Raphael Ducatez
email: ducatez@ceremade.dauphine.fr

Anderson localisation for infinitely many interacting particles in Hartree-Fock theory

Keywords: Anderson localisation, Hartree-Fock theory, multiscale analysis. MSC2010: 47A10, 81Q10, 81V70

We prove the occurrence of Anderson localisation for a system of infinitely many particles interacting with a short range potential, within the ground state Hartree-Fock approximation. We assume that the particles hop on a discrete lattice and that they are submitted to an external periodic potential which creates a gap in the non-interacting one particle Hamiltonian. We also assume that the interaction is weak enough to preserve a gap. We prove that the mean-field operator has exponentially localised eigenvectors, either on its whole spectrum or at the edges of its bands, depending on the strength of the disorder.

Introduction

In 1958, the physicist P.W. Anderson predicted that, in a random medium, diffusion could disappear and waves stay localised [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF]. Since then, Anderson localisation has played an important role to explain several properties of physical systems.

There are many works on the mathematical side, starting with the simpler one dimension case in the beginning of the 80s. In dimension 3, the first proof of Anderson localisation was provided by Fröhlich and Spencer [START_REF] Fröhlich | Absence of diffusion in the anderson tight binding model for large disorder or low energy[END_REF] who developed a method now called multiscale analysis. There are now other approaches which include for instance the fractional moment method proposed by Aizenman and Molchanov [START_REF] Aizenman | Localization at large disorder and at extreme energies: an elementary derivation[END_REF] or the techniques proposed by Imbrie [START_REF] Imbrie | Multi-Scale Jacobi Method for Anderson Localization[END_REF].

All these works are restricted to the one wave problem, which is well adapted to optics and acoustics. In condensed matter, the interaction between the particles is expected to play an important role and in this case one speaks of many-body localisation. This question is not fully understood and is very actively studied in physics [START_REF] Basko | Metal insulator transition in a weakly interacting many-electron system with localized single-particle states[END_REF][START_REF]Problems of Condensed Matter Physics[END_REF][START_REF] Aizenman | Localization Bounds for Multiparticle Systems[END_REF][START_REF] Gornyi | Interacting electrons in disordered wires: Anderson localization and low-t transport[END_REF]. A typical system of interest is a crystal composed of quantum electrons and classical nuclei placed on a random perturbation of a perfect lattice [START_REF] Blanc | Existence of the thermodynamic limit for disordered quantum Coulomb systems[END_REF][START_REF]The Reduced Hartree-Fock Model for Short-Range Quantum Crystals with Nonlocal Defects[END_REF]. This is a very complicated system since the Coulomb interaction is long range and there are of the order of 10 23 particles, usually mathematically treated as an infinite number. Due to screening effects, the Coulomb potential is often replaced by an effective short range interaction and this is what we are going to do in this work.

Finite interacting systems have been recently considered in several works [START_REF] Chulaevsky | Multi-particle Anderson Localisation: Induction on the Number of Particles[END_REF][START_REF] Aizenman | Localization Bounds for Multiparticle Systems[END_REF][START_REF] Klein | The bootstrap multiscale analysis for the multi-particle anderson model[END_REF][START_REF] Fauser | Multiparticle localization for disordered systems on continuous space via the fractional moment method[END_REF]] but infinite systems have not been studied thoroughly. One should however mention a series of works on superfluidity and Bose-Einstein condensation in the Lieb-Liniger 1D Bose gas in a strong random potential [START_REF] Seiringer | Disordered Bose-Einstein condensates with interaction in one dimension[END_REF][START_REF] Könenberg | Superfluid behavior of a Bose-Einstein condensate in a random potential[END_REF], and the very recent article of Seiringer and Warzel [START_REF] Seiringer | Decay of correlations and absence of superfluidity in the disordered Tonks-Girardeau gas[END_REF] on the 1D Tonks-Girardeau gas.

It is very hard to deal with the exact interacting Schödinger problem for an infinite system. A useful and widespread approximation is the Hartree-Fock model, where the particles are treated as independent objects but see a field which depends on their own states and is then self-consistently optimised. For random systems this model has been recently introduced by E. Cancès, S. Lahbabi and M. Lewin in [START_REF] Cancès | Mean-field models for disordered crystals[END_REF]. The authors were able to construct a solution of the random nonlinear Hartree-Fock equation for an infinite system of fermions with short range interaction, but the phenomenon of Anderson localisation was only investigated numerically in [START_REF] Lahbabi | Mathematical study of quantum and classical models for random materials in the atomic scale[END_REF].

In this work, we complete this program in the case of a discrete system with a periodic background. In short, we show that the unique solution of the Hartree-Fock equation

H min = -∆ + V 0 + V ω + W * γ(x, x) -W (x -y)γ(x, y) γ = 1 <µ (H min ), (1)
provides a mean-field operator H min which has exponentially localised eigenvectors, either on its whole spectrum or at the edges of its bands. Our main assumption is that the periodic potential V 0 is sufficiently strong to create a gap for -∆ + V 0 and that the random part V ω as well as the interaction W do not alter this gap. We take the chemical potential µ in this gap. Our argument is to adapt the well known multiscale analysis in order to include the nonlinear terms. The paper is organised as follows. In the next section, we properly define the discrete Hartree-Fock model and state our main localisation results. Sections 3 to 6 are devoted to the proof of our results. Then in section 7 we present some simple one dimensional numerical simulations in order to illustrate our findings. We also look at some cases which are not covered by our theorems and find, for instance, an interesting delocalisation phenomenon at the Fermi level in the absence of a gap, which deserves further investigations.

Model and results

We consider a system of fermions on a subset Λ of the lattice Z d and which are submitted to an external potential V . Without interactions the system is described by the one-particle Hamiltonian

H Λ = -∆ + V. (2)
In the canonical basis (δ x ) x∈Λ of 2 (Λ), the operator H Λ is defined by

∀x, y ∈ Λ, δ x , H Λ δ y =      V (x) if x = y, 1 if |x -y| = 1, 0 otherwise. ( 3
)
When Λ is the whole lattice Z d we will use the simpler notation H = H Z d . Our theorems will actually hold for finite as well as for infinite domains. Since H will later be perturbed by a non linear term describing the interactions between the particles, we will often call H the linear part.

The potential V describes a crystal lattice which is randomly perturbed. It is therefore assumed to be of the form

V = V 0 + V ω , (4)
where V 0 is a periodic potential (of an arbitrary period) and V ω is a random potential. We use the Anderson tight binding model where the value of V ω is chosen independently at each site of Λ with the same random probability law P:

V ω = i∈Λ v i (ω)δ i . (5)
Here the v i are iid random variables. Under suitable regularity assumptions on P, it is well known that H displays Anderson localisation. This means that there exist small intervals at the edges of the spectral bands where the spectrum is pure point with exponentially decaying eigenvectors ("Lifshitz tail"). Moreover, if the random potential V ω is strong enough (compared to -∆) then the whole spectrum is pure point with localised eigenvectors.

We will show that if the interactions between the particles are small enough then the same holds for the interacting model. Before introducing interactions, we first discuss the precise assumptions that we will use for P and V 0 .

(A1) Regularity of P. We assume that P has a bounded support and that it has a density ρ with respect to the Lebesgue measure which is Lipschitz by part.

We think that this assumption can be weakened in several possible ways but we will not discuss this for the sake of simplicity. With no loss of generality, we can assume that 0 ∈ Supp(P).

(A2) Gap. We assume that H has a gap [a, b] in its (deterministic) spectrum.

This assumption implies that the periodic potential V 0 is not constant and strong enough. Indeed the spectrum of H is almost surely equal to

σ(H) = σ(-∆ + V 0 ) + supp(P),
when the support of P is an interval. In particular it is enough to assume that -∆ + V 0 has a gap of size G 0 > 2|supp(P)|. When the domain Λ is large enough then H Λ will have a gap as well. We call the size of the gap G = ba, and choose a chemical potential µ in the middle µ = (a + b)/2. Now we turn to the definition of the interacting model. We assume that the interaction is translation-invariant and decays fast enough.

(A3) Short range interaction. We assume that there exists ν > 0 such that

|W (x -y)| ≤ Ce -ν|x-y| (6)
In the Hartree-Fock model (see for example [START_REF] Bach | Error bound for the Hartree-Fock energy of atoms and molecules[END_REF][START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF][START_REF] Lieb | The Hartree-Fock theory for Coulomb systems[END_REF][START_REF] Lions | Solutions of Hartree-Fock equations for Coulomb systems[END_REF][START_REF] Solovej | The ionization conjecture in Hartree-Fock theory[END_REF][START_REF] Bach | There are no unfilled shells in unrestricted Hartree-Fock theory[END_REF]) the system in the domain Λ is completely described by its one-particle density matrix which is an orthogonal projection γ on 2 (Λ). More precisely, for a finite system, the rank-N projection γ

= N i=1 |φ i φ i | corresponds to the N -particle wave function Ψ, Ψ(x 1 , x 2 ..., x N ) = 1 √ N ! det φ i (x j ) , (7)
called a Slater determinant. In a finite domain Λ, the many-body energy of this state is

  Ψ, N i -∆ i + V (x i ) + 1 2 i =j W (x i -x j ) Ψ   = Tr(H Λ γ) + 1 2 x,y∈Λ W (x -y)γ(x, x)γ(y, y) - 1 2 x,y∈Λ W (x -y)|γ(x, y)| 2 . ( 8
)
We define the effective interaction A eff by

A eff (γ)(x, y) = n W (n -y)γ(n, n)δ x=y -W (x -y)γ(x, y) (9)
for any γ positive bounded operator. It is the derivative of the interacting part of the energy. Any minimiser γ of the energy, with fixed rank N , is a solution of the non-linear equation which involve an effective Hamiltonian.

H min = -∆ + V + A eff (γ) γ = 1 <µ (H min ), (10)
In other words γ is the projection on the N first eigenfunctions of H min which itself depends of γ. To be more precise the equation [START_REF] Cancès | On the convergence of SCF algorithms for the Hartree-Fock equations[END_REF] hold under the condition that λ N +1 (H min ) > λ N (H min ) which is known to be automatically satisfied when W > 0 [START_REF] Bach | There are no unfilled shells in unrestricted Hartree-Fock theory[END_REF][START_REF] Bach | Generalized Hartree-Fock theory and the Hubbard model[END_REF].

We will first show that our problem is well defined for a bounded domain Λ as well as for the infinite domain Λ = Z d . Theorem 2.1 (HF ground states for infinitely many particles). Let V be a bounded function such that -∆ + V , defined on a subset Λ of Z d , has a gap [a , b ] in its spectrum, with G = ba and µ = (b + a )/2. If ||W || 1 < G /6, then there exists a unique solution of the system γ = 1 ≤µ (-∆ + V + A eff (γ)). If Λ is finite, the trace is preserved:

Tr(1 ≤µ (-∆ + V + A eff (γ)) = Tr(1 ≤µ (-∆ + V ))
and, because of uniqueness, our solution is as well the unique minimiser of the energy among all Hartree-Fock states with fixed particles number N = Tr 1 ≤µ (-∆ + V ) . This result will be shown in Section 3. The continuous case is more complicated and has been studied in [START_REF]The Reduced Hartree-Fock Model for Short-Range Quantum Crystals with Nonlocal Defects[END_REF]. When Λ is a large cube and V = V = V 0 + V ω then the number of particles Tr(1 ≤µ (-∆ + V )) is proportional to the volume, which is the interesting physical case. For Λ = Z d , H min (ω) exists and its spectrum, as in the linear model, does not depend on ω almost surely. The reason is that H min is stationary with respect to space translations, which follows from the uniqueness in the theorem.

The main result of our paper is the following:

Theorem 2.2 (Anderson localisation in Hartree-Fock theory). Under the assumptions (A1), (A2) and (A3), then the following holds:

1. There is such that if ||W || 1 < then there are small intervals at the edges of the bands of the spectrum of H min , where the spectrum is pure point with exponentially decaying eigenvectors.

2. In presence of strong disorder, meaning that ||ρ|| ∞ + ||ρ || ∞ small enough, and if ||W || 1 < G/6, then the whole spectrum of H min is pure point and its eigenvectors are exponentially decaying in space.

This theorem will be shown by using the multiscale analysis on H min . We will proceed in three step.

Step 1. We show that because of the gap, γ min only depends locally on the potential. From a practical point of view, in order to know how γ min looks like in a box of size L after solving the minimising problem for Z d , solving the minimising problem for a box of size 2L will be enough to have a very good approximation. In a more mathematical formulation: Theorem 2.3 (Locality). We assume that (A2) and (A3) hold. There exist a > 0 , ν > 0 and C > 0 such that, if ||W || 1 < aG, then for any modification of the potential V ω → V ω + δV so that V ω + δV ∈ supp(P), the change induced to the minimising projector given by Theorem 2.1 satisfies

sup y2∈Λ γ min (V + δV )(y 1 , y 2 ) -γ min (V )(y 1 , y 2 ) ≤ ||δV ||Ce -νd(y1,supp(δV )) . ( 11
)
Here d(y 1 , supp(δV )) is the distance between y 1 and the support of the perturbation δV . The constants do not depend on the choice of the domain Λ.

This theorem will be proved in Section 4.

Step 2. We then show in section 5 a kind of Wegner estimate. We denote by : (H min ) |Λ the submatrices

1 Λ H min 1 Λ , restricted to 2 (Λ) where Λ is a finite cube in Z d .
Theorem 2.4 (Wegner estimate). Assuming (A1), (A2), (A3), there exists a > 0 such that if ||W || 1 < aG then there exists a constant C so that

P d(σ[(H min ) |Λ ], λ) < ≤ C|Λ| √ | Supp(ρ)| -1/2 + ||ρ|| ∞ | Supp(ρ)| 1/2 + ||ρ || ∞ | Supp(ρ)| 3/2 , ( 12
) for any λ ∈ C.
This result says that there is no arbitrary small interval where we can find for sure an eigenvalue of (H min ) |Λ .

Step 3. In the last step we perform the multiscale analysis. This is explained in Section 6.

3 Construction of the mean-field Hamiltonian: proof of Theorem 2.1

The aim of this section is to prove that our operator H min is well defined. We will show that finding the unique solution γ min in presence of a gap can be done using a fixed point lemma.

In this subsection we solve our system

H min = -∆ + V + A eff (γ) γ = 1 <µ (H min ) (13)
under the assumption that µ is inside a gap [a , b ] in the spectrum, where

µ = (a + b )/2. We introduce G = b -a .
Let C be a loop in the complex plane surrounding a part I of the spectrum of a operator H. We make the assumption that the loop does not cross the spectrum (which implies that there exist gaps above and below I). Then

1 I (H) = 1 2iπ C (H -z) -1 dz (14)
is the projector on the spectral subspace associated with I.

Let us define an application that gives this projector.

Definition 3.1 (Fixed point map). Let C be a fixed loop in the complex plane. For all γ orthogonal projector and

H eff (γ) = -∆ + V + A eff (γ) , we define
F (γ) = 1 2iπ C (H eff (γ) -z) -1 dz. (15)
This application enable us to reformulate our system [START_REF] Chulaevsky | Multi-particle Anderson Localisation: Induction on the Number of Particles[END_REF] as

F (γ) = γ, (16)
where the loop C crosses the real axis at µ . Recall that H min is bounded so that we can always enclose all of its spectrum below µ . Because ||A eff || is bounded by 2||W || 1 , we always have

d σ H eff (γ) , µ ≥ b -a 2 -2||W || 1 . (17)
So (ba )/2 > 2||W || 1 is enough to ensure that C never crosses the spectrum of H eff and F is always well defined. In order to solve [START_REF] Fröhlich | Absence of diffusion in the anderson tight binding model for large disorder or low energy[END_REF] we will show that if ||W || 1 is small enough then F is a contraction.

Proof of Theorem 2.1. Let γ 1 and γ 2 be two orthogonal projectors. Then we have

F (γ 1 ) -F (γ 2 ) = - 1 2iπ C (H eff (γ 2 ) -z) -1 (H eff (γ 1 ) -H eff (γ 2 ))(H eff (γ 1 ) -z) -1 dz = - 1 2iπ C (H eff (γ 2 ) -z) -1 (A eff (γ 1 ) -A eff (γ 2 ))(H eff (γ 1 ) -z) -1 dz.
The map F does not depend on the choice of the surrounding loop provided it encloses the appropriate part of the spectrum. Expending it continuously to infinity, we can replace it in this formula by C = µ + iR and write z = µ + is. We estimate

A eff (γ 1 ) -A eff (γ 2 ) = A eff (γ 1 -γ 2 ) with 2||W || 1 ||γ 1 -γ 2 || and (H eff (γ 2 ) -µ + is) -1 ≤ (G /2 -2||W || 1 ) 2 + s 2 -1/2 . Therefore ||F (γ 1 ) -F (γ 2 )|| ≤ ||W || 1 ||γ 1 -γ 2 || π R (G /2 -2||W || 1 ) 2 + s 2 -1 ds, (18)
so that

||F (γ 1 ) -F (γ 2 )|| ≤ ||W || 1 ||γ 1 -γ 2 || (G /2 -2||W || 1 ) . ( 19
)
Now, if ||W || L 1 is smaller than G /6, then F is contracting, so it has a unique fixed point.

This concludes the first section, we have shown that in the presence of a gap, ( 13) has always a unique solution.

Local influence: proof of Theorem 2.3

The aim of this section is to show that under hypothesis (A2) and (A3) the random potential in a domain Λ 2 will only have a very small influence on A eff in Λ 1 if Λ 2 is far enough from Λ 1 . This implies a weak form of independence between the submatrices (H min ) |Λ1 and (H min ) |Λ2 which is necessary for the multiscale analysis. The key tool is a Combes-Thomas estimate.

Combes-Thomas estimate

Because we want to use it for more general operators than just the Laplacian, we have written again the details of the proof. Definition 4.1 (Exponential off-diagonal decay operator). We will say that an operator K on L 2 (Λ) has exponential off-diagonal decay if there exist a rate M > 0, and a constant C so that (δ x , Kδ y ) ≤ C exp(-M |x -y|) for all x, y ∈ Λ.

In our case, we note that -∆ + V + A eff is an exponential decay operator if W (xy) decays exponentially.

Lemma 4.1 (Combes-Thomas estimate). Let K be an exponential off-diagonal decay operator and Σ be its spectrum. Let λ ∈ C so that d(λ, Σ) > 0. Then there exist ν > 0 and

C > 0 such that (δ x , (K -λ) -1 δ y ) ≤ Ce -ν|x-y| . (20)
Proof. Let f (z) := e -ν|x-z| we have

(δ x , (K -λ) -1 δ y ) = δ x , f (z) -1 f (z)(K -λ) -1 f (z) -1 f (z)δ y = f (x) -1 f (y) δ x , f (z)(K -λ) -1 f (z) -1 δ y = e -ν|x-y| δ x , (f (z)Kf (z) -1 -λ) -1 δ y . (21)
Then we have

(f (z)Kf (z) -1 -K)u (n) = m∈Λ (e -ν(|x-n|-|x-m|) -1)K(m, n)u(m), (22)
so

||(f (z)Kf (z) -1 -K)u|| ≤ C m∈Λ (e ν(|n-m|) -1)e -M (n-m) |u(m)| < ∞,
and therefore

||(f (z)Kf (z) -1 -H)|| ≤ C||(e ν|x| -1)e -M |x| || 1 .
Because of the dominated convergence theorem, this converges to 0 with ν going to 0. So there exist ν > 0 so that ||(f (z)Kf (z) -1 -H)|| < d(λ, Σ) and

d λ, Σ(f (z)Kf (z) -1 ) ≥ d(λ, Σ) -||f (z)Kf (z) -1 -H|| > 0.
So there exists C such that

||(f (z)Kf (z) -1 -λ) -1 || ≤ 1 d λ, Σ(f (z)Kf (z) -1 ) = C
and using [START_REF] Lahbabi | Mathematical study of quantum and classical models for random materials in the atomic scale[END_REF] we find (δ x , (Kλ) -1 δ y ) ≤ C e -ν|x-y| .

Local influence

We prove here Theorem 2.3. We use again the map defined in Section 3.

F V (γ) = 1 2iπ C (-∆ + V + A eff (γ) -z) -1 dz, (23)
where C is the loop enclosing the whole part of the spectrum below the middle of the gap µ.

We will denote by γ min (V ) the solution of the system given by Theorem 2.1 and recall that

F V (γ min (V )) = γ min (V ).
Proof of Theorem 2.3. Because we can write δV = K k=1 δV /K and apply our theorem K times we can suppose δV arbitrary small. We are looking for the new fixed point for F V +δV which is the limit of (F V +δV ) n (γ). We start from γ = γ min (V ) and remark that

(γ min (V + δV ) -γ) = lim n→∞ (F V +δV ) n (γ) -γ = ∞ n=0 (F V +δV ) n+1 (γ) -(F V +δV ) n (γ) . ( 24
)
Step 1. We evaluate the first term of the sum

F V +δV (γ) -γ = F V +δV (γ) -F V (γ), as follows F V +δV (γ) -F V (γ) = 1 2iπ C 1 H eff (γ) + δV -z - 1 H eff (γ) -z dz = 1 2iπ C 1 H eff (γ) -z δV k≤0 (H eff (γ) -z) -1 δV k 1 H eff (γ) -z dz = 1 2iπ C 1 H eff (γ) -z B(δV ) 1 H eff (γ) -z dz,
where

B(δV ) = δV k≤0 (H eff (γ) -z) -1 δV k . ( 25
)
This sum converges as soon as δV < G 6 . Note that if the support of δV has a bounded support Ω so has B(δV ). We will use the Combes-Thomas estimate (20) for y 1 , y 2 outside Ω. Remark that if we choose the loop C correctly, ν does not depend on z but only on the size on the gap. We find

|(F V +δV (γ))(y 1 , y 2 ) -(F V (γ))(y 1 , y 2 )| = C (y 1 , 1 H eff (γ) -z B(δV ) 1 H eff (γ) -z y 2 )dz ≤ 1 2iπ C C 2 x,x ∈Ω e -ν|y1-x| (x, B(δV )x )e -ν|x -y2| dz ≤ C 2 |C|e -νd(y1,Ω) x,x ∈Ω |(x, B(δV )x )e -ν|x -y2| | ≤ C ||δV ||e -νd(y1,Ω)
where C is just a constant. With W small enough, there exists τ < 1 so that

[A eff (F V +δV (γ)) -A eff (γ))](y 1 , y 2 ) ≤ τ ||δV ||e -νd(y1,Ω) . ( 26
)
Step 2. We evaluate the remainder of the sum. We repeat the previous argument with

δA eff = A eff (F n V +δV (γ)) -A eff ((F n-1 V +δV (γ)) (27)
instead of δV . There is only one little difference: B does not have a bounded support any more but still has an off-diagonal exponential decay (proved by iteration with constant ν > ν > 0). We just check this does not bring more difficulties:

|(F V +δV ) n (γ))(y 1 , y 2 ) -(F V +δV ) n-1 (γ)(y 1 , y 2 )| ≤ C 2 x,x ∈Ω e -ν|y1-x| (x, B(δA eff )x )e -ν|x -y2| dy ≤ C 2 ||δA eff || x,x ∈Z 2 e -ν|y1-x| e -ν d(x,Ω) e -ν d(x ,Ω) e -ν|x -y2| ≤ C 2 ||δA eff ||e -ν d(y1,Ω)
x,x ∈Z 2 e ν d(y1,Ω)-ν|y1-x|-ν d(x,Ω)) e -νd(x ,Ω) e -ν|x -y2|

≤ C 2 ||(F n-1 V +δV (γ)) -F n-2 V +δV (γ)||e -ν d(y1,Ω)
, where C 2 is just another constant. We can conclude by iteration that

|A eff (F n V +δV (γ))(y 1 , y 2 ) -A eff (F n-1 V (γ))(y 1 , y 2 )| ≤ ||δV ||τ n e -ν d(y1,Ω) , (28)
and so

| A eff (γ min (V + δV )) -A eff (γ min (V )) (y 1 , y 2 )| = lim n→∞ |A eff (F n V +δV (γ))(y 1 , y 2 ) -A eff (γ)(y 1 , y 2 )| ≤ ||δV || 1 1 -τ e -ν d(y1,supp(δV )) ,
as we wanted.

5 Wegner estimate: proof of Theorem 2.4

In this section, we prove the Wegner-type estimate in Theorem 2.4. The idea of the proof is the following. At first sight we do not have any idea of how A eff looks like. But because of the gap, if

γ = 1 ≤µ -∆ + V + A eff (γ) (29) then γ = 1 ≤(µ+α) -∆ + V + A eff (γ) = 1 ≤(µ) -∆ + V -α1 Z d + A eff (γ) (30)
for any α ∈ R with 2|α| smaller than the gap. So if we could add α to the random potential with α a smooth random variable, in this case every eigenvalue of H min would just be offset by α no matter what the non linear part is and we are done. In our case, we will make the change of variable

V ω (x) → (α = 1 |Λ| x∈Λ V ω (x), V ω (x) -α) for
x ∈ Λ and we expect that the conditional density of α is smooth enough and that the change induced to γ is small. Proof of Theorem 2.4. Let Λ = Λ L (n) be the cube in Z d of size L with its center in n and Λ 2L (n) the cube twice bigger. Because of (30) and ( 11)

|| d dα (γ min )(V + α1 Λ 2L (n) |Λ L (n) || = || d dα (γ min ) |Λ L (n) (V + α1 Z d -α1 Λ 2L (n) c || = || d dα (γ min ) |Λ L (n) (V -α1 Λ 2L (n) c || ≤ Ce -νd Λ L (n),Λ 2L (n) c ≤ Ce -νL .
We suppose that L is large enough so that 2||W || 1 Ce -νL ≤ 1/2 and we obtain that

α → ||1 Λ L (n) A eff (V + α1 Λ 2L (n) )1 Λ L (n) || (31)
is 1/2 Lipschitz. Under this hypothesis, for any λ i (α) eigenvalue of

-∆ + V + α1 Λ 2L (n) + A eff γ min V + α1 Λ 2L (n) ) |Λ L (n) , we have d dα λ i (α) ≥ 1 -|| d dα (1 Λ L (n) A eff (V + α1 Λ 2L (n) )1 Λ L (n) )|| ≥ 1 2 . ( 32
)
Let λ ∈ R and > 0. Let D 0 = {d 1 < d 2 < ... < d k } be so that ρ(s) is Lipschitz on ]d n , d n+1 [. Let f and δ be two positive functions that will be chosen later. We define the following events

O x := {ω : ∀y such that |y -V ω (x)| < δ( ) : ρ(y) > f ( ) , and d(V ω (x), D 0 ) > δ( )} (33)
for any x ∈ Λ 2L (n). We now estimate

P d σ[(H min ) |Λ L (n) ], λ < ≤ P ∪ x∈Λ 2L (n) O c x + P ∩ x∈Λ 2L (n) O x ∩ d σ[(H min ) |Λ L (n) ], λ < . ( 34
)
We will deal with each term separately. Starting with the left term, we erase the indices because the probability does not depend of the position and argue as follows:

P ∪ x∈Λ 2L (n) O c x ≤ |Λ 2L (n)|P(O c x ) = 2 d |Λ|P(O c ) ≤ 2 d |Λ| P(d(V ω , D 0 ) < δ( )) + P(V ω : ∃y : |y -V ω | < δ( ), ρ(y) < f ( )) ≤ 2 d |Λ| P(d(V ω , D 0 ) < δ( )) + P(V ω : ρ(V ω ) < f ( ) + δ( )||ρ || ∞ ) ≤ 2 d |Λ| ||ρ|| ∞ 2δ( )#|D 0 | + (δ( )||ρ || ∞ + f ( ))|supp(φ)| . ( 35
)
The right term in (34) can be estimated by introducing the mean and the resolvent, using

1 [λ-,λ+ ] (α) ≤ 2 2 ((λ -α) 2 + 2 ) = 2 ( 1 λ -α + i ). (36)
We simplify a bit the notation using

∩ x O x instead of ∩ x∈Λ 2L (n) O x . We get P ∩ x O x ∩ d σ[(H min ) |Λ L (n) ], λ < = E 1 d(σ(H Λ min ),λ)< 1 ∩xOx ≤ 2 E Tr[ (H min ) |Λ L (n) -λ + i -1 ] 1 ∩xOx .
We now make a change of variable for V ω (x) ∈ Λ 2L (n) :

V ω (x) → α = 1 2 d |Λ| x∈Λ 2L (n) V ω (x), V ω (x) -α . (37)
We write Ṽω (x) = V ω (x)α and ξ Ṽ (α) for the conditional random density of the mean knowing Ṽ . We first integrate over α, then over Ṽ (we denote the expectation by E Ṽ ):

P ∩ x O x ∩ d σ[(H min ) |Λ L (n) ], λ < ≤ E Ṽ 2 T r[ (H min ) |Λ L -λ + i -1 ] 1 ∩xOx ξ Ṽi (α)dα ≤ 2| ||E Ṽ λi∈σ (Hmin) |Λ L ( 1 λ i (α) -λ + i )1 ∩xOx ξ Ṽi (α)dα .
We estimate the integral with a change of variable α = (λ i (α)λ), dα = ( d dα λ i )dα. Recall that

(α ) 2 + 2 ξ Ṽ (α)1 ∩xOx d dα λ i dα ≤ π sup |ξ Ṽ (α)1 ∩xOx d dα λ i . ( 38
)
So we have

P ∩ x O x ∩ d σ[(H min ) |Λ L (n) ], λ < ≤ 2π| ||Λ L |E Ṽ sup ξ Ṽ (α)1 ∩xOx d dα λ i .
Finally, because of (32), d dα λ i > 1/2 and we get

P ∩ x O x ∩ d σ[(H min ) |Λ L (n) ], λ < ≤ 4π| ||Λ L | sup[ξ Ṽi 1 ∩xOx ]. (39)
From now on, it is enough to have an estimate on ξ Ṽ 1 ∩xOx . A computation gives

ξ Ṽi (α)dα = P x V ω (x) ∈ [α, α + dα]| Ṽ = i∈Λ ρ( Ṽi + α) i∈Λ ρ( Ṽi + α )dα dα. ( 40
)
Let α 0 ∈ R. If we do not have Ṽω (x) + α 0 ∈ O x for all x, then ξ Ṽ (α 0 )1 ∩xOx = 0 and we have nothing else to do. So we can assume that ∀α, |α -

α 0 | < δ( ) ⇒ ρ( Ṽ (x) + α) > f ( ) for all x ∈ Λ 2L (n) and all |α -α 0 | < δ( ). So we have d dα xΛ 2L ρ( Ṽ + α) x∈Λ 2L ρ(V + α) = x∈Λ 2L ρ ( Ṽω (x) + α) ρ( Ṽω (x) + α) ≤ 2 d |Λ L | ||ρ || ∞ f ( ) . ( 41
)
From this differential equation we get

x∈Λ 2L ρ( Ṽω (x) + α) ≥ exp -(|α -α 0 |)2 d |Λ L | ||ρ || ∞ f ( ) x∈Λ 2L ρ( Ṽω (x) + α 0 )
and, after integrating, α0+δ( )

α0-δ( ) ρ( Ṽω (x) + α)dα ≥ 1 -exp -|δ( )|2 d |Λ L | ||ρ || ∞ f ( ) f ( ) 2 d |Λ L |||ρ || ∞ ρ( Ṽω (x) + α 0 ). Therefore ρ( Ṽω (x) + α 0 ) ρ( Ṽω (x) + α)dα ≤ 2 d |Λ L |||ρ || ∞ f ( )(1 -exp(-(|δ( )|)2 d |Λ L | ||ρ ||∞ f ( ) )
) and hence we have

ρ( Ṽω (x) + α 0 ) ρ( Ṽω (x) + α)dα ≤ 2 max( 2 d |Λ L |||ρ || ∞ f ( ) , 1 δ( ) ). (42)
We finally obtain

ξ Ṽi 1 ∩xOx ≤ 2 max( 2 d |Λ L |||ρ || ∞ f ( ) , 1 δ( ) ), (43)
for all V and all α 0 . To conclude, putting (34), ( 35), ( 39) and (43) together, we have

P d σ[(H min ) |Λ L (n) ], λ < ≤ 2 d |Λ| ||ρ|| ∞ δ( )#|D 0 | + δ( )||ρ || ∞ + f ( ) * |supp(ρ)| + 8π . max( |∆|||ρ || ∞ f ( ) , 1 δ( ) ,
from which we can conclude [START_REF] Carmona | Spectral theory of random Schrödinger operators[END_REF]

choosing f ( ) = √ /|supp(ρ)| -3/2 and δ( ) = √ * |supp(ρ)| 1/2 . Hence P d σ[(H min ) |Λ L (n) ], λ < ≤ C|Λ| |supp(ρ)| -1/2 + ||ρ|| ∞ |supp(ρ)| 1/2 + ||ρ || ∞ |supp(ρ)| 3/2 ,
where the constant C only depends on the cardinal of D 0 .

In particular, with the change V → V , we deduce that

P d σ[(H min ) |Λ L (n) ], λ < → 0 (44) when → ∞. Definition 6.1 (L-resonance). A number λ ∈ R is called L-resonant for the box Λ L (n) if there exists A c with ||A c || ≤ 2D exp(-νL) and d λ, σ[ Ĥ(n, L) + A c ] ≤ exp(- √ L), ( 49
)
where D is the constant defined in (48)

Remark that our definition of non-resonance is equivalent to

d λ, σ[ Ĥ(n, L)] > exp(- √ L) -2D exp(-νL). ( 50
)
We have added the operator A c in the above definition to handle the difference between (H min ) |Λ L (n) and Ĥ(n, L). This corresponds to Definition 9.1 in [START_REF] Disertori | Random Schrodinger Operators, Panoramas et synthèses[END_REF].

Definition 6.2 ((L, ζ,λ)-good box). The box Λ L (n) is called an (L,ζ,λ)-good box if 1. it is not L-resonant; 2. for any x ∈ Λ √ L (n), y / ∈ Λ L (n) and A c with ||A c || ≤ 2D exp(-νL), v | Ĥ(n, L) + A c -λ) -1 (x, v)| |Γ Ĥ(n,L)+Ac,Λ L (n) (v, y)| ≤ exp(-ζ|y -x|). (51)

From a scale to another

Let L 0 be not too small and set

L k = L α k 0 with 1 < α < 2.
In this subsection, we prove the following proposition. Proposition 6.2. If the following conditions are satisfied 1. for any 4 boxes of side length L k in Λ L k+1 (n), separated from each other by a distance of at least 2L k , there is at least one which is

(L k , ζ, λ)-good with ζ > 20/ √ L k , 2. no box in Λ L k+1 (n) of side length 4L k , 12L k ,20L k is L k -resonant; 3. the domain Λ L k+1 (n) is not L k+1 -resonant, then the cube Λ L k+1 (n) is (L k+1 , ζ k+1 , λ)-good with a decay satisfying ζ k+1 > 20/ √ L.
This proposition corresponds to Theorem 10.20 in [START_REF] Disertori | Random Schrodinger Operators, Panoramas et synthèses[END_REF].

Proof of Proposition 6.2. Let A c be so that

||A c || ≤ 2De -νL k+1 . Let ( Ĥ(n, L k+1 )) |Λ L k (m) be the restriction to the box Λ L k (m) of Ĥ(n, L k+1 ), with Λ k (m) ⊂ Λ k+1 (n). Because of Theorem 2.3, we have ||( Ĥ(n, L k+1 ))| Λ L k (m) -Ĥ(m, L k )|| ≤ De -νL k and so || Ĥ(n, L k+1 ) + A c ) |Λ L k (m) -Ĥ(m, L)|| ≤ 2De -νL k for L k big enough. Let K = Ĥ(n, L k+1 ) + A c
and for simplicity we will just write

Γ Λ instead of Γ K,Λ . Because of what we have just said, if Λ L k (m) is (L k , ζ, E) good then v (K Λ L k (m) -λ) -1 (x, v)|Γ(v, y)| ≤ exp(-ζ|y -x|) (52)
and

||(K Λ jL k (m) -λ) -1 || ≤ exp( jL k ) (53) if Λ jL k (m) is not jL k -resonant for j = 4, 12, 20.
The idea is to use equation ( 45) as many times as we want. For any v appearing in the equation (45), we can define another box Λ(v) with y / ∈ Λ(v) and repeat the formula with v instead of x. Proceeding this way again and again, we get after iteration

(K -λ) -1 (x, y) = (ui,vi)i=1..n (K Λ1 -λ) -1 (x, u 1 )Γ Λ1 (u 1 , v 1 )(K Λ2 -λ) -1 (v 1 , u 2 )Γ Λ2 (u 2 , v 2 )...(K -λ) -1 (v n , y). (54)
We will write the indices of the sum as a tree T of chains

X = u i , v i , Λ i i≤n with v i ∈ Λ i u i+1 ∈ Λ i , v i+1 /
∈ Λ i and y / ∈ Λ i . We first sum over the u i 's so as to reduce our chains to X = v i , Λ i i≤n and we introduce an upper bound R X such that

R X ≥ (ui)i=1..n |(K Λ1 -λ) -1 (x, u 1 )Γ Λ1 (u 1 , v 1 )(K Λ2 -λ) -1 (v 1 , u 2 )Γ Λ2 (u 2 , v 2 ) • • • Γ(u n , v n )|. (55)
Then, Equation (54) gives

|(K -λ) -1 (x, y)| ≤ ||(K -λ) -1 || X leaves of T R X . (56)
This formula is very general and is valid for any expansion. Different choices for the construction of the tree exit in the literature and we will follow that of [START_REF] Disertori | Random Schrodinger Operators, Panoramas et synthèses[END_REF]. The goal is to get at least one good box at each step. The choice of the Λ i and the construction of the tree T of X and R X are made according to the following algorithm. We start from x so we define v 0 = x and R = 1. The choice of Λ i+1 will depend on v i .

• If we get close to the boundary or to y, d(v i , ∂Λ L k+1 (n)) < L k or d(v i , y) < L k then we stop. The construction of this chain is over and we carry on with the other branches of the tree.

• Otherwise

-if Λ L k (v i ) is an (L k , ζ, E)-good box then R X (K -λ) -1 (v i , y) ≤ R X ui+1,vi+1 (K Λi -λ) -1 (v i , u i+1 )Γ Λi (u i+1 , v i+1 )(K -λ) -1 (v i+1 , y) ≤ vi+1 / ∈Λ L (vi) R X exp(-ζ|v i+1 -v i |)(K -λ) -1 (v i+1 , y) so for each v i+1 outside Λ L (v i ), we set R X +vi+1 = R X exp(-ζ|v i+1 -v i |) (57)
and carry on the algorithm with the new chain X + v i+1 ;

else if Λ L k (u i ) is not a good box, choose j = 4 or 12 or L k such that for every v in

Λ 2jL k \ Λ jL k , Λ L k (v)
Λ L k (m) ⊂ Λ L k+1 (m)) whom center is not included in M 1 are (L k , ζ, λ)-good.
We can assume than the good box decay is smaller than the off diagonal decay parameter ν, ζ < ν. We then have

R X (K -λ) -1 (v i , y) ≤ R X ui+1,vi+1 (K Λi+1 -λ) -1 (v i , u i+1 )Γ Λi+1 (u i+1 , v i+1 )(K -λ) -1 (v i+1 , y) ≤ R X ui+1,vi+1∈Λ 2jL k (vi) (K Λi+1 -λ) -1 (v i , u i+1 )Γ Λi+1 (u i+1 , v i+1 )(K -λ) -1 (v i+1 , y) + ui+1,vi+1 / ∈Λ 2jL k (ui) (K Λi -λ) -1 (v i , u i+1 )Γ Λi+1 (u i+1 , v i+1 )(K -λ) -1 (v i+1 , y) ≤ R X ui+1,vi+1∈Λ 2jL k (vi) u i+1 ,v i+1 / ∈Λ L k (vi+1) (K Λi+1 -λ) -1 (v i , u i+1 )Γ(u i+1 , v i+1 )× × (K Λ L k (vi+1) -λ) -1 (v i+1 , u i+1 )Γ Λ L k (vi+1 (u i+1 , v i+1 )(K -λ) -1 (v i+1 , y) + vi+1 / ∈Λ 2jL k (ui) (jL k ) d Ce √ jL k exp -ν(|v i+1 -v i | -jL k ) (K -λ) -1 (v i+1 , y) ≤ R X vi+1 (2jL k ) d e √ jL k exp -ζ max(|v i+1 -v i | -2jL k , L k ) (K -λ) -1 (v i+1 , y) + vi+1 / ∈Λ 2jL k (ui) (jL k ) d Ce √ jL k exp -ν(|v i+1 -v i | -jL k ) (K -λ) -1 (v i+1 , y).
Therefore we can set

R X +vi+1 = R X 2C(2jL k ) d exp( jL k ) exp -ζ max(L k , |v i+1 -v i | -2jL k ) (58)
and as previously, we carry on with the new chain X + v i+1 .

We have finished the description of the algorithm. We define the "good path length" P g (x, v) as the minimum length between x and v when any cube M = Λ 2jL k (v) containing bad boxes defined in the procedure can be crossed for free. We easily check that (57) and (58) imply that

R X ≤ exp(-ζ max(length(X )L k , P g (x, v i )). ( 59
)
for every chain X . From this, our algorithm gives us the following estimate:

Proposition 6.3. X R X ≤ 1 1 -(L k+1 ) d exp(-ζL k ) (L k+1 ) d L k+1 L k exp(-ζ(P g (x, y)). (60)
x y Pg(x,y)

x bad boxes bad boxes bad boxes Proof. We have

Λ L k+1 X good box
(X ) R X = ∞ N =1 X ,N =length(X ) R X ≤ ∞ N =1 X ,N =length(X ) exp(-ζmax(N L k , P g (x, y)) ≤ ∞ N =1 X ,N =length(X ) exp(-ζ(P g (x, y) + max(0, N - L k+1 L k )L k ) ≤ (L k+1 ) d L k+1 L k exp(-ζ(P g (x, y) ∞ N =0 L N d k+1 exp(-ζN L k ) ≤ 1 1 -(L k+1 ) d exp(-ζL k ) (L k+1 ) d L k+1 L k exp(-ζ(P g (x, y)),
because at each step i there are only L d k+1 possible v i 's. The fact that there are only 3 bad boxes implies that the good path length is close to the usual distance P g (x, y) ≥ |x -y| -20L k . We can now conclude. Let y be outside

Λ L k+1 . Then v |( Ĥ(n, L + 1) + A c -λ) -1 (x, v)||Γ(v, y)| ≤ v ||(( Ĥ(n, L + 1) + A c -λ) -1 ||. 1 1 -1 -(L k+1 ) d exp(-ζL k ) (L k+1 ) L k+1 /L k × × exp(-ζ(|x -v| -20))C exp(-ν|v -y|) ≤ C (L k+1 ) d exp(log(L k )L α-1 k ) exp( L k+1 ) exp(-ζ(|x -y| -20)
In order to conclude, we just remark that

√ L k + log(L k )L α-1 k + d log(L k ) = o(|x -y|). So we can choose ζ k+1 = ζ k - √ L k + log(L k )L α-1 + d log(L k ) L k , (61)
which finishes the argument.

The multiscale

(n 0 ), Λ L0 (m 0 ) separated by at least 2L 0 , P ∃λ ∈ I : Λ L0 (n) and Λ L0 (n) are not (L 0 , ζ, λ)-good ≤ 1 L 2p 0 , (62)
then for any k, and any cubes Λ L k (n k ), Λ L k (m k ) separated by at least 2L k , we have

P ∃λ ∈ I : Λ L k (n k ) and Λ L k (m k ) are not (L k , ζ, λ) good ≤ 1 L 2p k , (63)
where

L k+1 = L α k
This theorem is similar to Theorem 10.22 of [START_REF] Disertori | Random Schrodinger Operators, Panoramas et synthèses[END_REF].

Proof. The demonstration is done by iteration using Proposition 6.2. Suppose there exists λ ∈ I such that Λ L k+1 (n) and Λ L k+1 (m) are not (L k+1 , ζ, λ)-good. Then for each box one of the hypothesis of Proposition 6.2 fails. So either one of the boxes admits 4 separated bad sub-boxes, or there exists λ such that for the two boxes, one of their sub-boxes shows a resonance at λ. The probability that the hypothesis over the existence of 4 bad boxes is not true can be estimated by iteration. Indeed, 4 cubes means 2 pairs. Because of independence, the 4-cubes probability will be the 2-cubes probability squared and because there are only L d cubes, hence L 4d 4 cubes combinations we can estimate this probability by

≤ CL 4d 1 (L k ) 2p 2 ≤ C 1 (L k+1 ) 4 p α -4d ≤ 1 4 1 L 2p k+1 .
The last inequality is true for L k big enough because 2p < 4 p α -4d so α < 2p p+2d . The probability of the non resonance hypothesises is controlled by Wegner estimate [START_REF] Carmona | Spectral theory of random Schrödinger operators[END_REF] as it is done Theorem 10.22 of [START_REF] Disertori | Random Schrodinger Operators, Panoramas et synthèses[END_REF]

with O(L d √ e - √ L k ) = o(L -2p ).
From this and ( 12) we can deduce the following corollaries. The proof can be found again in [START_REF] Disertori | Random Schrodinger Operators, Panoramas et synthèses[END_REF]Part 9 and Part 11] . Corollary 6.1. In presence of strong disorder, meaning

|supp(ρ)| -1/2 + ||ρ|| ∞ |supp(ρ)| 1/2 + ||ρ || ∞ |supp(ρ)| 3/2
small enough, then H min has pure point spectrum and its eigenvectors are localised in space.

Furthermore the Lifshitz tail is not modified too much for ||W || very small because

P d(σ[(H min ) |Λ ], λ) < ≤ P d(σ[H Λ ], λ) < + 2||W || 1 , (64)
so we also get the following Corollary 6.2. There is such that if ||W || 1 < there are small intervals at the edges of the bands of the spectrum of H min , where the spectrum is pure point with exponentially decaying eigenvectors.

The two results conclude the proof of Theorem 2.2.

Numerical simulations

In this section, we present some simple numerical simulations in order to illustrate our theorems. Due to the computational cost we restrict ourselves to the one dimensional case, which however is known to present stronger localisation effects than in higher dimensions. It would be interesting to generalise our simulation to dimension 2 and 3.

We take Λ = [0, L] with L ranging from two hundred to a few thousands lattice sites, 200 ≤ L ≤ 2000. The discrete one-dimensional Laplacian is defined in [START_REF] Anderson | Absence of diffusion in certain random lattices[END_REF]. The deterministic potential V 0 is 2-periodic:

V 0 (n) = ξ if n is even -ξ if n is odd, (65)
where ξ > 0 is a parameter. The probability of the random potential V ω is the uniform law over the interval [0, ζ] where ζ is another parameter. For W (xy), we use a simple next-to-nearest neighbour interaction of the form

W (x -y) =          q if x -y = 0 q/2 if |x -y| = 1 q/4 if |x -y| = 2 or 3 0 otherwise, ( 66
)
where q is another parameter. Our model depends therefore on three parameters ξ, ζ and q. When ξ > 2 + ζ the spectrum of the linear Hamiltonian -∆ + V is composed of two distinct intervals. We then choose q such as to keep a gap in the spectrum and ensure that the map F is contracting (Theorem 2.1). In this model the particles fill half of the energy states, that is, there are N = L/2 particles.

Figure 1 :

 1 Figure 1: Schematic representation a typical chain X used in the proof of Proposition 6.2

Figure 2 :

 2 Figure 2: Values of ||F n+1 (γ 0 ) -F n (γ 0 )|| with γ 0 = 1 ≤µ (-∆ + V ) and ξ = 1, ζ = 1 q = 2, L = 500.

Figure 3 :

 3 Figure 3: The relative density γ min (V + δ 250 )(x, x)γ min (V )(x, x) with ξ = 1, ζ = 1 q = 2, L = 500.

Figure 4 :

 4 Figure 4: Average number of eigenvalues of H min in an interval I according to the size of I. We plot E(Tr(1 [2;2+ ] (H min ))) in terms of . The values of the parameters are ξ = 1, ζ = 1 and q = 2.

Figure 5 :

 5 Figure 5: Left: Standard deviation of all the eigenvectors of the mean field operator H min in terms of theirs eigenvalue. Here the size of the domain is L = 1000, hence a value of 10 shows localisation. Right: an eigenvector chosen at random. The values of the parameters are ξ = 1, ζ = 1 and q = 2.

Figure 6 :

 6 Figure 6: Standard deviation of the eigenvectors in terms of their eigenvalue, for ξ = 2, ζ = 3 and q = 2.

Figure 7 :

 7 Figure 7: Standard deviation of the eigenvectors in terms of their eigenvalue, for ξ = 2, ζ = 3 and q = 7.

Figure 8 :

 8 Figure 8: Standard deviation of the nonlinear eigenvectors in terms of their eigenvalue, with the periodic potential dropped out (ξ = 0). Here there are only L/4 particles. The Fermi level is µ = 3.5 and there is no gap. A small delocalisation seems to appear at the Fermi level. The other parameters are ζ = 4 and q = 4.

Figure 9 :

 9 Figure 9: Mean of the standard deviation for the eigenvectors of the linear problem (q = 0) according to the strength ξ = 0, .., 4 of the periodic potential, with a fixed random intensity ζ = 2.

  is a good box and Λ jL k is not resonant. It is always possible to do this because of the following remark: Either 3 boxes are far away from each other then there are 3 boxes M 1 , M 2 , M 3 of size 4L k separated by at least 2L k so that every cubeΛ L k (m) ⊂ Λ L k+1 (m) whom center is not included in ∪ i=1,2,3 M i are (L k , ζ, λ)-good.Or two of them are close and the other is far away then there are two boxes M 1 of size 12L k and M 2 of size 4L k separated by at least 2L k so that every cube in Λ L k (m) ⊂ Λ L k+1 (m) whom center is not included in M i are (L k , ζ, λ)-good. Or the three of them are together then there exist one box M 1 of size 20L k so that every cube

  Theorem 6.1. Assume that there exists a gap big enough in the spectrum, that the law of potential has a density Lipschitz by part and that ||W || L 1 is small enough compared to the gap. Then, let L 0 be large enough, λ ∈ R ζ > 1/ √ L 0 , p > 2d and 1 < α < 2p/(p + 2d). If for any cubes Λ L0

Acknowledgement I would like to thank Mathieu Lewin for his help and all the time he gave me during this work.

Multiscale analysis

We will now start the proof of the multiscale analysis. There will be very little differences with the proof we can found in [14, Part 10] and we will follow the method exposed there step by step. But because it is a more general case, we have written the proof again.

The setting

For any operator K with off-diagonal exponential decay and Λ ⊂ Z d , we define a border operator Γ by

The following proposition is a form of the Schur complement formula. Proposition 6.1. Let K with off-diagonal exponential decay, Λ a box of size L, and λ ∈ C \ R.

for any x ∈ Λ and any y / ∈ Λ, where

is the restriction of K to Λ.

Proof. We can divide K into the following three parts

where Λ c is the complement of Λ. We here use the resolvent formula

Just remark now that (K Λλ) -1 (x, y) = 0 and (K Λ cλ) -1 (x, y) = 0 if x ∈ Λ and y / ∈ Λ.

We now apply the multiscale method. Let Λ L (n) be the box of side length 2L + 1 centered at n ∈ Z d . We replace the random potential V ω by an arbitrary constant outside the box Λ 2L (n) in order to make the mean field Hamiltonian inside

Recall that 0 ∈ supp(P). From this potential we can obtain with Theorem 2.1 the minimiser

) and the mean-field Hamiltonian

). We denote its restriction to

.

We introduce this Hamiltonian because of two properties. First it is independent of what is happening outside Λ 2L (n). Second, it is a good approximation of (H min ) |Λ L (n) . Indeed, from Theorem 2.3 we have

where D does not depend on n and L.

7.1 Illustration of Theorems 2.1, 2.3, 2.4 and 2.2

In order to construct the solution γ min and the associated mean-field Hamiltonian H min , we use the fixed point algorithm employed in the proof of Theorem 2.1. In Figure 2 we display and confirm the exponential decrease of ||F n+1 (γ 0 ) -F n (γ 0 )|| with initial condition γ 0 = 1 ≤µ (-∆ + V ).

Next we have tested that adding a dirac at the site 250 to the potential V induces a perturbation in the non linear minimiser, which decays exponentially fast (Theorem 2.3). In Figure 3 we plot the relative density γ min (V + δ 250 )(x, x)γ min (V )(x, x).

In Figure 4 we have tested the Wegner-type estimate of Theorem 2.4, where we have obtained a bound in terms of √ instead of the usual that can be found in the literature [START_REF] Carmona | Spectral theory of random Schrödinger operators[END_REF]Prop.VIII.4.11]. In dimension 1, we observe that the usual bound should hold in the nonlinear case but we are unable to prove it so far. We conclude the illustration of our results with the case of Theorem 2.2. Since we deal with a one-dimensional system all the eigenvectors are localised, even in a regime of parameters which is not covered by the second part of Theorem 2.2. This is shown in Figure 5. It is an interesting open problem to prove a stronger localisation result in the one-dimensional Hartree-Fock model.

Closing the gap: insulators and metals

In Figure 6 and7 we have increased the intensity q of the interaction up to the point where the gap closes. For a large interaction the fixed point algorithm used to construct the solution in Theorem 2.1 does not work. Instead we have used the optimal damping algorithm of [START_REF] Cancès | On the convergence of SCF algorithms for the Hartree-Fock equations[END_REF] which works perfectly. In general, we observe that the eigenvectors are less localised except at the edges of the spectrum. There is no sign of a phase transition.

In Figure 8, we have erased the gap by choosing ξ = 0. A small delocalisation phenomenon seems to appear at the Fermi energy µ.

We hope to be able to understand these phenomena rigorously in the future.

7.3

The influence of the periodic potential V 0

The periodic potential seems to have an influence on the localisation phenomena even in the linear case. Our Figure 9 clearly illustrates that the periodic potential favours localisation. To our knowledge there are very few mathematical results about how a small random potential influences a highly varying periodic system. It is however an important question if we think of the absence of conductivity in ionic crystals.


J Gril

L Cocchi

P Dionisi-Vici

G Goli

B Marcon

P Mazzanti

M Togni

L Uzielli
email: luca.uzielli@unifi.it

The Mona Lisa Project: an update on the progress of measurement and monitoring activities

Introduction

Since 2004 an international research group of Wood Technologists has been given by the Louvre Museum the task of analysing the mechanical situation of the wooden panel on which Leonardo da Vinci painted his "Mona Lisa", possibly between 1503 and 15014. The general purpose of such study was to evaluate influences which could possibly derive from environmental fluctuations in the showcase where the painting is exhibited, and any measure which could improve its conservation conditions. In order to acquire data about the mechanical behaviour of the panel, and to feed and calibrate appropriate simulation models, the team has not only set up a continuous monitoring by means of automatic equipment, but has also performed manual measurements on the occasion of the annual openings of the showcase where the masterpiece is conserved and exhibited.

The "Mona Lisa" (Fig. 1) is painted on a panel of Poplar wood (Populus alba L.) ~79 x 53 cm, ~ 13 mm thick, is inserted in an Oak frame (châssis-cadre), and is slightly forced against it by means of four crossbars, which hold it flatter than it would be if unconstrained. Panel and châssis-cadre are inserted in a wooden gilded frame, the only visible by the public.

In 2006 a book was published [START_REF]Au coeur de la Joconde[END_REF] offering a wealth of scientific studies and researches concerning the "Mona Lisa", including the work performed by the Wood Technology Group. Also in 2006 a detailed report about finite-elements simulation model was presented [START_REF] Gril | saved by the Griffith theory: assessing the crack propagation risk in the wooden support of a panel painting[END_REF].

Development of measurement and monitoring techniques

The panel's shape has been determined through manual measurement of its deflection in several points, by means of a mechanical comparator (0,01 mm) and a reference bar. Although it was improved by adding some facilitating equipment, this technique is slow, and allows surveying only a limited number of points; some optical techniques (including Shadow Moiré Technique, Fringe Pattern Profilometry, and Monitoring Optical Markers) have been developed and tested, and their ability to replace manual measurements and provide numerous geometrical data has been demonstrated [START_REF] Brémand | Valle Mechanical structure analysis by Digital Image Correlation and Fringe Pattern Profilometry[END_REF].

Until 2013 the forces exerted between panel and crossbars have been measured manually; however the results were somewhat uncertain, due to the numerous disturbing factors. Accurate and reliable data were recently obtained thanks to the improvement of monitoring equipment, mentioned hereafter.

The deflection variations along time were initially measured by means of only one deformation transducer, located on a reference aluminum profiled crossbar fixed on the châssis-cadre. A special pivoting lever was designed to gain space (only 15 mm thickness allowed). Two further transducers have been added later, providing records of both (a) transversal deflection at the panel's center with reference to the lateral edges, and (b) longitudinal deflection with reference to the châssis-cadre.

Euromech conference "Theoretical, Numerical, and Experimental Analyses in Wood Mechanics" Dresden, Germany, 27-29.05.2015 Texte

Forces exerted by the crossbars on the rear face of the panel were initially monitored on two locations, at the two ends of the upper crossbar, by means of two sub-miniature load cells incorporated into the crossbar thickness, contacting the panel back face through a Teflon articulated head. Since 2013 four improved sub-miniature load cells replaced the previous ones at the four corners of the panel, providing a more complete, accurate and reliable monitoring. Fig. 3 shows the completed most recent assembly: the panel in the châssis-cadre in the frame, the new 2013 crossbars carrying four load cells, and the aluminium crossbar carrying data-loggers, transducers and instrumentation. Such system also provided opportunities for additional quite significant measurements, including (a) variation of forces on panel during its unhooking or hooking, handling, extracting or inserting in châssis-cadre and frame, (b) transition between exposure to different climatic conditions, at the end of the annual opening day, and (c) measurement of elastic parameters of the panel, by tightening or loosening the crossbar's action. Force monitoring could also be used for warning against excessive load on the Panel.

Contact forces between panel and châssis-cadre have been localized and estimated on the basis of local contact pressures, recorded by tests with a pressure-sensitive foil [START_REF] Goli | Uzielli Locating contact areas and estimating contact forces between the "Mona Lisa" wooden panel and its frame[END_REF].

The monitoring data are automatically read every 20 minutes, stored in a data-logger, and downloaded on the occasion of the annual openings of the showcase.

The results of the above mentioned measurements are being processed and fed into the simulation software which in turn is being developed to allow for an improved simulationand validation -of the panel's behavior under environmental fluctuations.

Fig. 1 -

 1 Fig. 1 -The painted face and the back face.

Fig. 2 -

 2 Fig. 2 -Panel, crossbars, châssis-cadre, frame.

Fig. 3 -

 3 Fig. 3 -Completed assembly since 2013 (see text).


Iris Brémaud

Joseph Gril

EFFECT OF TRANSITIONAL MOISTURE CHANGE ON THE VIBRATIONAL PROPERTIES OF VIOLIN-MAKING WOOD

Introduction

Wood is a very hygroscopic material. When submitted to different relative humidity (RH), which can typically be the case when an instrument is played in different places, the internal moisture content of wood will be changed. This is well known to result in dimensional swelling or shrinkage, and will also modify the wood mechanical or acoustical properties. The equilibrium moisture content (EMC) dependence of vibrational properties (specific modulus of elasticity E'/γ which is related to resonance frequencies, and damping coefficient tanδ) has been well-studied for spruce wood along the grain [START_REF] Obataya | The effects of adsorbed water on dynamic mechanical properties of wood[END_REF]] and for various wood species in the three main directions of wood [START_REF] Suzuki | Relationship between specific gravity and decrement of dynamic Young's modulus with water (in Japanese)[END_REF]]. However, equilibrium state is only reached after a relatively long time (weeks), whereas the humidity changes encountered when an instrument is played in different places involve short time scales (hours) and are typically out-of-equilibrium. Previous research have shown the existence of transitional effects, notably on damping [START_REF] Sasaki | Effect of moisture on the acoustical properties of wood (in Japanese)[END_REF][START_REF] Hunt | Evidence of a physical ageing phenomenon in wood[END_REF]. However these studies considered a single direction of anisotropy each, and wide steps of changes in relative humidity. The present work aims at observing transitional moisture effects on vibrational properties of violin-making woods (spruce and maple) over different steps and different histories of RH changes, in longitudinal and radial directions of wood.

Materials and methods

Material of Norway spruce (Picea abies [L.] Karst.) and of sycamore maple (Acer pseudoplatanus L.) was selected in order to reduce natural variability within the tested sampling. Longitudinal (12×2×150mm 3 , R×T×L) and radial (150×2×12 mm 3 , R×T×L) specimens were prepared with a fine planed surface. They were separated in 3 groups corresponding to 3 protocols of humidity changes: -Adsorption: samples were first oven-dried (48h at 60°C followed by 2h at 103°C), then stored in a desiccator for 2 weeks, then at 20°C and 35%RH, 50%RH, 65%RH, 85%RH, for at least 3 weeks each time. 8 radial and 8 longitudinal samples for each species served to measure properties at equilibrium, while 2 radial and 2 longitudinal for each species served to monitor transitional changes.

-Desorption: samples were first immersed in water for 3 weeks, then stored (3 weeks or more until stable) at 85% RH, 65% RH, 50%RH and 35%RH. Same protocol/number of specimens as above.

-Various humidity paths: 18 radial and 18 longitudinal specimens of each species were used to explore transitional changes with all possible steps between the above RH conditions. Vibrational properties were measured by noncontact forced vibration of free-free bars using a device and software developed at LMGC [START_REF] Brémaud | Diversity of woods used or usable in musical instruments making[END_REF][START_REF] Brémaud | Characterisation and categorisation of the diversity in viscoelastic vibrational properties between 98 wood types[END_REF]. Specific dynamic modulus of elasticity E'/γ and damping coefficient tanδ were measured at first resonance frequency in bending. Repeatability tests indicated an experimental error of ≤2%, and a point of measure could be taken every ≤1 minute.

Results and discussion

The equilibrium moisture content at different RH, as well as the hysteresis/difference between adsorption and desorption, was comparable between spruce and maple (Table 1). Regarding moisture dependence of vibrational properties at equilibrium (Figure 1), the general trends were comparable to previous results [START_REF] Obataya | The effects of adsorbed water on dynamic mechanical properties of wood[END_REF]]. The differences between adsorption and desorption were small (once taken into account the hysteresis in EMC), with the exception of longitudinal damping in maple (maybe due to some extraction during water saturation). Damping in maple was slightly less affected by high moisture than for spruce. Mostly, properties in radial direction were clearly more modified by elevated moisture than in longitudinal direction, resulting in a strong increase in the degree of anisotropy. When changing conditions of humidity, frequency and dimensions evolved together with internal moisture content of wood. On the contrary, a transitional increase in damping was always observed, whether the humidity decreased or increased (Figure 2). The peak value of this increase in damping occurred from 30minutes to a few hours after changing conditions (depending on RH steps and direction of specimens). Then damping slowly decreased when evolving towards equilibrium state, which took 2-4 weeks, that is, after moisture content itself was stabilized. Moreover, this transitional increase in damping had a bigger amplitude when smaller steps in ambient RH were applied (Figure 3). For relatively small RH steps (e.g. 35-50% RH or 50-65% RH), transitional changes in damping clearly exceeded the differences between initial and final equilibrium values.

EMC(%) at RH: 35%RH 50%RH 65%RH 85%RH

Conclusion

Differences in equilibrium moisture conditions reduce specific modulus of elasticity and increase damping, as is well known. Their anisotropy between longitudinal and radial direction is even more affected, which should modify vibration modes of instruments' plates. The transitional increase in damping occuring within a time scale of one/few hours seems even more relevant to instrument playing conditions, all the more so that its amplitude is more pronounced over smaller differences in RH, typically in the range of 35-65% that could be encountered in normal concert places.

Figure 1 :

 1 Figure 1: Dependence on Equilibrium moisture content EMC of specific modulus and damping and their longitudinal/radial anisotropy. (Triangles) spruce; (circles) maple; (filled symbols) adsorption; (open symbols) desorption.When changing conditions of humidity, frequency and dimensions evolved together with internal moisture content of wood. On the contrary, a transitional increase in damping was always observed, whether the humidity decreased or increased (Figure2). The peak value of this increase in damping occurred from 30minutes to a few hours after changing conditions (depending on RH steps and direction of specimens). Then damping slowly decreased when evolving towards equilibrium state, which took 2-4 weeks, that is, after moisture content itself was stabilized. Moreover, this transitional increase in damping had a bigger amplitude when smaller steps in ambient RH were applied (Figure3). For relatively small RH steps (e.g. 35-50% RH or 50-65% RH), transitional changes in damping clearly exceeded the differences between initial and final equilibrium values.

Figure 2 :

 2 Figure 2: Example of transitional variations in (circles) damping, (diamonds) frequency, (squares) mass and (triangles) radial dimension, along time after changing RH conditions. Longitudinal spruce specimens.

Figure 3 :

 3 Figure 3: Relative variation through time in resonance frequency, shrinkage and damping plotted against mass loss during desorption from 85%RH to (circles) 65%RH; (diamonds) 50%RH; (squares) 35%RH. Longitudinal spruce specimens.

Acknowledgements

The first author initiated these experiments in the end of 2012 when a Post-Doc at EMPA, Switzerland, and continued the work at CNRS-LMGC from 2013. We thank Francis Schwartze and René Steiger for providing access to EMPA climatic rooms, and Patrick Langbour and Daniel Guibal for providing access to climatic equipment in CIRAD Montpellier.