Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence
aa r X i v : . [ m a t h . M G ] N ov HAMILTONIANS ON DISCRETE STRUCTURES: JUMPS OF THEINTEGRATED DENSITY OF STATES AND UNIFORM CONVERGENCE
DANIEL LENZ AND IVAN VESELI´C
Abstract.
We study equivariant families of discrete Hamiltonians on amenable geometries andtheir integrated density of states (IDS). We prove that the eigenspace of a fixed energy is spannedby eigenfunctions with compact support. The size of a jump of the IDS is consequently givenby the equivariant dimension of the subspace spanned by such eigenfunctions. From this wededuce uniform convergence (w.r.t. the spectral parameter) of the finite volume approximants ofthe IDS. Our framework includes quasiperiodic operators on Delone sets, periodic and randomoperators on quasi-transitive graphs, and operators on percolation graphs. Introduction
The integrated density of states, in the following abbreviated as IDS, can be defined for avariety of models, ranging from Hamiltonians in the quantum theory of solids to Laplacians on p -cochains on CW-complexes. We refer e.g. to [39, 42, 25, 18, 2, 3, 34, 46, 19] which representbut a fraction of the literature devoted to this topic. While some of these references concernoperators on continuum configuration space, in the present paper we restrict ourselves to models ondiscrete spaces. Under certain geometric conditions the IDS can be approximated by its analoguesassociated to finite volume restrictions of the Hamiltonian. Here the approximation is a prioriunderstood in the sense of weak convergence of measures. We show that under an amenabilitycondition the following properties are universal for a wide range of models:(a) If λ is a point of discontinuity of the IDS, there exist compactly supported eigenfunctions to λ , and these compactly supported eigenfunctions actually span the whole eigenspace of λ .(b) The IDS can be approximated by its finite volume analogues uniformly in the spectral variable,i.e. with respect to the supremum norm.(c) The size of each jump of the IDS can be approximated by the jumps of the finite volumeanalogues.Some of our models are random, i.e. concern a whole family of operators. For such models thethree properties above hold almost surely . We present our results in a general setting. They can beapplied to a variety of models considered before, for instance, Anderson and quantum percolationmodels [40, 5, 4, 45, 20, 1], quasi-crystal Hamiltonians on Delone sets [15, 16, 29], Laplacians on p -cochains on complexes [7, 8, 10], Harper operators [41, 36, 35], random hopping models [22], andHamiltonians associated to percolation on tilings [17, 37].The geometric framework which allows us to treat all these models at once is given by an actionof an amenable group Γ on a metric space X such that the two are roughly isometric. It is notsurprising that the notion of rough isometry is fitting in this context since it has been proposedin [14] as a tool for studying geometric properties of a space ‘at infinity’. On the other hand, it iswell known that the IDS does not change under compactly supported perturbations. Due to thechosen setting we can treat in parallel situations where the underlying group is continuous anddiscrete.There is no model known to us where all of the features (a) – (c) were obtained before. Partialresults, however, were known for some specific models mentioned above. Let us now briefly discussthese earlier partial results recovered in our framework. Results about pointwise convergence were Key words and phrases. integrated density of states, discontinuities, random Schr¨odinger operators, percolationgraphs, tilings.November 15, 2018, jump2008-11-25.tex. obtained in, e.g., [15, 39, 36, 35]. Note that in the literature special attention was devoted toconvergence at the bottom of the spectrum [7, 8, 10], since it corresponds to approximation ofBetti-numbers. Results on uniform convergence of the IDS for models with finite local complexitywere obtained in [32, 27, 11]. Statement (a) was established for periodic models related to abeliangroups in [24, 23]. Weaker statements about the characterisation of the jumps of the IDS weresubsequently proven in [21] and [45]. Approximation of the jumps of the IDS by the finite volumeanalogues are contained in [35]. Let us stress that jumps of the IDS actually do occur for severalmodels — like quasi-periodic models [21] and percolation Hamiltonians [4, 45]. Thus uniformconvergence is by no means automatically implied by pointwise convergence.Our theorems show that uniform convergence of the IDS is an universal phenomenon anddoes not depend on specific features of the model, like number-theoretic properties or finite localcomplexity. We hope that this clarification will be helpful for the study of finer properties of theIDS, which may be indeed model-dependent. Among those are: the quantisation of jumps of theIDS [9, 43], their location [36, 6, 45], and the low energy behaviour of the IDS. Note that this lowenergy behaviour has been a recent focus of attention in particular for percolation models (seee.g. [20, 38, 1]) while surveys of results for other models can be e.g. found in [34, 19].In the context of uniform convergence, we would also like to emphasize that our method ofproof works without a uniform ergodic theorem. This is a fundamental difference to the earlierconsiderations of [32, 27] (see [26] as well). On the other hand the question whether such an ergodictheorem holds in the present contexts remains open. We consider this an interesting question.This paper is organized as follows: In Section 2 we present our results and fix the notation.Section 3 provides the necessary background information on rough isometries. The next foursections are devoted to the proofs of our three main results. Section 8 discusses how results onaperiodic order fit into our framework. In Section 9 we show how earlier results on periodicoperators on graphs can be recovered by our approach. Section 10 discusses percolation modelsand finally, Section 11 is devoted to models related to percolation on Delone sets.2.
Setting and results
Let (
X, d ) be a locally compact metric space with a countable basis of the topology. Let Γ bea locally compact amenable unimodular group with an invariant metric d Γ such that every ball isprecompact. The invariant Haar measure of a Borel mesurable set A ⊂ Γ is denoted by | A | . LetΓ act continuously by isometries on X such that the following two properties hold: • There exists a fundamental domain F ′ with compact closure F , which is a countable unionof compact sets. • The map Φ : X −→ Γ , x γ , whenever x ∈ γF ′ , is a rough isometry i.e. there exist b ≥ a ≥ a d Γ (Φ( x ) , Φ( y )) − b ≤ d ( x, y ) ≤ ad Γ (Φ( x ) , Φ( y )) + b for all x, y ∈ X .Note that all these assumptions are automatically satisfied if both Γ and X are discrete and Γacts cocompactly and freely on X . Models in such a geometric setting are considered in Sections9 and 10 below.Our operators will be families of operators indexed by elements of a certain topological space.This topological space will be considered next (see Appendix for details). We consider the followingfamily of uniformly discrete subsets of X D := { A ⊂ X | d ( x, y ) ≥ , for x, y ∈ A with x = y } . The lower bound d ( x, y ) ≥ Y is acompact space we can then equip the set of functions f : A −→ Y , where A ∈ D , with the vaguetopology and obtain a compact space D Y . In fact, there is a choice of Y which is in some senseuniversal, and which we discuss in the appendix. There we show in particular that the space˜ D := { ( A, h ) | A ∈ D , h : A × A → C ∗ } . UMPS OF THE INTEGRATED DENSITY OF STATES 3 is naturally equipped with the vague topolgy and moreover compact. Here C ∗ is an arbitrarycompactification of C . The mapping ( A, h ) P x,y ∈ A g ( x, y ) h ( x, y ) ∈ C is continuous for anycontinuous g : X × X → C with compact support. In particular, it is measurable for any continuous g : X × X → C . This measurablility then holds for as well for any nonnegative measurable g : X × X → C .The action of Γ on X induces an action on ˜ D by γ ( A, h ) := ( γA, γh ) where γA := { γa | a ∈ A } and ( γh )( x, y ) = h ( γ − x, γ − y ) for x, y ∈ γA . Let µ be an Γ-invariant probability measure on ˜ D ,whose topological support we denote by Ω. Then, Ω is a closed and hence compact subset of ˜ D .For ω = ( A, h ) ∈ ˜ D we use the notation X ( ω ) := A for the projection on the first component.The latter is a discrete metric subspace of X and gives rise to ℓ ( X ( ω )) with the natural countingmeasure δ X ( ω ) := P x ∈ X ( ω ) δ x . For ω ∈ ˜ D and Λ ⊂ X we denote by Λ( ω ) the intersection Λ ∩ X ( ω )and by ω (Λ) the cardinality | Λ( ω ) | = δ X ( ω ) (Λ).The measure µ induces a Γ-invariant measure m on X by m (Λ) := Z Ω ω (Λ) dµ ( ω ) . For any
S < ∞ , let M S be the maximal number of points with mutual distance at least 1 containedin a ball of radius S in X . Then M S is finite for all S by the very definition of D . Thus, themeasure m is bounded by M S on a ball of size S and hence m is finite on bounded sets. We assumethat Ω does not consist of the empty set only. This implies m ( X ) = 0, since the vague topologyon Ω is Hausdorff. As F ′ is a countable union of compact sets, the set IF ′ is measurable for anymeasurable I ⊂ Γ by standard monotone class arguments. The map Γ ⊃ I m ( IF ′ ) gives aninvariant measure on Γ. By uniqueness of the Haar measure we infer that dens( m ) := m ( IF ′ ) | I | is anonnegative constant and hence m ( IF ′ ) = dens( m ) | I | for all I ⊂ Γ compact with | I | 6 = 0. Since m ( X ) >
0, dens( m ) is actually strictly positive. A directcalculation using unimodularity now shows that u := | I | χ IF ′ satisfies(1) u has compact support and 1 = Z u ( γ − x ) dγ for all x ∈ X for any I ⊂ Γ compact with | I | > H the direct integral Hilbert space R ⊕ Ω dµ ( ω ) ℓ ( X ( ω )). Each ω = ( X ( ω ) , h ) ∈ ˜ D gives rise to an operator H ω : c ( X ( ω )) → ℓ ( X ( ω )) , ( H ω v )( x ) := X y ∈ X ( ω ) H ω ( x, y ) v ( y ) , with H ω ( x, y ) := h ( x, y )where c ( X ( ω )) denotes the complex valued functions with compact support in X ( ω ). If H ω : ℓ ( X ( ω )) → ℓ ( X ( ω )) is bounded by C ∈ R for all ω ∈ Ω, we call H : H → H , H = Z ⊕ Ω dµ ( ω ) H ω a bounded decomposable operator . Note that in this case the values of | H ω ( x, y ) | are bounded by C for all ω ∈ Ω and x, y ∈ X ( ω ). A decomposable operator is called of finite hopping range ifthere exists an R < ∞ such that for all ω ∈ Ω and x, y ∈ X ( ω ), d ( x, y ) ≥ R implies H ω ( x, y ) = 0.Now, let for every γ ∈ Γ a function s γ : X −→ { z ∈ C : | z | = 1 } be given. A decomposableoperator is called equivariant (w.r.t. the family s γ ) if and only if s γ ( x ) H γω ( γx, γy ) s γ ( y ) = H ω ( x, y )for all γ ∈ Γ, ω ∈ Ω, and x, y ∈ X ( ω ). Of course, this is equivalent to HT γ = T γ H, for all γ ∈ Γ with the unitary map T γ : H −→ H , ( T γ f ) ω ( x ) = s γ − ( x ) f γ − ω ( γ − x ). DANIEL LENZ AND IVAN VESELI´C
Remark.
It turns out that for the results and proofs of this paper it does not matter whether s γ is a non-trivial function or s γ ≡
1. The reason is that all calculations concern the diagonal matrixelements either of one single operator H or of a product of two operators HK . In this case, theterms s γ ( x ) and s γ ( x ) cancel and equivariance of operators H and K gives H γω ( γx, γx ) = H ω ( x, x )and ( HK ) γω ( γx, γx ) = X y ∈ X ( ω ) H ω ( x, y ) K ω ( y, x ) . Thus the function s γ can be immediately eliminated.We are interested in operators H satisfying (A) given as follows:(A) H : H → H is a bounded, selfadjoint, decomposable, equivariant operator H of finitehopping range.As usual we denote the spectral family of a selfadjoint operator T by E T . Theorem 2.1.
Let H satisfy (A). Let u satisfy (1) . Then, there exist a unique measure ν H on R with (2) ν H ( ϕ ) = 1dens( m ) Z Ω Tr( uϕ ( H ω )) dµ ( ω ) for all ϕ ∈ C c ( R ) . The measure ν H does not depend on u provided (1) is satisfied. It is a spectralmeasure for H , i.e. ν H ( B ) = 0 if and only if E H ( B ) = 0 . The distribution function of ν H is denoted by N H , i.e. N H : R −→ R , N H ( λ ) := ν H (( −∞ , λ ]) , and called the integrated density of states , IDS for short. Theorem 2.2.
Let H satisfy (A). Let λ ∈ R be arbitrary. Let P comp := ( P comp,ω ) , where P comp,ω is the projection onto the subspace of ℓ ( X ( ω )) spanned by compactly supported solutions of ( H ω − λ ) u = 0 . Then, (3) E H ( { λ } ) = P comp . The theorem does not assume ergodicity. If ergodicity holds, then the statement can be strength-ened to give the following corollary.
Corollary 2.3.
Let H satisfy (A). Let µ be ergodic. Let λ ∈ R be arbitrary. Then, the followingassertions are equivalent. (i) N H is discontinuous at λ . (ii) For a set of positive measure Ω ′ ⊂ Ω there exists a compactly supported nontrivial solution u ∈ ℓ ( X ( ω )) of ( H ω − λ ) u = 0 for each ω ∈ Ω ′ . (iii) For almost every ω ∈ Ω the space of ℓ solutions u to ( H ω − λ ) u = 0 is spanned bycompactly supported nontrivial solutions. Since Γ is by assumption amenable, there exists a Følner sequence in Γ. This is by definitiona sequence of compact, nonempty sets ( I n ) n , I n ⊂ Γ such that for any compact K ⊂ Γ and ǫ > | I n △ KI n | < ǫ | I n | if n is large enough. By passing to a subsequence one may assume that( I n ) n is tempered, i.e. that for some C ∈ (0 , ∞ ) and all n ∈ N the condition | S k Let ν ω,n be the measure whose distribution function is N ω,n . Theorem 2.4. Let H satisfy (A). Assume that µ is ergodic. Then, the sequence N ω,n ω (Λ n ) convergeswith respect to the supremum norm k · k ∞ to N H for µ -almost every ω ∈ Ω , Even without the assumption of ergodicity we obtain uniform convergence of the distributionfunctions N ω,n ω (Λ n ) , but the limit cannot be described explicitely, see Remark 7.4 for details.3. Rough isometry and linear algebra In this section, we collect some basic facts about rough isometries and subspace dimensions.They provide our working tools for the subsequent sections.Let the assumptions of the previous section hold. Let p ∈ X be fixed with Φ( p ) = id ∈ Γ. ForΛ ⊂ X , x ∈ X and r > d ( x, Λ) := inf { d ( x, y ) | y ∈ Λ } andΛ r := { x ∈ X | d ( x, Λ) < r } , Λ r := { x ∈ X : d ( x, X \ Λ) > r } , ∂ r Λ := Λ r \ Λ r . For I ⊂ Γ we use the analogous notation with d replaced by d Γ . Denote by B s ( γ ) the open ballaround γ ∈ Γ with radius s . If γ is the identity we simply write B s for B s ( γ ). Lemma 3.1. For r ≥ and any s ≥ ar + b , the inclusion F r ⊂ B s F ′ holds.Proof. For any x ∈ F r there exists y ∈ F ′ with d ( x, y ) < r and a unique γ ∈ Γ such that x ∈ γF ′ .Thus d Γ ( γ, id) ≤ ad ( x, y ) + b < ar + b , implying γ ∈ B s . (cid:3) Proposition 3.2. For each r > , let q ≥ ar + ab + b and s ≥ ar + b be given. Then for all I ⊂ Γ : ( IF ) r ⊂ I s F ′ and ( I q F ) ⊂ ( IF ′ ) r and ∂ r ( IF ) ⊂ ( ∂ q I ) F ′ . Proof. Since Γ acts on X by isometries, ( γF ) r = γF r . By the previous Lemma ( IF ) r = IF r ⊂ IB s F ′ = I s F ′ and the first inclusion holds. For x ∈ I q F ′ there exist unique γ ∈ I q , x ∈ F ′ suchthat x = γx . By definition d Γ ( γ, β ) > q for all β ∈ Γ \ I . Let y ∈ X \ IF ′ be arbitrary. Then thereare unique y ∈ F ′ , α ∈ Γ \ I with y = αy . Consequently d ( x, y ) = d ( γx , αy ) ≥ d Γ ( γ, α ) /a − b >q/a − b and thus x ∈ ( IF ′ ) r for r = q/a − b . Since I ar + ab + b F ⊂ I ar + ab + b B b F ′ ⊂ I ar + ab F ′ ⊂ ( IF ′ ) r we proved the second inclusion. The last inclusion is a combination of the previous twoinclusions. (cid:3) Recall that M S denotes the maximal number of points with distance at least one contained ina ball of radius S in X . Proposition 3.3. For any ρ > and C := M a ( ρ + b ) | B ρ | ω ( IF ′ ) ≤ C | I ρ | , for all I ⊂ Γ and all ω ∈ D . Recall that for I precompact | I ρ | is finite. Proof. Choose S > ab and define N X ( S, A ) to be the maximal number of points in A ⊂ X withdistance at least S between them and N Γ ( S, I ) to be the maximal number of points in I ⊂ Γ withdistance at least S between them. Then, ω ( IF ′ ) ≤ N X (1 , IF ′ ) ≤ M S N X ( S, IF ′ ) ≤ M S N Γ ( S/a − b, I ) . Set ρ := S/a − b > 0. Since | B ρ | N Γ ( ρ, I ) = (cid:12)(cid:12) S N Γ ( ρ,I ) i =1 B ρ ( γ i ) (cid:12)(cid:12) ≤ | I ρ | we obtain the claim. (cid:3) The proposition has the following consequence, which is crucial for our results. Proposition 3.4. Let ( I n ) be a Følner sequence. Then for arbitrary r ≥ and ω ∈ D , lim n →∞ ω (( ∂ r I n ) F ′ ) | I n | = 0 and lim n →∞ ω ( ∂ r Λ n ) | I n | = 0 . DANIEL LENZ AND IVAN VESELI´C Proof. Proposition 3.3 gives us ω (( ∂ r I ) F ′ ) ≤ M aρ + b | B ρ | | ( ∂ r I ) ρ | ≤ M aρ + b | B ρ | | ( ∂ r + ρ I ) | , thus the Følnerproperty of ( I n ) implies the first equality. Since r > ∂ r ( IF ) ⊂ ( ∂ q I ) F , for q ≥ ar + ab + b , the second equality follows immediately. (cid:3) Lemma 3.5. For ω ∈ D and Λ ⊂ X let U be a subspace of ℓ (Λ( ω )) . For S ≥ denote by U S the subspace consisting of all functions in U which vanish outside of Λ S . Then, ≤ dim( U ) − dim( U S ) ≤ ω ( ∂ S Λ) . Proof. Let Q : U −→ ℓ (cid:0) (Λ \ Λ S )( ω ) (cid:1) be the natural restriction map. Then,dim( U ) − dim(ker Q ) = dim(ran Q ) . As ker Q = U S and dim(ran Q ) ≤ ω (Λ \ Λ S ) ≤ ω ( ∂ S Λ), the statement follows. (cid:3) Covariant operators and their trace In this section we briefly discuss covariant operators and their natural trace. We will thenprovide a proof of Theorem 2.1.Let N be the set of all covariant decomposable bounded operators. Obviously, N is a vectorspace in the natural way. Moreover, by T ∗ := ( T ∗ ω ) and T S := ( T ω S ω ) it becomes a ∗ -algebra. Infact, it is even a von Neumann algebra. We will not use this in the sequel. We will use that forany selfadjoint T = ( T ω ) ∈ N the operator f ( T ) = ( f ( T ω )) for a bounded and measurable f : R → C defined by spectral calculus is an element of N as well. We start by having a look at functionswhich satisfy (1). Proposition 4.1. For every measurable precompact I ⊂ Γ with < | I | , the function | I | − χ IF ′ satisfies (1) .Proof. As I is precompact and the action of Γ is continuous by assumption, the function χ IF ′ has compact support. Let x ∈ X be arbitrary. Then, x can be uniquely written as x = αx with α ∈ Γ and x ∈ F ′ . A short calculation then shows χ IF ′ ( γ − x ) = χ αI − ( γ ) and the claim followseasily from unimodularity. (cid:3) For a function u on X and I ⊂ Γ we define u I by u I ( x ) := R I u ( γ − x ) dγ . Proposition 4.2. If u satisfies (1) , so does | I | − u I for any precompact measurable I ⊂ Γ with < | I | .Proof. This follows by a direct calculation using Fubini theorem and unimodularity. (cid:3) Theorem 4.3. Let u satisfy (1) . Then, the map τ : N −→ R , τ ( T ) := 1dens( m ) Z Tr( uT ω ) dµ ( ω ) does not depend on u and satisfies the following properties: • τ is faithful, i.e. for T ≥ , τ ( T ) = 0 if and only if T = 0 . • τ has the trace property, i.e. τ ( ST ) = τ ( T S ) for all T, S ∈ N .Proof. This can essentially be obtained from the groupoid theoretical considerations in [28]. Forthe convenience of the reader we give a direct proof (see [31, 30] for similar calculations as well). UMPS OF THE INTEGRATED DENSITY OF STATES 7 We first show that τ does not depend on u . Let u and v satisfying (1) be given. Then, Z Ω X x ∈ X ( ω ) u ( x ) T ω ( x, x ) dµ ( ω ) = Z Ω X x ∈ X ( ω ) u ( x ) T ω ( x, x ) (cid:18)Z Γ v ( γ − x ) dγ (cid:19) dµ ( ω )(Fubini) = Z Ω Z Γ X x ∈ X ( ω ) u ( x ) T ω ( x, x ) v ( γ − x ) dγdµ ( ω )(covariance) = Z Ω Z Γ X y ∈ γ − ω u ( γy ) T γ − ω ( y, y ) v ( y ) dγdµ ( ω )( µ invariant) = Z Ω Z Γ X y ∈ X ( ω ) u ( γy ) T ω ( y, y ) v ( y ) dγdµ ( ω )(Fubini) = Z Ω X y ∈ X ( ω ) (cid:18)Z Γ u ( γy ) dγ (cid:19) T ω ( y, y ) v ( y ) dµ ( ω )= Z Ω X y ∈ X ( ω ) v ( y ) T ω ( y, y ) dµ ( ω ) . and independence is proven. By independence of u and Proposition 4.1 we can replace u by | I | χ IF ′ for any precompact I ⊂ Γ. If T ≥ τ ( T ) = 0 we have χ B r F ′ T ω χ B r F ′ = 0 for almost all ω and any r > 0. Since the sequence B r F ′ exhausts X , T must be zero. Thus faithfulness is proven.We finally show τ ( T K ) = τ ( KT ). The calculation is similar to the one to show independence of u . The definition of τ gives after inserting 1 = R Γ u ( γ − y ) dγ and using Fubini τ ( T K ) = Z Ω Z Γ X x,y ∈ X ( ω ) T ω ( x, y ) K ω ( y, x ) u ( x ) u ( γ − y ) dγdµ ( ω )(covariance) = Z Ω Z Γ X x,y ∈ γ − ω T γ − ω ( x, y ) K γ − ω ( y, x ) u ( γx ) u ( y ) dγdµ ( ω )( µ invariant) = Z Ω Z Γ X x,y ∈ X ( ω ) T ω ( x, y ) K ω ( y, x ) u ( γx ) u ( y ) dγdµ ( ω )= τ ( KT ) . This finishes the proof. (cid:3) Definition 4.4. Let U be a subspace of H = R ⊕ ℓ ( X ( ω )) dµ ( ω ) such that the orthogonal projec-tion P = P U onto U belongs to N . Then, τ ( P ) is called the equivariant dimension of U .For later use we note the following consequence of Theorem 4.3 and Proposition 4.1. Corollary 4.5. For each measurable I ⊂ Γ with < | I | < ∞ the equation τ ( T ) = 1 | I | m ) Z Tr( χ IF ′ T ω ) dµ ( ω ) holds for every T ∈ N .Proof. Set J := B ∩ I and J n := ( B n +1 \ B n ) ∩ I for n ∈ N . Assume without loss of generalitythat | J n | > n . Then, I is the disjoint union of the J n , n ∈ N and each J n satisfies theassumption of Proposition 4.1. Hence, we obtain1 | I | Z Tr( χ IF ′ T ω ) dµ ( ω ) = 1 | I | X n ∈ N Z Tr( χ J n F ′ T ω ) dµ ( ω ) = 1 | I | X n ∈ N | J n | dens( m ) τ ( T ) = dens( m ) τ ( T ) . (cid:3) DANIEL LENZ AND IVAN VESELI´C Proof of Theorem 2.1. Set ν H ( ϕ ) = τ ( ϕ ( H )). Then ν H is a positive functional and hence definesa unique measure. By faithfulness of τ it is a spectral measure. By definition, (2) holds. (cid:3) Equivariant dimension of subspaces and jumps of the IDS This section is devoted to a proof of Theorem 2.2. Note that we do not need ergodicity toderive this result.Recall that every Følner sequence ( I n ) in Γ induces a sequence Λ n := I n F ′ of measurablesubsets of X . By Proposition 3.4 (Λ n ) is a van Hove sequence in X . We start with a technicalresult.Note that for A ⊂ A ⊂ X we can canonically regard elements of ℓ ( A ( ω )) as elements of ℓ ( A ( ω )), as well, by extending them by zero. Lemma 5.1. Let P ∈ N with P ≥ and τ ( P ) > be given. Let R > and a Følner sequence ( I n ) in Γ be given. Then, there exists an N ∈ N and a set e Ω in Ω of positive measure such thatfor all ω ∈ e Ω ran( p Λ N ( ω ) P ω ) ∩ ℓ (Λ N,R ( ω )) = { } . Proof. Without loss of generality we can assume that the density dens( m ) equals 1. As P bebelongs to N , the function ω 7→ k P ω k is essentially bounded. We can assume without loss ofgenerality that this constant is equal to 1. We set δ := τ ( P ) > 0. By Proposition 3.4, there exists N ∈ N with ω ( ∂ R Λ N ) ≤ δ | I N | for all ω ∈ Ω. As by Corollary 4.5, δ = τ ( P ) = Z | I N | Tr( χ Λ N P ω ) dµ ( ω ) , there exists a set e Ω of positive measure with1 | I N | Tr( χ Λ N P ω ) ≥ δ for all ω ∈ e Ω. This givesdim(ran( p Λ N ( ω ) P ω )) ≥ Tr( p Λ N ( ω ) P i Λ N ( ω ) ) ≥ δ | I N | for all ω ∈ e Ω. The statement follows from Lemma 3.5 with U = ran( p Λ N ( ω ) P ω ), since then U R = ran( p Λ N ( ω ) P ω ) ∩ ℓ (Λ N,R ( ω )). (cid:3) Lemma 5.2. Let H satisfy (A). Let λ ∈ R be arbitrary. Set P := E H ( { λ } ) and denote by P comp the projection on the closure of the linear hull of compactly supported eigenfunctions to λ . Then,the following holds: (a) P = P comp . (b) If P comp = 0 , then τ ( P ) > .Proof. (a) If P = 0 the statement is clear. We therefore only consider the case P = 0, i.e. τ ( P ) > 0. Assume P = P comp . Then, Q := P − P comp is a projection with Q = 0. Hence τ ( Q ) > τ is faithful. Set R to be twice the hopping range of H . By the previous lemma, there exist N ∈ N and a set e Ω of positive measure in Ω such that for each ω ∈ e Ωran( p Λ N ( ω ) Q ω ) ∩ ℓ (Λ N,R ( ω ) = { } . By definition of the hopping range this gives compactly supported eigenfunctions in the range of Q ω for all ω ∈ e Ω. This is a contradiction.(b) This follows as τ is faithful. (cid:3) Proof of Theorem 2.2. This is a direct consequence of the previous lemma. (cid:3) UMPS OF THE INTEGRATED DENSITY OF STATES 9 Proof of Corollary 2.3. As in Theorem 2.2, let P comp,ω be the projection onto the space of com-pactly supported solutions of ( H ω − λ ) u = 0 and P comp = ( P comp,ω ). Let Ω ′ be the set of ω ∈ Ωwith P comp,ω = 0. Then, Ω ′ is invariant and measurable, hence by ergodicity it has measure 0 or1. Now, by Theorem 2.2 and the faithfulness of τ , the function N H is discontinuous at λ (i.e. (i)holds) if and only if P comp = 0. By definition, P comp = 0 if and only if Ω ′ has positive measure(i.e. (ii) holds). As Ω ′ has either measure 0 or meassure 1, it has positive measure if and only ifit has measure 1 (i.e. (iii) holds). (cid:3) Some deterministic approximation results In this section we present three deterministic results. The first two results give estimates forthe difference χ Λ φ ( H ω ) − φ ( p Λ( ω ) H ω i Λ( ω ) )for suitable functions φ on R and Λ ⊂ X . In one way or other this type of result is entering allconsiderations on convergence of the IDS (see the first list of references in the Introduction). Onthe technical level our arguments are related to [10, 13, 35].The last result gives a lemma from measure theory on convergence of distribution functions formeasures on R . This type of lemma seems to have first been used in the present context in [12].There, one can also find a proof. For the convenience of the reader we include an alternative proof(and actually give a slightly strengthened result). Lemma 6.1. Let ( I n ) be a Følner sequence in Γ and Λ n := I n F ′ as before. Let H satisfy (A).Let φ be a continuous function on R . Then, for any ω ∈ D| Tr( χ Λ n φ ( H ω )) − ν ω,n ( φ ) || I n | −→ for n → ∞ .Proof. First we prove that the statement holds if φ ( x ) = x k for some k ∈ N . Note that ν ω,n ( φ ) =Tr( H kω,n ) is a sum over closed paths of lenght k , more precisely X x ∈ Λ n ( ω ) X x ,...,x k +1 ∈ Λ n ( ω ) H ω,n ( x , x ) . . . H ω,n ( x k , x k +1 )where in the second sum x = x = x k +1 . Since Tr( χ Λ n H kω ) can be written in a similar way, thedifference Tr( χ Λ n H kω ) − Tr( H kω,n ) is bounded in modulus by X x ∈ ∂ kR Λ n ( ω ) X x ,...,x k +1 ∈ X ( ω ) | H ω,n ( x , x ) | . . . | H ω,n ( x k , x k +1 ) |≤ X x ∈ ∂ kR Λ n ( ω ) ω ( B kR ) k k H k k ≤ ω ( ∂ kR Λ n )( M kR ) k k H k k . Here R denotes the finite hopping range of the operator H ω . Now one uses Proposition 3.4 andthe fact that I n , n ∈ N is a Følner sequence to conclude thatlim n →∞ ω ( ∂ kR Λ n ) | I n | ( M kR ) k k H k k = 0Now the following considerations extend the convergence result to all φ ∈ C ( R ):Note that the only relevant data of the function φ are its values on the spectrum of H ω , whichis a compact set as H ω is bounded. Let S be the family of functions for which the statement of thelemma holds. We just proved that all polynomials belong to S . Moreover, the set S is obviouslyclosed under uniform convergence. The statement follows from Stone-Weierstrass theorem. (cid:3) Lemma 6.2. Let ( I n ) be a Følner sequence in Γ and λ ∈ R arbitrary. Then, for any ω ∈ ˜ D| Tr( χ Λ n E ω ( { λ } )) − ν ω,n ( { λ } ) || I n | −→ for n → ∞ .Proof. Let R be the hopping range of H . Fix ω ∈ D . Let V n be the subspace of all solutions of( H ω − λ ) v = 0, which vanish outside Λ n,R . Let D n be the dimension of V n .We now apply Lemma 3.5 to the space U of all solutions of ( p Λ n ( ω ) H ω i Λ n ( ω ) − λ ) u = 0 andnote that U R = V n as the hopping range of H ω is R . This gives0 ≤ ν ω,n ( { λ } ) − D n ≤ ω ( ∂ R (Λ n )) . Moreover, obviously, D n ≤ Tr( χ Λ n E ω ( { λ } )) ≤ dim(ran( χ Λ n E ω ( { λ } ))) . We now apply Lemma 3.5 to U ′ := ran χ Λ n E ω ( { λ } ) and note that U ′ R = V n as the hopping rangeof H ω is R . This gives 0 ≤ Tr( χ Λ n E ω ( { λ } )) − D n ≤ ω ( ∂ R (Λ n )) . By Proposition 3.2, there exists ρ > ∂ R (Λ n ) ⊂ ( ∂ ρ I n ) F ′ for all n ∈ N . The statementfollows now from the triangle inequality and Proposition 3.4. (cid:3) Lemma 6.3. Let ν be a probability measure on R . Let ( ν n ) be a sequence of bounded measureson R which satisfy • ν n converge weakly to the measure ν , • ν n ( { λ } ) −→ ν ( { λ } ) for all λ ∈ R .Then, the distribution functions λ ν n (( −∞ , λ ]) of the ν n converge with respect to the supremumnorm to the distribution function λ ν (( −∞ , λ ]) of ν .Remark. Note that vague convergence ν n → ν , actually implies weak convergence, if one of thetwo following conditions hold: • all ν n are probability measures, or • there exists a compact interval such that the supports of ν and of ν n is contained in thisinterval for all n ∈ N . Proof. Let ǫ > ν c denote the continuous part of ν and ν p the point part of ν .Choose −∞ = t − = t < t < . . . < t L < t L +1 < t L +2 = ∞ such that( ∗ ) ν c (( t j , t j +1 ]) ≤ ǫ for j = 1 , . . . L , ν ( −∞ , t ) ≤ ǫ and ν ( t L +1 , ∞ ) ≤ ǫ .Such a choice is possible since ν is a probability measure. Let ( λ k ) be an enumeration of thepoints of discontinuity of ν . Assume without loss of generality that k runs through all of N .Choose N ∈ N with( ∗∗ ) P ∞ k = N +1 ν ( { λ k } ) ≤ ǫ .Choose continuous functions φ j with χ ( −∞ ,t j ] ≤ φ j ≤ χ ( −∞ ,t j +1 ] for j = 0 , . . . , L . Set φ − ≡ n we then have( ∗ ∗ ∗ ) | ν n ( φ j ) − ν ( φ j ) | ≤ ǫ for j = 1 , . . . , L + 1,( ∗ ∗ ∗∗ ) | ν n ( χ { λ j | j =1 ,...,N } ) − ν ( χ { λ j | j =1 ,...,N } ) | ≤ ǫ .For such n we prove now ν (( −∞ , λ ]) − ν n (( −∞ , λ ]) ≤ ǫ for all λ ∈ R as follows: Choose j ∈ , . . . , L + 1 with t j ≤ λ < t j +1 and define ψ by χ ( −∞ ,λ ] = φ j − + ψ UMPS OF THE INTEGRATED DENSITY OF STATES 11 be given. Since 0 ≤ ψ ≤ R and supp ψ ⊂ [ t j − , t j +1 ] we then obtain ν n (( −∞ , λ ]) = ν n ( φ j − ) + ν n ( ψ )( ∗ ∗ ∗ ) ≥ ν ( φ j − ) − ǫ + ν n ( ψ )= ν (( −∞ , λ ]) − ν ( ψ ) − ǫ + ν n ( ψ )( ∗ ) ≥ ν (( −∞ , λ ]) − ν p ( ψ ) − ǫ + ν n ( ψ )= ν (( −∞ , λ ]) − ǫ + ν n ( ψ ) − ν p ( ψ )( ∗∗ ) ≥ ν (( −∞ , λ ]) − ǫ + ν n ( ψ ) − ν p ( ψχ { λ j | j =1 ,...,N } ) ≥ ν (( −∞ , λ ]) − ǫ + ν n ( ψχ { λ j | j =1 ,...,N } ) − ν p ( ψχ { λ j | j =1 ,...,N } )( ∗ ∗ ∗∗ ) ≥ ν (( −∞ , λ ]) − ǫ. Note that the above inequalities hold also if we replace ν (( −∞ , λ ]) , ν n (( −∞ , λ ]) by ν (( −∞ , λ )) , ν n (( −∞ , λ )).Thus we have proven lim n →∞ sup λ ∈ R ν (( −∞ , λ ]) − ν n (( −∞ , λ ]) ≤ ν which is the reflection of ν around theorigin. It inherits all properties of ν which have been used in the previous calculation. Then˜ ν (( −∞ , λ )) = ν ([ − λ, ∞ )) = 1 − ν (( −∞ , − λ ])The above argument yields lim n →∞ sup λ ∈ R ˜ ν (( −∞ , λ )) − ˜ ν n (( −∞ , λ )) ≤ 0. Since ˜ ν (( −∞ , λ )) − ˜ ν n (( −∞ , λ )) = ν n (( −∞ , − λ ]) − ν (( −∞ , − λ ]), the proof is completed. (cid:3) Proof of Theorem 2.4 In this section, we prove Theorem 2.4. Since here ergodicity plays a role, we first discuss variousconsequences of ergodic theorems.For a function v on Ω and I ⊂ Γ with | I | > v I on Ω by v I ( ω ) := Z I v ( γ − ω ) dγ. For any v integrable with respect to µ and any tempered Følner sequence ( I n ) there exists aΓ-invariant ¯ v ∈ L ( µ ) such that lim n →∞ | I n | v I n ( ω ) = ¯ v ( ω )for almost every ω ∈ Ω and in L , see [33]. Moreover, if µ is ergodic ¯ v ( ω ) = R v ( ω ) dµ ( ω ) almostsurely.Our natural setting is not concerned with v I but rather with functions of the form χ IF ′ . Wenext show that these two types of functions are comparable. To do so, we recall that we havefixed a point p ∈ F ′ ⊂ X . Proposition 7.1. Let u be a measurable nonnegative function on X satisfying (1) . Let r > such that the support of u is contained in the open ball B r ( p ) around p with radius r . Then, | χ IF ′ ( x ) − u I ( x ) | ≤ χ ∂ r ( IF ′ ) ( x ) for any x ∈ X .Proof. Consider first x ∈ ( IF ′ ) r : Then, B r ( x ) ⊂ IF ′ . Any γ ∈ Γ with u ( γ − x ) = 0 satisfies d ( γ − x, p ) < r , hence γp ∈ B r ( x ) ⊂ IF ′ and thus γ ∈ I . For such x we therefore obtain Z I u ( γ − x ) dγ = Z u ( γ − x ) dγ = 1 . Consider now x / ∈ ( IF ′ ) r : Then, B r ( x ) ⊂ X \ IF ′ . Any γ ∈ Γ with u ( γ − x ) = 0 satisfies γp ∈ B r ( x ) ⊂ X \ ( IF ′ ) and hence γ / ∈ I . For such x we therefore obtain Z I u ( γ − x ) dγ = 0 . Consider now x ∈ ( IF ′ ) r \ ( IF ′ ) r : As u is nonnegative with R u ( γ − x ) dγ = 1, we obtain0 ≤ u ( x ) ≤ u .Having considered these three cases, we can easily obtain the statement. (cid:3) We now derive two consequences of the previous proposition and the ergodic theorem. Proposition 7.2. Let ( I n ) a tempered Følner sequence in Γ and T ∈ N be arbitrary. Then, foralmost every ω ∈ Ω , lim n →∞ | I n | Tr( χ Λ n T ω ) = g ( ω ) . where g ∈ L ( µ ) is Γ -invariant. The convergence holds also in L sense. If µ is ergodic, g =dens( m ) τ ( T ) almost surely.Proof. We show that ω | I n | Tr( χ Λ n T ω ) converges almost surely and in L for n → ∞ . ByProposition 3.4 combined with Proposition 7.1, it suffices to consider the sequenceTr( u I n T ω ) = X x ∈ X ( ω ) u I n ( x ) T ω ( x, x )with u satisfying (1), instead of consideringTr( χ Λ n T ω ) = X x ∈ Λ n ( ω ) T ω ( x, x )Define with such a u v ( ω ) := Tr( u T ω ) = X x ∈ X ( ω ) u ( x ) T ω ( x, x ) . By the ergodic theorem, v I n / | I n | converges almost surely to some Γ-invariant g ∈ L ( µ ). A directcalculation using equivariance shows v I n ( ω ) = X x ∈ X ( ω ) u I n ( x ) T ω ( x, x ) . Thus the first statement of the Proposition is proven. If µ is ergodic, then g = R Tr( u T ω ) dµ ( ω ) =dens( m ) τ ( T ) almost surely. (cid:3) We note the following special case of the previous proposition. Corollary 7.3. Let µ be ergodic and ( I n ) be a tempered Følner sequence in Γ . For almost every ω ∈ Ω , we have lim n →∞ ω (Λ n ) | I n | = dens( m ) = dens( m ) τ (Id) . In particular, τ (Id) = 1 .Proof. The previous proposition with T = Id shows pointwise and L convergence of the functions ω ω (Λ n ) | I n | to the constant dens( m ) τ (Id). Sincedens( m ) = m (Λ n ) | I n | = Z ω (Λ n ) | I n | dµ ( ω )for all n ∈ N , τ (Id) must be equal to 1 and the statement follows. (cid:3) Proof of Theorem 2.4. Since N ω,n ω (Λ n ) = N ω,n | I n | | I n | ω (Λ n ) and | I n | ω (Λ n ) converges to dens( m ) − almost surely,it suffices to show convergence of N ω,n | I n | . There are at most countably many points of discontinuityof ν H . Thus, Lemma 6.2 combined with Proposition 7.2, gives convergence in all points of dis-continuity of ν H almost surely. On the other hand, note that the space of continuous functionswith compact support on R is separable. Thus, weak convergence of probability measures followsfrom convergence on a countable dense subset of continuous functions on R with compact support.Thus, Lemma 6.1 combined with Proposition 7.2 gives weak convergence of the measures almostsurely. Now, the result follows from Lemma 6.3. (cid:3) UMPS OF THE INTEGRATED DENSITY OF STATES 13 Remark . Even in the case that µ is not ergodic we can prove a uniform convergence statement.By assumption (A) there is an R ′ ∈ R such that σ ( H ω ) ⊂ [ − R ′ , R ′ ]. Let ψ j , j ∈ N be a countabledense set in C ([ − R ′ , R ′ ]). By Proposition 7.2 there exists a set of full measure Ω j ⊂ Ω such thatfor all ω ∈ Ω j l ω ( ψ j ) := lim n →∞ | I n | Tr( χ Λ n ψ j ( H ω ))exists. This way one defines for all ω ∈ ˜Ω := ∩ j ∈ N Ω j a positive, bounded linear functional l ω ,i.e. a measure. In particular one may define a density for such ω by dens( ω ) = lim n →∞ ω (Λ n ) | I n | .For ω ∈ ˜Ω with dens( ω ) > N ω,n ω (Λ n ) = N ω,n | I n | | I n | ω (Λ n ) which thus converges to the distribution function of ν ω := l ω dens ω . If dens( ω ) = 0 we still see thatTr( χ Λ n ψ ( H ω )) ≤ k ψ k ∞ ω (Λ n ). Thus if X ( ω ) is not emptyTr( χ Λ n ψ ( H ω )) ω (Λ n ) ≤ k ψ k ∞ and we can conlude by the Banach-Alaouglu theorem that there is a subsequence along which N n k ω ω (Λ n k ) converges weakly. A posteriori, we can enhance this to uniform convergence using Lemma6.3. 8. Models with aperiodic order In this section, we discuss models with aperiodic order. In these cases Γ = X = R m is acontinuous group. We recover the main result of [21] concerning characterization of jumps of theIDS via compactly supported eigenfunctions. In fact, we obtain a strengthening of the result of[21] in three respects: We do not need an ergodicity assumption anymore, we do not need a finitelocal complexity assumption and we identify the size of the jumps as a equivariant dimension.(Note that the latter, however, can directly derived from the convergence statement in [32]). Wealso obtain a result on convergence of the IDS. This result, however, is strictly weaker than theresults of [32] as it neither holds for all ω ∈ Ω nor gives an explicit error bound on the speed ofconvergence.The setting is as follows: There is an obvious action of Γ = R m on X = R m by translation.A fundamental domain is given by the compact set { } and the map Φ : Γ −→ R m is just theidentity and therefore an isometry. Hence, the geometric assumptions of our setting are satisfied.The ball around x ∈ R m with radius r is denoted by B r ( x ). As before, D is the set of subsets of R m whose elements have Euclidian distance at least 1. We call a subset M of D a collection ofDelone sets if the following holds:– There exists an R ′ > A ∩ B R ′ ( x ) = ∅ for every A ∈ M and x ∈ X .It is said to be of finite local complexity if it also satisfies the following:– For each r > 0, the set { ( A − x ) ∩ B r (0) | A ∈ M , x ∈ A } is finite.Let µ = µ M be an invariant probability measure on D whose support Ω is a collection of Delonesets. In particular X ( ω ) = ω ∈ Ω is a discrete subset of R d . Assume that the density of m is 1.This setting gives a notion of an equivariant operator as a family ( H ω ) of operators H ω : ℓ ( X ( ω )) −→ ℓ ( X ( ω )) with H γ + ω ( γ + x, γ + y ) = H ω ( x, y ) . The natural trace is defined via τ ( H ) = Z Ω Tr( uH ω ) dµ ( ω ) , where the continuous u : R m −→ R is an arbitrary function with compact support and R m R u ( x ) dx =1. For an operator satisfying (A) and λ ∈ R our abstract results give: (i) Let U ω be the subspace of ℓ ( X ( ω )) spanned by compactly supported solutions to ( H ω − λ ) u = 0. Then, the value ν H ( { λ } ) = N H ( λ ) − N H ( λ − ) = τ ( E H ( { λ } ) is the equivariantdimension of the subspace R ⊕ U ω dµ ( ω ) of R ⊕ Ω ℓ ( X ( ω )) dµ ( ω ).(ii) If µ is furthermore assumed ergodic, then λ is a point of discontinuity of N H if and onlyif there exists a compactly supported eigenfunction of H ω to λ for almost every ω ∈ Ω. Inthis case, these compactly supported eigenfunctions span the eigenspace of H ω to λ .(iii) The normalized finite volume counting functions N ω,n | Λ n | converge almost surely with respectto the supremum norm towards the function N H , i.e.lim n →∞ (cid:13)(cid:13)(cid:13) | Λ n | N ω,n − N H (cid:13)(cid:13)(cid:13) ∞ = 0 . Periodic models on amenable graphs and CW-complexes In this section, we briefly discuss how the corresponding results of [6, 8, 10, 23, 35, 36] can berecovered in our framework. In these cases the group Γ as well as the space X are discrete andcountable.The geometric setting in the cited works is given either by a graph or a CW-complex. Let usfirst consider the case that a graph G = ( V, E ) with vertex set V and edge set E and a finitelygenerated amenable group Γ are given, such that each vertex degree is finite and Γ acts freelyand cocompactly on G by automorphisms. We set X := V and show that the assumptions ofour setting are satisfied: A finite set S of generators of Γ which is symmetric in the sense that S = S − := { γ − | γ ∈ S } defines a word metric d S on Γ. Obviously, d S is invariant under theaction of Γ on itself. If we choose a different (finite, symmetric) set of generators S it gives riseto another metric d S . This metric is equivalent to the metric d S and in particular the two spaces(Γ , d S ) and (Γ , d S ) are roughly isometric.As Γ acts cocompactly, there exists a compact (i.e. finite) fundamental domain F ′ of the actionof Γ. In particular, F ′ is equal to its closure F . As Γ acts freely, we obtain a well defined mapΦ : X −→ Γ , with x ∈ Φ( x ) F. We have to show hat this map is a rough isometry. To do so, we need of course a metric on X .Let us first assume that the graph is connected. Then X = V carries a natural metric d comingfrom finite paths between the points. This metric is obviously Γ-invariant. Set C := max s ∈ S ∪{ id } { d ( x, y ) | x ∈ F, y ∈ sF } < ∞ . Then, d ( x, y ) ≤ Cd Γ (Φ( x ) , Φ( y )) + C. As for the converse inequality, we need some more preparation. We say that γ ∈ Γ is a neighborof ρ ∈ Γ if there exists an edge connecting a vertex in ρF with a vertex in γF . Denote the set ofneighbors of id ∈ Γ by S . Since X is connected, S is a set of generators of Γ; since each vertexdegree is finite, S is finite; and by the properties of the action of Γ, S is symmetric. Thus d S (Φ( x ) , Φ( y )) ≤ d ( x, y ) . Since any word metric on Γ is roughly isometric with d S , this shows that Φ is indeed a roughisometry.If the graph is not connected, there is no natural choice of a metric on X . In this case, we caninduce a metric on X by the metric on Γ and Φ in two steps: the metric on the group Γ defines adistance between different fundamental domains. We can assume that this distance function takesvalues in N . Within the fundamental domains the distance between two points is defined usingshortest paths. Let us scale the latter distance function such that the diameter of a fundamentaldomain is bounded by one. This way one obtains a metric on X which is by construction roughlyisometric to Γ.Let us now turn to the case of CW-complexes. Thus let a CW-complex Y and a finitelygenerated amenable group Γ be given which acts freely on Y by automorphisms. We assume thatthe quotient Y / Γ is a CW-complex of finite type, i.e. all its skeleta are finite. For a j ∈ N denote UMPS OF THE INTEGRATED DENSITY OF STATES 15 by Y j the set of j -cells in Y . Two such cells are called adjacent if either the intersection of theirclosures contains a j − Y , or if both are contained in a the closure of a single j + 1 cell.Since we assumed that the quotient Y / Γ is of finite type, the number of cells adjacent to any givencell is finite. Now we fix j ∈ N and define a graph G j with vertex set V = Y j . Two elements V are connected by an edge iff they are adjacent. Each automorphism of the original CW-complexinduces a graph-automorphism on G j . In particular, Γ acts freely and cocompactly on G j . Thuswe are back in the setting which we discussed at the start of this section. (Note that for each j ∈ N we extract from the CW-complex Y a different graph G j and correspondingly the graphHamiltonians, which we define below, will also depend on j .)These considerations show that the geometric assumptions of our model are satisfied in thecited works.In the present setting the set X itself belongs to D and is in fact invariant under the action ofΓ. Thus, we can choose the measure µ on D to be supported on { X } . This means that Ω consistsof a single element which is just X . Thus, everything depending on the family ω ∈ Ω is replacedby a single object in the sequel. In particular, we certainly have all ergodic assumptions satisfied.In fact, all statements concerning almost sure convergence in Ω give deterministic statements.We will now deal with the operator theoretical side of things. In [36, 35] one is given a family s γ , γ ∈ Γ of maps s γ : V −→ { z ∈ C | | z | = 1 } . In the other cases one just sets s γ ≡ γ ∈ Γ. Thefamily of operators in question is then given by a single operator H = H X satisfying s γ ( x ) H ( γx, γy ) s γ ( y ) = H ( x, y ) . The natural trace becomes τ ( H ) = Tr( χ F H ) . As before denote by H n the restriction of the operator H to Λ n = I n F , where I n ∈ Γ is aFølner sequence. With the usual convention N H ( λ ) := τ ( E H ( λ )) and N H,n := Tr E H n ( λ ), ourresults can the be re formulated as follows:(i) For any λ ∈ R , the value ν H ( { λ } ) = τ ( E H ( { λ } ) is the equivariant dimension of thesubspace of ℓ ( X ) spanned by compactly supported solutions of ( H ω − λ ) u = 0.(ii) A number λ ∈ R is a point of discontinuity of N H if and only if there exists a compactlysupported eigenfunction of H to λ .(iii) The functions N H,n converge with respect to the supremum norm towards the function N H , i.e. lim n →∞ (cid:13)(cid:13)(cid:13) | Λ n | N H,n − N H (cid:13)(cid:13)(cid:13) ∞ = 0 . Anderson and percolation Hamiltonians In this section we discuss the application of our results to certain types of random models ongraphs. More precisely, we consider Anderson models, random hopping models, as well as siteand bond percolation models. They can be understood as randomized versions of the operatorsintroduced in Section 9. In particular we are again given a graph G = ( V, E ) with boundedvertex degree on which a finitely generated, amenable group Γ acts freely and cocompactly byautomorphisms.The application of our abstract theorems to these models recover and extend in particularthe results which concern the construction of the IDS by its finite volume analogues obtained in[44, 20, 27].Let us introduce the Anderson-Percolation Hamiltonian . We set X = V and assume that afunction h : X × X → C is given with h ( x, y ) = h ( y, x ) and h ( x, y ) = 0 ⇒ d ( x, y ) < R .Consider the setting from Section 2 and assume additionally that there exist a constant C and afunction V ω : X × X → [ − C, C ] with support on the diagonal D := { ( x, x ) | x ∈ X } sucht that forall ω = ( A, h ) ∈ Ω h = (cid:0) h + V ω (cid:1) χ A × A . The operator associated to ω ∈ Ω acts on the ℓ space of the diluted graph X ( ω ) = A . Moreprecisely for each v ∈ ℓ ( X ( ω )) and x ∈ X ( ω )( H ω v )( x ) = (cid:16) X y ∈ X ( ω ) h ( x, y ) v ( y ) (cid:17) + V ω v ( x ) . We can think of the first term as the kinetic energy or hopping term and of the second as a randompotential. In the special case that X ( ω ) = X for all ω ∈ Ω and h ( x, y ) = χ ( d ( x, y )) := ( d ( x, y ) = 1 , V ω ≡ h ( x, y ) = χ ( d ( x, y )) we have thesite-percolation Hamiltonian. Of course it is possible to choose the measure µ in such a way thatthe discussed models are i.i.d. with respect to the coordinates x ∈ X , see cf. [45].Let us now discuss the random hopping model, which includes the bond percolation model asa special case. Such models have bee considered for instance in [22, 20]. Here we require each( A, h ) ∈ Ω to satisfy additionally to the conditions in Section 2 that A = X and h ≡ D . The matrix coefficient h ( x, y ) may be considered as a hopping term between x and y (at least when h is non-negative). In the case that h ( x, y ) ∈ { , } we obtain a bond-percolationHamiltonian. Again, a suitable choice of the measure µ yields an i.i.d. model.Note that Dirichlet and Neumann boundary terms as considered in [20, 1] can be incorporatedinto a potential energy term V ω . Let us emphasize that the models discussed above do not havenecessarily finite local complexity.As before denote by N ω,n the eigenvalue counting functions associated to a Følner sequence I n in Γ. Our results from Section 2 can the be now reformulated as follows:(i) Let U ω be the subspace of ℓ ( X ( ω )) spanned by compactly supported eigenfunctions of H ω Then, ν H ( { λ } ) = N H ( λ ) − lim ǫ → N H ( λ − ǫ ) equals the equivariant dimension of the subspace R ⊕ U ω dµ ( ω ) of R ⊕ Ω ℓ ( X ( ω )) dµ ( ω ).(ii) If µ is furthermore assumed ergodic, then λ is a point of discontinuity of N H if and only ifthere exist compactly supported eigenfunctions to H ω and λ for almost every ω ∈ Ω. In thiscase, these eigenfunctions actually span the eigenspace of H ω to the eigenvalue λ for almostevery ω ∈ Ω.(iii) For µ -almost all ω , the distibution function N H can be approximated by N ω,n uniformly inthe energy variable : lim n →∞ (cid:13)(cid:13)(cid:13) | Λ n | N ω,n − N H (cid:13)(cid:13)(cid:13) ∞ = 0 . Percolation on Delone sets Consider the setting explained in Section 8. In particular, let the space X and the group Γequal R d . Let M ⊂ D be a collection of Delone sets of finite local complexity and h : R d → R abounded, measurable function of compact support satisfying h ( − x ) = h ( x ). For each A ∈ M let E ( A ) consist of pairs of subsets E , E of A × A satisfing the following(1) E and E are disjoint,(2) E i ∩ D = ∅ where as before D = { ( x, x ) | x ∈ R d } and ( x, y ) ∈ E i ⇒ ( y, x ) ∈ E i for i ∈ { , } .In other words E , E is a pair of disjoint sets of edges for the vertex set A . For such ( E , E ) ∈E ( A ) set Z A,E ,E := { ( A, h ) | h : A × A → R , h ( x, y ) ∈ { , h ( x − y ) } ,h ( x, y ) = h ( x − y ) for all ( x, y ) ∈ E , h ( x, y ) = 0 for all ( x, y ) ∈ E } . For p ∈ [0 , 1] fixed, define a measure µ A on the cylinder sets Z A,E ,E with E and E finite bysetting µ A ( Z A,E ,E ) := p | E | (1 − p ) | E | UMPS OF THE INTEGRATED DENSITY OF STATES 17 and extend it to { h : A × A → R } by uniqueness. Recall that the projection π : ˜ D → D , ( A, h ) A is measurable. Denote by µ M an invariant probability measure on D whose support is M . Nextwe define a measure µ on ˜ D . For this purpose we denote the pairs ( A, h ) by ω althought themeasure µ and thus its support Ω are yet to be identified. For a mesurable B ⊂ ˜ D we define µ ( B ) = Z M Z π − ( A ) χ B dµ A ( ω ) ! dµ M ( A ) . Note that if E and E form a partition of A × A , then Z A,E ,E contains a single element. Thusevery ω = ( A, h ) in the support Ω of µ can be identified with such an Z A,E ,E . The associatedoperator H ω has matrix coefficients H ω ( x, y ) = ( h ( x − y ) = h ( y − x ) if ( x, y ) ∈ E x, y ) ∈ E and defines a bond-percolation Hamiltonian on M .Since we are in Euclidean space we can choose I n = Λ n to be balls or cubes of diameter n ∈ N .Again we have the following results:(i) Let U ω be the subspace of ℓ ( X ( ω )) spanned by compactly supported solutions of ( H ω − λ ) u =0. Then, ν H ( { λ } ) = N H ( λ ) − lim ǫ → N H ( λ − ǫ ) equals the equivariant dimension of the subspace R ⊕ U ω dµ ( ω ) of R ⊕ Ω ℓ ( X ( ω )) dµ ( ω ).(ii) If µ is furthermore assumed ergodic, then λ is a point of discontinuity of N H if and only ifthere exists a compactly supported eigenfunctions to H ω and λ for almost every ω ∈ Ω. Inthis case, these eigenfunctions actually span the eigenspace of H ω to the eigenvalue λ foralmost every ω ∈ Ω.(iii) The distibution function N H can be approximated by N ω,n uniformly in the energy variable:lim n →∞ (cid:13)(cid:13)(cid:13) | Λ n | N ω,n − N H (cid:13)(cid:13)(cid:13) ∞ = 0almost surely. Appendix A. Topology on ˜ D and compactness We discuss the topology and compactness of ˜ D .We start with a slightly more general setting. Let Z be a locally compact space. Denote theset of measures on Z by M ( Z ) and the set of continuous functions with compact support on Z by C c ( Z ). The set M ( Z ) can be embedded into Q ϕ ∈ C c ( Z ) C viaΨ : M ( Z ) −→ Y ϕ ∈ C c ( Z ) C , Ψ( µ ) = ( ϕ µ ( ϕ )) . The product topology then induces the initial topology on M ( Z ), which is called vague topology.Assume now that U an open covering of Z . Define for C > M C, U := M C, U ( Z ) = { µ ∈ M ( Z ) : µ ( U ) ≤ C for all U ∈ U } . As U is an open covering, the support of any ϕ ∈ C c ( Z ) can be covered by finitely many elementsof U . Then, Tychonoff theorem easily yields that M C, U is contained in a compact subset of M ( Z ).As M C, U is closed it must then be compact as well.We now specialize these considerations to the situation outlined in Section 2:Thus, we are given a locally compact metric space X . As before, the set of uniformly discretesubsets of X with minimal distance 1 is denoted by D . Let Y be a compact space and set Z = X × Y . Let D Y be the set of all functions f : A −→ Y with A ∈ D . We can identify elementsof D Y with measures on Z via δ : D Y −→ M ( Z ) , ( f : A −→ Y ) X x ∈ A δ ( x,f ( x )) . This induces the initial topology on D Y . By a slight abuse of language we call this topologythe vague topology. Consider now the cover U of Z consisting of products of the form B / × Y with B / an open ball with radius 1 / X . Then, D Y is a closed and hence compact subset of M , U ( Z ).We can even consider a kind of universal Y as follows: Let C ∗ be an arbitrary compactificationof C . Then, Y := D C ∗ is a compact space by the preceeding considerations and so is then D Y .For an element h : A −→ C ∗ in Y , we set dom( h ) := A . It is not hard to see that˜ D := { f : A −→ Y | dom( f ( x )) = A for all x ∈ A } is a closed subset of the compact space D Y . Hence, ˜ D is compact as well. Any f : A −→ Y satisfying dom( f ( x )) = A gives rise to an h : A × A −→ C ∗ with h ( x, y ) := f ( x )( y ). We can andwill therefore naturally identify ˜ D with the set { ( A, h ) | A ∈ D , h : A × A → C ∗ } . For the use in the main text let us note that the topology of ˜ D is such that for any continuous φ : X × X × C ∗ → C of compact support, the map˜ D → C , ( A, h ) X x,y ∈ A φ ( x, y, h ( x, y ))is continuous. 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