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Quantum ergodicity for expanding quantum graphs in
the regime of spectral delocalization
Nalini Anantharaman, Maxime Ingremeau, Mostafa Sabri, Brian Winn
To cite this version:
Nalini Anantharaman, Maxime Ingremeau, Mostafa Sabri, Brian Winn. Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization. 2021. �hal-03135249�
IN THE REGIME OF SPECTRAL DELOCALIZATION
NALINI ANANTHARAMAN, MAXIME INGREMEAU, MOSTAFA SABRI, BRIAN WINN Abstract. We consider a sequence of finite quantum graphs with few loops, so that they converge, in the sense of Benjamini-Schramm, to a random infinite quantum tree. We assume these quantum trees are spectrally delocalized in some interval I, in the sense that their spectrum in I is purely absolutely continuous and their Green’s functions are well controlled near the real axis. We furthermore suppose that the underlying sequence of discrete graphs is expanding. We deduce a quantum ergodicity result, showing that the eigenfunctions with eigenvalues lying in I are spatially delocalized.
1. Introduction
In their far-reaching work [21], Kottos and Smilansky suggested that the ideas and results of quantum chaos should apply to quantum graphs. By quantum graphs, we mean a metric graph, equipped with a differential operator and suitable boundary conditions at each vertex. We refer the reader to Section2.1for a more precise definition.
In this paper, we study an analogue on quantum graphs of one of the most famous properties of quantum chaotic systems, namely quantum ergodicity. In its original context [26,13,29], quantum ergodicity says that, on a compact Riemannian manifold whose geo-desic flow is ergodic, the eigenfunctions of the Laplace-Beltrami operator become equidis-tributed in the high-frequency limit.
On a fixed quantum graph (with Kirchhoff boundary conditions), it was shown in [14] that quantum ergodicity generically does not hold in the high-frequency limit, unless the graph is very simple (homeomorphic to an interval or a circle).
However, instead of studying the asymptotics of eigenfunctions on a fixed quantum graph, one may study quantum ergodicity on sequences of quantum graphs whose size goes to infinity. Some positive results appeared in [9], [17], [20] [11] for several families of quantum graphs, while it was shown in [8] that quantum ergodicity does not hold on star graphs. We refer the reader to the introduction of [19] for an up-to-date survey of these recent developments.
All the preceding results study the high frequency behaviour of eigenfunctions. In [19], we proved a quantum ergodicity result for regular equilateral quantum graphs QN
converging to the regular equilateral tree Tq (in the sense of Benjamini-Schramm), in
the bounded energy regime, more precisely for energies lying in some bounded interval I ⊂ σac(HTq), assuming the underlying discrete graphs are expanders. Our aim in this
paper is to extend this result to the case of non-regular non-equilateral quantum graphs satisfying δ-conditions (§2.1).
As in [19], we suppose that the quantum graphs have few short cycles. This can be re-stated as supposing that our quantum graphs converge in the sense of Benjamini-Schramm to a measure P supported on the set of quantum trees. We introduced the notion of Benjamini-Schramm convergence for quantum graphs in [5], in analogy to the case of discrete graphs. We also assume the underlying discrete graphs are expanders.
2010 Mathematics Subject Classification. Primary 58J51. Secondary 34B45, Q1Q10. Key words and phrases. Quantum ergodicity, quantum graphs, delocalization, trees.
Our main assumption is that in the energy interval we consider, the spectrum at the limit is absolutely continuous. More precisely, we need a good control over the Green’s function near the real axis (see hypothesis (Green) below). Our results thus convert
spectral delocalization at the limit (AC spectrum) into spatial delocalization for the
eigen-functions (i.e. quantum ergodicity). We give in § 2.4 two important examples in which our assumptions hold.
The results of this paper can be regarded as a quantum graph counterpart of the results of [4] for discrete graphs, we refer the reader to that paper for a more detailed introduction of the problem of quantum ergodicity and its implications.
Let us discuss our results in more detail. Given a sequence QN of growing quantum
graphs, the aim is to show that for any orthonormal basis of eigenfunctions (ψ(N )j ) onQN,
the probability measure|ψ(N )j (x)|2dx approaches the uniform measure |Q1N|dx when N is
large enough. We only aim to prove this for most eigenfunctions in an interval I, so we consider Ces`aro means. More precisely, denoting by NN(I) the number of eigenvalues of
QN in I, we set to prove that
(1.1) lim N →∞ 1 NN(I) X λ(N)j ∈I hψ(N )j , fNψj(N )iL2(GN)− hfNi λ(N)j = 0
for any uniformly bounded observable fN ∈ L∞(GN), whereGN is the underlying metric
graph. Since hψj, f ψji = RGNf (x)|ψj(x)|2dx, if we had hfiλ(N) j
= |G1
N|
R
GNf (x) dx, i.e.,
if hfi were the uniform averages independently of λ(N )j this would show that |ψj(x)|2dx
approaches |GN|1 dx in some weak sense.
It turns out that such perfect uniform distribution can only be true in very special cases, cf. [19]. In fact, since we consider a regime of bounded energies (lying in a fixed I, not the high frequency regime), we expect the potential we put on the edges to have some influence over the probability of finding the wavefunction in various places of the graph, which is given by|ψ(N )j (x)|2dx. The true “limiting measure” is thus not the uniform measure in general, but one with a possibly non-constant density. The density is very satisfactory as it is directly related to the spectral density of the limiting quantum tree. In fact, our results show in a weak sense that |ψj(N )(x)|2dx tends to the measure
Im ˜gλ (N) j +i0 N (˜x,˜x) R GNIm ˜g λ(N)j +i0 N (˜y,˜y) dy dx, where ˜gzN(˜x, ˜x) is the Green’s function of the universal cover ofQN (˜gNz(˜x, ˜x) approaches
the Green’s function of the limiting tree when N → ∞, see Appendix C). Accordingly, the meanhfNiλ(N)
j
above will actually depend on the energy1 λ(N )j . We now discuss the main steps of the proof:
(1) In Sections 4 and 5 we reduce (1.1) to proving that analogous Ces`aro means defined on the discrete graph (which we call quantum variances) vanish as N −→ ∞. In this process the L2 scalar producthψj, f ψji =RGN|ψj(x)|2f (x) dx is replaced by ℓ2 scalar
products of the form Pv∈VN|ψj(v)|2Kf,j(v) or Pv∈VN
P
w∼vψj(v)ψj(w)Mf,j(v, w),
for some auxiliary (energy-dependent) observables Kf,j, Mf,j built from f . Such
discretization philosophy is well established in the quantum graphs literature, espe-cially when the quantum graph is equilateral, in which case the restrictions ψj|VN
become eigenfunctions of some nice adjacency matrix. It is known however that when the graph is not equilateral, the discretization produces a complicated energy-dependent Schr¨odinger operator. In this paper we circumvent this problem by using
1Note that in the special case where ˜gz(x, x) is independent of x we get the uniform measure 1 |GN|dx. Roughly speaking, the general quotient detects the inhomogeneities in the limit object, a tree in our setting.
non-backtracking eigenfunctions fj, fj∗ living on the directed edges of the graph, an
idea that already proved fruitful in discrete graphs [3, 4], and it is quite remarkable that it also works for quantum graphs. This new construction is explained in Section4. The eigenvalue equation for fj, fj∗ implies that the corresponding quantum variance is
invariant under simple averaging operators, weighted by the Green’s functions of the quantum universal cover (Proposition 4.2). In the usual proof of quantum ergodicity on manifolds, we would instead be using the invariance of eigenfunctions under the wave propagator.
(2) We can bound the quantum variance by Hilbert-Schmidt norms. For this we follow the general scheme of [4], but the procedure is complicated by two problems: first, neither the restrictions ψj|V of the eigenfunctions to the vertices, nor the non-backtracking
eigenfunctions fj, fj∗ form an orthonormal basis; second we have less a priori bounds
on the auxiliary observables Kf,j, Mf,j. This calls for several technical innovations
(Sections 6–7).
(3) Applying the averaging operators from step (1), and developing the Hilbert-Schmidt norms (Section 8) reduces the proof to some contraction estimates on a family of sub-stochastic operators. These estimates require a careful analysis involving the expanding properties of the graphs (Section 9). Compared to [4], this part contains several novelties which are necessary due to the more complicated recursion relations satisfied by the Green’s functions in the quantum setting (Section3).
Each of the preceding steps is not “exact” in the sense that it holds modulo error terms. To control them, we derive in Appendix C some important implications of Benjamini-Schramm convergence and spectral delocalization.
Acknowledgments. N.A. was supported by Institut Universitaire de France, by the ANR
project GeRaSic ANR-13-BS01-0007 and by USIAS (University of Strasbourg Institute of Advanced Study).
M.I. was supported by the Labex IRMIA during part of the realization of this project. M.S. was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH. He thanks the Universit´e Paris Saclay for excellent working conditions, where part of this work was done.
2. Main results
2.1. Quantum graphs. Let G = (V, E) be a graph with vertex set V and edge set E. For each vertex v∈ V , we denote by d(v) the degree of v. If v1, v2 ∈ V , we write v1 ∼ v2 if
{v1, v2} ∈ E. We let B = B(G) be the set of oriented edges (or bonds), so that |B| = 2|E|.
We assume that there is at most one edge between two vertices, so we will view B as a subset of V × V . If b ∈ B, we shall denote by ˆb the reverse bond. We write ob for the
origin of b and tb for the terminus of b. We will also write e(b)∈ E for the edge obtained
by forgetting the orientation of b.
For us, a quantum graph Q = (V, E, L, W, α) is the data of: • A connected combinatorial graph (V, E).
• A map L : E → (0, ∞). If b ∈ B, we denote Lb := L(e(b)).
• A potential W = (Wb)b∈B ∈Lb∈BC0([0, Lb]; R) satisfying for x∈ [0, Lb],
(2.1) Wbb(Lb− x) = Wb(x) .
• Coupling constants α = (αv)v∈V ∈ RV.
The underlying metric graph is defined by
where (b, xb)≃ (b′, x′b′) if b′ = ˆb and x′b′ = Lb− xb. In the sequel, we will sometimes write
x ∈ b to indicate that x = (b, xb) ∈ G. Condition (2.1) then simply ensures that W is
well-defined onG.
In general a function f : G −→ R can be identified with a collection of maps (fb)b∈B
such that fb(Lb − ·) = fˆb(·). We say that f is supported on e for some e ∈ E if fb ≡ 0 unless e(b) = e. In the sequel, we will often write f (xb) instead of f ((b, xb)) or fb(xb).
If each fb is positive and measurable, we defineRGf (x)dx := 12Pb∈B
RLb
0 fb(xb)dxb. We
may then define the spaces Lp(G) for p ∈ [1, ∞] in the natural way (see below). In the sequel, we will always make the following assumptions:
(Data) There exist 0 < m < M and D ∈ N such that for all v ∈ V and all b ∈ B: 3≤ d(v) ≤ D
|αv| ≤ M
m≤ Lb≤ M
Wb ∈ Lip([0, Lb]) and max (kWbk∞, Lip(Wb))≤ M
Wb(Lb− ·) = Wb(·).
This last assumption says that the potential on each edge is symmetric (in particular, this is the case if W ≡ 0). The symmetry condition is used in the “trigonometric” relations (2.6), but we believe one should be able to adapt our proof to non-symmetric potentials — at the expense of using a modified version of (2.6).
Here Lip(I) is the set of Lipschitz-continuous functions on I and Lip(f ) is the Lipschitz constant of f .
Let Q be a quantum graph. Consider the Hilbert space2 L2(G) :=n(fb)b∈B ∈ M b∈B L2(0, Lb) fbb(Lb− ·) = fb(·) and X b∈B kfbk2L2(0,Lb)<∞ o
and its subset
H2(G) :=n(fb)b∈B ∈ M b∈B H2(0, Lb) fbb(Lb− ·) = fb(·) and X b∈B kfbk2H2(0,Lb)<∞ o . We say that a function f ∈ H2(G) satisfies the δ-boundary conditions at v ∈ V if
(2.2)
∀b, b′ ∈ B such that ob = ob′ = v, we have fb(0) = fb′(0) =: f (v),
P
b∈B, ob=v
fob′ (0) = αvf (v).
The set of such functions is denoted by:
HQ:=nf ∈ H2(G) ∀v ∈ V, f satisfies (2.2)o. We then define an operator HQ acting on ψ = (ψb)b∈B ∈ HQ by
(2.3) (HQψb)(xb) =−ψb′′(xb) + Wb(xb)ψb(xb).
Our assumption (Data) implies in particular that the operator HQ :HQ → L2(G) is
then self-adjoint [7, Theorem 1.4.19].
We say that the quantum graph is finite if V and E are finite. In this case we denote L(Q) =Pe∈EL(e). If Q is finite, HQ has compact resolvent ([7, Theorem 3.1.1]). So for
2In the sequel, all the scalar products in Hilbert spaces will be linear in the right variable, and anti-linear
finite QN, there exists an orthonormal basis (ψ(N )j )j≥1 of L2(GN) made of eigenfunctions
of HQN. We denote by λ (N ) j
j≥1 the corresponding eigenvalues. Recall that we write
NN(I) := #
n
j≥ 1 | λ(N )j ∈ Io.
Since our Schr¨odinger operators have real coefficients, we may—and will—restrict our attention to real valued eigenfunctions. This restriction was already present in the work [4] on discrete graphs. It is only necessary in AppendixB.2. More precisely, what we need to assume is that
ψj(N )(ob)ψj(N )(tb)∈ R
for any N , j and b∈ BN.
2.2. Eigenfunctions on the edges. LetQ be a quantum graph. Given an oriented edge b∈ B and γ ∈ C, let Cγ,b(xb) and Sγ,b(xb) be a basis of solutions of the equation
(2.4) − ψ′′(xb) + Wb(xb)ψ(xb) = γψ(xb) satisfying Cγ,b(0) Sγ,b(0) Cγ,b′ (0) Sγ,b′ (0) = 1 0 0 1 .
If Wb ≡ 0, these are the familiar cosine and sine functions, hence our notation. Note that
any solution ψ of (2.4) satisfies ψ(Lb) ψ′(Lb) = Mγ(b) ψ(0) ψ′(0) where Mγ(b) = Cγ,b(Lb) Sγ,b(Lb) Cγ,b′ (Lb) Sγ,b′ (Lb) . For all b∈ B and all z ∈ C, we have
(2.5) Sγ,b = Sγ,b,
since both functions satisfy the same differential equation with the same boundary condi-tions.
For any xb ∈ [0, Lb], the maps γ7→ Cγ,b(xb) and γ7→ Sγ,b(xb) are holomorphic functions.
A proof of this fact can be found, for instance, in [24, Chapter 1].
Similarly, using the symmetry of the potential, we note that for any γ ∈ C, we have the “trigonometric” relations (similar to the usual ones satisfied by cos and sin)
(2.6) Sγ,b(Lb)Cγ,b(xb)− Cγ,b(Lb)Sγ,b(xb) = Sγ,b(Lb− xb) , Sγ,b′ (Lb)Cγ,b(xb)− Cγ,b′ (Lb)Sγ,b(xb) = Cγ,b(Lb− xb) ,
since the two pairs of functions satisfy the same equation with the same boundary con-ditions (see [19, §2.1] for more details). In the sequel, we will write Sγ(xb), Cγ(xb) for
Sγ,b(xb), Cγ,b(xb) to lighten notations.
2.3. Main result. Let QN = (VN, EN, LN, WN, αN) be a sequence of quantum graphs,
each with N vertices. We denote by GN = (VN, EN) the underlying discrete graphs.
Our first two assumptions concern the geometry of GN = (VN, EN):
(EXP) The sequence (GN) forms an expander family. That is, ifAGN is the adjacency
matrix3 on GN, then the operator PN = d1NAGN, has a uniform spectral gap in ℓ2(VN).
More precisely, the eigenvalue 1 of PN is simple, and the spectrum of PN is contained in
[−1 + β, 1 − β] ∪ {1}, where β > 0 is independent of N. (BST) For all r > 0, lim N →∞ |{x ∈ VN : ρGN(x) < r}| N = 0 ,
3Recall that the adjacency matrix acts on f ∈ ℓ2(V ) by Af(v) = P
where ρGN(x) is the injectivity radius at x, i.e. the largest ρ such that the ball BGN(x, ρ) is
a tree. Here, because of our uniformity assumption (Data), it does not matter to choose a discrete distance or a continuous one for this property.
We will in addition suppose that QN converges in the sense of Benjamini-Schramm to
some probability measure P on the set of rooted graphs satisfying (Data) — as we may always do, up to extracting a subsequence [5, Corollary 3.6]. Assumption (BST) becomes equivalent to asking that P is supported on the set of quantum trees, i.e. quantum graphs without cycles. In a more probabilistic language, QN converges to a random rooted
quantum tree.
The next two hypotheses can be seen as a condition of spectral delocalization. Indeed, they imply that P-almost all quantum trees have purely absolutely continuous spectrum in I, see [6, Theorem A.6].
(Green) There exists a bounded open interval I1⊂ R such that for all s > 0,
sup λ∈I1,η∈(0,1) EP Im ˆR+λ+iη(ob) −s+ Im ˆRλ+iη+ (oˆb) −s <∞ , where ˆR+ is the Weyl-Titchmarsh function (defined in Section3.1below).
This assumption implies in particular that the Green’s functions of the Schr¨odinger operator on the infinite tree has finite moments; see §C.2.
As the proof will show, we actually only need (Green) to hold for all 0 < s < s0 for
some finite s0 which can in principle be made explicit and is not too big; we chose the
above formulation for comfort.
Our last assumption is that I1 avoids the “Dirichlet spectrum”.
(Non-Dirichlet) Let D=[ N [ b∈BN {λ ∈ R : Sλ(Lb) = 0} . Then we assume I1∩ D = ∅ ,
so any compact I⊂ I1 is isolated from all Dirichlet values. In the applications we have in
mind, D is a discrete subset of R, or an ǫ-neighborhood of a discrete subset. It follows that there exists CDir > 0 such that for all λ∈ I, we have
|Sλ(Lb)| ≥ CDir.
By continuity, this implies the existence of CDir′ > 0 and 0 < ηDir ≤ 1 such that for all
λ∈ I, η ∈ [0, ηDir],
(2.7) | Re Sλ+iη(Lb)| ≥ CDir′ .
We also note that Sγ(Lb) 6= 0 for any γ ∈ C+ = {z : Im z > 0}, since otherwise the
self-adjoint operator Df =−f′′+Wbf on [0, Lb] with Dirichlet boundary conditions would
have a complex eigenvalue γ corresponding to Sγ(x).
Theorem 2.1. Let QN be a sequence of quantum graphs satisfying (Data) for each N ,
such that (EXP), (BST), (Green) and (Non-Dirichlet) hold true on the interval I1.
Fix an interval I such that I⊂ I1.
Let (ψj(N ))j∈N be an orthonormal basis of eigenfunctions of HQN. Then for any sequence
of functions fN ∈ L∞(GN) satisfying kfNk∞≤ 1, we have
(2.8) lim η↓0N →∞lim 1 NN(I) X λ(N)j ∈I hψ(N )j , fNψ(N )j iL2(GN)− hfNi γj(N) = 0 ,
where hψj, fNψjiL2(GN)=RG NfN(x)|ψj(x)| 2dx, γ(N ) j = λ (N ) j + iη and (2.9) hfNiγj = R GNfN(x) Im ˜g γj N(x, x) dx R GNIm ˜g γj N(x, x) dx . Here, ˜gNγ(x, y) is the Green kernel of the operator HQe
N, defined on the universal cover of
QN. It will be defined precisely in Section3.3. We will see in Section3.1that its imaginary
part is always positive. In the special case of regular equilateral quantum graphs treated in [19], the universal covering quantum tree is the regular equilateral quantum tree Tq,
independent of N , so the Green function ˜gγN(v, w) = GγTq(v, w) itself is independent of N . Remark 2.2. By the Cauchy-Schwarz inequality, if fN ≥ 0, we have
Z GN (fN(x))1/2dx 2 ≤ Z GN fN(x) Im ˜gγNj(x, x) dx Z GN Im ˜gγj N(x, x) −1 dx . By Corollary C.8, there exists C > 0 independent of N such that
sup λ∈I1,η∈(0,ηDir) lim sup N −→∞ 1 N Z GN Im ˜gγN(x, x)±1 dx ≤ C. This implies that, when fN is non-negative,
lim inf N −→∞hfNiγj ≥ 1 C2 1 N Z GN (fN(x))1/2dx 2 . For example, if fN = χΛN with |ΛN| = αN, this gives a lower bound α
2
C2 > 0. As ΛN is
arbitrary (for fixed cardinality), Theorem2.1is really a delocalization result: for most ψj,
we cannot have|ψj(x)|2 concentrating on a portion of GN of cardinality o(N ).
Quantum ergodicity for integral operators. In a weak sense, the previous theorem asserts
that |ψj(N )(x)|2 behaves asymptotically like
Im ˜gNλj(x,x) R
GNIm ˜g λj N(x,x) dx
. More generally, we have a quantum ergodicity result involving integral operators. The aim here is to study the eigenfunction correlator ψj(N )(x)ψ(N )j (y).
In general, an integral operator K on L2(G) takes the form
(Kψ)b(xb) = X b′∈B Z Lb′ 0 Kb,b ′(xb, yb′)ψb′(yb′) dyb′,
with the condition that for all b, b′ ∈ B and almost all xb ∈ [0, Lb], yb′ ∈ [0, Lb′], we have
Kˆb,b′(Lb− xb, yb′) =Kb,b′(xb, yb′)
and similarly for the second argument. In the sequel, we will denote by Bk the set of
non-backtracking paths of length k (see (3.16) below for a precise definition), and work with the spaces of operators (indexed by k∈ N)
Kk :=nK : L2(G) −→ L2(G) | ∀b, b′ ∈ B, Kb,b′ ≡ 0
unless there exists (b1, . . . , bk)∈ Bk with b1 = b, bk = b′
o .
Thus, Kk is the space of operators with kernel supported on edges connected by a
non-backtracking path of length k. Note that any integral operator with a bounded compactly supported kernel can be written as a finite linear combination of operators in Kk for
various k. When we want to insist that these operators live on the graphQN indexed by
Theorem 2.3. Let k ≥ 0. Under the assumptions of Theorem 2.1, let (KN)N ∈N be a
sequence of operators with KN ∈ Kk(N ), such that |KN,b,b′(x, y)| ≤ 1 for every N ∈ N.
Then (2.10) lim η↓0N →∞lim 1 NN(I) X λ(N)j ∈I hψj(N ),KNψ (N ) j iL2(G N)− hKNiγ(N)j = 0 , wherehψj,Kψji = P (b1;bk)∈Bk RLb1 0 RLbk 0 K(xb1, ybk)ψj(xb1)ψj(ybk) dxb1dybk, γj(N )= λ (N ) j +iη, and hKiγ= P (b1;bk) R R K(xb1, ybk) Im ˜gγ(xb1, ybk) dxb1dybk R GNIm ˜gγ(x, x) dx .
This says that in a weak sense, when N gets large, the eigenfunction correlator ψj(x)ψj(y)
looks like the quotient of spectral densities Im ˜g
λj N(x,y) R GNIm ˜g λj N(x,x) dx
on the universal cover.
2.4. Examples.
2.4.1. N-lifts. An important example is whenQN is some (connected) N -lift of a compact
quantum graphQ1. In other words, the underlying graph4GN is an N -fold covering over
G1 and the data is lifted naturally L(v,w) = L(πNv,πNw), W(v,w) = W(πNv,πNw), αv = απNv,
where πN : GN −→ G1 is the covering projection.
It is known that N -lifts – when picked randomly – are typically connected and most of their points have a large injectivity radius – see [10, Lemma 24], [12, Lemma 9]. More precisely, condition (BST) holds generically. It is also known that they are typically expanders; see [16,25]. Thus, our assumptions are generic.
It is known that such (QN) converge in the Benjamini-Schramm sense to a deterministic
limit, namely the universal covering tree T = eQ1 with a random root (see [5]). More
precisely QN converges to the random rooted quantum tree defined by the measure
P= P 1 b∈B1Lb X b∈B1 Z Lb 0 δ[T, ˜L1, ˜W1, ˜α1,(˜b,x b)]dxb,
where (L1, W1, α1) is the data on the base graphG1 and T = eG1 is the combinatorial tree
underlyingT . In particular EP(| Im ˆR±λ+iη(ob)|−s+| Im ˆR±λ+iη(oˆb)|−s) = 2 P b∈B1Lb X b∈B1 Lb| Im ˆR±λ+iη(ob)|−s.
Also note that in this example, D =∪b∈B1{λ ∈ R : Sλ(Lb) = 0}.
We showed in [6] that the spectrum of HT consists of bands of pure AC spectra along
with a possible set of discrete eigenvalues (outside the bands). We also showed that within the bands, the limits ˆR±λ+i0(ob) exist, are finite and satisfy Im ˆR±λ+i0(ob) > 0. It follows
that (Green) is satisfied on any compact I⊂ I1, where I1 is some AC band.
Theorem 2.3 thus tells us that such (QN) are quantum-ergodic. This result can be
regarded as a non-regular, non-equilateral generalization of [19].
2.4.2. Random quantum graphs. We may also consider weak random perturbationsQωN
of the previous example. We leave the precise definition to [5, Section 8.4], but essentially one endows the graphs with random independent, identically-distributed (i.i.d.) lengths Lω
e and i.i.d. coupling constants αωv. Note that here only the combinatorial graph GN
covers G1, the data on each QωN is entirely random, so each has its own universal cover.
Assuming (BST) holds (which is true generically as previously mentioned), we calculated the limit measure in [5]. In particular, we get
EP(| Im ˆR± λ+iη(ob)| −s+ | Im ˆR±λ+iη(oˆb)|−s) = 2 |B1|E(Lωb) X b∈B1 E(Lωb| Im ˆR±λ+iη(ob)|−s),
where E is the expectation with respect to the random data (Lωb, αω) and the distributions are assumed to be the same for each b∈ B1. In other words, this is a random perturbation
of an equilateral model onQN (more general situations can be considered).
We showed in [6] that if the perturbation is weak enough, then the bands of AC spectra remain stable, and (Green) holds in such bands. Note that here we assume there is no edge potential W . So D =∪n≥0[ π
2n2 (L−ǫ)2, π
2n2
(L+ǫ)2], where ǫ is the small disorder window around the
unperturbed length L. We technically need the coupling constants to be nonnegative with a H¨older distribution. This can be seen as a result of Anderson (spectral) delocalization, strengthening earlier results in [1].
Theorem 2.3 implies that almost surely, the eigenfunctions of QωN are quantum
er-godic. In other words, Anderson (spatial) delocalization holds, in addition to spectral delocalization.
3. Preliminary constructions and notation
3.1. The Green function on a quantum tree. Let T = (V, E, L, W, α) be a quantum tree, i.e. a quantum graph such that (V, E) is a tree, with underlying metric graph T . We describe here the functional equations satisfied by the Green function on T, due to the topological fact that trees are disconnected by removing a point. While such relations are well-known for discrete laplacians on trees, they have been less exploited for quantum trees. This paragraph builds on the work of Aizenman-Sims-Warzel [1].
If b ∈ B(T) we denote T±b the two subtrees obtained by removing b, more precisely
tb ∈ T+b while ob ∈ Tb−. Let T±b be the induced quantum trees and xb ∈ [0, Lb]. If
x = (b, xb) is the corresponding point in T, we define T+x as the quantum tree [xb, tb]∪ T+b
see [6] for a more precise definition. The quantum tree T−x is defined in a similar fashion. Let us define Hmax
T±x on T ±
x to be the Schr¨odinger operator −∆ + W with domain
D(HTmax±
x ), the set of ψ ∈ H 2(T±
x) satisfying δ-conditions on inner vertices of T±x. Note
that Hmax
T±x is not self-adjoint, due to the absence of boundary condition at the root x.
By [1, Theorem 2.1], for any γ ∈ C+ = {z ∈ C; Im z > 0}, there are unique
eigen-functions Vγ;x+ ∈ D(Hmax T+x ), U − γ;x ∈ D(HTmax− x ) of H max
T±x , for the eigenvalue γ, satisfying
Uγ;x− (x) = V+
γ;x(x) = 1.
One can use the functions Uγ−, V+
γ to construct the Green’s function Gγ of HT, see [6,
Lemma 2.1]. For our purposes, we use them to define the Weyl-Titchmarch functions [1] as follows: if x∈ T+ o ∩ T−v, (3.1) Rγ+(x) = (V + γ;o)′(x) Vγ;o+(x) and Rγ−(x) = −(U − γ;v)′(x) Uγ;v− (x) . This does not depend on o, v. Given an oriented edge b = (ob, tb), we define
(3.2) ζγ(b) = G γ(o b, tb) Gγ(o b, ob) .
This quotient of Green kernels will appear in the definition of the non-backtracking eigenfunctions. See [6, § 2] for more comments on this quantity.
Given an oriented edge b, let Nb+ be the set of outgoing bonds from b, and let Nb− be the set of incoming bonds to b i.e.
(3.3) N + b := n b′∈ B; ob′ = tb, b′ 6= ˆb o Nb−:= n b′∈ B; tb′ = ob, b′ 6= ˆb o .
(Later these definitions will apply to more general graphs than trees.)
The following lemma gives a quantum graph analog for the classical recursive identities of Green’s functions on discrete trees. It tells us that the functions ζγ, C
γ and Sγ can
be used as building blocks to understand the function Gγ. In particular, (3.10) is the well-known multiplicative property of the Green function on a tree. See [6, Section 2] for a proof, and AppendixA for a complement.
Lemma 3.1. Let γ ∈ C+. We have the following relations between ζγ and the WT
functions R±γ: (3.4) ζγ(b) = Cγ(Lb) + Rγ+(ob)Sγ(Lb) , ζγ(ˆb) = Sγ′(Lb) + R−γ(tb)Sγ(Lb) , (3.5) R+γ(tb) = S′γ(Lb) Sγ(Lb) − 1 Sγ(Lb)ζγ(b) , R−γ(ob) = Cγ(Lb) Sγ(Lb) − 1 Sγ(Lb)ζγ(ˆb) . Moreover, (3.6) 1 ζγ(b)S γ(Lb) + X b+∈N+ b ζγ(b+) Sγ(Lb+) = X b+∈N+ b Cγ(Lb+) Sγ(Lb+) + S ′ γ(Lb) Sγ(Lb) + αtb, (3.7) 1 ζγ(b)− ζ γ(ˆb) = Sγ(Lb) Gγ(t b, tb) , ζ γ(ˆb) ζγ(b) = Gγ(ob, ob) Gγ(t b, tb) , (3.8) −1 Gγ(o b, ob) = R+γ(ob) + R−γ(ob) and (3.9) X b+∈N+ b Cγ(Lb+) Sγ(Lb+) + S ′ γ(Lb) Sγ(Lb) + αtb = X tb′∼tb ζγ(b′) Sγ(Lb′) + 1 Gγ(t b, tb) ,
where b′ = (tb, tb′). Given a non-backtracking path (v0; vk)∈ T, if bj = (vj−1, vj), then
(3.10) Gγ(v0, vk) = Gγ(v0, v0)ζγ(b1)· · · ζγ(bk) = Gγ(vk, vk)ζγ(ˆb1)· · · ζγ(ˆbk) .
Finally, for any path (v0; vk)∈ T,
(3.11) Gγ(v0, vk) = Gγ(vk, v0) .
The following lemma is an important result on the properties of the Weyl-Titchmarsh functions (3.1) and the fact that they are involved in “currents” passing through the edges from some fixed arbitrary source∗ (the “current” is Iλ
∗(b) =|Gλ+i0(∗, ob)|2Im Rλ+i0+ (ob)).
Lemma 3.2. The functions F (γ) = R+γ(ob), R−γ(tb) and Gγ(v, v) are Herglotz functions:
Im F (γ)≥ 0 for γ ∈ C+. Moreover, we have the following “current” relations:
(3.12) X b+∈N+ b Im Rγ+(ob+)≤ Im R+γ(ob) |ζγ(b)|2 and X b−∈N− b Im R−γ(tb−)≤ Im R−γ(tb) |ζγ(ˆb)|2 .
More precisely, we have (3.13) X b+∈N+ b Im R+γ(ob+) = Im R+ γ(ob) |ζγ(b)|2 − Im γ |ζγ(b)|2 Z Lb 0 |ξ γ +(xb)|2dxb, (3.14) X b−∈N− b Im R−γ(tb−) = Im R−γ(tb) |ζγ(ˆb)|2 − Im γ |ζγ(ˆb)|2 Z Lb 0 |ξ γ −(xb)|2dxb, where (3.15) ξ+γ(xb) = Vγ;o+(xb) Vγ;o+(ob) = Cγ(xb) + R+γ(ob)Sγ(xb) ξ−γ(xb) = Uγ;v− (xb) Uγ;v− (tb) = Cγ(Lb− xb) + Rγ−(tb)Sγ(Lb− xb) .
See Appendix Afor a proof.
3.2. Operators on graphs. Let now Q be a finite quantum graph (typically one in our familyQN) with vertex set V and bond set B. WhenQ = QN the corresponding notions
will be indexed by N (e.g. BN, VN), but we will often tend to drop the index N from the
notation.
In the study of quantum ergodicity, we need to go back and forth between “classical observables” (i.e. functions on a classical phase space) and “quantum observables” (i.e. operators on a Hilbert space). Here this will be done in a simple-minded way: starting from a function on the set of non-backtracking paths, we explain how to build an operator on ℓ2(B) or ℓ2(V ).
If k ≥ 1, we denote by Bk the set of non-backtracking paths in Bk of length k:
(3.16) Bk:=
(b1, . . . , bk)∈ Bk ; ∀i = 1, . . . , k − 1, tbi = obi+1 and tbi+1 6= obi
. If b0 ∈ B, we will also write
Bb0k :=(b1, . . . , bk)∈ Bk such that (b0, . . . , bk)∈ Bk+1 Bk,b0 := (b−k, . . . , b−1)∈ Bk such that (b−k, . . . , b0)∈ Bk+1 . Note that Bb01 =Nb0+ and B1,b0 =Nb0−.
In the sequel, we will often write (b1; bk) instead of (b1, . . . , bk) to lighten the notation.
For all k≥ 1, we also define
Hk:= CBk.
If K ∈ Hk, then K is a map from Bk to C, and we extend it to a map Bk → C by zero
on Bk\ Bk. This extension will still be denoted by K.
If K ∈ Hk, we define an operator KB : ℓ2(B)−→ ℓ2(B) by
(3.17) ∀f ∈ ℓ2(B),∀b1 ∈ B, (KBf )(b1) =
X
(b2;bk)∈Bb1k−1
K(b1; bk)f (bk).
In particular, if K ∈ H1, then KB is a diagonal operator.
For k = 0, we will write B0 := V , and H0:= CV.
For all k≥ 0, we also define KG: ℓ2(V )−→ ℓ2(V ) by
∀h ∈ ℓ2(V ),∀v ∈ V, (KGh)(v) := X (b1;bk)∈Bk ob1=v K(b1; bk)h(tbk) if k ≥ 1 (KGh)(v) := (Kh)(v) = K(v)h(v) if k = 0.
3.3. Green functions notation. In the paper we consider a variety of quantum graphs, and we need to adopt a notation for the Green function of each of them: the sequence QN, their universal covers eQN, the limiting random quantum tree T .
Let us first define notations pertaining to universal covers. Let Q be a quantum graph. Let ˜G = ( ˜V , ˜E) be the universal cover of the combinatorial graph G. We endow ˜G with the lifted data ˜Lb := Lπb, ˜Wb := Wπb and ˜αv = απv, where π : ˜G → G is the covering
map. This yields a quantum tree eQ := ( ˜V , ˜E, ˜L, ˜W , ˜α), which is called the universal cover of Q. The underlying metric graph of eQ will be denoted by eG. We then have a natural projection π : eG −→ G.
Throughout the paper, if v, w ∈ VN and z ∈ C \ R, we will write
gNγ(v, w) := (HQN− γ)−1(v, w)
for the Green function of the compact quantum graphQN.
Let ˜gNγ be the Green’s functions of HQe
N. Throughout the paper we will encounter
quantities of the form ˜gNγ(obi, tbj) where (b1, . . . , bk) is a fixed non-backtracking path in
GN and i, j ≤ k. We define this as follows.
Given (b1; bk)∈ Bk(QN), choose any lift ˜b1 ∈ B( eQN) and let (˜b2; ˜bk)∈ B˜b0k−1( eQ) be the
path such that π(˜b1, . . . , ˜bk) = (b1, . . . , bk). Then we define
˜
gNγ(ob1, tbk) := (HQgN − γ)−1(o˜b1, t˜bk) .
This depends on the full path (b1, . . . , bk) (although not apparent in our notation),
however it does not depend on the choice of the lift (˜b1, . . . , ˜bk). We define ˜gγN(obi, tbj) :=
(HQg
N − γ) −1(o
˜bi, t˜bj) for i, j ≤ k, where (˜b1, . . . , ˜bk) is the lift we fixed. The definition
extends naturally to ˜gNγ(x, y) with x∈ bi and y∈ bj.
Throughout the paper, we always let for b∈ BN,
(3.18) ζγ(b) := ˜g
γ N(ob, tb)
˜
gNγ(ob, ob)
thus suppressing the index N , which should cause no confusion.
Similarly, the Weyl-Titchmarsh functions (3.1) denoted by R±γ(x) will stand (without index N ) for the WT-functions of the universal covering tree eQN.
For the Benjamini-Schramm limiting random tree T , we use the notation Gγ(x, y) = (HT − γ)−1(x, y)
for x, y∈ T . For ζ and the WT-functions, we simply add a hat. More precisely, we let ˆ ζγ(b) := G γ(o b, tb) Gγ(o b, ob)
for b∈ B(T ), and similarly denote the Weyl-Titchmarsh functions of T by ˆR± γ(x).
3.4. A scalar product expression for boundary values of eigenfunctions. For each γ∈ C and b ∈ BN, let us define
Σ1(γ; b) := Z Lb 0 |Sγ (xb)|2dxb Σ2(γ; b) := Z Lb 0 Sγ(Lb− xb)Sγ(xb)dxb.
Note that, by the Cauchy-Schwarz inequality, Σ2
1(γ; b)− |Σ2(γ; b)|2 > 0. As the lower
bound only depends on Lb, Wb, γ, we have Σ21(γ; b)− |Σ22(γ; b)| ≥ c > 0 in the compact set
m≤ L ≤ M, and f is a Lipschitz function on [0, L] with both norm and Lipschitz constant ≤ M.
For each γ ∈ C and each b ∈ BN, we denote by Sγ,+b and Sγ,−b the functions on GN
defined respectively as xb 7→ Sγ(xb) and xb 7→ Sγ(Lb− xb) on the edge b, and vanishing
on the other edges.
If ψγ satisfies HQψγ= γψγ, and if the potentials are symmetric, then we have
ψγ(xb) = Sγ(Lb− xb) Sγ(Lb) ψγ(ob) + Sγ(xb) Sγ(Lb) ψγ(tb), so that Sγ(Lb)hSγ,+b , ψγi = ψγ(ob)Σ2(γ; b) + ψγ(tb)Σ1(γ; b) Sγ(Lb)hSγ,−b , ψγi = ψγ(ob)Σ1(γ; b) + ψγ(tb)Σ2(γ; b). We therefore have (3.19) ψγ(ob) = Sγ(Lb) |Σ2 2(γ; b)| − Σ21(γ; b) D Σ2(γ; b)Sγ,+b − Σ1(γ; b)Sbγ,−, ψγ E =:hYγb, ψγiL2(GN) ψγ(tb) = Sγ(Lb) |Σ2 2(γ; b)| − Σ21(γ; b) D Σ2(γ; b)Sγ,−b − Σ1(γ; b)Sbγ,+, ψγ E =:hZγb, ψγiL2(GN)
Note that, thanks to hypothesis (Data), we have sup N max b∈BN sup γ∈I+i[0,1] max(kYγbkL2(GN),kZγbkL2(GN))≤ CI,M. That γ 7→ Zb
γ is not analytic poses a technical obstacle. Since we will be using
holo-morphic tools, we prefer to use c Σ1(γ; b) := Z Lb 0 Sγ(xb)2dxb, Σc2(γ; b) := Z Lb 0 Sγ(Lb− xb)Sγ(xb)dxb,
in other words the analytic functions that coincide with Σ1, Σ2 on the real line.
By Cauchy-Schwarz and by continuity, we may find ηI, cI > 0 such that for all N ,
b∈ BN, γ∈ ΩI := I + i[−ηI, ηI], we have
ReΣc21(γ; b)− cΣ22(γ; b)> cI.
Therefore, we may define, for γ ∈ ΩI, the functions
b Zγb := Sγ(Lb) c Σ22(γ; b)− cΣ21(γ; b) c Σ2(γ; b)Sbγ,−− cΣ1(γ; b)Sγ,+b .
Note that ΩI ∋ γ 7→ bZγb ∈ L2(G) is holomorphic, coincides with Zγb when γ ∈ I, and
(3.20) bZγb− Zγb L2(GN)≤ bZγb − bZRe(γ)b L2(GN)+ Zγb − ZRe(γ)b L2(GN)≤ C ′ Iη0 uniformly in γ∈ ΩI, b∈ BN, N ∈ N.
3.5. Notation for the remainders. In all the paper, we will be dealing with quantities depending on N , and on a complex parameter γ, and we will use the following notation. Let fN : C+ −→ C be a sequence of functions. We will write that fN = ON →+∞,γ(1) if
sup η0∈(0,ηDir) lim sup N −→∞ sup λ∈I1|fN (λ + iη0)| < ∞, with ηDir as in (2.7).
Similarly, we will write fN = ON →+∞,γ(Im γ) if fNIm γ(γ) = ON →+∞,γ(1). In fact, most of
If fN depends on an additional parameter κ, we write fN = ON →+∞,γ(κ) (1).
If the quantity fN we consider depends on N and η0, but not on λ, we will write
fN = ON →+∞,η0(1) or fN = ON →+∞,η0(η0), with the same definition.
Remark 3.3. Our main statements, Theorems2.1and2.3say that some quantity, divided by NN(I) goes to zero as N −→ +∞ followed by η ↓ 0. We will recall in Appendix C,
and more precisely in (C.2) and (C.3) that, under the assumptions we make, there exist constants C1, C2> 0 such that for all N large enough,
C1N ≤ NN(I)≤ C2N.
Therefore, in the course of the proof, when trying to show that a quantity divided by NN(I) goes to zero, we will sometimes replace NN(I) by N .
4. Non-backtracking eigenfunctions
The quantum variance (2.8) involves functions living on the metric graph GN. Through
the main part of the paper, we shall prefer to work with quantum variances defined on the combinatorial graph GN. It is shown in Section 5 how to pass from one to the other.
Such discretization is generally nontrivial for non-equilateral quantum graphs. We will show however that we can construct functions on the directed edges BN which, quite
miraculously, are eigenfunctions of a simple non-backtracking operator denoted below by ζγB. This reduction from continuous to discrete will use the quantum Green’s functions
identities derived in Section 3.1, and may be relevant to other problems on quantum graphs.
Let Q be a quantum graph, whose set of oriented edges is denoted by B (later, the following construction will be applied to Q = QN, and all the objects depend on N ).
From now on, we fix η0> 0 which will go to zero in the sequel. Recall that ζγ was defined
in Section3.3using the universal cover ofQ.
Let ψj be an eigenfunction of HQ with eigenvalue λj. We define fj, fj∗ ∈ CB by
(4.1) fj(b) = ψj(tb) Sλj(Lb) − ζγj(b)ψj(ob) Sλj(Lb) , fj∗(b) = ψj(ob) Sλj(Lb) − ζγj(bb)ψj(tb) Sλj(Lb)
where γj = λj+ iη0 and ζγj(b) is as in (3.18).
Note that ψj(ob) and ψj(tb) can be recovered from fj(b) and fj∗(b) as follows: ψj(ob) = Sλj(Lb) 1−ζγj(b)ζγj(bb) f ∗ j(b) + ζγj(bb)fj(b) and ψj(tb) = Sλj(Lb) 1−ζγj(b)ζγj(bb) fj(b) + ζ γj(b)f∗ j(b) . These expressions are well defined, since by (3.7) we have
1− ζγ(b)ζγ(ˆb) = Sγ(Lb)ζ γ(b) Gγ(t b, tb) = Sγ(Lb)ζ γ(ˆb) Gγ(o b, ob) , which does not vanish using (3.6).
Recall that Nb+ was introduced in (3.3). We define the non-backtracking operator B : CB → CB by
(Bf)(b) = X
b+∈N+ b
We observe that ψj(tb) = ψj(ob)Cλj(Lb) + ψ′j(ob)Sλj(Lb), so X b+∈N+ b ψj(tb+) Sλj(Lb+) = ψj(tb) X b+∈N+ b Cλj(Lb+) Sλj(Lb+) + X b+∈N+ b ψ′j(ob+) = ψj(tb) X b+∈N+ b Cλj(Lb+) Sλj(Lb+) + ψ′j(tb) + αtbψj(tb) = ψj(tb) X b+∈N+ b Cλj(Lb+) Sλj(Lb+) +S ′ λj(Lb)ψj(tb)− ψj(ob) Sλj(Lb) + αtbψj(tb) ,
where we used the δ-conditions and the fact that ψλ(ob) ψ′λ(ob) = Mλ(b)−1 ψλ(tb) ψ′λ(tb) . We thus get (4.2) (Bfj)(b) = ψj(tb) X b+∈N+ b C λj(Lb+) Sλj(Lb+) − ζγj(b+) Sλj(Lb+) +S ′ λj(Lb) Sλj(Lb) + αtb − ψj(ob) Sλj(Lb) = 1 ζγj(b)fj(b) + Oψj,η0(b) ,
with Oψj,η0(b) = ψj(tb)Oγj(b) where, by (3.6), we have
Oγj(b) = X b+∈N+ b C λj(Lb+)− ζγj(b+) Sλj(Lb+) − Cγj(Lb+)− ζγj(b+) Sγj(Lb+) +S ′ λj(Lb) Sλj(Lb) − Sγj′ (Lb) Sγj(Lb) + 1 ζγj(b)S γj(Lb) − 1 ζγj(b)S λj(Lb) .
Similarly, since fj∗ = ιfj, where ι is the edge-reversal, and since B∗ = ιBι, we get
B∗f∗
j = ιζ1γjf ∗
j + ιOψj,η0(b), with ιOψj,η0(b) = ψj(ob)ιOγj(b).
Note that, by Corollary C.8and analyticity of Cz(Lb), Sz(Lb), we have for all s > 0
(4.3) 1
NkOγjk
s
ℓs(BN) = ON →+∞,γ(s) (η0).
If K = Kγ ∈ H
k for some k ≥ 1, is some operator, possibly depending on γ ∈ C+, we
define its non-backtracking quantum variance by VarInb,η0(K) := 1 NN(I) X λj∈I hfj∗, K λj+iη0 B fji (4.4) = 1 NN(I) X λj∈I X (b1,...,bk)∈Bk fj∗(b1)Kλj+iη0(b1, . . . , bk)fj(bk) . (4.5)
The advantage of this non-backtracking variance is that it is invariant under a spectral averaging operator, whose kernel is relatively simple due to non-backtracking.
More precisely, we have by (4.2) and its analog for fj∗ that
hfj∗, KBfji = h(ιζγjB∗)n−rfj∗, KB(ζγjB)rfji + error
(4.6)
for any n ≥ 1 and 0 ≤ r ≤ n, where the error depends on Oψj,η0 and will be developed
in (4.9) and (4.11). But before starting the calculation, we would like to express the “sandwich” (Bιζγj)n−rK
B(ζγjB)r as a new observable in phase space5.
5Very roughly, this heuristic can be likened to the classical proof of quantum ergodicity on manifolds
First note that [(ζγB)rf ](b1) = X (b2,...,br+1)∈Brb1 ζγ(b1)· · · ζγ(br)f (br+1) [(ιζγB∗)kf ](b1) = X (b−k+1,...,b0)∈Br,b1 ζγ(ˆb1)· · · ζγ(ˆb−k+2)f (b−k+1).
Recalling (3.17) this yields:
Proposition 4.1. If we defineRγjn,r : Hk−→ Hn+k by (4.7) (Rγjn,rK)(b1; bn+k) = ζγj(ˆb 2)· · · ζγj(ˆbn−r+1)K(bn−r+1; bn−r+k)ζγj(bn−r+k)· · · ζγj(bn+k−1) , we have h(ιζγjB∗)n−rfj∗, KB(ζγjB)rfji = hfj∗, (R γj n,rK)Bfji.
Thus, Rγjn,rK is the “observable” we seek.
We now derive the expression of the error in (4.6) more precisely. We have
(4.8) X (b1;bn+k)∈Bn+k f∗ j(b1)(R γj n,rK)(b1; bn+k)fj(bn+k) = X (b1;bn+k−1)∈Bn+k−1 h fj∗(b1)ζγj(ˆb2)· · · ζγj(ˆbn−r+1)K(bn−r+1; bn−r+k) × ζγj(bn−r+k)· · · ζγj(bn+k−2)· [ζγjBfj](bn+k−1) i = X (b1;bn+k−1)∈Bn+k−1 f∗ j(b1)(R γj n−1,r−1K)(b1; bn−1+k)[fj+ ζγjOψj,η0](bn+k−1) =hfj∗, (Rγjn−1,r−1K)B[fj+ ζγjOψj,η0]i.
where we used (4.2). Iterating this equation r times, we obtain (4.9) hfj∗, (R γj n,rK)Bfji = hfj∗, (R γj n−r,0K)Bfji + r X ℓ=1 hfj∗, (R γj n−ℓ,r−ℓK)BζγjOψj,η0i.
By a computation similar to (4.8), we obtain that for all m > 1,
(4.10) X (b1;bm+k)∈Bm+k f∗ j(b1)(Rγjm,sK)(b1; bm+k)fj(bm+k) = X (b2,...,bm+k)∈Bm+k−1 fj∗(b2) + ιζγjιOψj,η0(b2)(Rγjm−1,sK)(b2; bm+k)fj(bm+k) =hfj∗+ ιζγjιOψj,η0, (Rγjm−1,sK)Bfji.
Applying (4.10) n− r times to the first term in the left-hand side of (4.9), we obtain (4.11) hfj∗, (R γj n−r,0K)Bfji = hfj∗, KBfji + n−r X ℓ′=1 hιζγjιO ψj,η0, (Rγjn−r−ℓ′,0K)Bfji.
As (4.9) and (4.11) hold for any r≤ n, we thus get the desired (approximate) invariance of the quantum variance under the averaging operatorRγn,r:
theorem. We see that even if a(x) is a function, one must consider the phase space observable a ◦ Gt(x, ξ).
Proposition 4.2. (4.12) VarInb,η0(K)≤ VarInb,η0 1 n n X r=1 Rγn,rK +En,η0(K), where (4.13) En,η0(K) = 1 n n X r=1 1 NN(I) X λj∈I r X ℓ=1 hfj∗, (R γj n−ℓ,r−ℓK)BζγjOψj,η0i + n−r X ℓ′=1 hιζγjιO ψj,η0, (Rγjn−r−ℓ′,0K)Bfji . The quantityEn,η0(K) is negligible thanks to the following lemma proven in Section6.1.
Lemma 4.3. For any n ∈ N, k ≥ 0 and any K ∈ Hk possibly depending on γ, but
satisfying (5.1), we have
En,η0(K) = ON →+∞,η0(n) (η0).
5. Reduction to non-backtracking variances
From now on, all the constructions take place on the graphQN, and N varies and goes
to +∞. So it should be understood that all objects depend on N, although it is not always apparent in the notation.
In this section, we will show that the statement of Theorem 2.3 can be reduced to an estimate on the non-backtracking variances (4.4). To this end, we will start from an integral operatorK ∈ Kk, and build several new operators JKγ ∈ Hk′ from it. These JKγ
will depend on the parameter γ ∈ C+ (and N ), but this dependence will always satisfy
the following hypothesis.
Definition 5.1. Let k≥ 0 and let C+∋ γ 7→ Kγ= Kγ
N ∈ Hk. We say that Kγ satisfies
hypothesis (Hol) if
• For all 0 < η0 < ηDir and (b1, . . . , bk) ∈ Bk, the map R ∋ λ 7→ Kλ+iη0(b1, . . . , bk)
has an analytic extension Kz
η0(b1, . . . , bk) to the strip {z : | Im z| < η0/2}.
• For all s > 0, we have
(5.1) sup η0∈(0,ηDir) lim sup N →+∞ sup η1∈(−η02,η02 ) sup λ∈I1 1 N X (b1,...,bk)∈Bk |Kη0λ+iη1(b0, . . . , bk)|s < +∞.
• For almost all λ ∈ I1, for all s > 0 and all t∈ (−1/2, 1/2), we have
(5.2) lim η0↓0lim supN →+∞ 1 N X (b1,...,bk)∈Bk |Kη0λ+tiη0(b0, . . . , bk)− Kλ+iη0(b0, . . . , bk)|s = 0.
Beware that, unless γ 7→ Kγ(b
1, . . . , bk) is analytic, the quantities Kλ+iη0+iη1(b1, . . . , bk)
and Kη0λ+iη1(b1, . . . , bk) are in general different for η1 6= 0.
Remark 5.2. Note that if Kγ and Jγ satisfy (Hol), so does their sum and product. If
Kγsatisfies (Hol), then Kγ also satisfies (Hol). Indeed, its restriction to a horizontal line
is real-analytic, so it can be extended holomorphically to a strip by an extension satisfying the desired properties.
Last but not least, thanks to Corollary C.8 and Proposition C.9, the functions in the class Lkγ introduced in Definition C.2all satisfy (Hol).
Given a linear operator Fγ: ℓ2(V )−→ ℓ2(V ) possibly depending on γ∈ C+, we define VarIη0(Fγ) := 1 NN(I) X λj∈I |h˚ψj, Fλj+iη0ψ˚jiℓ2(V )| ,
where ˚ψj is the restriction of ψj to the vertices, which is well-defined thanks to (2.2)
˚
ψj(v) = ψj(v).
Let us write Ψγ,v(w) := Im ˜gNγ(v, w) and define, for K ∈ Hk (possibly depending on
γ∈ C+): (5.3) hKiγ:= P 1 v∈V Ψγ,v(v) X (b1,...,bk)∈Bk K(b1, . . . , bk)Ψγ,ob1(tbk) if k≥ 1 hKiγ:= P 1 v∈V Ψγ,v(v) X v∈V K(v)Ψγ,v(v) if k = 0.
Finally, we denote by 1 = 10 ∈ H0 the constant function equal to one on every vertex.
In the sequel, we will also denote by ˆ1 the function on GN which is constant equal to one.
As a first step towards the reduction to non-backtracking variances, we control the quantities appearing in Theorems 2.1 and 2.3 by some discrete variances of the form VarI
η0.
Proposition 5.3. (1) Let fN ∈ L∞(GN) be a sequence of functions withkfNk∞≤ 1. We
may build (Jf,pγ )p=1,...,8, each belonging to H0 or H1 and satisfying (Hol), such that
lim η0↓0lim supN →∞ 1 NN(I) X λj∈I |hψj, f ψji − hfiγj| ≤ C 8 X p=1 lim
η0↓0lim supN →∞ Var I η0 (Jf,pγ )G− hJf,pγ iγ1 . (2) Let k ≥ 1 and let KN ∈ Kk be a sequence of non-backtracking integral operators
satisfying |KN(x, y)| ≤ 1 for all x, y ∈ GN.
We may build (JK,pγ )p=1,...,6, each belonging to Hkp for k−2 ≤ kp ≤ k, and satisfying
(Hol), such that lim η0↓0lim supN →∞ 1 NN(I) X λj∈I |hψj,Kψji−hKiγj| ≤ C 6 X p=1 lim
η0↓0lim supN →∞ Var I η0 (JK,pγ )G− hJK,pγ iγ1 . The idea of the proof is quite simple: expand the eigenfunction ψj in the basis of
solutions Sλ, Cλ given in § 2.2, then split the averagehfiγj into convenient pieces, so that
each operator in the discrete variance has zero mean. The scheme is quite similar to the equilateral case [19]. We give the technical details in AppendixB.1
Our next step is to show that the discrete variance of J − hJiγ1 can be bounded by
some non-backtracking variances, plus some terms which will be negligible. Before stating the proposition, we need to introduce some notation. In the sequel, we will often write hJiγ instead ofhJiγ1 in the computations of the variances.
If d(x) is the degree of x∈ V , we denote P = 1dAG. Let
Nγ(x) = Im ˜gγ(˜x, ˜x) , Pγ = d Nγ PNγ d . We defineST,γ: ℓ2(V )→ ℓ2(V ) and eST,γ: ℓ2(V )→ ℓ2(V ) by ST,γJ = 1 T T −1X s=0 (T − s)PγsJ and SeT,γJ = 1 T T X s=1 PγsJ .
We also define Lγ : ℓ2(V )→ ℓ2(B) andE γ: ℓ2(V )→ ℓ2(V ) by (LγJ)(b) = S 2 Re γ(Lb)|˜gγ(tb, tb)|2(1 + ζγ(b)ζγ(bb)) Re Sγ(Lb)|ζγ(b)|2 J(tb) Nγ(ob) − J(ob) Nγ(tb) , (EγJ)(ob) = X tb∼ob Im Sγ(Lb) Re ˜gγ(tb, tb) Re Sγ(Lb) J(tb) Nγ(ob) − J(ob) Nγ(tb) . Proposition 5.4. (1) Let J ∈ H0. For any T ∈ N, we have
(5.4) Var
I
η0(J− hJiγ)≤VarInb,η0[Lγd−1ST,γ(J − hJiγ)]
+ VarIη0[Eγd−1ST,γ(J − hJiγ)] + VarIη0( eST,γ(J − hJiγ)) .
(2) Let J∈ Hksatisfy (Hol). There exists P1, P2∈ N depending only on k, and operators
(LγpJ)1≤p≤P1 and (FpγJ)1≤p≤P2 with LγpJ ∈ H0 and FpγJ ∈ Hkp for some 1≤ kp ≤ k,
all satisfying (Hol), such that
VarIη0(JG− hJiγ)≤ P1 X p=1 VarIη0(LγpJ − hLγpJiγ) + P2 X p=1 VarInb,η0 FpγJ+ ON →+∞,γ(η0).
This proposition will be applied to the J = Jf,pγ , JK,pγ appearing in Proposition5.3. The proof is given in AppendixB.2.
To finish this section, we will show that the error variances VarIη0[Eγd−1ST,γ(J− hJiγ)]
and VarIη0( eST,γ(J − hJiγ)) from (5.4) vanish asymptotically. Note that if we apply part
(1) of Proposition 5.4to the first sum of variances in part (2), then all that remains will be to control non-backtracking variances.
If K ∈ CV ∼= H0, we define the “weighted Hilbert-Schmidt norm”
(5.5) kKk2γ:= 1 |V | X v∈V |K(v)|2Nγ(x)2.
Then we have the following bound proven in Section6.1. An analogous bound for non-backtracking variances will be given in Proposition7.1. These two propositions tell us the quantum variance(s) are dominated by weighted Hilbert-Schmidt norms.
Lemma 5.5. Let Kγ ∈ H0 satisfy hypothesis (Hol) from Definition 5.1. We have
lim
η0↓0lim supN →∞ Var I η0(Kγ)2 .lim η0↓0lim supN →∞ Z IkK λ+iη0k2 λ+iη0dλ.
This lemma makes it easy to deal with VarIη0[Eγd−1ST,γ(J − hJiγ)]. Indeed, for any
fixed T ∈ N, Eγd−1ST,γ(Kγ− hKγiγ) ∈ H0 satisfies (Hol) by Remark 5.2. Therefore,
from the expression ofEγ, which involves Im Sγ(Lb) = O(η0), we see using (5.1) that
(5.6) lim
η0↓0N →∞lim Var I
η0 Eγd−1ST,γ(Kγ− hKγiγ= 0.
The error VarIη0( eST,γ(J − hJiγ)) is dealt with thanks to the following lemma.
Lemma 5.6. Let K ∈ H0 satisfy (Hol). We then have
lim
T →∞η0∈(0,ηsupDir)lim supN →∞ supλ∈I
eST,γ(Kγ− hKγiγ)
γ= 0.
Proof. Let us write (YγK)(v) = Nγ(v)d(v) P
w∈VP Nγ(w)K(w)
w∈Vd(w) , so that YγK =
hNγKiU
hdiU Nγd , where
hJiU := N1 Pv∈V J(v) is the average with respect to the uniform scalar product.
Noting that for any s∈ N, we have Ps
γ = Nγd Ps N γ
eST,γKγ− YγKγ 2 γ= 1 N X v∈V T1 T X s=1 d(v)PsNγK γ d (v)− hNγK γi U hdiU d(v) 2 ≤ D N T T X s=1 PsNγ Kγ d − hNγKγiU hdiU 1 2 ℓ2(V,d) ≤ D N T T X s=1 (1− β)2s NγKγ d − hNγKγiU hdiU 1 2 ℓ2(V,d) ≤ D βN T NγK γ d − hNγKγiU hdiU 1 2 ℓ2(V,d) ≤ 4D βN T kNγK γk2 ℓ2(V,d),
where we used the fact that NγKd γ − hNγKγiU
hdiU 1 is orthogonal to constants in the space
ℓ2(VN, dN), the assumption (EXP), and the fact that d≥ 1.
Now, since Kγ satisfies (Hol), we know by Remark5.2that NγKγ satisfies hypothesis
(Hol), so that N1 kNγKγk2ℓ2(V )= ON →+∞,γ(1). Therefore,
(5.7) lim
T →∞η0∈(0,ηsupDir)lim supN →∞ supλ∈I
eST,γKγ− YγKγ
γ = 0.
Now, the result follows by applying (5.7) to ˚Kγ := Kγ− hKγiγ1, and by noting that
YγK˚γ = 0, since YγhKγiγ1 =hKγiγ hNhdiγiUUNγd and, by definition,hKγiγ= hNγK γi
U
hNγiU .
Combining Propositions 5.3 and 5.4, Lemma 5.5, Lemma 5.6 and equation (5.6) we obtain the following corollary.
Corollary 5.7. Let QN be a sequence of quantum graphs satisfying (Data) for each N ,
such that (EXP), (BST), (Green) and (Non-Dirichlet) hold true on the interval I1.
Suppose that for any Kγ ∈ H
k satisfying (Hol), k≥ 0, we have
(5.8) lim
η0↓0N →∞lim Var I
nb,η0(Kγ) = 0.
Then, if fN ∈ L∞(GN) is a sequence of functions satisfying kfNk∞ ≤ 1, (2.8) holds.
Furthermore, for any k≥ 0, if KN ∈ Kk is a sequence non-backtracking integral operator
satisfying|KN(x, y)| ≤ 1 for all x, y ∈ GN, then (2.10) holds.
The rest of the paper will be devoted to proving (5.8), thus establishing Theorem2.3. 6. Contour integrals and complex analysis
In this section, we shall prove Lemmas 4.3 and 5.5 and develop tools from complex analysis that will be used later on. The quantities we wish to estimate are expressed as sums over the eigenvalues of the quantum graph QN, which are the poles of the Green
function gN. Thanks to Cauchy’s formula, these sums will be expressed as contour integrals
involving the Green functions. The manipulation of the unknown eigenfunctions ψ(N )j is thus replaced by manipulation of gN. Later on, this will be replaced by the Green function
˜
gN of the universal cover, and finally we will use that it converges in distribution to the
Green function of the limiting random treeT .
If χ ∈ Cc∞(R), we will denote by ˜χ an almost analytic extension of χ, i.e., a smooth function ˜χ : C7→ C such that ˜χ(z) = χ(z) for z∈ R, ∂ ˜∂zχ(z) = O((Im z)2), and
(6.1) supp ˜χ⊂ {z; Re z ∈ supp χ}.
Here ∂z∂ = 12(∂x∂ + i∂y∂ ). For instance, one can take ˜χ(x + iy) = χ(x) + iyχ′(x)−y22χ′′(x). We refer the reader to [15, § 2.2] for more details about almost analytic extensions.
Recall that if Kγ∈ H
ksatisfies (Hol), then for any 0 < η0< ηDirand (b1, . . . , bk)∈ Bk,
λ 7→ Kλ+iη0(b1, . . . , bk) admits a holomorphic extension to {| Im z| < η0/2}, which is
denoted by Kz
η0(b1, . . . , bk).
Proposition 6.1. Let χ∈ C∞
c (I1). Define ˜χ as above. Let k, k′ ∈ N and let Kγ ∈ Hk,
and K′γ ∈ Hk′ satisfy (Hol). Then for any 0 < η0< min(ηDir
2 , ηI1), we have (6.2) 1 N X j≥1 X (b1;bk) X (b′ 1;b′k′) b′ 1=b1 χ(λj)Kγj(b1; bk)K′γj(b1′; b′k′)ψj(obk)ψj(ob′ k′) =−1 N X (b1;bk) X (b′1;b′k′) b′ 1=b1 1 2πi Z Γη0/4 ˜ χ(z)gzN(obk, ob′k′)K z η0(b1; bk)Kη0′z(b1′; b′k′)dz+ON →+∞,η0(η0),
where Γη is the boundary of Ωη = I1+ i[−η, η].
The same formula holds if obk and/or ob′k′ are replaced by tbk and/or tb′k′ in both terms.
Proof. Step 1 : From a sum to an integral
Let v, w ∈ VN, and let h be a holomorphic function in a strip {z ∈ C; | Im z| < ηDir2 }.
The function h may depend on N , v, w. Using (3.19), and noting that Zλb = Zb
λ for λ∈ R, we have X j≥1 χ(λj)h(λj)ψj(v)ψj(w) = X j≥1 χ(λj)h(λj)hZλjbv, ψjihZλjbw, ψji,
where we chose bv 6= bw such that v = tbvand w = tbw. This is possible since d(v), d(w)≥ 2
by assumption (Data).
We may define holomorphic functions bZzb as in the end of Section3.4, on some open set ΩI1 := I1+ i[−ηI1, ηI1] which does not depend on N ∈ N or b ∈ BN.
Let 0 < η1< min(ηI1,ηDir2 ). Consider the rectangle Ωη1 := I1+ i[−η1, η1] and the curve
Γη1 := ∂Ωη1. Cauchy’s integral formula f (λ) = 2πi1
R
Γ f (z)
z−λdz for analytic f generalizes to
f (λ) = 1 2πi Z Γη1 f (z) z− λdz + Z Ωη1 ∂f /∂z z− λ dz∧ dz
for λ∈ Ωη1, where dz∧ dz = −2i dxdy, see e.g. [18, Theorem 1.2.1].
We apply this result to λ = λj and
f (z) = ˜χ(z)h(z)h bZzbv, ψjih bZzbw, ψji.
Noting that z ∈ ΩI 7→ h(z)h bZzbv, ψjih bZzbw, ψji is holomorphic, we obtain
χ(λj)h(λj)ψj(v)ψj(w) = 1 2πi Z Γη1 ˜ χ(z)h(z)h bZbv z , ψjih bZzbw, ψji z− λj dz + Z Ωη1 h(z)h bZzbv, ψjih bZzbw, ψji z− λj ∂ ˜χ ∂z dz∧ dz .
Next, we want to sum this expression over j. Since (ψj) is an orthonormal basis of
L2(G N), we obtain that X j≥1 h bZbv z , ψjih bZzbw, ψji λj− z =X j≥1 h bZbv z , ψjihψj, bZzbwi λj− z =h bZzbv, (HQN − z)−1Zbzbwi .
Therefore, we have X j χ(λj)h(λj)ψj(v)ψj(w) = −1 2πi Z Γη1 ˜ χ(z)h(z)h bZzbv, (HQN − z)−1Zbzbwi dz + Z Ωη1 h(z)h bZzbv, (HQN− z) −1Zbbw z i ∂ ˜χ ∂z dz∧ dz . Now, we know that
h bZzbv, (HQN − z) −1Zbbw
z i ≤ C
Im z, where C does not depend on N ∈ N or on v, w.
We deduce that X j χ(λj)h(λj)ψj(v)ψj(w)− −1 2πi Z Γη1 ˜ χ(z)h(z)h bZzbv, (HQN − z) −1Zbbw z idz ≤ C′η1 Z Ωη1|h(z)|dz ∧ dz.
Step 2 : Using the properties of Zz We would like to replace the bZz by Zz in the
previous formula, for the following reason. Since bv 6= bw, the map y7→ (HQN− z)
−1(x, y) is an eigenfunction on bw, so that (6.3) (HQN − z) −1Zbw z (x) = (HQN − z) −1(x, w) by (3.19). The map x7→ (HQN − z) −1(x, w) is an eigenfunction on b v, so that, by (3.19) again, we have hZzbv, (HQN− z)−1Zzbwi = (HQN − z)−1(v, w).
To estimate the cost of replacing bZz by Zz, we write
(6.4) D b Zzbv, (HQN − z)−1Zbzbw E =DZzbv, (HQN − z)−1Zzbw E +DZbzbv− Zzbv, (HQN − z)−1Zzbw E +DZzbv, (HQN − z) −1Zbbw z − Zzbw E +DZbzbv− Zzbv, (HQN − z) −1Zbbw z − Zzbw E By (3.20), the last term is easy to estimate
DZbzbv− Zzbv, (HQN − z)−1
b
Zzbw− ZzbwE ≤ CI1,M(Im z2)k(HQN − z)−1k
≤ CI1,M| Im z|.
As to the second term on the right-hand side of (6.4), we use (3.20), the Cauchy-Schwarz inequality and (6.3), to see that its modulus is bounded by some constant times
| Im z| (HQN − z) −1( ·, w) L2(bv):=| Im z| Z Lbv 0 (HQN− z) −1(x bv, w) 2 dxbv 1/2 . Since (HQN − z) −1(x bv, w) = Sz(Lbv− xbv) Sz(Lbv) gzN(obv, w) + Sz(xbv) Sz(Lbv) gNz(tbv, w),
we deduce from (Data) and (Non-Dirichlet) that the second term is bounded by CI,M,CDir| Im z|(|gNz (obv, w)| + |gNz(v, w)|).
We have a similar estimate for the third term. Therefore, we have (6.5) X j χ(λj)h(λj)ψj(v)ψj(w)− −1 2πi Z Γη1 ˜ χ(z)h(z)(HQN − z)−1(v, w)dz ≤ Cη1 Z Ωη1|h(z)|dz ∧ dz + Z Γη1|h(z)| |g z N(obv, w)| + 2|gzN(v, w)| + |gzN(v, obw)| + 1 dz . Step 3 : Using (Hol) Take η1 = η04 for η0 ∈ (0, ηI1). For each (b1; bk), (b1, b′k′), we
apply (6.5) with h(z) = Kz
η0(b1; bk)Kη0′z(b′1; b′k′).
When summing over (b1; bk), (b1, b′k′) and dividing by N , the first term in the remainder
is bounded by C′η0 sup λ∈I1,η∈(−η04,η04 ) 1 N X (b1;bk) X (b′ 1;b′k′) b′ 1=b1 |Kη0λ+iη(b1; bk)Kη0′λ+iη(b′1; b′k′)|,
which is a ON →+∞,η0(η0) by the Cauchy-Schwarz inequality and (5.1).
Concerning the second term, using Cauchy-Schwarz and (5.1), it can be bounded by C′η0 sup λ∈I1,η∈(−η04 ,η04 ) 1 N X (b1;bk) X (b′ 1;b′k′) b′ 1=b1 |gNz(oˇbk, ob′ k′)| 2+|gz N(obk, ob′ k′)| 2 +|gzN(obk, oˇb′ k′ )|2+ 1 1/2 , where ˇbk, ˇb′k′ are chosen so that tˇb
k = obk and tˇb′k′ = ob′k′ but ˇbk 6= ˇb ′ k′. Applying Corol-lary C.8 to Fz(b 1; bk) = P (b′ 1;b′k′) b′ 1=b1
(...), we deduce this is ON →+∞,η0(η0). Proposition 6.1
follows.
We deduce the following corollary, which we will use several times. Corollary 6.2. Let χ∈ C∞
c (I1), let Kγ ∈ Hk, and K′γ ∈ Hk′ satisfy (Hol). Then
1 N X j≥1 X (b1;bk) X (b′ 1;b′k′) b′ 1=b1 χ(λj)Kγj(b1; bk)K′γj(b′1; b′k′)ψj(obk)ψj(ob′ k′) = ON →+∞,η0(1).
The same result holds if obk and/or ob′
k′ are replaced by tbk and/or tb ′ k′.
Proof. By Proposition6.1, up to a term which is ON →+∞,η0(η0) the modulus of the
quan-tity we want to estimate is bounded by C N λ∈I1sup X (b1;bk) X (b′ 1;b′k′) b′ 1=b1 gλ±i η0 4 N (obk, ob′ k′) Kλ±i η0 4 η0 (b1; bk) K ′λ±iη04 η0 (b′1; b′k′) .
Using H¨older’s inequality, (5.1) and Corollary C.8this is ON →+∞,γ(1).
Remark 6.3. The very same proof as that of Proposition6.1 gives us that, if Kγ∈ H 0
satisfies (Hol), then up to an error ON →+∞,η0(η0) we have
1 N X j≥1 X v∈V χ(λj)Kγj(v)|ψj(v)|2=− 1 N X v∈V 1 2πi Z Γη0/4 ˜ χ(z)gNz(v, v)Kη0z (v)dz,
