ABSOLUTELY CONTINUOUS SPECTRUM FOR MULTI-TYPE GALTON WATSON TREES MATTHIASKELLER Abstract. Weconsidermulti-typeGaltonWatsontreesthatareclosetoatreeoffiniteconetype indistribution. Moreover,weimposethateachvertexhasatleastoneforwardneighbor. Then, we show that the spectrum of the Laplace operator exhibits almost surely a purely absolutely 2 continuous component which is included in the absolutely continuous spectrum of the tree of 1 finiteconetype. 0 2 n a J 5 1. Introduction and main result 2 AftertheseminalworkofKlein[Kl1,Kl2]therehasbeenalotofeffortintherecentyearstodevelop ] h varioustechniquestoshowpreservationofabsolutelycontinuousspectrumforrandomoperatorson p treelikegraphs,see[ASW,AW,FHH,FHS2,FHS3,Hal,Ke,KLW2,KS1,KS2]. Whilemostofthe - h workisconcernedwithdiagonalperturbationsbyarandompotential,weconsiderarandomization t of the geometry. a m In particular, we study the absolutely continuous spectrum of the discrete Laplace operator on [ a multi-type Galton Watson tree. We show that parts of the spectrum are almost surely purely 2 absolutely continuous, whenever the distribution is close to a deterministic one. Trees associated v to deterministic distributions are called trees of finite cone type or periodic trees. The spectrum 9 of the Laplacian on such deterministic trees consists of finitely many bands of purely absolutely 5 continuousspectrum,see[Ke,KLW1]. Animportantassumptiononthedistributionoftherandom 9 trees is that each vertex has at least one forward neighbor. Without this assumption it can be 1 . easily seen that the operator exhibits eigenvalues with finitely supported eigenfunctions that are 9 spread all over the spectrum, compare [V] for a related result on Zd. 0 1 Theresultofthispaperstandsinclearcontrasttoresultsfortreeswithradialsymmetricbranching. 1 : In [BF] it is shown among other things, that such a tree has absolutely continuous spectrum if v and only if the branching function is eventually constant. Hence, if the branching function is i X random, then the spectrum is almost surely singular. In [HP], Anderson localization was shown r for a corresponding model on quantum graphs. These results rely on the fact that the models are a in some sense one dimensional by the radial symmetry. A similar contrast already occurs for random Schr¨odinger operators on regular trees. While for smalldisorderinducedbyarandompotentiallargepartsoftheabsolutelycontinuousspectrumare preserved,[ASW,AW,FHS2,KLW2,Kl1,Kl2],thespectrumisalmostsurelypurepointwhenever the disorder is induced by an arbitrary small radial symmetric random potential, [ASW]. Our model is closely related to what is studied in the physics literature under the name quantum percolation. There the randomness of the geometry comes from deleting edges of a given graph independently with some fixed probability. For trees this can be also modeled by a branching process. In particular, the distribution of those branching processes is such that the vertex degree is bounded and vertices have zero forward neighbors with positive probability. While in classical percolation one asks for existence of an infinite cluster, see e.g. [L], in quantum percolation one is concerned with the transition from localized states to extended states, see e.g. [Har2, Har3]. In [Har2]anasymptoticformulaforacriticalprobabilityisderivedatwhichthetransitionissupposed to happen. 1 2 MATTHIASKELLER The question of quantum percolation has also attracted some attention in the mathematics com- munity over the past years. For bond percolation on Zd, spectral properties of the Laplacian and spectral asymptotics in the sub- and supercritical regime can be found in [KM, MS1]. In [V], the set of eigenvalues with compactly supported eigenfunctions and the integrated density of states of Anderson Hamiltonians for site percolation graphs of Zd are studied. Related questions for amenable Cayley graphs are considered in [AV]. A nice introductory survey and a discussion of recentdevelopmentscanbefound[MS2](forfurtherreferencesseealsotherein). Itisalsointerest- ing to note the connection of the spectrum in the subcritical percolationregime and the spectrum of certain lamplighter groups, see [LNW]. 1.1. The model. Let T be a locally finite rooted tree with root o. We do not distinguish in notation between the tree and its vertex set. Likewise we do not distinguish between subgraphs andsubsetsofvertices. Letℓ2(T)betheHilbertspaceofsquaresummablecomplexvaluedfunctions on the vertices. We study the Laplace operator ∆=∆(T) on D(∆)= ϕ ℓ2(T) (x (ϕ(x) ϕ(y))) ℓ2(T) { ∈ | 7→ − ∈ } y∼x X acting as ∆ϕ(x)= (ϕ(x) ϕ(y)). − y∼x X The operator ∆ is selfadjoint and its restriction to the functions of finite support is essentially selfadjoint, see [W, KL]. Note that the operator is bounded if and only if the vertex degree is bounded. Let be a finite set. We refer to its elements as labels or vertex types. Let b be a multi-type A GaltonWatsonbranchingprocesswithtypesin . Forbackgroundonbranchingprocessesseethe A monographs [AN, Har1]. For the sake of brevity, we will speak in the following only of branching processes meaning multi-type Galton Watson branching processes. For a vertex type j let ∈ A Θ = Θ(b) be the subset of rooted trees that are realizations of the process b with root of vertex j j typej. Eachtreeθ Θ isequippedwithalabeling function a :θ , assigningtoavertexin j j j j ∈ →A θ its vertex type. We let P =P(b) be the probability measure on Θ induced by b. Furthermore, j j j j let (Θ,P) = (Θ(b),P(b)) = ( Θ , P ). A labeling function a on θ = (θ ) Θ is defined j∈A j j∈A j j ∈ via its restrictions as Q N a:θ , a =a . →A |θj j We impose the following assumption on the realizations of the branching processes: (F) Each vertex has at least one forward neighbor. If the realizations of a branching process b satisfy (F) almostsurely, then we say b satisfies (F). In particular, this assumption guarantees non extinction of the tree. For θ =(θ ) Θ, we let ∆(θ) on ℓ2(θ) be the direct sum of the operators ∆(θ ) on ℓ2(θ ), j . j j j ∈ ∈A Let us introduce a distance for multi-type Galton Watson branching processes. We call a function s : N a configuration of forward neighbors of a vertex, i.e., s(k) is the number of forward 0 neigAhb→ors with label k. Moreover, we denote s = s(k), s NA, which is the total k k k∈A ∈ 0 numberofforwardneighbors. Forabranchingprocessb,denotebyP(b)(s)the probabilitythatthe P j configuration of the forward neighbors of a vertex of type j is given by s NA. For p 1, let ∈ 0 ≥ := b branching process that satisfies (F) and max P(b)(s) s p < . Wp { j∈A j k k ∞} sX∈NA0 We let d : [0, ) be the metric given by p p p W ×W → ∞ d (b ,b )=max P(b1)(s) P(b2)(s) s p. p 1 2 j∈A j − j k k sX∈NA0 (cid:12) (cid:12) (cid:12) (cid:12) If inf n N P(b)(s)=0 for all s NA with s >n and all j < , then we say that b has { ∈ | j ∈ 0 k k ∈A} ∞ bounded branching. AC SPECTRUM FOR MULTI-TYPE GALTON WATSON TREES 3 Next, we introduce the deterministic trees to which we compare the random trees. Let a matrix M : N, (j,k) M j,k A×A→ 7→ begiven. Weassumethatif consistsofonlyoneelement,thenM satisfiesM 2. Thiscondition A ≥ is necessary in order to exclude the one dimensional situation. This gives rise to a deterministic branching process b=b by letting the probability that a vertex of type j has M neighbors of M j,k type k be exactly one for all j,k . Then, Θ(bM) consists of only one element T=(Tj)j∈A. We saythe treesT aregeneratedby∈thAesubstitutionmatrixM over . These treesareoftenreferred j to as trees of finite cone type or periodic trees, see [L, NW]. AsAabove, ∆(T) denotes the direct sum of operators ∆(T ) on ℓ2(T)= ℓ2(T ). j∈A j j∈A j The operators ∆(LTj)+ βo(j), where βo(j)L= ,δo(j) δo(j) and δo(j) is the delta function of the h· i root o(j), have purely absolutely continuous spectrum which consists of finitely many intervals. For a proof for the case of the adjacency matrix see [KLW1] and for the general case of label invariantoperatorssee[Ke]. Inparticular,positivityoftheentriesofM ensuresthatthespectrum is independent of the label of the root. Moreover,in [KLW1] it is also proven that the densities of thespectralmeasurearecontinuousoutsideofafinitesetΣ . Theoperatorβ canbeconsidered 0 o(j) asaboundaryconditionattheroot. By thegeneraltheoryofrankoneperturbations,see[S1],the absolutely continuous spectrum of the operators ∆(T ) and ∆(T )+β coincides. j j o(j) 1.2. Main result. The absolutely continuous spectrum of an operator H is denoted by σ (H). ac We will prove the following theorem. Theorem. There exists a finite set Σ R such that for all compact I σ (∆(T)) Σ and 0 ac 0 ⊂ ⊆ \ p >1 there is δ =δ(I,p)> 0 such that for all b with d (b,b )< δ the spectrum of ∆(θ) is p p M ∈W purely absolutely continuous in I for almost all θ Θ(b). ∈ Indeed, Σ is the finite set mentioned above. If we restrict ourselves to the case of bounded 0 branching, then we obtain the following immediate corollary. Corollary. There exists a finite set Σ such that for all compact I σ (∆(T)) Σ there is 0 ac 0 ⊆ \ q (0,1) such that for all b with bounded branching where the probability for a vertex of p ∈ ∈ W type j to have M forward neighbors of type k is larger than q for all j,k , it follows that the j,k ∈A spectrum of ∆(θ) is almost surely purely absolutely continuous in I. Remark. (a)A specialcaseof ourmodelis the binary tree,where one deletes for eachvertexone of the forward neighbors with small probability. In [FHS3], a proof for preservation of absolutely continuous spectrum is sketched for this case. However, our model does not only allow for a finite number of vertex types, but it is also not restricted to the removal of edges. In particular, the distributionsconsideredinthemaintheoremdonotevenassumeaboundonthenumberofforward neighbors of a vertex. (b)Thevalidityofthetheoremdoesnotdependonthechoiceoftheoperator∆. Indeed,theresult holds for any selfadjoint label invariant operator in the sense of [Ke]. In particular, it is also true for the normalized Laplacian or the adjacency matrix. However, selfadjointness of the adjacency matrix is an issue if the vertex degree is unbounded. Nevertheless, in the setting of the corollary the adjacency matrix is a bounded operator and therefore selfadjoint. (c) All our estimates are explicit. So, one can get upper bounds for the critical probability closely related to quantum percolation on a K-regular tree. Let us be more specific: Consider a K- regular tree and let p (0,1). For each vertex delete every forward neighbor except for one ∈ with probability (1 p) each. We consider the adjacency matrix at zero energy. We can give − upper bounds on the critical probability p such that for p > p some absolutely continuous K K spectrum is preserved almost surely. Since the focus of this paper is primarily on a qualitative result, the estimates are certainly not optimal. In particular, the estimates as given in the proofs yieldanupper bound K−1 1 2−32K−22/(2K 1)!(nevertheless,withsomeminoradaptions,see − − the remark after Proposition 1, one can get an upper bound p K−√11 2−33K−22). Since p K ≤ − our upper bound tends to 1 as K it does not give information about the behavior expected →∞ by physicists. In particular, [Har2] derives an asymptotic formula for the critical probability 4 MATTHIASKELLER p a/K, a 1.42153, for the quantum percolation model, where the transition from localized K ∼ ≈ to extended states is supposed to take place. (d) The theorem deals only with a purely absolutely continuous component of the spectrum. It would be very interesting to know more about the spectrum as a set and the nature of the whole spectrum. Since we deal with rooted trees, the model is not ergodic, so, already non-randomness of the spectrum as a set is an issue. For the spectrum as a set it is very likely that it stands in a close relationship to the union of the spectra of trees of finite cone type that are realizations of the distribution. For the nature of the spectrum, one might expect from the perspective of random Schr¨odinger operators on trees, see [AW], far more absolutely continuous spectrum than the one of the deterministic tree. Indeed, considering the results of [AW], one might ask whether singular spectrum can be excluded almost surely whenever the distributions is close enough to a deterministic one. 1.3. Outline of the proof. The proof of the theorem is based on techniques developed in [Ke, KLW2] to treat randomoperators on trees of finite cone type. These techniques itself are inspired byideasof[FHS1,FHS2]usinghyperbolicgeometryandafixpointanalysisinordertogetbounds on moments of the Green functions. The fundamental difference to the situation in this work is that [Ke, KLW2] deal with small random perturbations appearing everywhere while here we deal with very large perturbations (removal and addition of edges) that occur with small probability. The Greenfunctions satisfy a recursionrelationwhichwe use this to comparethe Greenfunctions of the random tree to the ones of the deterministic tree. Our aim is to bound the difference in the expected value in terms of a function γ which is related to the standard hyperbolic metric of the upper half plane H and is given by ξ ζ 2 γ(ξ,ζ)= | − | , ξ,ζ H. ImξImζ ∈ WeobtainthisboundofthedistanceoftheGreenfunctionsbyprovingavectorinequality: Denote by Eγ the vector in [0, )A where the j-th component is the expected value of the γ-distances for ∞ the root with label j. We prove the vector inequality Eγ (1 δ)PEγ+C, ≤ − with a stochastic irreducible matrix P, δ >0 and a vector C with positive entries. For the precise statement see Lemma 5 in Section 3.5. The idea is to condition first on the event that (a subset of) the first two spheres of the random tree agrees with the deterministic tree. By assumption, this event occurs with large probability. For this case, we apply a two step expansion estimate and a uniform contraction estimate from [KLW2]. For the other cases, we prove a different two step expansion estimate, where we use an estimate for additive perturbations in one argument of γ. This estimate takes the role of the triangle inequality which does not hold for γ. At the end it turns out that the error term from this estimate is compensated by the low probability of these events. Havingthevectorinequalityabove,weconcludebythePerronFrobeniustheoremthateachcompo- nentofEγ isbounded. Wefinishtheproofbyargumentssimilartotheonesfoundin[FHS2,KLW2] and a variant of the limit absorbtion principle. The paper is structured as follows. In the next section, we start with some preliminaries. In Section 3 we provide the crucial estimates such as the two step expansion estimate, the uniform contraction estimate and the vector inequality. These inequalities are used in Section 4 to prove the main theorem. 2. Preliminaries InthissectionwerecallsomebasicfactsabouttheGreenfunctionontrees. Moreover,weintroduce an equivalence relation for the random trees that is determined by the branching in the first two spheres. Finally, we consider conditioned expectations with respect to the equivalence classes and related invariance properties of the Green function. AC SPECTRUM FOR MULTI-TYPE GALTON WATSON TREES 5 Let us start with some remarks about notation. The roots of the components θ of θ Θ are j ∈ denoted by o(j). By o′(j), we denote a fixed forward neighbor of o(j) of label j whenever such a vertexexists. WedenotegeneralrootedtreesbyT andtheirrootsbyo. Moreover,weassumethat T is always equipped with a labeling function a. We let H = z C Imz > 0 be the complex upper halfplane andalwayslet z be anelement of H which dec{om∈pose|sas z =E}+iη with E R ∈ andη >0. Moreover, λ, foracomplexnumberλ,denotesthe modulusofλ,while A,forafinite | | | | set A, denotes the cardinality of A. 2.1. Basic facts about the Green function. Let δ be the function on a rooted tree T that x gives value one to the vertex x and zero to all other vertices and let µ be the spectral measure of x ∆(T) with respect to δ . Then, the Green function z G (z,∆(T)) on H is given by the Borel x x 7→ transform of µ , i.e., x 1 G (z,∆(T)):= dµ = δ ,(∆(T) z)−1δ , z H. x x x x t z h − i ∈ Zσ(∆(T)) − It is well known that the Green function is analytic and maps H to H (for background see e.g. [DK]). As there is a natural ordering of the vertices in a rooted tree given by their distance to the root, there is a natural notion of forward neighbors of vertices. In particular, we denote by T the x forward tree of x and we denote the Green function of the operator ∆(T )+β on ℓ2(T ) by x x x Γ (z,∆(T)):=G (z,∆(T )+β ), z H, x x x x ∈ where β = ,δ δ canbe consideredas boundary termsince by consideringthe forwardtree the x x x h· i backwardedge is ’missing’. (In the case of the adjacency matrix which considered in [KLW1] this is not necessary due to the zero diagonal of the operator.) It is well known, see for instance [ASW, Ke, KLW1, Kl1], that the Green function satisfies the following recursion equation 1 ( ) =z deg (x)+ Γ (z,∆(T)), z H, ♣ −Γ (z,∆(T)) − T y ∈ x yX∈SxT where deg (x) is the vertex degree of x in T if x = o and deg (o) is the vertex degree of o plus T 6 T one which is due to adding β to ∆(T). Moreover, ST denotes the set of forward neighbors of x. o x For a branching process b and Θ = Θ(b), we denote the Green functions z Γ (z,∆(θ)), θ Θ, x 7→ ∈ x θ, for short by ∈ Γθ :=Γ (z,∆(θ)), z H, x x ∈ andwewriteΓ(·)forthefunctionθ Γθ. InthecaseoftreesTgeneratedbyasubstitutionmatrix, we have Γ (z,x∆(T)) = Γ (z,∆(T)7→) forxvertices x,y that carry the same label, i.e. a(x) = a(y). x y We define the vector Γ=(Γ ) HA by letting j j∈A ∈ Γ =Γ (z,∆(T)), z H, j . j o(j) ∈ ∈A Similarly,deg depends only onthe label ofavertexandit is givenby 1+ M for avertex T k∈A j,k of label j . So, we write deg(a(x)) = deg (x) = 1+ M , for x T. With this notation, t∈heArecursion relation for the Green fuTnctions ∆(T) cka∈nAbear(xe)d,kucPed to t∈he finite system P of equations given by 1 =z deg(j)+ M Γ , j . j,k k −Γ − ∈A j k∈A X In [Ke, Theorem 3.1] (compare also [KLW1, Theorem 6]) it is shown that there is a finite set Σ 0 such that the truncated Green functions Γ = (Γ ) can be extended continuously to a function j Σ H H, where ∪ → Σ=σ (∆(T)+β) Σ , ac 0 \ and β = ,δ δ with o(j) being the root of T , j . In particular, for all E Σ and j∈Ah· o(j)i o(j) j ∈A ∈ j the limits Γ (E,∆(T)) = lim Γ (E +iη,∆(T)) exist, are continuous functions in E and j η↓0 j ∈ A P 6 MATTHIASKELLER ImΓ (E,∆(T)) > 0. Since the measures ImG (E+iη,∆(T))dE converge weakly to the spectral x x measure µ , we have Σ σ (∆(T )+β) for all x T and, therefore, Σ σ (∆(T)+β) (as x ac x ac T =T ). ⊂ ∈ ⊂ j o(j) 2.2. Equivalence classes of random trees. For θ Θ and x θ, we write Sθ for the set of ∈ ∈ x forward neighbors of x without specifying in which component θ , j , of θ the vertex x is j ∈ A actually contained in. Letθ,θ′ Θ. WesaythattwosubsetsA θ,A′ θ′areisomorphic ifthereisagraphisomorphism ∈ ⊆ ⊆ betweenAandA′ thatleavesthelabelinginvariant. Whenevertwosetsareisomorphic,weidentify the correspondingverticesin notation. Forexample,for fixedj allelements ofΘ havea root j ∈A of label j. Forallθ ,wefixavertexo′(j)inSθ oflabelj wheneversuchavertexexists. Inthiscase,wesay j o(j) that o′(j) exists. Otherwise, when there is no such vertex,we say that o′(j) does not exist and we let Sθ = . In T , the vertex o′(j) always exists by positivity of the entries of the substitution o′(j) ∅ j matrix. We define S(θ ):=Sθ Sθ o′(j) . j o′(j)∪ o(j)\{ } Hence, in the case where o′(j) does not exists, we get S(θ )=Sθ . j o(j) For each j , we define an equivalence relation on Θ by j ∈A θj ∼=θj′ whenever S(θ ) and S(θ′) are isomorphic. In particular, if o′(j) exists, then the subsets Sθ j j o′(j) ∪ Sθ o′(j) and Sθ′ Sθ′ o′(j) have to be isomorphic and otherwise only Sθ and Sθ′ o(j)\{ } o′(j)∪ o(j)\{ } o(j) o(j) have to be isomorphic. We denote the equivalence classesby[θj], j ∈A. Moreover,wewrite θ ∼=θ′ if θj ∼=θj′ for allj ∈A and denote the equivalence classes by [θ] correspondingly. There is a one to one map from the equivalence classes [θ ] θ Θ to NA NA, where { j | j ∈ j} 0 × 0 [θ ] (n,m) is such that there are n vertices of label k in Sθ and m vertices of label k in j 7→ k o(j) k Sθ , k . (In the case, where no vertex of label j exists in Sθ , we have m =0 for all k o′(j) ∈A o(j) k ∈A since Sθ = .) Hence, the set of equivalence classes is countable. o′(j) ∅ Wheneverthedependanceofasetoraquantityonθisindeedonlyadependanceon[θ],weindicate thisbyreplacingθ by[θ]innotation. Forexample,bytheidentificationoftheverticesvialabeling invariant graph isomorphisms, the set Sθ does not depend on the choice of θ [θ′] for θ′ Θ for v ∈ ∈ v o(j),o′(j) j . Therefore, we will write S[θ] = Sθ and S([θ ]) = S(θ ). Indeed, S[θ] ∈ { | ∈ A} v v j j v depends only on the component of θ where the vertex v lies in. Let us stress the importance of this equivalence relation: Let two functions f :θ C, f′ :θ′ C → → for θ,θ′ Θ be given. Since the vertex sets of θ and θ′ can be totally different except for their ∈ roots, it is not clear how to compare these functions. However, if θ = θ′, then we can compare ∼ these functions on S (θ) and S (θ′), j by identification of the vertices on these sets. j j ∈A 2.3. Conditioned expectations. For a measurable set A Θ and an integrable function f, we ⊆ write 1 E (f A)= f(θ )dP (θ ), j , j | P (A) j j j ∈A j ZA if P (A)>0 and E (f A)=0 otherwise. For A=Θ , we write E (f)=E(f Θ ). j j j j j | | Lemma 1. Let θ Θ, j and x S(θ ). Then, for every function g such that θ g(Γθ) is ∈ ∈ A ∈ j 7→ x integrable E (g(Γ(·)) [θ ])=E g(Γ(·) ) j x | j o(a(x)) o(a(x)) (cid:0) (cid:1) AC SPECTRUM FOR MULTI-TYPE GALTON WATSON TREES 7 and for y S(θ ) with a(x)=a(y) and every function f such that θ f(Γθ,Γθ) is integrable ∈ j 7→ x y E f(Γ(·),Γ(·)) [θ ] =E f(Γ(·),Γ(·)) [θ ] . j x y | j j y x | j (cid:0) (cid:1) (cid:0) (cid:1) Proof. By( )thevalueofthetruncatedGreenfunctionofavertexdependsonlyonthebranching ♣ ofthe forwardtree. Since the distributionofthe branchinginaforwardtree ofavertexxdepends only on a(x), we conclude the first statement. Moreover, the random variables Γθ and Γθ are x y identicallydistributedfora(x)=a(y). Furthermore,sincethe forwardtreesofx,y S(θ ), x=y, j ∈ 6 do not coincide, the distribution of their branching is independent. We conclude that the random variables Γθ and Γθ are independent. Thus, the second statement follows. (cid:3) x y 3. The fundamental inequalities In this section we prove the crucial inequalities which are the ingredients for the proof of the theorem. 3.1. The substitute for the triangle inequality. The distance function γ on H defined in the introduction as γ(ξ,ζ) = ξ ζ 2/(ImξImζ), ξ,ζ H, does not satisfy the triangle inequality. | − | ∈ Nevertheless, we can give an estimate for additive perturbations in one argument,where the error term does not depend on the other argument. A similar estimate is found in [KLW2, Lemma 1]. Lemma 2. Let ζ H, λ C such that ζ+λ H. Then, for all ξ H, ∈ ∈ ∈ ∈ γ(ξ,ζ) c (ζ,λ)(γ(ξ,ζ +λ)+1), 0 ≤ where 2 Imλ 2λ c (ζ,λ)= 1+ 1+ | | . 0 Imζ Im(ζ+λ) (cid:18) (cid:19)(cid:18) (cid:19) Proof. Let r=Imζ/Im(ζ+λ). We distinguish two cases: If ξ (ζ+λ) Im(ζ+λ)/2, then | − |≥ 2 2 λ 2λ rγ(ξ,ζ) 1+ | | γ(ξ,ζ+λ) 1+ | | γ(ξ,ζ+λ). ≤ ξ (ζ+λ) ≤ Im(ζ +λ) (cid:18) | − |(cid:19) (cid:18) (cid:19) If, on the other hand, ξ (ζ +λ) Im(ζ+λ)/2, then Imξ Im(ζ+λ)/2. So, we obtain | − |≤ ≥ 2λ ξ (ζ +λ) + λ2 λ Im(ζ +λ)+ λ2 rγ(ξ,ζ) γ(ξ,ζ+λ)+ | || − | | | γ(ξ,ζ+λ)+2| | | | . ≤ ImξIm(ζ+λ) ≤ (Im(ζ+λ))2 The statement now follows from the definition of c . (cid:3) 0 3.2. The one step expansion estimate. In this subsection we prove an inequality that allows to estimate the γ-distance of two Green functions at a vertex by their distances attained at the forward neighbors. Inthefollowing,g alwaysdenotesanarbitraryelementofH,butitcanbethoughtasΓ (z,∆(T)), x x x T. Recall that Γ =Γ (z,∆(T)) are the truncated Green functions of ∆(T) indexed by the j o(j) lab∈els j , where T=(T ) are the trees generated by a substitution matrix. j ∈A For v T, x Sv, let the weights qx :HSv (0,1) be given by ∈ ∈ → Img q (g):= x , g HSv, x Img ∈ u∈Sv u and note that x∈Svqx ≡ 1. MoreovePr, let Qx,y : H{x,y} → [0,1] and cosαx,y : H{x,y} → R, x,y T, be given by ∈ P Img Img ImΓ ImΓ γ(g ,Γ )γ(g ,Γ ) x y a(x) a(y) x a(x) y a(y) Q (g):= , x,y 1(Img ImΓ γ(g ,Γ )+Img ImΓ γ(g ,Γ )) 2p x a(y) y a(y) y a(x) x a(x) cosα (g):=cos arg(g Γ )(g Γ ) , x,y x a(x) y a(y) − − for g H{x,y} such that gx =(cid:0)Γa(x), gy = Γa(y). Then, Q(cid:1)x,y is the quotient of a geometric and ∈ 6 6 an arithmetic mean which implies that it takes values between 0 and 1. If g = Γ or g = Γ , x x y y 8 MATTHIASKELLER we put Q (g) = cosα (g) = 0. Sometimes it will be convenient to put a higher dimensional x,y x,y vector such as g HSv into the functions qy, Qx,y and cosαx,y without indicating explicitly that ∈ we actually consider the appropriate restriction of g. Hence, by what we discussed so far, we have ( ) q (g)Q (g)cosα (g) 1, g HSv. y x,y x,y ♠ ≤ ∈ (cid:12)yX∈Sv (cid:12) (cid:12) (cid:12) The functions Q , α (cid:12)depend on the unperturbed(cid:12)Green functions Γ , j , and thus on z, x,y x,y j ∈ A but we suppress this dependance on z to ease notation. In any case, by continuity of Γ all the j estimates that follow will be uniform in z for z I+i(0,1] and I Σ compact. ∈ ⊂ For x T, let NT be the number of forward neighbors of x in T of type j , i.e., ∈ x,j ∈A NT = y ST a(y)=j . x,j |{ ∈ x | }| Next, we prove the one step expansion formula. A similar statement can be found in [KLW1, Lemma 5] and [KLW2, Lemma 2]. Lemma 3. (One step expansion estimate) Let I Σ be compact, T be a rooted tree that satisfies ⊂ (F) and v T. If NT =M for all j , then, for all z I +i(0,1] ∈ v,j a(v),j ∈A ∈ ImΓ γ ΓT,Γ a(x) q (ΓT)Q (ΓT)cosα (ΓT) γ(ΓT,Γ ), v a(v) ≤ M ImΓ  y x,y x,y  x a(x) (cid:0) (cid:1) xX∈SvT j∈A a(v),j j yX∈SvT P   where ΓT =Γ (z,∆(T)), x T. Otherwise, there exists c >0 such that for all z I+i(0,1] x x ∈ 1 ∈ ImΓ γ ΓT,Γ c a(x) γ(ΓT,Γ )+ ST . v a(v) ≤ 1 M ImΓ x a(x) | v| (cid:0) (cid:1) (cid:16)xX∈SvT j∈A a(v),j j (cid:17) P Proof. We start the proof with two observations: Firstly, by a direct calculation one checks that for ξ,ζ,z H, c,d R ∈ ∈ 1 1 γ , γ(ξ c,ζ d)=γ(ξ,ζ+c d). −z c+ξ −z d+ζ ≤ − − − (cid:16) − − (cid:17) Secondly, for g HSvT, we calculate directly ∈ 2 g Γ = ImΓ Img cosα (g)Q (g) γ(g ,Γ ). x a(x) a(x) y x,y x,y x a(x) − (cid:12)xX∈SvT xX∈SvT (cid:12) xX∈SvT (cid:16)xX∈SvT (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) Combined with ( ), these observations yield the first statement. ♣ For the other statement, let k =a(v). We use ( ) and apply the first observation to get ♣ γ ΓT,Γ γ ΓT, (M Γ +NT M ) , v a(v) ≤ x k,j j v,j − k,j (cid:0) (cid:1) (cid:16)xX∈SvT jX∈A (cid:17) where we additionally used that deg (v) = 1+ N and deg(k) = 1+ M . We apply T j v,j j k,j Lemma2withξ = ΓT,ζ = (NT M +M Γ )andλ= (NT M )(Γ 1), (clearly ζ+λ H), txo∈oSbvTtaixn j∈A v,j− Pk,j k,j j j∈APv,j− k,j j− ∈ P P P ImΓ γ(ΓT,Γ ) c γ ΓT, NT Γ +1 c a(x) γ(ΓT,Γ )+1 , v v ≤ 0 x v,j j ≤ 0 ImΓ x a(x) (cid:16) (cid:16)xX∈SvT jX∈A (cid:17) (cid:17) (cid:16)xX∈SvT u∈SvT a(u) (cid:17) P whereweusedthesecondobservationandinequality ( )inthe secondstep. The constantc from 0 ♠ Lemma 2 is a product c =c′c′′ with c′ =(1+Imλ/Imζ) and c′′ =(1+2λ/Im(ζ+λ))2. With 0 0 0 0 0 | | our choice of ζ and λ the constant c is given by 0 NT ImΓ 2 (NT M )(Γ 1) 2 c =c′c′′ = j v,j j 1+ | j v,j − a(v),j j − | 0 0 0 PjMa(v),jImΓj! P jNvT,jImΓj ! P P AC SPECTRUM FOR MULTI-TYPE GALTON WATSON TREES 9 and, therefore, since ImΓ = NT Γ , u∈ST a(u) j∈A v,j j v γP(ΓT,Γ ) c′′ P ImΓa(x) γ(ΓT,Γ )+c . v v ≤ 0 M ImΓ x a(x) 0 xX∈SvT j∈A a(v),j j Let us estimate c′ and c′′ to finish thePproof. By definition of Σ and compactness of I there is 0 0 r 1 such that (Γ +1)/ImΓ r for all k,l and all z I +i(0,1]. The first term, c′, can ≥ | k| l ≤ ∈A ∈ 0 be estimated by NT ImΓ c′ = j v,j j rST . 0 M ImΓ ≤ | v| Pj a(v),j j For the second term, c′′, we first note tPhat NT M ST + M 2 since T and T 0 k| v,k− j,k|≤| v| k j,k− have at least one forward neighbor in common by (F). We estimate P P 2 (NT M )(Γ 1) M 2 c′′ =1+ | j v,j − a(v),j j − | 1+2r 1+ j a(v),j − 2r M , 0 NT ImΓ ≤ ST ≤ a(v),j P j v,j j (cid:18) P | v| (cid:19) j∈A p X where we used the inePquality 1 +(m 2)/n m 1 for n 1, m 2 with n = ST 1 − ≤ − ≥ ≥ | v| ≥ (by (F)) and m = M 2 (by non one-dimensionality) in the last step. Letting c := j a(v),j ≥ 1 4r3max ( M )2, we get the statement. (cid:3) k∈A j k,Pj P 3.3. The two step expansion estimate. Next, we adapt the one step expansion to our model to get the two step expansion by iteration. We first introduce some notation. Let the root o of the tree T have label j and recall that ∈ A S(T)=So′∪So\{o′} if there is a vertex o′ oflabel j in So andS(T)=So otherwise. For x∈SoT, let ImΓ a(x) p := j,x M ImΓ k∈A j,k k and for x∈SoT′, let P ImΓa(o′)ImΓa(x) p := , j,x ( M ImΓ )2 k∈A j,k k whereΓj,j ∈A,aretheGreenfunctionsofP∆(T)asabove. ForT =Tj,wehave x∈S(Tj)pj,x =1. Thus, for z H Σ, the matrix P =P(z): [0, ) given by ∈ ∪ A×A→ ∞ P P := p , j,k , j,k j,x ∈A x∈S(TXj),a(x)=k defines a stochastic matrix. We define the contraction quantities c :HS[T] [ 1,1] for x S[T] by x → − ∈ o(j) c (g):= q (g)Q (g)cosα (g) x y x,y x,y y∈XS[T] o(j) and for x S[T] by ∈ o′(j) cx(g):= qy(g)Qx,y(g)cosαx,y(g) qy(g)Qo′(j),y(g)cosαo′(j),y(g) , (cid:16)y∈XSo[T(]j) (cid:17)(cid:16)y∈XSo[T′](j) (cid:17) where go′ is defined in the spirit of ( ) as ♣ 1 go′(j) :=−z−deg(j)+ x∈SoT′(j)gx and q , Q and α are taken from the previous subPsection. By inequality ( ), we see that c y x,y x,y x ♠ takes indeed values in [ 1,1]. − 10 MATTHIASKELLER We now come to the two step expansion estimate. For the case θ [T ] the following estimate j j ∈ is similar to Proposition 1 of [KLW2]. In the other cases, we get a similar expression with a multiplicative and additive error term. Lemma 4. (Two step expansion estimate) Let I Σ be compact and j . For all z I+i(0,1] and θ [T ], ⊂ ∈A ∈ j ∈ γ(Γθ ,Γ ) p c (Γθ)γ(Γθ,Γ ). o(j) o(j) ≤ j,x x x a(x) x∈XS(Tj) Moreover, for all p>1 there is c >0 such that for all z I+i(0,1] and all θ Θ 2 ∈ ∈ E γ(Γ(·) ,Γ )p [θ ] c S[θ] p+ S[θ] p P E γ(Γ(·) ,Γ )p +1 . j o(j) o(j) | j ≤ 2 | o(j)| | o′(j)| j,k k o(k) k (cid:16) (cid:17) (cid:0) (cid:1)(cid:16)kX∈A (cid:0) (cid:1) (cid:17) Proof. The firststatementfollowsdirectlyby combining the firststatementofthe one stepexpan- sion estimate, Lemma 3, for v =o(j) and v =o′(j). For the second statement let θ Θ. By the second statement of Lemma 3 applied for v =o(j) in ∈ the first estimate and for v =o′(j) in the second estimate, we get p E γ Γ(·) ,Γ p [θ ] cpE ImΓa(x) γ(Γ(·),Γ )+ S[θ] [θ ] j o(j) o(j) j ≤ 1 j Pk∈AMj,kImΓk x a(x) o(j) j (cid:16) (cid:0) (cid:1) (cid:12)(cid:12)(cid:12) (cid:17) (cid:16)(cid:16)x∈XSo[θ(]j) (cid:12)(cid:12) (cid:12)(cid:12)(cid:17)p(cid:12)(cid:12)(cid:12) (cid:17) c2pE p γ(Γ(·),Γ )+ S[θ] + S[θ] [θ ] . ≤ 1 j j,x x a(x) o(j) o′(j) j Notethatinthecase,whereo′(j)doesn(cid:16)o(cid:16)txe∈xXSis([tθjt]h)esecondestimateis(cid:12)(cid:12)trivia(cid:12)(cid:12)llyt(cid:12)(cid:12)rue. F(cid:12)(cid:12)o(cid:17)rx(cid:12)(cid:12)(cid:12) (cid:17)S([θ ]), j ∈ let v(x)=o(j) if x S[θ] and v(x)=o′(j) if x S[θ] . Since p Mj,a(x) 1, we get ∈ o(j) ∈ o′(j) x∈S([θj]) j,xN[θ] ≤ v(x),a(x) by Jensen’s inequality and the inequality (m+n)p 2p(mp+nPp) for m,n 0, ≤ ≥ ... 2pc2p p Nv[θ(x]),a(x) p−1E γ(Γ(·),Γ )p [θ ] + S[θ] p+ S[θ] p . ≤ 1 j,x M j x a(x) j o(j) o′(j) j,a(x) (cid:16)x∈XS([θj]) (cid:16) (cid:17) (cid:0) (cid:12) (cid:1) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) By Lemma 1 and with c :=2pc2p 2 1 ...=c P E γ(Γ(·) ,Γ )p + S[θ] + S[θ] p , 2 j,k k o(k) k o(j) o′(j) (cid:16)kX∈A (cid:0) (cid:1) (cid:0)(cid:12) (cid:12) (cid:12) (cid:12)(cid:1) (cid:17) e (cid:12) (cid:12) (cid:12) (cid:12) N[θ] whereP := p ( v(x),k)p−1,j,k . Sincep dependsonlyona(x)andv(x) j,k x∈S([θj]),a(x)=k j,x Mj,k ∈A j,x e P N[θ] p−1 N[θ] p v(x),k v(x),k P = p = p . j,k j,x j,x M M x∈S([θXj]),a(x)=k (cid:16) j,k (cid:17) x∈S([TXj]),a(x)=k (cid:16) j,k (cid:17) e We finish the proof by estimating P /P (N[θ] )p S[θ] p+ S[θ] p j,k j,k ≤ x∈S([Tj]),a(x)=k v(x),k ≤ o(j) o′(j) since x∈Sv[θ]Nv[θ,a](x) = Sv[θ] and keMj,k ≥1. P (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12)(cid:3) P (cid:12) (cid:12) P (cid:12) (cid:12) 3.4. The averaged contraction coefficient and the uniform contraction estimate. In [KLW2, Proposition 2] a uniform contraction estimate is proven, which also plays an important role in the proof our main theorem. We recall the corresponding definitions and the statement from [KLW2]. Definition. (Label invariant permutations) For o T , we define the set of label invariant per- j mutations Π:=Π of S(T ) by ∈ j j Π:= π :S(T ) S(T ) bijective and a(π(x))=a(x) for all x S(T ) . j j j { → | ∈ }