THE COLORED BENJAMINI-SCHRAMM TOPOLOGY

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1 THE COLORED BEJAMII-SCHRAMM TOPOLOGY MOSTAFA SABRI Abstract. These are lecture notes on the Benjamini-Schramm topology. We discuss basic notions, examples and implications for the spectral measures.. Introduction In these notes we discuss the basic properties of the Benjamini-Schramm topology, carry out some examples, and prove that convergence in the sense of Benjamini-Schramm implies convergence of the associated empirical measures. othing presented here is new; our purpose is simply to give complete proofs for newcomers. What Benjamini and Schramm introduced in [8] is a notion of convergence for a sequence of finite graphs (G n ). An important aspect is that if this sequence has a uniformly bounded degree, then there always exists a it object up to passing to a subsequence. Moreover, it gives a correct notion of convergence from the point of view of spectral theory. For example it has been known at least since [2] that if one is interested in the spectral properties of the (q + )-regular tree T q, then studying what happens on a sequence of growing balls around some fixed origin is not a good idea, because the spectral behavior of the it object is completely different from those of T q. And indeed, such tree balls do not Benjamini-Schramm converge to T q. On the other hand, a sequence of (q + )-regular graphs with few short cycles do converge to T q in the sense of Benjamini-Schramm, and it is known that the mean spectral measures of the corresponding adjacency matrices converge to the density of states of the adjacency matrix on T q. This is known as the law of Kesten-McKay, and it turns out to be a general phenomenon. As previously mentioned, an important advantage of this notion of convergence is the existence of a it object. A very broad question is whether specific information about the it object implies something on the convergent sequence when n gets large. It is one such question that is considered in [6], where we show that if (G n, W n ) is a sequence of colored graphs of uniformly bounded degree and coloring, and if (G n, W n ) has a local weak it ρ supported on colored trees, then roughly speaking, AC spectrum of the it Schrödinger operator implies quantum ergodicity for the sequence. A lot of different spectral questions have also been studied in the literature, and relations have been found with conjectures in group theory [3]. 2. Basic definitions A colored rooted graph is a triple (G, o, W ), where G = (V, E) is a graph, o is a marked vertex in G called the root, and W is a map from V R which we see as a coloring ; it can also be regarded as a potential on l 2 (V ). This is a special case of what is called a network in [3], but the discussion of this note applies to this general setting as well. All graphs are assumed to be locally finite, i.e. each vertex has a finite degree. If G is connected, we denote by B G (x, r) the r-ball {y V : d G (x, y) r}, where d G is the length of the shortest path between x and y in G. We say there is a rooted isomorphism Date: April 8, 207. Thanks to alini Anantharaman for clarifications!

2 2 MOSTAFA SABRI ϕ between two balls B G (x, r) and B G2 (x 2, r), and denote ϕ : B G (x, r) B G2 (x 2, r) if ϕ : B G (x, r) B G2 (x 2, r) is a graph isomorphism with ϕ(x ) = x 2. We define a distance between colored connected graphs by (2.) d loc [(G, o, W ), (G 2, o 2, W 2 )] = where + α,2, α,2 = sup r > 0 : ϕ : B G (o, [r]) B G2 (o 2, [r]) with W 2 (ϕ(v)) W (v) < /r v B G (o, [r]). Lemma 2.. The function d loc is a pseudo-metric on the set of colored graphs. Proof. We prove the triangle inequality. Denote W j ϕ W i = sup x BGi (o i,t) W j (ϕ(v)) W i (v), where the radius t of the ball will be clear from the context. We need to show that +α,2 +α,3 + +α 2,3. Assume α,2 = r and let s > r. Then there is no ϕ : B G (o, [s]) B G2 (o 2, [s]) having W 2 ϕ W < /s. We have two cases : if there is no ψ : B G (o, [s]) B G3 (o 3, [s]), then α,3 < s. In this case, d(g, G 2 ) = +r = sup s>r +s +α,3 = d(g, G 3 ). Similarly, if there is no ψ : B G2 (o 2, [s]) B G3 (o 3, [s]), then α 2,3 < s, implying d(g, G 2 ) d(g 2, G 3 ). In either case we get d(g, G 2 ) d(g, G 3 ) + d(g 2, G 3 ). So assume now that there exists ψ : B G (o, [s]) B G3 (o 3, [s]) and ψ 2 : B G2 (o 2, [s]) B G3 (o 3, [s]). Then ψ2 ψ : B G (o, [s]) B G2 (o 2, [s]). So by assumption, we must have W 2 ψ2 ψ W /s. Hence, s W W 2 ψ2 ψ W W 3 ψ + W 3 ψ W 2 ψ2 ψ = W W 3 ψ + W 3 W 2 ψ2 = W 3 ψ W + W 3 ψ 2 W 2. So W 3 ψ W + W 3 ψ 2 W 2 s and thus + s + W 3 ψ W + W 3 ψ 2 W 2 = W 3 ψ W + W 3 ψ W + W 3 ψ 2 W 2 + W 3 ψ 2 W 2 + W 3 ψ W +. + W 3 ψ 2 W = W 3 ψ W + W 3 ψ 2 W 2 + W 3 ψ W + W 3 ψ 2 W 2 Finally, we have α,3 W 3 ψ W and α 2,3 W 3 ψ 2 W by definition (because the r must satisfy r < W j ϕ W i ). We thus showed that +s +α,3 + +α 2,3, and this completes the proof as before. We say that two colored graphs (G, o, W ) and (G 2, o 2, W 2 ) are equivalent if there is a graph isomorphism ϕ : G G 2 such that ϕ(o ) = o 2 and W 2 ϕ = W. We denote the equivalence class of (G, o, W ) by [G, o, W ]. We denote by G the set of equivalence classes of connected colored rooted graphs. Lemma 2.2. The function d loc induces a metric on G.

3 THE COLORED BEJAMII-SCHRAMM TOPOLOGY 3 Proof. The value of α,2 is independent of the choice of the representative in the equivalence class. Suppose d loc [(G, o, W ), (G, o, W )] = 0. We show that [G, o, W ] = [G, o, W ]. For any r, there is ϕ r : B G (o, r) B G (o, r) with W ϕ r W BG (o,r) < /r. : B G (o, n) B G (o, r) for r n. Actu- Let ϕ (n) r ally, ϕ (n) r = ϕ r BG (o,n) for r n. Then ϕ (n) r : B G (o, n) B G (o, n) for all r n, because ϕ r is a graph isomorphism and thus preserves neighbors. So ϕ (n) r : B G (o, n) Ran ϕ (n) r is a graph isomorphism. So Ran ϕ (n) r = B G (o, n) = B G (o, n), where the last equality holds because ϕ n is a graph isomorphism. It follows that Ran ϕ (n) r = B G (o, n). Hence, ϕ r (n) : B G (o, n) B G (o, n) for all r n. Since G and G are locally finite, ϕ (n) r has a convergent (in fact stationary) subsequence ϕ (n) r j. Denote its it by ϕ (n). Then ϕ (n) : B G (o, n) B G (o, n). ow ϕ (n+) BG (o,n) = ϕ (n+) r j BG (o,n) = ϕ (n) r j = ϕ (n). So ϕ (m) BG (o,n) = ϕ (n) for all m n. Hence, if for v G, say v B G (o, n) for some n, we put ϕ(v) := ϕ (n) (v), then ϕ is well-defined. Moreover, ϕ is bijective. In fact, if w G, then w = ϕ (n) (v) for some n and v, so w = ϕ(v); injectivity is similar. Finally ϕ is a graph isomorphism since given v v, say both in B G (o, n), we have {ϕ(v), ϕ(v )} = {ϕ (n) (v), ϕ (n) (v )} is an edge in G. We thus showed that ϕ : (G, o) (G, o ). It remains to check the coloring. ote that for any n and r n, we have W ϕ r (n) W BG (o,n) < /r, since ϕ r satisfies this on the bigger ball B G (o, r). Since ϕ (n) is the it of ϕ (n) r j, we get W ϕ (n) W BG (o,n) = 0. This shows that W ϕ = W. Hence, [G, o, W ] = [G, o, W ] as required. Lemma 2.3. The metric space (G, d loc ) is a Polish space, i.e. separable and complete. Proof. Consider the family C n = ([n],, W ), where [n] = {,..., n} and W takes values in Q. This is a countable family, since the set of maps W is just Q [n]. We denote C = n C n. Given [G, o, W ] G and ε > 0, choose r such that +r < ε. Since G is locally finite, the number of vertices in B G (o, r) is some n = n(r). If ϕ : B G (o, r) ([n], ) is a rooted graph isomorphism, we may choose W n : [n] Q such that W n ϕ W < /r. Then d loc [(G, o, W ), ([n],, W n )] +r, so C is dense in G. ext, suppose ([G n, o n, W n ]) is Cauchy in G. Assuming V (G n ) = V n, we consider equivalently the sequence ([[V n ],, W n]), where W n = W n ϕ n for some ϕ n : G n [Vn ]. We may check that for any r there exists n r and ψn m : B [Vn](, r) B [Vm](, r) such that W m ψn m W n < /r for all n, m n r. For any v [V nr ], the sequence {a,..., a nr, W n r (v), W n r+ ψnr+ n r (v), W n r+2 ψnr+2 n r (v),... }, a j =, is thus Cauchy in R and converge to some A v. Since [B [Vn](, r)] is stationary in n for any r, it converges to some [B [V ] (, r)], possibly [V ] =. If we define A : [V ] R by A(v) = A v, it follows that d loc [([V n ],, W n), ([V ],, A)] +r for n n r. As r is arbitrary, the sequence converges to [[V ],, A]. We next define G D,A to be the subset of equivalence classes [G, o, W ] such that G has degree uniformly bounded by D and W takes values in [ A, A]. Lemma 2.4. The metric space (G D,A, d loc ) is compact. Proof. Clearly, G D,A is closed in G and thus complete. So it suffices to show that it is totally bounded. Suppose this is not true. Then for some ε > 0, there is no finite ε-net. So we may construct a sequence ([G n, o n, W n ]) such that d loc [(G n, o n, W n ), (G m, o m, W m )] ε for all n m. Hence, there exists r such that α n,m < r for all n m, where α n,m is as in (2.). ow observe that there are only finitely many equivalence classes of rooted balls B Gn (o n, r), since the degree is uniformly bounded by D. So if α n,m < r for all n m, there must be a sequence of isomorphic balls B Gnj (o nj, r) on which the coloring is distant,

4 4 MOSTAFA SABRI say W np ψ p q W nq /r for all p q, where ψ p q : B Gnq (o nq, r) B Gnp (o p, r) is a rooted isomorphism. So the sequence {W n, W n2 ψ 2, W n 3 ψ 3,... } on B G n (o n, r) has no convergent subsequence. But this is a sequence in [ A, A] B Gn (o n,r), which is compact for the product topology, which coincides with the topology of pointwise convergence, i.e. the topology endowed by W = sup v BGn W (v). This contradiction completes (on,r) the proof. So far we have defined a metric on connected colored rooted graphs. We now introduce a notion of convergence for unrooted graphs (G n, W n ), which are not necessarily connected. The idea is to consider the convergence of the law of G n under uniform rooting. Since G is a Polish space, we may consider the set of probability measures on G, denoted by P(G ). The latter is also a Polish space. If (G, W ) is a finite colored graph, G = (V, E), we denote (G(v), W ) the subgraph spanned by the vertices in the connected component on v. We then define U (G,W ) P(G ) by U (G,W ) = δ V [G(v),v,W ]. v V This captures the idea of choosing the root v uniformly at random in V. If (G n, W n ) is a sequence of finite colored graphs, we say that ρ P(G ) is the local weak it of (G n, W n ) if U (Gn,Wn) converges weakly to ρ in P(G ). This notion of convergence was first introduced in [8] and later generalized in [3]. It is often called the Benjamini- Schramm convergence. Let C(G D,A ) be the set of continuous functions f : G D,A R. Recall that a linear subspace A C(G D,A ) is called an algebra if is is closed under multiplication and contains the constant function. We say that A separates points if for any [G, o, W ] [G, o, W ] G D,A, there is some f A such that f([g, o, W ]) f([g, o, W ]). Lemma 2.5. Let (G n, W n ) be a sequence of finite colored graphs, G n = (V n, E n ), with degree uniformly bounded by D and coloring W n : V n [ A, A] for all n. Then () (G n, W n ) has a subsequence which converges in the sense of Benjamini-Schramm, i.e. U (Gnj,W nj ) converges weakly to some µ P(G D,A ). (2) (G n, W n ) has a local weak it ρ iff there is an algebra A C(G D,A ) which separates points, such that for all f A, n V n f ([G n (v), v, W n ]) = v V n G D,A f ([G, o, W ]) dρ ([G, o, W ]). Proof. Both items follow from the compactness of G D,A, see [2, Chapter 3]. The previous lemma gives a convenient criterion to prove that (G n, W n ) has a local weak it. However, it may not be very clear how a continuous function on G D,A looks like. We start with the special case where A = 0, i.e. without coloring. Lemma 2.6. Let (G n ) be a sequence of finite graphs, G n = (V n, E n ), with degree uniformly bounded by D. Then (G n ) has a local weak it ρ iff for any r-ball B F (o, r), #{x : B Gn(x)(x, r) = B F (o, r)} = ρ ({[H, x] : B H (x, r) n V n = B F (o, r)}). Here, B Gn(x)(x, r) = B F (o, r) means there exists ϕ : B Gn(x)(x, r) B F (o, r). Proof. Let C [Fr,o] = {[H, x] : B H (x, r) = B F (o, r)}. We first note that C [Fr,o] is a clopen subset of G D,0. Indeed, given [H, x ] C [Fr,o] and [H 2, x 2 ] G D,0, denote α,2 = sup{s : B H (x, s) = B H2 (x 2, s)}. Then for any [H 2, x 2 ] with α,2 r, we have [H 2, x 2 ]

5 THE COLORED BEJAMII-SCHRAMM TOPOLOGY 5 C [Fr,o]. Hence, d loc ([H, x ], [H 2, x 2 ]) +r implies [H 2, x 2 ] C [Fr,o], so C [Fr,o] is open. ext, suppose ([H n, x n ]) C [Fr,o] converges to some [H, x]. Then if α n = sup{s : B Hn (x n, s) = B H (x, s)}, we may find n r such that α n r for all n n r. In particular, B H (x, r) = B Hnr (x nr, r) = B F (o, r), so [H, x] C [Fr,o]. Hence C [Fr,o] is closed. ote that U Gn (C [Fr,o]) = #{x:b Gn(x)(x,r) =B F (o,r)} V n. Since C [Fr,o] is clopen, U Gn ( C [Fr,o]) = 0, so if U Gn converges weakly to ρ, then U Gn (C [Fr,o]) ρ(c [Fr,o]) for any B F (o, r). For the converse, note that since C [Fr,o] is clopen, its indicator function χ C[Fr,o] is continuous. ext, C [Fr,o] C [F r2,o ] = C [Fr,o] if B F (o, r ) = B F (o, r 2 ) and is empty otherwise, so χ χ C[Fr,o] C = χ [F r2,o ] C [Fr or 0. ow if the it in the lemma holds for any B,o] F (o, r), this means that in Lemma 2.5(2), the it is true for any χ C[Fr,o]. This implies is also holds for linear combinations thereof. Finally, it trivially holds for the constant functions and 0. So the it holds for the algebra of functions A = {αχ + βχ C[Fr,o] C [F } {0, }. r2,o ] ote that A separates points: if [G, o] [G, o ], then d loc ([G, o], [G, o ]) > 0, so we may find r such that B G (o, r) is not isomorphic to B G (o, r) as rooted graphs. Taking B F (o, r) = B G (o, r), we get χ C[Fr,o] ([G, o]) = but χ C[Fr,o] ([G, o ]) = 0. It now follows from by Lemma 2.5(2) that U Gn converges weakly to ρ. We now discuss the general case. We first have a partial analogy with Lemma 2.6. Lemma 2.7. Let (G n, W n ) be a sequence of finite colored graphs, G n = (V n, E n ), with degree uniformly bounded by D and coloring W n : V n [ A, A] for all n. If (G n, W n ) has a local weak it ρ, then for any r and any r-ball (B F (o, r), o, W F ), #{x : ϕ x n : B Gn(x)(x, r) B F (o, r) with W F ϕ x n W n BGn(x) (x,r) < /r} n V n = ρ Ä {[H, x, W ] : ϕ : B H (x, r) B F (o, r) with W F ϕ W BH (x,r) < /r} ä. Proof. Given an r-ball (B F (o, r), o, W F ) with r, let Then C F = {[H, x, W ] : ϕ : B H (x, r) B F (o, r) with W F ϕ W BH (x,r) < /r}. C F = {[H, x, W ] : d loc ([F, o, W F ], [H, x, W ]) }. + r Hence, C F is closed. It is also open: if [H, x, W ] C F, then there exists ϕ : B F (o, r) B H (x, r) with W ϕ W F BF (o,r) < /r. Choose s, s > r, such that 0 < s < r W ϕ W F BF (o,r). If d loc ([H, x, W ], [H, x, W ]) < +s, there exists ψ : B H(x, s) B H (x, s) with W ψ W BH (x,r) < /s. As W ψ W BH (x,r) = W ψ ϕ W ϕ BF (o,r), it follows that W ψ ϕ W F BF (o,r) < /s + W ϕ W F BF (o,r) < /r. But ψ ϕ : B F (o, r) B H (x, r) since s > r. Hence, [H, x, W ] C F and C F is open. ote that U (Gn,W n)(c F ) = #{x:[b Gn(x)(x,r),x,W ] C F } V n. Since C F is clopen, U (Gn,W n)( C F ) = 0, so if U (Gn,W n) converges weakly to ρ, then U Gn (C F ) ρ(c F ). To obtain a converse, one needs to assume moreover that the it holds for all elements of the form C F C F2, in order to argue as before. Under the hypotheses of the lemma, it is also true that #{x : B Gn(x)(x, r) = B F (o, r)} = ρ ({[H, x, W ] : B H (x, r) n V n = B F (o, r)}), since the sets on the RHS are still clopen. So as expected, if (G n, W n ) converge as colored graphs, they converge in particular as graphs without coloring.

6 6 MOSTAFA SABRI 3. Examples We start with simple examples without coloring. 3.. Cycle graphs. The cycle graph with n vertices C n converges to Z in the sense of Benjamini-Schramm. More precisely, C n has the local weak it δ [Z,o], where o Z is an arbitrary root. Indeed, given r, if B F (o, r) is isomorphic to an r-ball in Z, then #{x:b Cn (x,r) =B F (o,r)} = for all n > r. If B F (o, r) is not isomorphic to an r-ball in Z, then n #{x:b Cn (x,r) =B F (o,r)} n = 0 for all n > r. So the it of #{x:b Cn (x,r) =B F (o,r)} n is (or 0) if B F (o, r) is isomorphic to an r-ball in Z (or not). Since δ [Z,o] ({[H, x] : B H (x, r) = B F (o, r)}) has the same values then the claim follows from Lemma Lattice cubes. The cubes Λ n = {,..., n} d converge to Z d in the sense of Benjamini- Schramm. Indeed, if B F (o, r) is isomorphic to an r-ball in Z d, then #{x:b Λn (x,r) =B F (o,r)} (n 2r) d n d n d =. Otherwise, #{x:b Λn (x,r) =B F (o,r)} (2r)d 0. It follows as before that U n d n d Λn has the local weak it δ [Z d,o], where o Z d is arbitrary Regular graphs with few cycles. Let G = (V, E ) be a sequence of (q + )- regular connected graphs with V =. As in [4, 5], we define the property (BST) For all R > 0, {x V : ρ G (x) < R} = 0, where ρ G (x) is the injectivity radius at x, i.e. the largest ρ such that B G (x, ρ) is a tree. This property holds in particular if the girth of G grows to infinity. We claim that (G ) satisfies (BST) iff (G ) converges to the (q + )-regular tree T q in the sense of Bejamini-Schramm, i.e. iff (G ) has the local weak it δ [Tq,o], where o T q is an arbitrary root. Indeed, let B F (o, r) be an r-ball and assume (G ) satisfies (BST). If B F (o, r) is isomorphic to a ball in T q, then #{x:b G (x,r) =B F (o,r)} V = #{x:ρ G (x) r}. If B F (o, r) is not isomorphic to a ball in T q, then #{x:b G (x,r) =B F (o,r)} V #{x:ρ G (x)<r} 0. It follows as before that (G ) has the local weak it δ [Tq,o]. Conversely, if (G ) has the local weak it δ [Tq,o], given R > 0, pick a ball B F (o, R) in T q, Then #{x:ρ G (x) R} = #{x:b G (x,r) =B F (o,r)} V, so (BST) follows Graphs with bounded degree. If we assume that G = (V, E ) is a sequence of graphs, V =, with degree uniformly bounded by D, then (BST) no longer guarantees convergence. For instance if G 2 are 3-regular and G 2+ are 4-regular, and if (G ) satisfies (BST), then G 2 will converge to T 2 while G 2+ will converge to T Tree balls. Fix a root o in the (q + )-regular tree T q and let G = B Tq (o, ). Then (G ) does not converge to T q in the sense of Benjamini-Schramm. Indeed, if B F (o, r) is isomorphic to a ball in T q, then #{x:b G (x,r) =B F (o,r)} V = V r V q, since V r n = + (q + ) n j= q j = + (q + ) qn q, so that V r V = (q )q +(q+)(q r q ) q r. This (q )q +(q+)( q ) already shows the local weak it cannot be T q. For B F (o, r) which are not isomorphic to an r-ball in T q, the value of #{x:b G (x,r) =B F (o,r)} V is 0 if B F (o, r) is also not isomorphic to any B G (x, r) with x S n, r + n, where S n is the n-th sphere. If B F (o, r) is isomorphic to B G (x, r) with x S j+, then #{x:b G (x,r) =B F (o,r)} V = S j+ V = (q+)q j + q+ q (q ) q q j. Based on this information, we construct a random rooted tree (T q, o) called the canopy tree, cf. [2] and [, Chapter 4] :

7 THE COLORED BEJAMII-SCHRAMM TOPOLOGY 7 p 0 = q q p = q q 2 p 2 = q q 3 p 3 = q q 4 p 4 = q q 5 L 0 L L 2 L 3 L 4 Figure. The canopy tree, as introduced in [2]. This is a fixed infinite tree Tq ; the randomness comes from the root. More precisely, this tree is not transitive, so the position of the root matters. We divide the tree into infinite levels (L j ) j=0 as in Figure. For fixed j, all trees (T q, o) with o L j are equivalent. By some abuse of notation we let [Tq, o] = L j in this case. We then define ρ P(G D,A ) by ρ = q j=0 δ q j+ Lj. This is indeed a probability measure, since q j=0 =. By q j+ construction, B T q (o, r) is isomorphic to an r-ball B F (o, r) in T q precisely when o is in any level L j with j r. In other words, ρ({[h, x, W ] : B H (x, r) = B F (o, r)}) = q j=r = q j+ q j=0 q q j = q q r+ q r+ q = q, which is the iting value we obtained along G r. Similarly, if B F (o, r) is not isomorphic to any B G (x) with x S n, r + n, we find ρ({[h, x, W ] : B H (x, r) = B F (o, r)}) = 0, and if B F (o, r) is isomorphic to B G (x, r) with x S j+, then ρ({[h, x, W ] : B H (x, r) = B F (o, r)}) = q δ q j Lj (L j ) = q. q j This completes the proof that ρ is the local weak it of (G ). We now turn to colored graphs Colored graphs with few cycles. Let (G, W ) be a sequence of colored connected graphs G = (V, E ) with degree uniformly bounded by D, coloring W : V [ A, A] for all and V =. Let T D,A be the subset of colored rooted trees in G D,A. We show that if (G, W ) has a local weak it ρ which is concentrated on T D,A, then (G ) satisfies (BST). Conversely, if (G ) satisfies (BST), and if (G j, W j ) is a subsequence with a local weak it ρ, then ρ is concentrated on T D,A. Indeed, we may assume the R in (BST) is in. We observe that #{x:ρ G (x)<r} = #{x:b G (x,r) is not a tree } = U (G,W )({[H, x, W ] : B H (x, R) is not a tree }). Let C R = {[H, x, W ] : B H (x, R) is not a tree }. The C R is clopen. Indeed, if [H, x, W ] C R and d loc [(H, x, W ), (H, x, W )] < +R, then B H (x, R) = B H (x, R), so [H, x, W ] C R. Hence, C R is open. If [H n, x n, W n ] C R converges to some [H, x, W ], then B H (x, R) = B Hn (x n, R) for all n n R and thus [H, x, W ] C R. So C R is closed. If (G, W ) has a local weak it ρ concentrated on T D,A, then U (G,W )(C R ) ρ(c R ) = 0. Hence (G ) satisfies (BST). Conversely, if (G ) satisfies (BST) and ρ is the local weak it of a subsequence, then we need to show that for any M G D,A, we have ρ(m ) = ρ(m T D,A ). For this, note that (T D,A ) c = R C R, with C C Hence, ρ[(t D,A ) c ] = R ρ(c R ). But by hypothesis, ρ(c R ) = j U (Gj,W j )(C R ) = j #{x:ρ j (x)<r} j = 0. So ρ is concentrated on T D,A.

8 8 MOSTAFA SABRI 3.7. The Anderson model. In this example we closely follow our presentation in [7]. Let Ω = [ A, A] Zd and define P on Ω by P = v Z d ν for some probability measure ν on [ A, A]. Given ω = (ω v ) Ω, define W ω (v) = ω v for v Z d. Then the {ω v } v Z d are i.i.d. random variables with common distribution ν. Let Λ n = {,..., n} d. Given ω Ω, we define Wn ω (v) = ω v for v Λ n. We will show that for P-a.e. ω, the sequence of graphs (Λ n, Wn ω ) has a local weak it ρ which is concentrated on {[Z d, 0, W ω ] : ω Ω}, and acts by taking the expectation w.r.t. P. More precisely, denoting D = 2d, f([g, o, W ]) dρ([g, o, W ]) = E[f([Z d, 0, W ω ])]. G D,A Let A = r A r, where A r = f C(G D,A ) : f([g, o, W ]) = f([g, o, W ]) if [B G (o, r), o, W ] = [B G (o, r), o, W ]. Then A is an algebra of continuous functions containing. To see that it separates points, let [G, o, W ] [G, o, W ]. Then we may find r with d loc ([G, o, W ], [G, o, W ]) > +r. Define C G = {[H, x, V ] : d loc ([G, o, W ], [H, x, V ]) +r }. We showed in Lemma 2.7 that χ CG is continuous. Moreover, χ CG ([H, x, V ]) = χ CG ([H, x, V ]) if [B H (o, r), o, V ] = [B H (o, r), o, V ]. Indeed, if χ CG ([H, x, V ]) =, there is ϕ : B H (x, r) B G (o, r) with W ϕ V BH (x,r) < /r. If ψ : B H (x, r) B H (o, r) with V ψ = V, then ϕ ψ : B H (x, r) B G (o, r) has W ϕ ψ V < /r and thus χ CG ([H, x, V ]) =. Similarly, if χ CG ([H, x, V ]) = 0, no such ϕ exists and χ CG ([H, x, V ]) = 0. Hence, χ CG A r. Finally, χ CG ([G, o, W ]) = while χ CG ([G, o, W ]) = 0. We thus showed that A separates points. Using Lemma 2.5, it now suffices to show that there exists Ω 0 Ω with P(Ω 0 ) = such that for any ω Ω 0 and any f A, we have (3.) n n d f([λ n, x, Wn ω ]) = E[f([Z d, 0, W ω ])]. For this, we first adapt the strong law of large numbers in [0, Theorem 2.3.5]. Given f A r, let Y x = Y x (n) = f([λ n, x, Wn ω ]) E[f([Λ n, x, Wn ω ])] and S n = n d Y v. Then E[Y x ] = 0. Moreover, Y x only depends on (ω z ) z BΛn (x,r), since f([λ n, x, W ω n ]) = f([λ n, x, W ω n ]) if W ω n = W ω n on B Λn (x, r). It follows that Y x and Y y are independent if d Λn (x, y) > 2r. ow ñ E Y x ô 4 = E(Yx 4 ) E(Yx 2 Y 2 x,y Λ n x,y,z Λ n E(Y x Y y Y 2 z + Y x Y 2 y Y z + Y 2 y ) + 4 E(Y x Yy 3 + Y y Yx 3 ) x,y Λ n x Y y Y z ) + 24 E(Y x Y y Y z Y t ). x,y,z,t Λ n The first three sums are O(n d ) and O(n 2d ). For the fourth, note that E(Y x Y y Yz 2 ) = 0 if d(x, y) > 4r, since either d(x, z) > 2r and Y x is independent of the pair (Y y, Y z ), or d(y, z) > 2r and Y y is independent of (Y x, Y z ). Thus, we have either E(Y x Y y Yz 2 ) = E(Y x ) E(Y y Yz 2 ) = 0 or E(Y y Y x Yz 2 ) = E(Y y ) E(Y x Yz 2 ) = 0. Hence, x,y,z Λ n E(Y x Y y Yz 2 ) n 2d (4r) d (2 f ) 4. The other terms of this sum are treated similarly. Finally, for E(Y x Y y Y z Y t ) to be non zero, each point must be at distance 2r from one of the three others. Hence, we must have [d(x, y) 2r and d(z, t) 2r] (or a permutation

9 THE COLORED BEJAMII-SCHRAMM TOPOLOGY 9 thereof) or [d(x, ) 8r for = y, z, t.] It follows that x,y,z,t Λ n E(Y x Y y Y z Y t ) 3n 2d (2r) 2d (2 f ) 4 + n d (8r) 3d (2 f ) 4. In any case E( S n 4 ) C r,f,d n 2d. By the Borel-Cantelli Lemma, if A ε n,f = { S n > ε}, then P(A ε n,f i.o.) = 0. Thus, if Ω ε 0,f = {A ε n,f occurs finitely often}, we have P(Ω ε 0,f ) =. Since C(GD,A ) is separable, A is separable, and we may choose a countable dense subset {f j } A. We then let Ω 0 = ε Q + j Ω ε 0,f j. Then P(Ω 0 ) =. Let ω Ω 0. Given j and ε > 0 let 0 < ε < ε, ε Q +. Then ω Ω ε 0,f j, so there is n ω such that S n ε < ε for any n > n ω. Hence, S n 0 for any ω Ω 0. ow if f A, say f A r, we have n d f([λ n, x, Wn ω ]) E[f([Z d, 0, W ω ])] S n + n d E[f([Λ n, x, Wn ω ])] E[f([Z d, 0, W ω ])]. Assume n > r. If x {r +,..., n r} d =: Cn, r there is ϕ : B Λn (x, r) B Z d(0, r). In fact, we take ϕ(v) = v x. Denoting Wx ω (v) = W ω (v + x), we get [B Λn (x, r), x, Wn ω ] = [B Z d(0, r), 0, Wx ω ], so f([λ n, x, Wn ω ]) = f([z d, 0, Wx ω ]). By usual measure-preserving transformations, we check that E[f([Z d, 0, Wx ω ])] = E[f([Z d, 0, W ω ])]. Hence, n d f([λ n, x, Wn ω ]) E[f([Z d, 0, W ω ])] S n + n d E[f([Λ n, x, Wn ω ])] E[f([Z d, 0, W ω ])] S n + (2r)d n d (2 f ). x/ C r n Taking n, it follows that if ω Ω 0, then (3.) is true for any f {f j }, the dense subset of A. Arguing as in [2, Corollary 5.3], the proof is complete. 4. Convergence of spectral measures Our aim in this section is to show that the Benjamini-Schramm convergence implies the convergence of the mean spectral measures. This can be interpreted as the assertion that the integrated density of states of the it operator has a finite-volume approximation. Though this can be proved directly, we will first prove a convergence result for rooted spectral measures as in [3, Chapter 2], which is of independent interest. Let [G, o, W ] G D,A, let A be the adjacency matrix on G and define the (Schrödinger) operator H = A + W. This is a bounded self-adjoint operator. We sometimes denote H = H (G,W ) to avoid confusion. We define the rooted spectral measure µ (G,W o ) by µ (G,W ) o (J) = δ o, χ J (H)δ o for Borel J R. Lemma 4.. Suppose [G n, o n, W n ] G D,A, d loc ). Then µ (Gn,Wn) (G D,A we have converges to [G, o, W ] in the metric topology of o n converges weakly to µ (G,W o ). So for any continuous ϕ : R R, δ o n n, ϕ(h (Gn,Wn))δ on = δ o, ϕ(h (G,W ) )δ o. Proof. Since all operators H n = A n + W n and H = A + W are uniformly bounded by some A + D, the supports of the spectral measures is compact, so it suffices to show that for any k, t k dµ on (t) t k dµ o (t); see [2, Chapter 3].

10 0 MOSTAFA SABRI Given k, choose an arbitrary integer r k. Then we may find n r such that for n n r, there exists ϕ r : B Gn (o n, r) B G (o, r) with W ϕ r W n BGn (o,r) < /r. ow t k dµ on (t) = δ on, Hnδ k on = H n (o n, u 0 )H n (u 0, u )... H n (u k, o n ), u 0,...,u k and H n (v, w) = (A n δ w )(v) + W n (v)δ w (v). So the RHS only depends on B Gn (o n, k) and its coloring. As r k and ϕ r : B Gn (o n, r) B G (o, r), if we let H n = A + W n ϕ r on G, we get δ on, Hnδ k on = δ o, Hnδ k o. So for n n r, t k dµ on (t) t k dµ o (t) = δ o, (Hn k H k k )δ o = δ o, Hn k i (H n H)H i δ o C k,d,a W n ϕ r Since r k is arbitrary, this completes the proof. i= W BG (o,r) C k,d,a r If (G, W ) if a finite colored graph, G = (V, E), with degree uniformly bounded by D and coloring in [ A, A], we define the mean spectral measure µ (G,W ) = µ x (G,W ). V x V Corollary 4.2. Suppose (G n, W n ) is a sequence of finite colored graphs with degrees uniformly bounded by D and coloring W n : V n [ A, A] for all n. If (G n, W n ) has a local weak it ρ, then µ (Gn,Wn) converges weakly to G D,A continuous ϕ : R R, we have n V n tr[ϕ(h (G n,w n))] = G D,A. µ (G,W ) o dρ([g, o, W ]). So for any δ o, ϕ(h (G,W ) )δ o dρ([g, o, W ]). Proof. Given continuous ϕ : R R, define the transform ϕ : G D,A R by ϕ([g, o, W ]) = (G,W ) ϕ(t) dµ o (t). Then ϕ is continuous on G D,A by Lemma 4.. It is also bounded since G D,A is compact. By hypothesis, U (Gn,Wn) converges weakly to ρ. It follows that ϕ du(gn,w n) ϕ dρ, i.e. V n x V n ϕ([g n, x, W n ]) ϕ([g, o, W ]) dρ([g, o, W ]). Since ϕ([g n, x, W n ]) = δ x, ϕ(h (Gn,Wn))δ x, the assertion follows. References [] M. Abért, A. Thom and B. Virág, Benjamini-Schramm convergence and pointwise convergence of the spectral measure, author homepage. [2] M. Aizenman, S. Warzel, The canopy graph and level statistics for random operators on trees, Math. Phys. Anal. Geom. 9 (2006), [3] D. Aldous, R. Lyons, Processes on unimodular random networks, Electron. J. Probab. 2 (2007) [4]. Anantharaman, E. Le Masson, Quantum ergodicity on large regular graphs, Duke Math. Jour. 64 (205) [5]. Anantharaman, Quantum ergodicity on regular graphs. To appear in Comm. Math. Phys. [6]. Anantharaman, M. Sabri Quantum Ergodicity on Graphs : From Spectral to Spatial Delocalization, preprint. [7]. Anantharaman, M. Sabri Quantum Ergodicity for the Anderson model on regular graphs, preprint. [8] I. Benjamini, O. Schramm, Recurrence of distributional its of finite planar graphs, Electron. J. Probab. 6 (200) 3 pp. [9] C. Bordenave, Lecture notes on random graphs and probabilistic combinatorial optimization, author homepage. [0] R. Durrett, Probability. Theory and Examples, Fourth Edition, Cambridge University Press 200. [] P. Gabor, Probability and Geometry on Groups, author homepage. [2] A. Klenke, Probability Theory. A Comprehensive Course, Second edition, Springer 204.

11 THE COLORED BEJAMII-SCHRAMM TOPOLOGY [3] J. Salez, Some implications of local weak convergence for sparse random graphs, PhD Thesis, Hal Id: tel IRMA, Université de Strasbourg, 7 rue René Descartes, 67084, Strasbourg, France. Department of Mathematics, Faculty of Science, Cairo University, Cairo 263, Egypt. address: sabri@math.unistra.fr