ∈ N . Then there exists n pairwise disjoint D p -massive subsets D , D , . . . , D n of V if and only if | ∂ p (Γ) | ≥ n . By combining this theorem with Corollary 2.7 of [11] we obtain
Corollary 2.4.
Let < p ∈ R , n ∈ N and let Γ be a graph of bounded degree. Ifthere exists n pairwise disjoint D p -massive subsets of V , but there does not exist n + 1 disjoint D p -massive subsets of V , then BHD p (Γ) can be identified with R n . Proof of Theorem 2.3
The following lemma will be needed for the proof of Theorem 2.3. For a proofof the lemma see the first part of the proof of [2, Lemma 5.7]
Lemma 3.1.
Let h be a nonconstant function in BHD p (Γ) , and let U be an infiniteconnected subset of V . Let a and b be real numbers such that inf x ∈ U h < a < b < sup x ∈ U h. Then each component of the set { x ∈ U | h ( x ) > b } and each component of { x ∈ U | h ( x ) < a } is D p -massive. Proof of Theorem 2.3.
We are now ready to prove Theorem 2.3. Let D , D , . . . , D n be a collection of pairwise disjoint D p -massive subsets of V . Foreach k , with 1 ≤ k ≤ n , let u k be an inner potential for D k . We may and doassume u k = 0 on V \ D k . Also, D k ∩ ∂ p (Γ) = ∅ by Proposition 2.2. For each k we will produce an element χ k ∈ D k ∩ ∂ p (Γ) for which χ k ( u k ) = 0 and χ k ( u j ) = 0if j = k . This will establish | ∂ p (Γ) | ≥ n . Extend u k to a continuous function on Sp ( BD p (Γ)). By [11, Theorem 2.6] there exists a p -harmonic function h k on V such that h k = u k on ∂ p (Γ). The maximum principle ([11, Theorem 4.7]) says that0 < h k < V . Let B k = { x ∈ D k | h k ( x ) > − ǫ } , where 0 < ǫ < . Sincesup u k = 1 on D k , B k = ∅ . Let C k be a component of B k . By Lemma 3.1 C k is D p -massive. Thus C k ∩ ∂ p (Γ) = ∅ . Select χ k ∈ C k ∩ ∂ p (Γ). Because h j = u j on ∂ p (Γ) , χ k ( h j ) = χ k ( u j ). Consequently, χ k ( u k ) = 1 and χ k ( u j ) = 0 if k = j . Hence, | ∂ p (Γ) | ≥ n if there exists n pairwise disjoint D p -massive subsets of V .Conversely, let χ , χ , . . . , χ n be distinct elements from ∂ p (Γ). By Urysohn’slemma there exists a continuous function f : Sp ( BD p (Γ)) → [0 ,
1] with f ( χ ) = 1and f ( χ k ) = 0 if k = 1. Let M = f − (1). For each integer k with 2 ≤ k ≤ n we can inductively define a continuous function f k : Sp ( BD p (Γ)) → [0 ,
1] with thefollowing properties: f k ( x ) = x = χ k x = χ i , i = k x ∈ ∪ k − i =1 M i where M k = f − k (1).By the density of BD p (Γ) in C ( Sp ( BD p (Γ))), we can assume f k ∈ BD p (Γ) foreach k . Using Theorems 4.6 and 4.8 of [11], we obtain a unique h k ∈ BHD p (Γ)with h k = f k on ∂ p (Γ) for each k . Also, 0 < h k < V . Observe that if h k ( χ ) = 1 = h j ( χ ) for some χ ∈ ∂ p (Γ), then k = j . Let ǫ > M. J. PULS the set A k,ǫ = { x ∈ V | h k ( x ) > − ǫ } . For each k let D k,ǫ be a component of A k,ǫ . Furthermore, choose the D k,ǫ so that D k,ǫ ⊆ D k,ǫ if 0 < ǫ < ǫ . Lemma3.1 yields that D k,ǫ is D p -massive. The proof will be complete if there exists an ǫ > D k,ǫ ∩ D j,ǫ = ∅ if k = j . Assume for the purposes of contradictionthat this condition is not true. Then there exists j, k with D k,ǫ ∩ D j,ǫ = ∅ forall ǫ >
0. Let i ∈ N . Denote by C i a component of D k, − i ∩ D j, − i . By thecomparison principle [2, Theorem 3.14] C i is infinite. Using Lemma 3.1 we canproduce a D p -massive subset of C i . An appeal to Proposition 2.2 produces a ψ i ∈ C i ∩ ∂ p (Γ). Clearly ψ i ( h j ) > − − i and ψ i ( h k ) > − − i . The sequence( ψ i ) in ∂ p (Γ) has a convergent subsequence that converges to some ψ in ∂ p (Γ).Consequently, ψ ( h k ) = 1 = ψ ( h j ). This contradicts our earlier observation that if h k ( χ ) = 1 = h j ( χ ) for some χ ∈ ∂ p (Γ), then k = j . Therefore, there exists an ǫ > D k,ǫ ∩ D j,ǫ = ∅ for each j, k with 1 ≤ j, k ≤ n . The proof of the theoremis now complete. 4. A result of Kim and Lee
In this section we elaborate on how Theorem 2.3 improves the main result of [9].We start by giving some needed definitions.Recall that E represents the edge set of a graph Γ. Denote by F ( E ) the set ofall real-valued functions on E and let F + ( E ) be the subset of F ( E ) that consistsof all nonnegative functions. For f ∈ F ( E ) set ξ p ( f ) = X e ∈ E | f ( e ) | p . The edge set of a path γ in Γ will be denoted by Ed ( γ ) . Let Q be a set of pathswith no self-intersections in Γ. Indicate by A ( Q ) the set of all f ∈ F + ( E ) thatsatisfy ξ p ( f ) < ∞ and P e ∈ Ed ( γ ) f ( e ) ≥ γ ∈ Q . The extremal length oforder p for Q is defined by λ p ( Q ) − = inf { ξ p ( f ) | f ∈ A ( Q ) } . The number λ p ( Q ) − is commonly known as the p -modulus of the path family Q .We shall say that a property holds for p -almost every path in a collection of pathsif the set of paths for which the property does not hold has infinite extremal length(or p -modulus zero).Let A ⊆ V , write Γ A for the largest subgraph of Γ that has vertex set A .Let γ be a one-sided infinite path in Γ. For a real-valued function f on V , set f ( γ ) = lim n →∞ f ( x n ) as n → ∞ along the vertices of γ . Let P A be the set ofall one-sided infinite paths with no self-intersections contained in Γ A . We define areal-valued function f to be asymptotically constant on A if there exists a constant c such that f ( γ ) = c for p -almost every path γ ∈ P A . We shall say that an infinite connected set U has property AC if each function in BHD p (Γ) is asymptotically constant on U .An infinite connected subset S of V is said to be p -hyperbolic if there exists anonempty finite subset A of V for which Cap p ( A, ∞ , S ) = inf u I p ( u, S ) > , -HARMONIC BOUNDARY AND D p -MASSIVE SUBSETS 7 where the infimum is taken over all finitely supported functions u on S ∪ ∂S suchthat u = 1 on A . If S is not p -hyperbolic, then it is said to be p -parabolic . Thequantity Cap p ( A, ∞ , S ) is known as the p -capacity of S .Motivated by [18, Theorem 3.1], Kim and Lee prove the following result in [9,Theorem 1.1] Theorem 4.1.
Let n ∈ N and let Γ be a graph with n p -hyperbolic ends. Supposeeach p -hyperbolic end has property AC . Then given any real numbers a , a , . . . , a n ∈ R , there exists an unique h ∈ BHD p (Γ) such that h ( γ ) = a i for p -almost every path γ ∈ P F i for each i = 1 , , . . . , n , where F , F , . . . , F n are the p -hyperbolic ends of Γ . We see immediately that if a graph Γ satisfies the hypothesis of this theorem,then
BHD p (Γ) can be identified with R n , which is the same conclusion as Corollary2.4. However, the hypothesises of Theorem 4.1 are quite strong. The number ofends of a graph Γ is independent of p , and the AC property is also very restrictive.For example, let G denote a co-compact lattice in the real rank 1 simple Lie groups Sp ( n, , n ≥
2. The Cayley graph of the group G has one end, but there arenonconstant p -harmonic functions with finite p -Dirichlet sum on G exactly when p > n + 2. See [10, Section 4] for the details.When the cardinality of ∂ p (Γ) is finite, Theorem 2.3 completely characterizesthe number of elements in ∂ p (Γ) in terms of pairwise disjoint D p -massive sets. Itis the case that D p -massive sets are also p -hyperbolic. The reason we are able todrop the property AC assumption from Theorem 4.1 in our Theorem 2.3 is givenin Proposition 4.3 below. Before we prove the proposition we need the following Lemma 4.2.
Let Γ be a graph with bounded degree and let < p ∈ R . Suppose F isan infinite connected subset of V with property AC . For h ∈ BHD p (Γ) , denote by c h the constant for which h ( γ ) = c h for p -almost every path in P F . If χ ∈ F ∩ ∂ p (Γ) ,then χ ( h ) = c h .Proof. Let h ∈ BHD p (Γ). Suppose c h < χ ( h ). Let ǫ > c h < χ ( h ) − ǫ .Define A = { x ∈ F | h ( x ) > χ ( h ) − ǫ } and let C be a component of A . Observethat λ p ( P C ) = ∞ due to h ( γ ) > c h for each γ ∈ P C . By Lemma 3.1, C is D p -massive. Proposition 5.3 of [12] now yields the contradiction λ p ( P C ) < ∞ . A similarargument shows that it is also not the case χ ( h ) < c h . Therefore χ ( h ) = c h . (cid:3) Denote by V ( γ ) the vertex set of an infinite path γ in Γ. Write V ( γ ) for theclosure of i ( V ( γ )) in Sp ( BD p (Γ)). The set of extreme points of γ is given by Ex ( γ ) = V ( γ ) \ i ( V ( γ )) . Proposition 4.3.
Let < p ∈ R and let Γ be a graph of bounded degree. Let F bea p -hyperbolic subset of V . Then F has property AC if and only if | F ∩ ∂ p (Γ) | = 1 .Proof. Because F is p -hyperbolic, it is the case λ p ( P F ) < ∞ . Lemma 5.2 of [12]implies F ∩ ∂ p (Γ) = ∅ . Now suppose χ and χ are distinct elements from F ∩ ∂ p (Γ).Since BD p (Γ) separates points in Sp ( BD p (Γ)), there exists an f ∈ BD p (Γ) forwhich χ ( f ) = χ ( f ). Combining Theorems 4.6 and 4.8 of [11] we obtain an h ∈ BHD p (Γ) with the property f = h on ∂ p (Γ). Thus χ ( h ) = χ ( h ), contradictingLemma 4.2. Hence, | F ∩ ∂ p (Γ) | = 1. M. J. PULS
Now assume | F ∩ ∂ p (Γ) | = 1 and let χ be the unique element in F ∩ ∂ p (Γ). Selectan h ∈ BHD p (Γ) and let c h = χ ( h ). We will now show that h ( γ ) = c h for p -almostevery path in P F . Denote by P ∞ the set of all γ ∈ P F for which h ( γ ) does not exist.Let γ = x x . . . x n . . . ∈ P ∞ . The identity h ( x n ) = h ( x ) − P nk =1 ( h ( x k − ) − h ( x k ))implies P ∞ k =1 | h ( x k − ) − h ( x k ) | = ∞ . It now follows [7, Lemma 2.3] that λ p ( P ∞ ) = ∞ . For each n ∈ N , set P /n = { γ ∈ P F \ P ∞ | | h ( γ ) − c h | > /n } . Now suppose λ p ( P /n ) < ∞ for some n ∈ N . By [12, Lemma 5.2]( ∪ γ { Ex ( γ ) | γ ∈ P /n } ) ∩ ∂ p (Γ) = ∅ . Let ψ be an element in this intersection. The definition of P /n implies that ψ ( h ) = c h . Combining the fact P /n ⊆ P F with the hypothesis | F ∩ ∂ p (Γ) | = 1 yields ψ = χ , contradicting the fact χ ( h ) = c h . Hence λ p ( P /n ) = ∞ for all n ∈ N . Let P U = ∪ ∞ n =1 P /n . Lemma 2.2 of [7] says that λ p ( P U ) = ∞ , and λ p ( P U ∪ P ∞ ) = ∞ .Let P h = { γ ∈ P F | h ( γ ) = c h } . Then P F = P h ∪ P U ∪ P ∞ . Another appeal to[7, Lemma 2.2] shows that λ p ( P h ) < ∞ since λ p ( P F ) < ∞ . Thus h ( γ ) = c h for p -almost every path in P F . Therefore, h is asymptotically constant on F . (cid:3) It follows immediately from this proposition that if a graph Γ satisfies the as-sumptions of Theorem 4.1, then | ∂ p (Γ) | = n . References [1] Alicia Cant´on, Jos´e L. Fern´andez, Domingo Pestana, and Jos´e M. Rodr´ıguez. On harmonicfunctions on trees.
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