Paul De Buyer
email: debuyer@math.cnrs.fr

A NOTE ON HELFFER-SJÖSTRAND REPRESENTATION FOR A GINZBURG-LANDAU PROCESS

Keywords: 1991 Mathematics Subject Classification. 60K35, 82C22, 82C41 Helffer-Sjöstrand Representation, Convergence towards equilibrium, Stochastic Process, Conservative Systems

In this work, we explore a link between an unbounded spin system and a random walk. This allows us to study the decay of the (co)variance of functions with respect to time. We extend here the previous work of T. Bodineau and G. Graham [1] to a more general class of graph and potential.

Introduction

A model of Ginzburg-Landau is a conservative model defined from a system of stochastic differential equations whose drift is given by a gradient (in the discrete sense) of a function, called the potential function. On each vertex x of a graph G, we assign a real value η x called a mass which evolves according to the value its neighbors and random part given by Brownian motion. These kind of dynamics can be seen as hydrodynamic limit of particle system, see [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF], and in this context has been studied in the 80's with a series of article [START_REF] Fritz | On the hydrodynamic limit of a one-dimensional Ginzburg-Landau lattice model. The a priori bounds[END_REF][START_REF] Fritz | On the hydrodynamic limit of a scalar Ginzburg-Landau lattice model: the resolvent approach[END_REF][START_REF] Fritz | On the hydrodynamic limit of a Ginzburg-Landau lattice model. The law of large numbers in arbitrary dimensions[END_REF][START_REF] Funaki | Hydrodynamic limit for Ginzburg-Landau type continuum model[END_REF][START_REF] Funaki | Derivation of the hydrodynamical equation for onedimensional Ginzburg-Landau model[END_REF].

Central limit theorem has been broadly discussed, and we refer the reader to the Ph.D. thesis of J. Sheriff [START_REF] Sheriff | Central Limit Theorem for Ginzburg-Landau Processes[END_REF] which is fully dedicated to this subject. The spectral gap and the log-Sobolev inequality is also studied in particular in [START_REF] Bodineau | Helffer-sjostrand representation for conservative dynamics[END_REF][START_REF] Chafai | Glauber versus kawasaki for spectral gap and logarithmic sobolev inequalities of some unbounded conservative spin systems[END_REF][START_REF] Landim | Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems[END_REF][START_REF] Ledoux | Logarithmic sobolev inequalities for unbounded spin systems revisited[END_REF].

In this work, following the approach of [START_REF] Bodineau | Helffer-sjostrand representation for conservative dynamics[END_REF][START_REF] Cancrini | A note on the helffer-sjöstrand representation for conservative unbounded spin systems in dimension one[END_REF], we use the Helffer-Sjöstrand representation to link the evolution of the spin system with a random walk in random dynamic environment. This representation, see [START_REF] Helffer | Remarks on decay of correlations and witten laplacians ii. analysis of the dependence on the interaction[END_REF][START_REF] Helffer | Remarks on decay of correlations and witten laplacians iii. application to logarithmic sobolev inequalities[END_REF][START_REF] Helffer | Semiclassical analysis, Witten Laplacians, and statistical mechanics[END_REF][START_REF] Helffer | On the correlation for kac-like models in the convex case[END_REF], relies on the commutation of an operator with a gradient giving rise to a second operator, called the Witten-Laplacian, which has, under some assumptions, a probabilistic interpretation. A closely related model, where this method has been used, is the ∇ϕ interface model, in [START_REF] Dembo | Stochastic interface models[END_REF], which is the same model as ours in dimension 1.

Furthermore, the space-time correlation in Z d between two masses, in [START_REF] Spohn | Large scale dynamics of interacting particles[END_REF], is conjectured to behave in the following way:

Cov (η x (0) ; η y (t)) ≃ C 1 t d/2 exp -|x -y| 2 C 2 t
where C 1 and C 2 are explicit. Using the connection between timeevolution of the covariance of masses and a random walk should lead to theses estimates. As a short intuition, one can understand that the behavior of a random walk is encoded by the behavior of the diffusion of the mass; therefore, the behavior of the masses should be encoded by the behavior of a particle (of the masses).

In this work, we solve the difficulty encountered in [START_REF] Bodineau | Helffer-sjostrand representation for conservative dynamics[END_REF] while trying to use the Helffer-Sjöstrand representation on general graph by introducing a site approach. The paper is organized in the following way. In the second section, we introduce the notations, the model and the results. In the third section, we give the representation and prove the main theorem. In the last section, we prove the auxiliary results.

Notations and Results

In this section, we introduce the notations, the model and the result. Let G = (V, E) be a connected graph, where V is the set of vertices and E the set of unoriented edges. We assume that the degree of each vertex is uniformly bounded by a constant d 0. Furthermore, we fix a vertex and call it origin, denoted 0. We note d G the graph distance and |•| the distance of a point to the origin. For practical reasons, we give an arbitrary orientation to the edges and note -→ B the set of these edges; we note ← -B the set of reversed oriented edges, i.e. ← -

B = (y, x) : (x, y) ∈ -→ B . Furthermore, we introduce B = ← - B ∪ -→ B .
To standardize the notations, we write e = {x, y} for an unoriented edges and b = (x, y) for an oriented one (belonging to -→ B or ← -B ). For two vertices x and y, we note x ∼ y iff the edge {x, y} ∈ E. Moreover, for all x ∈ V , we define sgn x : B → {-1; 0; 1} the function which assign to an oriented edge b = (y, z) associates :

sgn x (b) =      -1 if y = x 1 if z = x 0 else
We call an environment, an element η = (η x ) x∈V ∈ Ω = R V and introduce the partial order : for any two environments σ and η, σ η if and only if ∀x ∈ V, σ x η x . Thereby, we define the notion of increasing function, meaning that f is an increasing function if ∀η, σ :

σ η ⇒ f (σ) f (η)
Let (V x ) x∈V be a family of functions such that exp (-V x (t)) dt = 1 which satisfied the following assumptions:

Assumption 1 (Mean). There exists a constant M such that for all

x ∈ V , R t × exp(-V x (t))dt = M. Assumption 2 (Potential C 2 ). for all x ∈ V , V x ∈ C 2 .
Assumption 3 (Strict convexity). There exists two constants

C + C -> 0 such that for all η ∈ Ω, x ∈ V , C + V ′′ x (η) C -.
We define the function H : Ω → R, called the potential function, given by ∀η, H (η) = x V x (η).

Finally, we introduce operators of partial derivatives for all x ∈ V ,∂ x = d dηx , and for all b = (x, y) ∈ B, ∂ b = ∂ x -∂ y . The dynamic of Ginzburg-Landau is defined by the following system of differential equations:

(2.1)

dη x (t) = b∈ -→ B sgn x (b) ∂ b H (η) dt + √ 2dB b (t) , x ∈ S
where (B b (t)) b∈ -→ B is a family of independent Brownian motion indexed by the set of oriented edges. From these SDE, we extract the infinitesimal generator L e which describes the evolution of an environment.

The subscript e has been chosen to signify that the operator describes the evolution of the environment. The generator is given by:

L e f (η) = - 1 2 x,y:x∼y (∂ x -∂ y ) 2 f (η) + 1 2 x,y:x∼y (∂ x -∂ y ) H (η) × (∂ x -∂ y ) f (η) = - x,y:x∼y ∂ x (∂ x -∂ y ) f (η) + x,y:x∼y ∂ x H (η) × (∂ x -∂ y ) f (η) (2.2) = - b∈ -→ B ∂ b ∂ b f (η) + ∂ b H (η) ∂ b f (η)
where f is a local function, twice differentiable and with a finite norm f = x∈S ∂ x f ∞ ; these three assumptions on the functions will always be assumed in the rest of the article. We note for all generator L, the associated semigroup P L t = e -Lt t 0 , in particular, when there is no ambiguity, we will omit the superscript for the semigroup associated to the generator L e , meaning that (P t ) t 0 : = P Le t t 0 . A reversible measure of the Ginzburg-Landau process on a finite graph is the Gibbs measure µ given by:

dµ (η) = exp (-H (η)) dη = x∈S exp (-V x (η x )) dη
This definition can be easily extended in the case of infinite graph under assumption 1, 2 and 3. For simplicity, we will use the probabilistic notations, meaning that P will denote the reversible measure, E the associated mean, and

Cov (f ; g) = E [(f -E [f ]) (g -E [g])]
. Since the model is conservative, note that P ρ the law conditioned on a mass density ρ := ρ (η) = 1

|V |

x η x (defined for finite graph), is also a reversible measure.

We define the random walk (X (t)) t 0 on the vertices whose jump rates are dependent of the environment (η (t)) t 0 which evolve according to the Ginzburg-Landau process. For all time t, the random walk X (t) ∈ S and the jump rate from its position to one of its neighbor is given by V ′′

x (η x (t)) t 0 . The law describing the joint evolution of the walker and the environment is noted P σ x (•) = P (.|X (0) = x; η (0) = σ). We write L η p the generator of this random walk is therefore given by:

L η p f (x) = z∼x V ′′ (η x ) (f (z) -f (x)) ∀ (x, η) ∈ S × Ω
The subscript p has been chosen to signify that the generator acts on the position. Finally, we define the generator L = Id ⊗ L e + L p ⊗ Id describing the joint evolution of the walker and the environment id acting on the functions F : S × Ω → R in the following way:

(2.3) LF (x, η) = (Id ⊗ L e ) F (x, η) + (L p ⊗ Id) F (x, η) , ∀ (x, η) ∈ S × Ω
To simplify the notations, we will write L e instead of Id ⊗ L e and L p instead of L p ⊗ Id so that we can write L = L e + L p .

The main theorem of this article is the following:

Theorem 2.1. Let G = (V, E) be an infinite graph with bounded degree. Under assumptions 1, 2 and 3, ∀x, y ∈ V :

Cov (η x ; P t η y ) 1 C - E [P η x (X (t) = y)] Cov (η x ; P t η y ) 1 C + E [P η x (X (t) = y)]
Remark 2.2. If we have space and time bounds on the coefficients P η x (X (t) = y), then, these bounds apply immediately to the Ginzburg-Landau model. Note that bounds can be (and some have been obtained in [START_REF] Giacomin | Equilibrium fluctuations for ∇ϕ interface model[END_REF]) on Z d following [START_REF] Eric A Carlen | Upper bounds for symmetric markov transition functions[END_REF][START_REF] Mourrat | Variance decay for functionals of the environment viewed by the particle[END_REF]. Furthermore, if C -= C + , so that the potential is Gaussian, we obtain the equality instead of inequality above, and the walk is a simple random walk.

Remark 2.3. The uniform bounded degree assumption on the graph may not be necessary as well as the assumptions 1 and 3. Indeed, they guarantee the well definition of the model but they can be extended.

The main result has the following corollary: Corollary 2.4. Under the assumptions 1, 2 and 3, then for all function f : Ω → R such that

(2.4) f = x∈S ∂ x f ∞ < +∞
the following inequality holds:

Cov (f ; P t f ) 1 C - f 2 sup x∈S E [P η x (X t = x)]
Furthermore, if f and g are two increasing functions then,

Cov (f ; P t g) f × g C - sup x∈supp(f ) y∈supp(g) E [P η x (X t = y)]
The next proposition, taken from [START_REF] Bodineau | Helffer-sjostrand representation for conservative dynamics[END_REF], compares the spectral gap of the Ginzburg-Landau process with the spectral gap of the random walk. The proof is given for the sake of completeness.

L

The spectral gap has been derived especially in [START_REF] Caputo | Spectral gap inequalities in product spaces with conservation laws[END_REF][START_REF] Chafai | Glauber versus kawasaki for spectral gap and logarithmic sobolev inequalities of some unbounded conservative spin systems[END_REF][START_REF] Landim | Spectral gap and logarithmic Sobolev inequality for unbounded conservative spin systems[END_REF]. In particular, in (Z/NZ) d , one can easily obtain a spectral gap of order C/N 2 with a constant C that depends on C -and C + using the Helffer-Sjöstrand representation.

3. The Representation of Helffer-Sjöstrand by Site in this section, we prove start by proving the main theorem using two key lemmas which will be proved later.

3.1. Proof of the Theorem 2.1. The two following key lemmas are inspired by [START_REF] Bodineau | Helffer-sjostrand representation for conservative dynamics[END_REF]: Lemma 3.1. For all t 0 and all x, y ∈ S, we have the following equality:

Cov (V ′ x (η x ) ; P t η y ) = E [P η x (X (t) = y)] Lemma 3.2.
For all increasing functions f and g, we have the:

Cov (f ; P t g) 0
From these two key lemmas, the proof of the main theorem is immediate:

Proof of the theorem 2.1. Consider two vertices x and y of G and the functions f : Ω → R and g : Ω → R defined by:

f (η) = 1 C - V ′ (η x ) -η x g (η) = η y
Since f and g are increasing, using lemma 3.2, we obtain:

Cov (f ; P t g) 0 ⇒ Cov (η x ; P t η y ) 1 C - Cov (V ′ (η x ) ; P t η y )
Then, using lemma 3.1 which concludes the proof of the first part of the theorem. Using the same reasoning with f :

η → η x -1 C + V ′ (η x )
, we obtain the second inequality which concludes the proof of 2.1.

3.2.

Proof of Lemma 3.1. In this section, we introduce the Helffer-Sjöstrand representation, a probabilistic interpretation of the intertwining technique. We recall briefly that the intertwining technique is the commutation of a generator and an operator. Here, we will commute L e and ∂ x so we get a second generator satisfying (informally) the relation ∂ x L e = L∂ x . Recall that L is the infinitesimal generator describing the joint evolution of the random walk and the environment. This relation is the key to interpret the evolution of the mass with the evolution of the random walk.

The Helffer-Sjöstrand representation is the following:

Lemma 3.3. Define G (x, η) = ∂ x g (η)
of some function g We have the following inequality:

∂ x L e g (η) = LG (x, η)
where L is defined by (2.3). Consequently:

∂ x P Le t g (η) = P L t G (x, η) Proof. By definition: ∂ x L e g (η) = - b ∂ b ∂ b ∂ x g (η) + b ∂ b H (η) × ∂ b ∂ x g (η) + b ∂ b ∂ x H (η) × ∂ b g (η) = L e ∂ x g (η) + y∼x V ′′ (η x ) × (∂ x g (η) -∂ y g (η)) = (L e + L p ) ∂ x g (η) = LG (x, η)
The consequence is due to the fact P L t = exp (-Lt). We begin the proof of the lemma by recalling the formula of integration by part. :

(3.1) E [f × L e g] = E
x,y:x∼y

∂ x f × (∂ x -∂ y ) g
Applying this formula with the functions f : η → V ′ x (η x ) and g : η → η y -E [η y ], defining the function G : (x, η) → ∂ x g (η) and noting that for all h : Ω → R we have E [L e h] = 0, we obtain:

∂ t E [f × P t g] = E y:x∼y V ′′ x (η x ) × (∂ x -∂ y ) P t g = E y:x∼y V ′′ x (η x ) × P L t G (x, η) -P L t G (y, η) = E L p P L t G (x, η) = E LP L t G (x, η) = ∂ t E P L t G (x, η) Which immediately implies that E [f × P t g] = E P L t G (x, η)
. By the definition of G, we have for all z and η that G (z, η) = 1 y=z , therefore P L t G (x, η) = P η x (X (t) = y) which concludes the proof of the lemma.

3.3.

Proof of Lemma 3.2. In this section, we prove Lemma 3.2.

Given the independence assumption on the potential (H (η) = V x (η x )), the Harris FKG-inequality would be enough if one can prove that the partial order is preserved with time. Indeed, in this case, if g is an increasing function, then P t g is an increasing function and therefore the lemma is proved. This lemma is already proven in [1, Lemma 4.8], so we give here a improved version on a more general class of potential for the time preservation of the partial order following the same ideas.

In the next lemma, we consider that the potential there exists two families of functions of

C 2 (R, R) (V x ) x∈S and (V x,y ) x,y∈V,y∼x such that H (η) = x,y∼x V x (η x ) + V x,y (η x + η y ).
Lemma 3.4. Assume that there exists four constants C 1,-, C 2,-, C 1,+ and C 2,+ such that:

C 1,-V ′′ x C 1,+ C 2,-V ′′ x,y C 2,+
Then the partial order is time preserved if:

inf x inf y∼x z∼y C 2,--C 2,+ + C 1,- 0 C 2,- 0
Proof. We define two solutions (η (t)) t 0 and (σ (t)) t 0 of the system of SDE (2.1) driven by the same family of Brownian motions.and such that η (0) σ (0). We define for all x ∈ V the function φ x : R + → R, defined by φ x (t) = η x (t) -σ x (t). Of course, this is a continuous function φ x . Then, we define the function

Φ : R + → R by Φ (t) = x∈S (2d) -|x| φ 2 x (t) 1 {φx(t)<0}
where d is the uniform upper bound on the degree of the vertices of V . Note that Φ is continuous, positive, Φ (0) = 0 and well defined, see [START_REF] Shiga | Infinite dimensional stochastic differential equations and their applications[END_REF]Théorème 2.1]. Intuitively, this function Φ measure the number of negative φ x . Using Gronwall Lemma, we will show that the function Φ is equal to the zero constant function.

To simplify the notations, we will omit to write the dependence on t of the processes η (t) and σ (t) when there are no ambiguity and rather write η and σ. In the same way, we will write η x and σ x instead η x (t) and σ x (t).

To establish the comparison lemma, we differentiate each summand of Φ :

dφ x (t) =d (η x (t) -σ x (t)) = y∼x z∼y V ′ y,z (η y + η z ) -V ′ y,z (σ y + σ z ) -V ′ x,y (η x + η y ) -V ′ x,y (σ x + σ y ) + y∼x V ′ y (η y ) -V ′ y (σ y ) -V ′ x (η x ) -V ′ x (σ x )
Consider the case φ x (t) < 0, meaning that η x (t) σ x (t). Consequently, dφ x (t) is an increasing function inσ x so we can bound it from below by setting σ x = η x . For each term, we measure its contribution in term of φ . (t). Observe that:

V ′ x,y (η x + η y ) -V ′ x,y (η x + σ y ) = ηy σy V ′′ x,y (η x + s) ds V ′ y,z (η y + η z ) -V ′ y,z (σ y + σ z ) =V ′ y,z (η y + η z ) -V ′ y,z (σ y + η z ) + V ′ y,z (σ y + η z ) -V ′ y,z (σ y + σ z ) = ηy σy V ′′ y,z (s + η z ) ds + ηz σz V ′′ y,z (σ y + s) ds V ′ y (η y ) -V ′ y (σ y ) = ηy σy V ′′ y (s) ds
Thus:

dφ x (t) y∼x ηy σy z∼y V ′′ y,z (s + η z ) -V ′′ x,y (η x + s) + V ′′ y (s) ds + y∼x z∼y ηz σz V ′′ y,z (σ y + s) ds
Under the assumption of the lemma, the signs of the integrands are positive. Under the ellipticity assumptions on the potential, we get:

dφ x (t) y∼x φ y (t) 1 φy(t)<0 z∼y C 2,+ -C 2,-+ C 1,+ + y∼x z∼y φ z (t) 1 φz(t)<0 C 2,+
Recall that Φ is given by:

Φ (t) = x∈V 2 -|x| φ 2 x (t) 1 {φx(t)<0}
By differentiating and writing C max = max x,y:y∼x z∼y C 2,+ -C 2,-+ C 1,+ , we obtain:

d dt Φ (t) x∈V 2 1-|x| φ x (t) × 1 φx(t)<0 × C max y∼x φ y (t) 1 φy(t)<0 + z∼y φ z (t) 1 φz(t)<0
Using the trivial inequality ∀a, b ∈ R, 2ab a 2 + b 2 , then there exists a constant C := C (C max , d) such that:

d dt Φ (t) C × x∈V (2d) -|x| φ 2 x (t) 1 φx(t)<0 C × Φ (t)
Finally, using Gronwall Lemma, we obtain:

Φ (t) Φ (0) × exp (C × t)
Which concludes the proof of the lemma since Φ (0) = 0.

We can now prove the second key Lemma.

Proof of Lemma 3.2. As written above, It is enough to show that the Harris-FKG inequality holds. It relies on the sufficient condition of the theorem 3 of [START_REF] Cj Preston | A generalization of the fkg inequalities[END_REF]. For any two configurations σ and η, noting

(η ∨ σ) x = max {η x , σ x } and (η ∧ σ) x = min {η x , σ x }, if (3.2) µ (η ∨ σ) µ (η ∧ σ) µ (σ) µ (η)
then the Harris-FKG inequality holds. This criterion if immediately verified since we are in the independent case, i.e.

V x ((η ∨ σ) x ) + V x ((η ∧ σ) x ) = V x (η x ) + V x (σ x ) , ∀x ∈ V
Thus, the inequality (3.2) is an equality in our case, which concludes the proof of the lemma.

Remark 3.5. One can easily verified that the inequality (3.2) is not true when H can be written H (η) = x V x (η x ) + y∼x V x,y (η x + η y ), where V x,y is a strictly convex function. Moreover, in this case, one can show that E [η x η y ] < 0 for x ∼ y and therefore the Harris-FKG can't be established.

Proof of Auxiliary Results

We start by proving corollary 2.4:

Proof. Because of the condition (2.4) on f , we have that f is Lipschitz in all its coordinate. We write

L f (x) = ∂ x f ∞ to get: |f (η) -f (η x )| L f (x) |η x -η x
x | where η x is the configuration which all coordinates are equal to the ones of η except (potentially) in x, i.e. ∀y = x we have η x y = η y . From this, we obtain for the functions g + and g -defined by g + (η) =

x L f (x) η x + f (η) and g -(η) = x L f (x) η x -f (η) are increasing functions and using lemma 3.2:

Cov (P t g -; g + ) 0 ⇔ Cov P t x L f (x) η x ; y L f (y) η y Cov (P t f ; f )
which gives by the main theorem:

Cov (f ; P t f ) x,y L f (x) L f (y) Cov (η x ; P t η y ) x,y L f (x) L f (y) sup x Cov (η x ; P t η x )
Using Cauchy-Schwartz inequality and concavity of the square root function, this proves the first part of the corollary. If f and g are two increasing functions, by lemma 3.2:

Cov x L f (x) η x ; P t y L g (y) η y Cov f ; P t y L g (y) V ′ y (η y ) Cov (f ; P t g)
Expanding the sum, and taking the supremum over the x and y such that L f (x) = 0 and L g (y) = 0 and the main theorem, concludes the corollary.

We then prove proposition 2.5.

Proof. Recall that λ * L is the spectral gap of the random walk defined by:

λ * L = inf f =0 -∂ t=0 P L t f 2 2 f 2 where f 2 2 = f 2 (x). Therefore P L t f 2 2 f 2 2 e -λ * L t
, for all function f . In this way,

Var P L t f = R + -∂ s Var (P s f ) ds = R + E x,y ((∂ x -∂ y ) P s f ) 2 ds = R + E x,y P L s (∂ x -∂ y ) f 2 ds R + e -λ * L s E x,y ((∂ x -∂ y ) f ) 2 ds (λ * L ) -1 × (-∂ t Var (P t f ))
Taking the infimum over the function f , one obtains:

λ * L inf f -∂ t=0 Var (P t f ) Var (f ) = λ * e
Which ends the proof.

reversed orientation ←b = (y, x), one can note that ∂ b = -∂ ← -b so we get:

b∈ -→ B ∂ b ∂ b ′ H × ∂ b f = b∈B b∼b ′ ∂ b ∂ b ′ H (∂ b f -∂ b ′ f ) + (∂ b ′ ∂ b ′ H - b∈B b∼b ′ ∂ b ∂ b ′ H)∂ b f = L ′ p ∂ b ′ f + (∂ b ′ ∂ b ′ H - b∈B b∼b ′ ∂ b ∂ b ′ H)∂ b ′ f (4.2) Where b ∼ b ′ means b ′ = ← - b and∂ b ∂ b ′ H < 0. On Z, note that (∂ b ′ ∂ b ′ H - b ′ ∼b∈B ∂ b ∂ b ′ H) = 0.
Furthermore one can see that L ′ p is the generator of a random walk on the set of oriented edges B. To continue the discussion, we can restrict ourselves to the case where the graph is Z 2 and the Gaussian potential "H (η) = 1 2 η 2

x ". In this case, the walker follows the law of a simple random walk on the oriented "kite" graph, drawn with unoriented edges in figure 4.1. t F (b, •) Where F (b, •) = ∂ b f . Before stating the next proposition, we define (X ′ (t)) t the simple random walk on the oriented kite graph of generator L ′ p . A result obtained in [START_REF] De | Vitesse de convergence de systémes de particules en interaction[END_REF], where the proof can be found, is the following: The result can be generalized to a more general class of potential, i.e. the potential that can be written H (η) = V 0 (η x ) where V 0 ∈ C 2 (R, R), V ′′ 0 C -∈ R and lim |t|→∞ V 0 (t) = +∞. However, one can see that this quantity is hard to exploit since we need to compensate the exponential term.

Proposition 2 . 5 .

 25 The spectral gap λ * e of the Ginzburg-Landau Process given by the generator L e is lower bounded by the spectral gap λ * L of the random walk given by the generator L: λ * e λ *

Figure 4 . 1 .

 41 Figure 4.1. Z 2 and the kite graph

Proposition 4 . 1 . 2 x

 412 On Z 2 with a Gaussian potential, for any oriented edge b ∈ B, we have the following relation: = e 4t P b (X ′ (t) = b) -P b X ′ (t) = ←b

Appendix A. Helffer-Sjöstrand Representation by edge

In the appendix, we develop an extension of the original approach of [START_REF] Bodineau | Helffer-sjostrand representation for conservative dynamics[END_REF], where the authors commuted the generator L e and ∂ b with b ∈ -→ B . We give the approach, its difficulties and some associated results. A discussion of this approach can be found in the Ph.D. thesis of the author, see [START_REF] De | Vitesse de convergence de systémes de particules en interaction[END_REF] (in french). Commuting L e and ∂ b , one gets:

The authors argued that the second summand of (4.1) can only be interpreted as the (positive) generator of a random walk on the directed edges (of -→ B ) on the torus (or Z) since the Hessian of the potential function would have non negative off-diagonal term and positive diagonal term. However, noting for every b = (x, y) ∈ B the edge with a


Paul De

Jean-Christophe Mourrat
email: jean-christophe.mourrat@ens-lyon.fr

DIFFUSIVE DECAY OF THE ENVIRONMENT VIEWED BY THE PARTICLE

Keywords: Mathematics Subject Classification. 60K37, 60J27, 76M50 Random Walks in Random Environments, Algebraic convergence to equilibrium, Homogenization, Functional inequalities

We prove an optimal diffusive decay of the environment viewed by the particle in random walk among random independent conductances, with, as a main assumption, finite second moment of the conductance. Our proof, using the analytic approach of Gloria, Neukamm and Otto, is very short and elementary.

Introduction

In this paper we are interested in the speed of convergence of the environment viewed by the particle in the context of random walk among random independent conductances. We refer to [START_REF] Biskup | Recent progress on the random conductance model[END_REF] for an introduction. We will first introduce the model and then state our main theorem.

Consider the d-dimensional lattice (Z d , B d ), d ≥ 1, with B d the set of unoriented edges connecting any two points of Z d at Euclidian distance 1 and set Ω := [0, +∞) B d . Given an environment ω = (ω x,y ) {x,y}∈B d ∈ Ω we shall consider the associated Markov process (X ω t ) t 0 with jump rate between x and y given by the conductance ω x,y and write P ω x for its law starting from x ∈ Z d . In many places we may simply write (X t ) t 0 for simplicity. The environment itself will also be a random variable. In fact, throughout the paper, we will assume that the conductances ω x,y , {x, y} ∈ B d , are i.i.d. with common law µ, whose support is included in [1, ∞). The law of the environment is therefore the product measure P := µ B d and we denote by E the associated expectation. Since P is a product measure, standard percolation arguments guarantee that that the Markov process (X t ) t 0 is well defined for almost all ω ∈ Ω, for all times, see e.g. [START_REF] Kipnis | Central limit theorem for additive functionals of reversible markov processes and applications to simple exclusions[END_REF][START_REF] De Masi | An invariance principle for reversible markov processes. applications to random motions in random environments[END_REF][START_REF] Mourrat | Variance decay for functionals of the environment viewed by the particle[END_REF]. Moreover it is reversible with respect to the counting measure, i.e. for all x, y ∈ Z d , it holds P ω x (X t = y) = P ω y (X t = x). Now, define the translation operators (θ z ) z∈Z d given by (θ z ω) x,y := ω x+z,y+z . Then the Markov process (ω (t)) t 0 := (θ Xt ω) t 0 , called the environment viewed by the particle, is reversible with respect to P.

The aim of this short paper is to give an optimal quantitative bound on the decay to equilibrium of (ω (t)) t 0 .

In order to state our theorem, we need some more notations. A function f : Z d ×Ω → R is said to be translation invariant if f (x, ω) = f (0, θ x ω) for all x ∈ Z d and local if f (0, •) depends only on a finite set of conductances. The smallest set satisfying that property is called the support of f and is denoted by supp(f ), while #supp (f ) stands for the size of the support (i.e. the number of sites of Z d contained in supp(f )). For any translation invariant function f , we set

E[f ] := E[f (0, •)].
Our main theorem is the following.

Theorem 1.1. Let d ≥ 3. Assume that the law µ of the conductance has support in [1, ∞) and finite second moment E ω 2 •,• < ∞. Then there exists a constant C > 0 that depends only on d such that for all local translation invariant function f :

Z d × Ω → R with E [f ] = 0, all x ∈ Z d , it holds (1.1) E E ω x [f (X t , ω)] 2 CE ω 2 •,• #supp (f ) 2 E f 2 t d/2 ∀t > 0.
Let us comment on Theorem 1.1. We first observe that similar results already exist in the literature. One of us [Mou11] proved a polynomial decay in dimension 1 with exponent 1/2, in dimension 2 with behavior log t/t, in dimension 3 to 6 with t -1 and in dimension 7 and higher with t -d 2 +2 . Moreover, in the right hand side of (1.1) appears a much stronger norm than only the L 2 norm (the sum of the L ∞ norm of f and the so-called triple norm |||f ||| that involves the infinite sum of L ∞ -norm of local gradients of f , a natural norm largely used in the statistical mechanics literature, see for example [START_REF] Martinelli | Lectures on Glauber dynamics for discrete spin models[END_REF] and also [START_REF] Bertini | Coercive inequalities for Kawasaki dynamics. The product case. Markov Process[END_REF][START_REF] Janvresse | Relaxation to equilibrium of conservative dynamics. I. Zero-range processes[END_REF][START_REF] Roberto | Slow decay of Gibbs measures with heavy tails[END_REF]). On the other hand, the results of [START_REF] Mourrat | Variance decay for functionals of the environment viewed by the particle[END_REF] hold without the assumption on the finiteness of the second moment of the conductance. More recently, Gloria, Neukamm and Otto [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF] obtained, among other results, a polynomial decay t -d 2 -1 when the function f is the divergence of some other function. Because we follow closely their approach, our result is not far from theirs, although they need to assume that conductances are bounded, i.e. that µ has support contained in a compact set and that the function f considered is in L p , where p is very large and not explicit. Finally, we mention that [Yur86, CK08, GO11, GO12] are related papers that deal with estimates on the diffusion matrix and we refer to [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Armstrong | A regularity theory for random elliptic operators[END_REF] and references therein for recent results on homogenization of random operators.

Then, we observe that the power in (1.1) is the best possible, and we note that it can be obtained under stronger assumptions in [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]. On the other hand, we believe that the result should hold without the assumption on the finiteness of the second moment of the conductance, which appears as a technical fact in our proof. The second moment E f 2 is also best possible. Indeed, a smaller norm would imply some regularization effect that would in turn imply a spectral gap estimate which is known not to hold. For the same reason, there must be some dependence, in the right hand side of (1.1), on the size of the support of f . Finally, we observe that the assumption on the support of µ is also necessary since it is known, for some specific choice of function f , with conductances taking small values, that the decay of the heat-kernel coefficients can be very slow, see [START_REF] Berger | Anomalous heat-kernel decay for random walk among bounded random conductances[END_REF], which would interfer with the decay in (1.1).

As for the proof, diffusive scaling is usually obtained, including other settings on graphs such as interacting particle systems (Kawasaki dynamics), using functional inequalities of Nash type, see e.g. [START_REF] Liggett | L2 rates of convergence for attractive reversible nearest particle systems: the critical case[END_REF][START_REF] Bertini | Coercive inequalities for Kawasaki dynamics. The product case. Markov Process[END_REF][START_REF] Mourrat | Variance decay for functionals of the environment viewed by the particle[END_REF], Harnack [START_REF] Delmotte | Parabolic Harnack inequality and estimates of Markov chains on graphs[END_REF] or weak Poincaré inequalities [START_REF] Bobkov | Distributions with slow tails and ergodicity of Markov semigroups in infinite dimensions[END_REF][START_REF] Roberto | Slow decay of Gibbs measures with heavy tails[END_REF], see also [START_REF] Bakry | Sobolev inequalities in disguise[END_REF][START_REF] Saloff-Coste | Lectures on finite Markov chains[END_REF]. Induction techniques can also be used [START_REF] Janvresse | Relaxation to equilibrium of conservative dynamics. I. Zero-range processes[END_REF][START_REF] Cancrini | Diffusive long-time behavior of Kawasaki dynamics[END_REF], or specific aspects of the model such as attractivness [START_REF] Deuschel | Algebraic L 2 decay of attractive critical processes on the lattice[END_REF]. However, none of these techniques seem, to the best of our knowledge, to apply to our setting. Here instead, we will mainly follow the PDEs ideas from [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF], but with many simplifications due to our non specific choice of class of functions.

As a result we believe that our approach of the decay to equilibrium of the environment viewed by the particle improves upon known results into different directions (class of function, assumption on the conductance, short and elementary proof), but, as a counterpart, we are not able to extend the result of Gloria, Neukamm and Otto [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF] to unbounded conductances for divergence functions.

The paper is organized as follows. In the next subsection we will give some more notations. Then, we will give the proof of Theorem 1.1. Such a proof will rely on a series of lemmas that we will prove later on. Finally, in the last section, using the theory of completely monotonic functions, we comment on a possible short way of extending the results of Gloria, Neukamm and Otto [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF] to our setting.

1.1. Notations. In this section we give some more notations.

Given an environment ω ∈ Ω, the infinitesimal generator of the process (X t ) t≥0 and its associated semi-group, acting on functions f :

Z d ×Ω → R are given, for any x, ω ∈ Z d ×Ω by L ω f (x, ω) = |z|=1 ω x,x+z (f (x + z, ω) -f (x, ω)) ,
respectively by

P ω t f (x, ω) = exp (L ω t) f (x, ω) = y∈Z d P ω x (X t = y) f (y, ω) .
In some situations, we may want to work with a given fixed environment. In that case, and to emphasize this fact, we shall use the letter m instead of ω and call that given environment the walk scheme. A walk scheme is well-defined when there is a threshold such that the set of conductances above it does not percolate. This will be used in particular to evaluate the behaviour of (X m t ) t 0 . In many places we shall use the following equality P θxω 0 (X t = y) = P ω x (X t = y + x) that holds for all x, y ∈ Z d and all ω ∈ Ω.

For simplicity of notation, we set f t (x, ω) where E ω x is the mean associated to P ω x , and in many places we shall omit the dependence in ω when there is no ambiguity.

:= P ω t f (x, ω) = E ω x f (X t , ω), t ≥ 0, x ∈ Z d , ω ∈ Ω,
Next, we define three different gradients. Denote by e 1 , . . . , e d the canonical orthonormal basis of Z d (i.e. for all i, e i = (0, . . . , 0, 1, 0, . . . , 0) with 1 at the i th coordinate), set e -i = -e i , i = 1, . . . , d and e 0 := (0, . . . , 0) for the origin. With that notation in hand, we define the following local gradient of g : Z d → R:

D i g(y) := g (y + e i ) -g (y) -d i d, y ∈ Z d .
In particular, if f :

Z d × Ω → R, then D i f (y, ω) = f (y + e i , ω) -f (y, ω), ω ∈ Ω,
and D i P ω x (X t = y) stands for the gradient applied to the mapping y → P ω x (X t = y). Similarly,

∇ i P m x (X t = y) := P m x+e i (X t = y) -P m x (X t = y) -d i d, x, y ∈ Z d .
Therefore, the infinitesimal generator of a random walk with scheme m can be written as

L m f (x, ω) = -d i d m x,x+e i D i f (x, ω) x ∈ Z d , ω ∈ Ω. Finally, for x ∈ Z d , let a (x) = {ω x,x+e i ; 1 i d}, a (x) = {ω e , e ∈ B d } \ a(x) = ∪ y =x
a (y) and E (x) be the conditional expectation given a (x). Then, for y ∈ Z d , we set

∂ x f (y, ω) := f (y, ω) -E (x) [f (y, ω)] so that E (x) [(∂ x f ) 2 ]
is nothing else than the variance of f with respect to the conditional expectation E (x) . Two sites x, y are neighbors, a property we denote by x ∼ y, if {x, y} ∈ B d . Also, given f : Ω → R, we say that x is a neighbor of y with respect to f , and write x ∼ f y, if a (xy) ∩ supp (f ) = ∅ (observe that this is not an equivalence relation). In particular, observe that, if x is not a neighbor of y with respect to a local translation invariant function

f (i.e. if a (x -y) ∩ supp (f ) = ∅) then ∂ x f (y, ω) = 0.
2. Variance decay for unbounded conductances: proof of Theorem 1.1

In this section we prove Theorem 1.1. The idea, following [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF], is to decompose the variance E E ω

x [f (X t , ω)] 2 , using Efron-Stein's inequality, into an infinite sum of terms of the type E (∂ y P t f ) 2 , which, by Duhamel's formula, are split into two different terms that need to be analyzed separately (the core of the proof). The point in using Duhamel's Formula is to commute the operators P t and ∂ y . The proof ends by applying some sort of Gronwall Lemma.

Proof. [Proof of Theorem 1.1] Let f : Ω → R be a local translation invariant function with E[f ] = 0, and assume, by homogeneity and for simplicity, that E[f 2 ] = 1. Following [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF], we apply Efron-Stein's Inequality and the Duhamel formula (that we recall below, in Lemma 2.1, for completeness) to bound E f 2 t :

E f 2 t E   y∈Z d (∂ y P t f ) 2   = E   y∈Z d P t ∂ y f + ˆt 0 P t-s h s (0, y, ω) ds 2   2E   y∈Z d (P t ∂ y f ) 2   + 2E   y∈Z d ˆt 0 P t-s h s (0, y, ω) ds 2   . (2.1) where h s (x, y, ω) := E (y) [Lf s (x, ω)] -LE (y) [f s (x, ω)], x, y ∈ Z d , ω ∈ Ω, s ≥ 0.
Next we analyze each term of the right hand side of the latter separately and start with the first one.

First recall that if a (y

-x) ∩ supp (f ) = ∅ then ∂ y f (x, ω) = 0. Hence E   y∈Z d (P t ∂ y f ) 2   = E    y   x∈Z d P ω 0 (X t = x) ∂ y f (x, ω)   2    #supp(f ) y x:y∼ f x E P ω 0 (X t = x) 2 ∂ y f (x, ω) 2 . (2.2)
By invariance by translation we have

E P ω 0 (X t = x) 2 ∂ y f (x, ω) 2 = E P θ -x ω 0 (X t = x) 2 ∂ y f (x, θ -x ω) 2 = E P ω 0 (X t = -x) 2 ∂ y-x f (0, ω) 2 . Therefore, changing variables (set x ′ = y -x and y ′ = x ′ -y), it holds E   y∈Z d (P t ∂ y f ) 2   #supp(f ) y x:y∼ f x E P ω 0 (X t = -x) 2 ∂ y-x f (0, ω) 2 #supp(f ) y x ′ :x ′ ∼ f 0 E P ω 0 X t = x ′ -y 2 ∂ x ′ f (0, ω) 2
#supp(f )

x ′ :x ′ ∼ f 0 y ′ E P ω 0 X t = y ′ 2 ∂ x ′ f (0, ω) 2 .
Finally, using Lemma 2.2 below and the fact that

E ∂ x ′ f (0, •) 2 2E f (0, •) 2 = 2E f 2 = 2,
we conclude that, for some constant C that depends only on d,

(2.3) E   y∈Z d (P t ∂ y f ) 2   C #supp (f ) 2 (t + 1) d/2
Next we focus on the second term in the right hand side of (2.1). Using Lemma 2.3 we have

E   y∈Z d ˆt 0 P t-s h s (0, y, ω) ds 2   = E   y∈Z d ˆt 0 d i=1 D i P ω 0 (X t-s = y) g s (y, y, ω, i) ds 2 
 so that, by Minkowski's integral inequality and the invariance by translation, it holds

E   y∈Z d ˆt 0 P t-s h s (0, y, ω) ds 2   1 /2 √ d ˆt 0 y i E (D i P ω 0 (X t-s = y)) 2 g s (y, y, ω, i) 2 1 /2 ds = √ d ˆt 0 y i E ∇ i P ω 0 (X t-s = y) 2 g s (0, 0, ω, i) 2 1 /2 ds √ 2d ˆt 0 y i E ∇ i P ω 0 (X t-s = y) 2 E (0) [ω 0,e i D i f s (0, ω)] 2 + E ω 2 0,e i ∇ i P ω 0 (X t-s = y) 2 E (0) [D i f s (0, ω)] 2 1 /2 ds (2.4) where g s (x, y, ω, i) := E (x) [ω y,y+e i D i f s (y, ω)]-ω y,y+e i E (x) [D i f s (y, ω)], s ≥ 0, x, y ∈ Z d , ω ∈ Ω and i = 1, . . . , d. Therefore, using twice that (a + b) 2 2a 2 + 2b 2 , Lemma 2.2
and Lemma 2.4 guarantee that, for some constant C that depends only on d, it holds

E   y∈Z d ˆt 0 P t-s h s (y, 0, ω) ds 2   1 /2 C ˆt 0 (t -s + 1) -d 2 i E E (0) [ω 0,e i D i f s (0, ω)] 2 + (t -s + 1) -d/2 i E ω 2 0,e i E (0) [D i f s (0, ω)] 2 1/2 ds √ 2C ˆt 0 (t -s + 1) -d /4 E ω 2 0,e 1 1/2 -∂ s E |f s (0, ω) | 2 1/2 ds.
(2.5) Plugging (2.3) and (2.5) into (2.1), we end up with

E f 2 t 1 2 C ′ #supp (f ) E ω 2 0,e 1 1 2 (t + 1) -d 4 + ˆt 0 (t -s + 1) -d 4 -∂ s E f 2 s 1 2 ds
for some constant C ′ that depends only on d. The expected result will finally follow from Lemma 2.6 with a (t) := E f 2 t 1 2 and α = d 4 . Indeed, a(t) is non-increasing since, using classical computations for reversible Markov processes,

∂ t E f 2 t = 2E [f t Lf t ] = -i E ω 0,e i (D i f t ) 2 ≤ 0. This ends the proof.
In the proof of Theorem 1.1 we used the following series of lemma. The first two lemmas are well known results from Probability Theory and Analysis. The others are technical.

Lemma 2.1. [Efron-Stein's Inequality and the Duhamel formula] The following holds. (Efron-Stein's Inequality):

Let n > 1 and f be a function of X 1 , ..., X n , n independent variables, then

Var (f ) n i=1 E Var (i) (f )
where Var (i) is the conditional variance given {X 1 , ..., X n } \ {X i }. (Duhamel's Formula): For all t 0 and almost all ω ∈ Ω it holds

∂ y P t f (x, ω) = P t ∂ y f (x, ω) + ˆt 0 P t-s h s (x, y, ω) ds where h s (x, y, ω) := E (y) [Lf s (x, ω)] -LE (y) [f s (x, ω)], x, y ∈ Z d , ω ∈ Ω, s ≥ 0.
Proof. Efron-Stein's Inequality, also called tensorisation of the variance (see e.g. [ABC + 00, Proposition 1.4.1]), is a well known result following from Cauchy-Schwarz' inequality. See [START_REF] Lata | Between Sobolev and Poincaré[END_REF] for an extension to general φ-entropy, see also [START_REF] Massart | Concentration inequalities and model selection[END_REF].

As for the Duhamel formula, we observe that

∂ y P t f (x, ω) = P t ∂ y f (x, ω) + ˆt 0 ∂ s P t-s ∂ y P s f (x, ω) ds
which leads to the expected result, since h s (x, y, ω) = (∂ y L -L∂ y ) P s f and ∂ t P t = LP t = P t L.

Lemma 2.2. There exists a constant C (that depends only on d)

such that for all welldefined walk scheme m and for all t > 0 it holds

y∈Z d P m 0 (X t = y) 2 C (t + 1) -d/2 .
Proof. The proof of this inequality is a consequence of the reversibility and the fact that the invariant measure is the uniform measure. Indeed

y∈Z d P m 0 (X t = y) 2 = y∈Z d P m 0 (X t = y) P m y (X t = 0) = P m 0 (X 2t = 0)
which gives the desired result combining [CKS87, Theorem 2.1] and [Mou11, Proposition 10.2] in its first arXiv version.

Lemma 2.3. Given f :

Z d ×Ω → R, define h s (x, y, ω) := E (y) [Lf s (x, ω)]-LE (y) [f s (x, ω)] and g s (x, y, ω, i) = E (y) [ω x,x+e i D i f s (x, ω)] -ω x,x+e i E (y) [D i f s (x, ω)], x, y ∈ Z d , ω ∈ Ω, s ≥ 0 and i = 0, . . . , d.
Then, for all s, t > 0 and all x, y it holds

P t h s (x, y, ω) = - d i=1 D i P x (X t = y) g s (y, y, ω, i) .
Proof. Recall the definition of D i . On the one hand, by definition, we have

E (y) [Lf s (x, ω)] = d i=1 E (y) [ω x,x+e i (D i f s (y, ω))] -E (y) [ω x,x-e i (D i f s (x -e i , ω))]
and

LE (y) [f s (x, ω)] = d i=1 ω x,x+e i E (y) [(D i f s (x, ω))] -ω x,x-e i E (y) [(D i f s (x -e i , ω))] .
On the other hand, ω x,x+e i is ā (y)-measurable iff x = y and ω x,x-e i is ā (y)-measurable iff xe i = y. Therefore,

h s (x, y, ω) =        d i=1 g s (y, y, ω, i) if x = y -g s (y, y, ω, i) if x -e i = y, i = 1, . . . , d 0 otherwise.
It finally follows that

P t h s (x, y, ω) = z∈Z d P x (X t = z) h s (z, y, ω) = P x (X t = y) h s (y, y, ω) - d i=1 P x (X t = y + e i ) h s (y + e i , y, ω) = d i=1 (P x (X t = y) -P x (X t = y + e i )) g s (y, y, ω, i)
which is the desired result.

Lemma 2.4. Let f :

Z d × Ω → R.
Then, for all s > 0 it holds

d i=1 E E (0) [ω 0,e i D i f s (0, ω)] 2 -E ω 2 0,e 1 ∂ s E f s (0, ω) 2 , d i=1 E ω 0,e i E (0) [D i f s (0, ω)] 2 -E ω 2 0,e 1 ∂ s E f s (0, ω) 2 .
Remark 2.5. The proof actually leads to a better bound in the first inequality above. Indeed, one can replace the second moment of the conductance by the first moment. This refinement will anyway not be useful for our purpose.

Proof. [Proof of Lemma 2.4] Using Cauchy-Schwarz' inequality and the fact that ω 0,e i is a (0)-measurable, we have

E E (0) [ω 0,e i D i f s (0, ω)] 2 E E (0) [ω 0,e i ] E (0) ω 0,e i (D i f s (0, ω)) 2 = E [ω 0,e i ] E ω 0,e i (D i f s (0, ω)) 2 .
Using that

∂ t E f 2 t = ∂ t E f t (0, ω) 2 = -d i=-d E ω 0,e i (D i f s (0, ω))
2 (a classical consequence of the reversibility) and summing over i = 1, . . . , d, we get

d i=1 E E (0) [ω 0,e i D i f s (0, ω)] 2 -E [ω 0,e 1 ] ∂ s E f s (0, ω) 2
which leads to the first inequality since ω 0,e 1 ≥ 1.

For the second inequality by conditioning and Jensen's inequality, we have 2 which leads to the desired result and ends the proof of the lemma.

E ω 0,e i E (0) [D i f s (0, ω)] 2 = E E (0) ω 2 0,e i E (0) [D i f s (0, ω)] 2 = E ω 2 0,e i E D i f s (0, ω) 2 . Now, since ω 0,e 1 ≥ 1, one has d i=1 E D i f s (0, ω) 2 ≤ d i=1 E ω 0,e 1 D i f s (0, ω) 2 = -∂ s E f s (0, ω)
The last lemma is related to Lemma 15 of [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]. However, due to our specific setting, considering p = 1/2, its proof is a bit simpler. We give it for completeness. Lemma 2.6. [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]] Let α > 1/2 and a : R

+ → R + be a C 1 non-increasing function. Let b(t) = -2a ′ (t)a(t), t ≥ 0. Assume that a (t) C (t + 1) -α + ˆt 0 (t -s + 1) -α b (s) ds ∀t ≥ 0
for some constant C. Then there exists a constant C α that depends only on α such that

a (t) C α max(C, a(0)) (t + 1) -α .
Proof. Throughout the proof we use that u v if there exists a constant A that depends only on α such that u ≤ Av. The expected result will follow from the fact that, for some t o > 0 that will be chosen later on, [t o , ∞) ∋ t → Λ (t) := sup to≤s t (s + 1) α a (s) is a bounded function, bounded by C ′ max(C, a(0)) for some C ′ depending only on α.

Indeed, since a is non-increasing a(s) ≤ a(0) for any s ∈ [0, t o ] which, together with the bound on Λ would lead to the desired conclusion. Our starting point is the following inequality obtained using that a is non-increasing.

a (t) 2 t ˆt t 2 a (u) du 2C t ˆt t 2 du (u + 1) α + 2 t ˆt t 2 ˆu 0 b(s)dsdu (u + 1 -s) α ≤ C ( t 2 + 1) α + 2 t ˆt t 2 ˆT 0 b(s)dsdu (u + 1 -s) α + 2 t ˆt t 2 ˆu/2 T b(s)dsdu (u + 1 -s) α + 2 t ˆt t 2 ˆu u/2 b(s)dsdu (u + 1 -s) α = C ( t 2 + 1) α + I + II + III (2.6)
where T ∈ [1, t/4] is a parameter that will be chosen later on. In order to bound I, II and III, we will repeatedly use the following bounds, that holds for all T 1 < T 2 and whose proof is given below

(2.7) ˆT2 T 1 b (s) ds    √ T 2 -T 1 a (T 1 ) Λ (T 2 ) T 1 2 -α 1 if T 1 > 0.
To prove the first inequality, we use Cauchy-Schwarz' inequality. Namely

1 T 2 -T 1 ˆT2 T 1 b (s) ds 2 ˆT2 T 1 b 2 (s) ds = - ˆT2 T 1 d ds a 2 (s) ds a (T 1 ) 2 .
To prove the second inequality, we repeatedly use the latter. set N = ⌈log 2 ( T 2/T 1 )⌉, then we have

ˆT2 T 1 b (s) ds = N -1 n=0 ˆ2n+1 T 1 2 n T 1 b (s) ds N -1 n=0 2 n T 1 a (2 n T 1 ) N -1 n=0 (2 n T 1 ) 1 2 -α Λ (2 n T 1 ) Λ (T 2 ) T 1 2 -α 1 N -1 n=0 2 1 2 -α n Λ (T 2 ) T 1 2 -α 1 .
Now, since T ≤ t/4 and thanks to (2.7), we have

I := 2 t ˆt t 2 ˆT 0 b(s)dsdu (u + 1 -s) α 2 t ˆt t 2 ˆT 0 b(s)dsdu ( t 4 + 1) α = ´T 0 b(s)ds ( t 4 + 1) α √ T a(0) (t + 1) α .
Again using (2.7) we have

II := 2 t ˆt t 2 ˆu/2 T b(s)dsdu (u + 1 -s) α ≤ 2 t ˆt t 2 ´u/2 T b(s)ds ( t 4 + 1) α du ≤ 2 t ´t t 2 Λ( u 2 )T 1 2 -α du ( t 4 + 1) α Λ(t)T 1 2 -α (t + 1) α .
In order to bound the third term, which is more intricate, we first use the Fubini Theorem, noting that 1 t/2 u t 1 u/2 s u 1 t/4 s t 1 s u t , and that α > 1/2 to get

III 2 t ˆt t 4 ˆt s (u + 1 -s) -α dub (s) ds = 2 t ˆt t 4 ˆt-s 0 t ′ + 1 -α dt ′ b (s) ds 2 t ˆt 0 t ′ + 1 -α dt ′ × ˆt t 4 b (s) ds (t + 1) -1 2 -α Λ (t)
Therefore, plugging the previous bounds on I, II and III into 2.6, it holds

(1 + t) α a(t) ≤ A √ T max(C, a(0)) + AΛ(t) T 1 2 -α + 1 √ t
for some constant A that depends only on α. Now, since α > 1/2, there exist t o ≥ 4 and T ≥ 1 such that for all t ≥ t o , 1/ √ t ≤ 1/(4A) and T 1 2 -α ≤ 1/(4A) so that, taking the supremum and using the monotonicity of Λ, it holds Λ(t) ≤ A √ T max(C, a(0)) + 1 2 Λ(t) for all t ≥ t o which leads to the desired conclusion. The proof of the lemma is complete.

Additional remarks

3.1. Completely monotonic functions. In this section, we prove some results on P m x (X t = y), for a given walk scheme m, using the notion of completely monotonic functions. Recall that a function f : (0, ∞) → R is said to be completely monotonic if it possesses derivatives f (n) of all orders and if (-1) n f (n) (x) ≥ 0 for all x > 0 and all n = 0, 1, 2, . . . (see e.g. [START_REF] Feller | An introduction to probability theory and its applications[END_REF][START_REF] Miller | Completely monotonic functions[END_REF]). Proposition 3.1. Let C, α > 0. Assume that f : (0, ∞) → R is a completely monotonic function satisfying for all t > 0, f (t)

C t α . Then, for all t > 0, it holds -f ′ (t) ≤ C ′ t α+1
for some constant C ′ that depends only on C and α. Remark 3.2. At the price of some technicalities, the above result can be extended to more general decay (i.e. replacing 1/t α by some general completely monotonic function g with lim ∞ g = 0).

Proof. Without loss of generality assume that f (0) = 1. It is well known (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF][START_REF] Widder | The Laplace Transform[END_REF]) that f is the Laplace transform of a positive random variable X, namely that, for all t > 0, f (t) = E e -tX where E denotes the mean of the law of X. Using the Markov Inequality we get for all λ > 0 and all x > 0 P (X x) E e -λX e λx C exp (λxα log λ) .

Optimizing over the λ > 0 we get (since inf λ>0 {λxα log λ} = αα log α + α log x (the minimum being reached at λ = α/x)) P (X x) Ce α x α α . Therefore, using Fubini's theorem, for all t > 0 it holds

-f ′ (t) = E Xe -tX = E X ˆ∞ X te -tx dx = ˆ∞ 0 te -tx E [X1 X≤x ] dx ≤ ˆ∞ 0 txe -tx P(X ≤ x)dx ≤ C α t α ˆ∞ 0 (tx) α+1 e -tx dx = C α Γ(α + 2) t α+1
which ends the proof.

Corollary 3.3. There exists a constant C such that for all well-defined walk scheme m and all t > 0 it holds y∈Z

d ,|z|=1 (P m 0 (X t = y) -P m 0 (X t = y + z)) 2 C/ (t + 1) d 2 +1 .
Proof. For p ≥ 1, set f p p := x |f (x) | p and, given an operator P acting on functions, P p→q = sup P f q , where the supremum is taken over all f with f p = 1.

First, we observe that the quantity is bounded as a consequence of Lemma 2.2. As a matter of fact, for all y ∈ Z d and t 0, x P m y (X t = x) 2 = P m y (X t = y) 1. Then, note that t → P m t f 2 2 is a completely monotonic function. It is due to the fact that L m is self-adjoint (< f, L m g >=< L m f, g >) and defines a negative quadratic form (< L m f, f > 0). Indeed, for any function f Observe that for f = 1 {0} (the function that equals 1 at 0 and 0 elsewhere), we have f ∈ ℓ 1 and P m t f (x) = y∈Z d P m x (X t = y)f (y) = P m 0 (X t = x) which plugged in (3.1) gives the desired result.

, ∂ t < P m t f, P m t f >= 2 < LP m t f, P m t f > 0, ∂ t ∂ t < P m t f, P m t f >= 4 < L m P m t f, L m P m t f > 0,
3.2. Gloria, Neukamm and Otto with a fixed walk scheme. Using the monotonic function approach above, we can prove a stronger decay of the variance of the environment view by the particle, when the function f is the divergence of an other function, but only when the walk scheme m is fixed. This is a result (much weaker but) in the spirit of [START_REF] Gloria | Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics[END_REF]. (P m 0 (X t = x) -P m 0 (X t = x + e i )) 2 E g 2 which, combined with Corollary 3.3 gives the expected result.

3.3. Discussion about the polynomial decay. In the introduction, we told that t -d/2 is the optimal decay for local functions in L 2 . Indeed, in [START_REF] De Masi | An invariance principle for reversible markov processes. applications to random motions in random environments[END_REF], they proved that, under the annealed law, the walker X t converges to a Brownian motion, suggesting a diffusive behaviour and this rate of decay as optimal. In [START_REF] Mourrat | Variance decay for functionals of the environment viewed by the particle[END_REF], it is suggested that, using spectral theory, we can construct a non-local function f such that E[f 2 t ] decays as fast or as slow as we want. We note here that one can construct a local function with a faster decay than t -d/2 . Indeed, because E[f 2 t ] is a completely monotonic function, it is a consequence of corrolary 3.1. For example, consider a function g ∈ L 2 such that f = Lg, then

E[f 2 t ] = E[(Lg t ) 2 ] = ∂ t ∂ t E[g 2 t ] C g t -d/2-2
Iterating the process, considering f = L n g, we have that E[f 2 t ] C g t -d/2-2n . Note that in this case, f might not be in L 2 , even if f t would.

Proposition 3. 4 .

 4 There exists a constant C > 0 such that for almost all walk scheme m, all t 0 and all function f = D i g, where-d i d, g is local, translation-invariant and E[g 2 ] < ∞, it holds E (P m t f (0, ω)) 2 C#supp(g) 2 E[g 2 ]Using the Cauchy-Schwarz Inequality and that E [g (x, ω) g (y, ω)] = 0 as soon as supp (g(x, ω)) ∩ supp (g(y, ω)) = ∅ we haveE (P m t f (0, ω)) 2 = E 0 (X t =xe i ) -P m 0 (X t = x)) g (x, ω)

  and because L m commutes with P m t , we can conclude repeating these last arguments with the function L m f . Hence, using Proposition 3.1, there exists a constant C ′ that depends only on d and C such that for all f ∈ ℓ 1 , d

	dt P m t f 2 2 On the other hand, by definition of L m and P m C ′ /t d 2 +1 . t , we have
	(3.1)	d dt	P m t f 2 2 = -	x,|z|=1	m x,x+z (P m t f (x + z) -P m t f (x)) 2

Acknowledgement. We thank Cyril Roberto and Julien Bureaux.


Gilles Hug

Chengjie Lu

Liangbo Sun

Jie Zhang
email: hitzhangjie@hit.edu.cn

Qin Qi

Brazing mechanisms of the Ti 2 AlC joints using a pure Al filler metal

Keywords: Ti 2 AlC, aluminum filler, brazing, mechanical property, electrical conductivity

Brazing mechanisms of the Ti 2 AlC joints using a pure

Al filler metal

filler has weak effect on the stability of Ti 2 AlC substrate, and only a small amount of decomposition products including the TiC x and TiAl 3 compounds can be observed in the joints. In addition, the formation of an inter-diffusion layer in the Ti 2 AlC substrate is considered as the major brazing mechanism. The maximum shear strength of the Ti 2 AlC joints using the pure Al filler metal is 95 MPa, with an electrical conductivity of 2.21×10 6 S•m -1 , after holding the sample at 1123 K for 10 min.

Introduction

The MAX phases, which refer to a group of nanolaminate ternary ceramics with general formula M n+1 AX n (M, transition metal; A, A-group element; X, carbon or nitrogen), have attracted increasing attention in recent years due to their unique combination of metallic and ceramic properties [START_REF] Barsoum | The M n+1 AX n phases: a new class of solids; thermodynamically stable nanolaminates[END_REF][START_REF] Radovic | MAX phases: bridging the gap between metals and ceramics[END_REF]. Among them, the Ti 2 AlC phase is one of the most fascinating members with typical 211 structure (i.e. n=1), since it possesses the lowest density of all compounds in this class [START_REF] Sun | Progress in research and development on MAX phases: a family of layered ternary compounds[END_REF], an excellent electrical conductivity [START_REF] Bai | Microstructures, Electrical, Thermal, and Mechanical Properties of Bulk Ti 2 AlC Synthesized by Self-Propagating High-Temperature Combustion Synthesis with Pseudo Hot Isostatic Pressing[END_REF] and a small friction coefficient [START_REF] Gupta | Tribological behavior of select MAX phases against Al 2 O 3 at elevated temperatures[END_REF]. Such characteristics qualify the Ti 2 AlC ceramics for many structural and functional applications as, for example, the heat/electron exchanger or high temperature solid lubricant. Numerous efforts have been carried out to synthesize Ti 2 AlC ceramics, via various techniques like combustion synthesis (CS) [START_REF] Thomas | Thermodynamic predictions for the manufacture of Ti 2 AlC MAX-phase ceramic by combustion synthesis[END_REF], spark plasma sintering (SPS) [START_REF] Kulkarni | Synthesis of Ti 2 AlC by spark plasma sintering of TiAl-carbon nanotube powder mixture[END_REF] and self-propagating high-temperature synthesis (SHS) [START_REF] Thomas | Effect of particle size on the formation of Ti 2 AlC using combustion synthesis[END_REF]. However, it is still difficult to process dense Ti 2 AlC ceramics with high purity, large dimensions and complex shapes, because of the narrow single Ti 2 AlC phase domain in the Al-C-Ti ternary phase system [START_REF] Witusiewicz | Thermodynamic description of the Al-C-Ti system[END_REF], which severely limits its applications. Joining technique is one of the convenient and promising solutions to overcome this problem. By achieving the Ti 2 AlC joints, the limitation of dimensions can be overpassed, and the component with complex shapes can be obtained by joining small bulks.

Only few papers reporting about the joining of the MAX ceramics can be found in the past years, among which the diffusion bonding technique was mostly adopted.

Inspired by the phenomenon that the MAX phases could form various solid solutions like (M, M')AX and M(A, A')X, Ganguly et al. [START_REF] Ganguly | Interdiffusion between Ti 3 SiC 2 -Ti 3 GeC 2 and Ti 2 AlC-Nb 2 AlC diffusion couples[END_REF] could successfully bond Ti 3 SiC 2 -Ti 3 GeC 2 and Ti 2 AlC-Nb 2 AlC couples using a direct diffusion method.

Interestingly, they discover that the Ti 3 SiC 2 -Ti 3 GeC 2 diffusion couple could form in the temperature range 1473 K -1773 K, by forming a continuous gradient of Ti 3 (Si, Ge)C 2 solid solutions. However, the diffusion temperature to obtain the Ti 2 AlC-Nb 2 AlC couple was much higher, about 1723 K -1773 K, which was mainly attributed to the strong M-X bonds in the MAX structure that need to break and reconstruct during the diffusion process. Similarly, Yin et al. [START_REF] Yin | Diffusion bonding of Ti 3 AlC 2 ceramic via a Si interlayer[END_REF] managed to join two samples of Ti It can be concluded from the above analysis that extremely high temperature is required to achieve the MAX phase diffusion couple by forming solid solutions, which is considered to be even higher than the decomposition temperature of the ceramic substrate under vacuum [START_REF] Pang | Iop, Comparison of thermal stability in MAX 211 and 312 phases[END_REF]. To decrease the processing temperature, as well as to overcome some disadvantages of the diffusion technique, we have performed a study of brazing Ti 2 AlC ceramics using a pure Ag filler metal [START_REF] Wang | Microstructure evolution and brazing mechanism of Ti 2 AlC-T 2 AlC joint by using pure-silver filler metal[END_REF]. The results suggested that the silver filler could diffuse into the Ti 2 AlC substrate along the grain boundary, and then formed an Ag 3 (Ti, Al) solid solution in the interaction area. However, no TiC x compound, the typical decomposition product of the Ti 2 AlC ceramics, was detected in the XRD pattern, suggesting that the interaction effect between the silver filler and the Ti 2 AlC substrate would not induce a decomposition behavior of Ti 2 AlC ceramics [START_REF] Lu | Substitution Behavior of Ag Atoms in the Ti 2 AlC Ceramic[END_REF].

Finally, the maximum shear strength of the Ti 2 AlC joints using pure Ag filler metal was measured to be 132 MPa, and was obtained for a brazing temperature of 1030 °C holding for 5 min.

The goal of this work is to further decrease the brazing temperature of Ti 2 AlC ceramics, by using a pure Al filler metal. In addition, it is necessary to characterize the interaction behavior of the filler elements with the Ti 2 AlC substrate, to better understand the intrinsic structural features of MAX phases.

Experimental procedure

The bulk polycrystalline Ti 2 AlC selected in this study was processed by spark plasma sintering (SPS) using Ti, Al and graphite powders. In general, minority phases, including the TiC x , TiAl x and Ti 3 AlC 2 compounds are always observed in Ti 2 AlC compounds, therefore, a detailed characterization of the raw material was performed. A backscattered electron (BSE) micrograph of the Ti 2 AlC substrate reveals its morphology as displayed in Fig. 1(a). Two phases with different contrasts can be observed. Firstly, the grey phase labeled A in the figure possesses an average atomic percentage of Ti 49%, Al 24% and C 27% according to the EDS analysis. It is confirmed to be the Ti 2 AlC main phase. Secondly, the light phase labeled B is identified as the TiAl x intermetallic, with an average atomic percentage of Ti 72% and Al 28%. Afterwards, an X-ray diffraction (XRD) characterization was performed, and the corresponding result is shown in Fig. 1(b). The major phase showing up in the XRD pattern is the Ti 2 AlC compound, and the minor phase is determined to be the Ti 3 Al compound, corresponding to the light phase B discovered in the BSE morphology. In order to obtain the Ti 2 AlC brazing joints with excellent heat/electron conductivity, the pure Al filler metal has been selected in the present work. In addition, the brazing filler with high Al content might be favorable for maintaining the stability of the Ti 2 AlC substrate during the brazing process, since the depletion of Al atoms in the Ti 2 AlC structure is considered to be the major cause of the decomposition behavior [START_REF] Pang | In situ diffraction study of thermal decomposition in Maxthal Ti 2 AlC[END_REF]. Although Ti 2 AlC joints brazed using a pure Al filler metal might not be applicable in many high-temperature areas, it is still a convenient way to achieve the Ti 2 AlC ceramic components which serve at room or intermediate temperature.

The bulk Ti 2 AlC ingots were cut into samples with dimensions of 4×4×4 mm 3 for microstructural observation, and samples of 4×4×10 mm 3 for shear strength testing and electrical conductivity test. The surface of the specimens was ground using the 800-grit SiC paper, followed by the mechanical polishing using the 0.5 μm diamond paste.

Commercial pure Al foils 270 μm thick were cut into pieces with dimensions of 4×4 mm 2 , and then etched in a 10 wt. % NaOH solution and a 5 vol. % HNO 3 solution, to remove the native Al 2 O 3 oxide scale. Afterwards, the Al foils were cleaned with water, and then preserved in ethanol. Two Ti 2 AlC rectangular specimens together with a pure Al foil were assembled in a sandwich structure: Ti 2 AlC/Al/Ti 2 AlC, and then placed in a graphite jig, on which a constant load of 5 MPa was applied to promote the contact of brazing filler with the ceramic substrates. A set of brazing experiments were performed under vacuum in a furnace, at brazing temperatures of 1023 K, 1073 K, 1123 K and 1173 K, and holding times of 0 min, 10 min and 30 min. A heating rate of 10 K•min -1 was used while a slower cooling rate of 5 K•min -1 was set in order to reduce the thermal stress in the joints that may form during the cooling period.

After the brazing experiments, shear strength tests were firstly conducted to obtain the optimal brazing parameters. The measurements were performed using a universal testing machine (Instron 5569) at room temperature, with a constant loading speed of 0.5 mm•min -1 . Three samples were tested for each brazing parameter. Furthermore, a special designed jig was designed for this work [START_REF] Lu | Microstructure evolution and brazing mechanisms of the Ti 2 AlC/Ni joints using nickel based filler alloy[END_REF] in order to minimize the influence of friction on the results of the tests: the jig was able to adjust the gap width accordingly.

Considering the excellent conductivity of both the Ti 2 AlC substrate and the Al filler, the electrical conductivity of Ti 2 AlC joints at different brazing parameters was measured using the four-probe technique. The brazing mechanisms were analyzed by observing the morphologies and characterizing the phases in the joints. The morphology of the joints were investigated using a Sigma 500, Zeiss scanning electron microscope (SEM), which is equipped with an energy-dispersive spectrometer (EDS) to determine the elemental composition of the phases observed in the joints. In addition, the crystallographic structures of the bulk Ti 2 AlC substrate, as well as the phases in the joints, were characterized using an X'pert pro MPD, Panalytical X-ray diffraction (XRD) using the CuKα radiation and steps of 0.05°at a scanning rate of 5°•min -1 . improvement might be attributed to the reinforced interaction effect between the Ti 2 AlC substrate and the pure Al brazing filler: with the increasing brazing temperature, the wetting process, as well as the inter-diffusion or reaction process is promoted. However, when the brazing temperature reaches 1173 K, the shear strength of the Ti 2 AlC joints declines to 75 MPa. The decreasing of mechanical properties at high temperature might be attributed to many factors, including a severe transformation of the ceramic substrate due to the changing phase equilibrium, the formation of continuous reaction products and the consumption of brazing filler in the joints. Overall, the optimal brazing temperature of the Ti 2 AlC joint is determined to be 1123 K in the present work. Fig. 2(b) shows the shear strength of the Ti 2 AlC joints brazed at the optimal temperature, with holding time of 0 min, 10 min and 30 min, respectively. It can be concluded from the figure that extending the holding time results in a similar tendency whatever the brazing temperature is. When the holding time prolongs from 0 min to 10 min, the shear strength of the Ti 2 AlC joints increases from 53 MPa to 95 MPa, which may be attributed to the reinforced interaction effect as well; when the holding time further extends up to 30 min, however, the shear strength decreases to 30 MPa. Overall, the optimal holding time is determined to be 10 min in this work. The microstructure is considered to have a significant effect on the mechanical properties of the joints.

Results and discussion

Shear strength test and electrical conductivity measurement

Therefore, a detailed observation of the morphology has been performed in the following sections to further illustrate how the microstructure evolution affects the shear strength of the Ti 2 AlC joints.

The electrical conductivity measurement results are also displayed in Fig. 2. Using the four-probe technique, the electrical conductivity of the Ti 2 AlC substrate in this work is measured to be 2.38×10 6 S•m -1 , close to that reported in other works [START_REF] Barsoum | Processing and characterization of Ti 2 AlC, Ti 2 AlN, and Ti 2 AlC 0.5 N 0.5[END_REF]. According to the diagrams, changing the brazing parameters has little effect on the electrical conductivity of the Ti 2 AlC joints: with the rising brazing temperature, the electrical conductivity increases from 2.12×10 6 S•m -1 at 1023 K, to 2.23×10 6 S•m -1 and 2.21×10 6 S•m -1 at 1073 K and 1123 K, respectively, then decreases to 2.13×10 6 S•m -1 at 1173 K; with the extending holding time, the electrical conductivity changes from 2.17×10 6 S•m -1 for 0 min to 2.21×10 6 S•m -1 for 10 min, and finally declines to 2.10×10 6 S•m -1 for similarly with the brazing parameters, suggesting that the two properties might be driven by the same microstructure characteristics of the Ti 2 AlC joints.

In summary, the optimal brazing temperature and holding time in the present work have been determined to be 1123 K and 10 min, respectively. The maximum shear strength of the Ti 2 AlC joints using the pure Al filler metal is measured to be 95 MPa, with the electrical conductivity of 2.21×10 6 S•m -1 , accounts for 43% and 93% of the Ti 2 AlC substrate, respectively.

Microstructure observation and phase characterization

The morphologies of the Ti 2 AlC joints brazed at different temperatures are displayed in Fig. 3. In the diagrams, two phases with different morphologies can be observed in the filler area. Firstly, a phase with a light grey contrast is labeled A in the figure. It has a rod-like shape and it generally originates from the interaction surface. Secondly, a dark grey phase labeled B in the figure, makes up the major part of the filler area. The EDS results suggest that the elemental composition of the phase A is Ti 24.2 Al 75.8 (in atomic percentage), and that of the phase B is Ti 0.9 Al 99.1 (in atomic percentage). Therefore, the light grey phase A is suggested to be the TiAl 3 intermetallic compound, while the dark grey phase B is supposed to correspond to the Al filler metal.

It is worthwhile pointing that the formation of the TiAl 3 intermetallic compound has also been discovered in the Al alloy/Ti 2 AlC composites produced by infiltration of sponge of MAX phase with molten Al metal (or alloy) [START_REF] Hu | High-Performance Metal/Carbide Composites with Far-From-Equilibrium Compositions and Controlled Microstructures[END_REF], in spite of the short infiltration time of approximately 30 seconds. Therefore, it is reasonable to suggest that Both the two reactions are reasonable considering thermodynamic aspect, which has been verified by Witusiewicz et al [START_REF] Witusiewicz | Thermodynamic description of the Al-C-Ti system[END_REF], and the reaction products differ depending on the Al concentration and temperature. However, the other by-products of the reactions ( 1) and ( 2), either the Ti 3 AlC 2 or the TiC x compound, are not observed in the BSE micrographs. It is worthwhile pointing out that the close average atomic number between TiC compound (𝑍 ̅ TiC =14) and Al metal (𝑍 ̅ Al =13), as well as that between Ti 2 AlC phase (𝑍 ̅ Ti 2 AlC =15.75) and Ti 3 AlC 2 phase (𝑍 ̅ Ti 3 AlC 2 =15.17) renders it difficult to discriminate these phases from BSE images. Therefore, the presence of these by-products should be investigated using other characterization techniques.

Nevertheless, a microstructure evolution with the temperature can be evidenced from the micrographs. With the increasing temperature, an apparent rise in the content of the TiAl 3 compound can be observed in the joints, which, again, may be attributed to the heavier decomposition degree of the Ti 2 AlC substrate. Therefore, the reinforced wetting and interaction process are considered to be the major factors contributing to the [START_REF] Barsoum | The M n+1 AX n phases: a new class of solids; thermodynamically stable nanolaminates[END_REF] (2) improved mechanical property of the Ti 2 AlC joints. In addition, the morphology of the TiAl 3 compound changes as well: at low brazing temperature, the TiAl 3 compound with low aspect ratio forms in the joints, originates from the wetting surface; while brazed at high temperature, the aspect ratio of the TiAl 3 compound increases significantly, and the grains even cross the whole joint at 1173 K. Since the TiAl 3 intermetallic compound possesses poorer deformation ability than the Al metal filler, cracks may easily be initiated from the TiAl 3 compound, which contributes to the declining shear strength of the Ti 2 AlC joints at 1173 K. Furthermore, a measurement performed under the SEM suggests that the width of the Ti 2 AlC joints brazed at 1023 K, 1073 K and 1123 K are very close to each other, as they are determined to be 31.8 μm, 34.4 μm and 29.6 μm, respectively. However, when the brazing temperature reaches up to 1173 K, the width remarkably reduces to 16.4 μm, which is considered to be another factor responsible for the poor mechanical property of the joints at high temperature.

Fig. 4 shows the morphology of the Ti 2 AlC joints brazed at 1123 K, holding for 0 min and 30 min, respectively. It can be concluded from the figure that the extending holding time has similar effect on the microstructure evolution of the brazing joints with the increasing brazing temperature. When the holding time is 0 min, the wetting process, as well as the interaction process is not sufficient. Therefore, a small amount of TiAl 3 intermetallic compound with small aspect ratio forms on the interface. Afterwards, when the holding time prolongs, more TiAl 3 compound can be observed in the joints and the shear strength remarkable improves, which is attributed to the reinforced interaction between Ti 2 AlC substrate and pure Al metal filler. In addition, the holding time is supposed to have more significant effect on the width of the Ti 2 AlC joints. It can be observed from the figures that the joints width remarkably decreases from 60.5 μm to 29.6 μm when the holding time is extended from 0 min (Fig. 4(a)) to 10 min (Fig. 3(c)), and then further declines to 20.9 μm for 30 min (Fig. 4(b)). The major factor responsible for the narrowing width of the Ti 2 AlC joints is assumed to be the squeezed out liquid Al filler during the brazing process, due to the improved wettability and fluidity with extending holding time, which is verified by the macro morphology observation.

According to the conclusion above, the decreasing width of the joints would lead to an increase of the relative fraction of the TiAl 3 intermetallic compound in the Ti 2 AlC joints, which is considered to be unfavorable for the mechanical property. Correspondingly, the shear strength declines to a low level when the holding time is 30 min.

The pure Al metal filler adopted in the present work is considered to be favorable for restraining the outward-diffusion of Al atoms in the Ti 2 AlC structure, thereby preventing the transformation of the ceramic substrate. However, a small amount of decomposition products can still be observed in the Ti 2 AlC joint. In order to determine the reaction products in the joints, thereby revealing the decomposition mechanisms of the Ti 2 AlC ceramic during the brazing process, some supplementary characterizations were performed. According to the reaction (1), the increasing Al content in the substrate would contribute to the transformation of the Ti 2 AlC phase to the Ti 3 AlC 2 phase, accompanied with the formation of the TiAl 3 intermetallic compound. Therefore, the reaction product Ti 3 AlC 2 might distribute in the Ti 2 AlC substrate if the reaction proceeds following the equation ( 1). An EDS line scan was carried out across the joining interface, and the results are displayed in Fig. 5. Fig. 5(a) shows the magnified morphology of the interaction area in the Ti 2 AlC brazing joints. It is worthwhile pointing out that the grain boundary of the Ti 2 AlC ceramic can be apparently observed in the interaction area, which is mainly attributed to the diffusion of the Al filler into the ceramic substrate along the grain boundary. In addition, the yellow line in the figure illustrates the scanning path, and the corresponding EDS line scan result is shown in Fig. 5(b). It can be concluded from the figure that an inter-diffusion layer with a thickness of 2.6 μm forms in the interaction area (the corresponding position is labeled in the Fig. Although it is difficult to characterize the phases in the inter-diffusion layer accurately using the SEM and XRD techniques, the formation of such a thin layer would not be the major factor responsible for large quantities of the TiAl 3 compound in the joints.

According to the reaction (2), part of the Ti 2 AlC substrate would decompose during the brazing process, and then forms the TiAl 3 and TiC x compounds. In general, the TiC x phase with nanolaminate microstructure can be dissolved by the liquid filler, therefore, the reaction product TiC x might be able to be found in the filler area if the reaction proceeds following the equation ( 2). Afterwards, an XRD analysis of the brazing joints was performed in the following way: the Ti 2 AlC brazing couple was cut along one of the substrate sides and then cautiously ground until the joint was exposed. The corresponding XRD pattern is displayed in Fig. 5. It can be observed from the diagram that the Al filler metal is the major phase in the Ti 2 AlC joints. In addition, the TiC x compound, as well as the TiAl 3 intermetallic compound is detected, suggesting that the decomposition of Ti 2 AlC substrate during the brazing process contributes to the formation of TiAl 3 and TiC x compounds, following the reaction [START_REF] Radovic | MAX phases: bridging the gap between metals and ceramics[END_REF].

Conclusions

In the present work, joining of the Ti 2 AlC ceramics has been successfully performed using the pure Al filler metal, at the brazing temperature of 1023 K, 1073 K, 1123 K and 1173 K, holding for 0 min, 10 min and 30min, respectively. The results lead to the following conclusions:

(1). The maximum shear strength of the Ti 2 AlC joints brazed using the pure Al filler metal is measured to be 95 MPa, together with an electrical conductivity of 2.21×10 6 S•m -1 . This optimal characteristic is obtained at the brazing temperature of 1123 K and holding for 10 min. The shear strength of the brazing joints accounts for 43% of the shear strength of the Ti 2 AlC substrate, while the electrical conductivity accounts for 93%.

(2). During the brazing process, an inter-diffusion layer forms in the Ti 2 AlC substrate, in which the Ti atoms diffuses into the brazing filler while the Al atoms diffuses into the ceramic substrate. The outward-diffusing Ti atoms can be rapidly consumed by forming the TiAl 3 compound in the joints, while the inward-diffusing Al atoms, however, mainly distribute on the grain boundary of the Ti 2 AlC substrate.

(3). Since the outward-diffusion of Al atoms in the Ti 2 AlC structure has been restrained by utilizing the brazing filler with high Al content, only a small amount decomposition products, including the TiAl 3 and TiC x compounds can still be observed in the Ti 2 AlC joint. The weak interaction between Ti 2 AlC ceramic and pure Al filler is considered to be favorable for restraining the thermal stability of the Ti 2 AlC brazing joints.

3

 3 AlC 2 ceramics by inserting a Si interlayer. The experiments were performed in the temperature range 1573 K -1673 K, for 120 min. The presence of a Ti 3 (Si, Al)C 2 solid solution domain was confirmed by XRD, and a maximum flexural strength of the joints of 285 MPa, was obtained after holding the sample at 1673 K for 120 min, accounting for 80% of the strength of the Ti 3 AlC 2 substrate.

Fig. 2 (

 2 Fig. 2(a) shows the shear strength of the Ti 2 AlC joints brazed at different

the Ti 2

 2 AlC ceramics can readily react with molten Al during the brazing process. The formation of the TiAl 3 intermetallic compound can be explained by the equilibrium in the Al-C-Ti phase diagram. Indeed a transformation of the Ti 2 AlC ceramics might occur under the presence of Al. The following reactions can provide a source of Ti atoms according to: 2Ti 2 AlC + 2Al → Ti 3 AlC 2 + TiAl 3 Ti 2 AlC + 2Al → TiC + TiAl 3

5

  (a)), in which the Ti atoms diffuses into the brazing filler while the Al atoms diffuses into the ceramic substrate. The outward-diffusing Ti atoms can be rapidly consumed by reacting with the Al filler metal, and then forms the TiAl 3 intermetallic compound which nucleates on the interface because of the small temperature gradient. Meanwhile, the Al atoms diffuse into the Ti 2 AlC substrate since the pure Al metal filler possesses higher Al concentration. However, the Al content of the Ti 2 AlC substrate in the interaction area remains unchanged according to the EDS results. Considering the morphology observation in Fig.5(a), it is reasonable to suppose that the inward-diffusion Al atoms mainly distribute on the grain boundary of the Ti 2 AlC ceramic, and would not lead to the transformation from Ti 2 AlC phase to Ti 3 AlC 2 phase.

Fig. 1 .

 1 Fig. 1. Characterization of the raw Ti 2 AlC ceramics: (a). The BSE morphology; (b). The

Fig. 2 .

 2 Fig. 2. The shear strength and electrical conductivity of the Ti 2 AlC joints: (a) achieved

Fig. 3 .

 3 Fig. 3. The microstructure of the Ti 2 AlC joints, brazed at (a). 1023 K; (b). 1073 K; (c).

Fig. 4 .

 4 Fig. 4. The microstructure of the Ti 2 AlC brazing joints holding for (a). 0 min and (b). 30

Fig. 5 .

 5 Fig. 5. The EDS line scanning results across the interaction area: (a). The BSE

Fig. 6 .

 6 Fig. 6. The XRD pattern collected in the filler area of the Ti 2 AlC brazing joints.

Acknowledgement

This work was financially supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51372049, 51621091 and U1537206. The authors gratefully acknowledge the financial support from the China Scholarship Council (CSC).