Raphaël Cerf

Sana Louhichi

Dynamical coupling between Ising and FK percolation

We investigate the problem of constructing a dynamics on edge-spin configurations which realizes a coupling between a Glauber dynamics of the Ising model and a dynamical evolution of the percolation configurations. We dream of constructing a Markov process on edge-spin configurations which is reversible with respect to the Ising-FK coupling measure, and such that the marginal on the spins is a Glauber dynamics, while the marginal on the edges is a Markovian evolution. We present two local dynamics, one which fulfills only the first condition and one which fulfills the first two conditions. We show next that our dream process is not feasible in general. We present a third dynamics, which is non local and fulfills the first and the third conditions. We finally present a localized version of this third dynamics, which can be seen as a contraction of the first dynamics.

Introduction

The Ising model, invented almost a century ago, is one of the most studied model of statistical mechanics. A fundamental tool to analyze the Ising model is the random cluster model, or FK model, invented by Fortuin and Kasteleyn around 1969 (see the reference book [START_REF] Grimmett | The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften[END_REF]). Results on this dependent percolation model can be transferred towards the Ising model via a coupling construction due to Edwards and Sokal [START_REF] Edwards | Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm[END_REF]. This machinery works extremely well, but so far, it has essentially been employed to study the systems at equilibrium. Yet another rich facet of the Ising model is the dynamics. There exist several microscopic dynamics on spin configurations, which give rise to the so-called stochastic Ising model, and whose equilibrium are described by the Ising Gibbs measure: the Metropolis dynamics, the heat-bath dynamics, the Kawasaki dynamics, to name a few of them. On one hand, these dynamics provide a basic model to study fundamental questions on dynamics, for instance to model the metastability phenomenon [START_REF] Olivieri | Large deviations and metastability, volume 100 of Encyclopedia of Mathematics and its Applications[END_REF]. On the other hand, there is a hope that the understanding of the dynamics will shed light on the equilibrium measure.

Percolation models possess also a natural dynamics. For the Bernoulli percolation model, it consists in updating independently the edges, and it leads to beautiful difficult problems [START_REF] Häggström | Dynamical percolation: early results and open problems[END_REF][START_REF] Steif | A survey of dynamical percolation[END_REF]. For the FK model, the construction of a dynamics is more subtle and it involves typically non-local computations [START_REF] Grimmett | The stochastic random-cluster process and the uniqueness of random-cluster measures[END_REF]. One naturally wonders whether the stochastic Ising model and dynamical percolation could help to understand each other. Our note is a little investigation into the possibility of building a coupling for the dynamics on Ising and percolation models, which could help to understand the dynamics of the Ising model. We focus here on the case of a Glauber type dynamics, that is a local dynamics which modifies at most one spin at a time. The precise definition of such dynamics, together with several examples, are given in section 3. We dream of building a coupling dynamics on edge-spins configurations such that:

• The marginal on the spins is a Glauber dynamics.

• The marginal on the edges is a simple Markovian evolution.

• The coupling dynamics on edge-spins configurations is reversible with respect to the coupling measure between the Ising and the FK models.

Notice that if we drop the first constraint, then we could simply consider a time continuous Markov process on the edge-spins configurations which, after an exponential time of parameter one, jumps on a new independent edge-spins configuration, drawn according to the Ising-FK coupling measure. However the first constraint is essential for us, indeed our hope is to build a dynamical coupling which would help to study a Glauber dynamics of the Ising model.

The definition of the Ising-FK coupling measure is recalled in section 2. The stochastic Ising model is defined in section 3. The FK dynamics is defined in section 4. The above conditions are precisely stated in section 5. Unfortunately, we did not succeed in building a dynamics which fulfill these three conditions, even if we weaken partially the last one. However, we manage to build couplings which satisfy the third condition. The most basic coupling is a dynamics which changes at most one object at each step, i.e., either one spin or one edge is modified at a time. We prove that, for this type of coupling, the first condition can never be satisfied. We build such a coupling dynamics in section 6. Another natural dynamics consists in changing simultaneously the spin at one vertex, together with the edges incident to this vertex. Such a dynamics is presented in section 7. The good news is that the marginal of this dynamics on the spins is a Glauber dynamics. However the marginal of this type of dynamics on the edges is not Markovian. In section 8, we show that our dream process is not feasible in general. We present a third dynamics in section 9, which is non local and fulfills the first and the third conditions. We finally present in section 10 a localized version of this third dynamics, which can be seen as a contraction of the first dynamics.

Ising-FK coupling measure

In this section, we recall some classical notation and we define the Ising-FK coupling measure IP . We consider a finite graph (V, E), where V is the set of the vertices and E the set of the edges. The edges are unoriented, and E is a subset of the set of pairs of points of V . Throughout the paper, we suppose that the set of edges E is not empty. An edge configuration η is an element of {0, 1} E (where 0 stands for closed and 1 for open). A spin configuration σ is an element of {-1, 1} V . An edge e with endvertices x and y is written as e = x, y or e = y, x . We define

∀σ ∈ {-1, 1} V ∀e = x, y ∈ E δ σ (e) = 1I σ(x)=σ(y) .
An edge configuration η and a spin configuration σ are said to be compatible if we have

∀e ∈ E η(e) ≤ δ σ (e) ,
i.e., if the endvertices of any open edge in η have the same spins in σ. We denote by C the set of the pairs of compatible configurations:

C = (η, σ) ∈ {0, 1} E × {-1, 1} V : η(e) ≤ δ σ (e) for any e ∈ E .
Let p be a fixed number in [0, 1]. The Ising-Percolation measure IP is the probability measure on the product space {0, 1} E × {-1, +1} V defined as follows. For any (η, σ) in

{0, 1} E × {-1, +1} V , we set IP (η, σ) = 1 Z e∈E p1I η(e)=1 δ σ (e) + (1 -p)1I η(e)=0 ,
where Z is the partition function, i.e., the normalising constant that makes IP a probability measure on {0, 1} E × {-1, +1} V , given by

Z = (η,σ)∈{0,1} E ×{-1,+1} V e∈E p1I η(e)=1 δ σ (e) + (1 -p)1I η(e)=0 .
The Ising-Percolation measure can also be written as

IP (η, σ) = 1 Z p |{e∈E: η(e)=1}| (1 -p) |{e∈E: η(e)=0}| 1I (η,σ)∈C , (1)
where the absolute value of a set denotes its cardinality. In fact, the measure IP is the Bernoulli product measure on edge-spin configurations with parameters (p, 1/2) conditioned to the set C of the compatible configurations. It is well-known that the probability measure IP is a coupling of the Ising measure µ β defined on {-1, +1} V , together with the random-cluster measure φ p,2 defined on {0, 1} E , with the relation p = 1 -e -β . In fact, the first marginal measure φ p,2 on {0, 1} E is given by

∀ η ∈ {0, 1} E φ p,2 (η) = σ∈{-1,+1} V IP (η, σ) = 1 Z RC e∈E p η(e) (1 -p) 1-η(e) 2 k(η) . (2)
Here k(η) denotes the number of connected components (or open clusters) of the graph having for vertices V and for edges the open edges in the configuration η (recall that an edge e is said to be open in η if η(e) = 1 and closed if η(e) = 0). Naturally, Z RC is the partition function that makes φ p,2 a probability measure on {0, 1} E . The second marginal measure is given by

∀ σ ∈ {-1, +1} V η∈{0,1} E IP (η, σ) = 1 Z (1 -p) |{ e∈E:δσ(e)=0 }| . (3)
Notice that

|{ e ∈ E : δ σ (e) = 0 }| = e∈E 1 -δ σ (e) = |E| - e∈E δ σ (e) .
Let β > 0 be such that p = 1 -e -β . Rewriting the formula (3) with these notations, we see that the second marginal is the measure µ β on {-1, +1} V given by

∀ σ ∈ {-1, +1} V µ β (σ) = 1 Z I exp β e∈E δ σ (e) , (4)
where the partition function Z I is

Z I = Z exp(β|E|) .
Let us rewrite the Hamiltonian in a more classical way. For an edge e with endpoints x, y, we have

δ σ (e) = 1 2 1 + σ(x)σ(y) . (5)
Summing over e ∈ E, we get e∈E δ σ (e) =

x,y∈V {x,y}∈E

1 2 1 + σ(x)σ(y) = 1 2 |E| + 1 2
x,y∈V {x,y}∈E σ(x)σ(y) .

The standard Hamiltonian of the Ising model is defined as

∀σ ∈ {-1, +1} V H(σ) = - 1 2 x,y∈V {x,y}∈E σ(x)σ(y) . (6)
In the end, we have

∀ σ ∈ {-1, +1} V µ β (σ) = 1 Z β exp (-βH(σ)) , (7)
where the partition function Z β is given by

Z β = σ∈{-1,+1} V exp (-βH(σ)) = Z I exp - 1 2 β|E| .
We refer to the book of Grimmett [START_REF] Grimmett | The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften[END_REF] and the references therein for more details about the coupling measure IP between random-cluster and Ising measures, its history and usefulness. We refer to the paper of Schonmann [START_REF] Roberto H Schonmann | Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region[END_REF] for a nice presentation of the Gibbs measure of the Ising model and the associated dynamics.

The stochastic Ising model

In this section, we define Glauber type dynamics for the Ising model. Each of these dynamics defines a Markov process (σ t ) t≥0 on the spin configurations which is called a stochastic Ising model. A Markov process (σ t ) t≥0 on the spin configurations is classically defined through its infinitesimal generator L (see [START_REF] Thomas | Interacting particle systems[END_REF]). This generator L acts on functions f of the spin configuration and it is of the following form. For any function f : {-1, +1} V → IR, we have

∀ σ ∈ {-1, +1} V Lf (σ) = σ ′ ∈{-1,+1} V c(σ, σ ′ )(f (σ ′ ) -f (σ)) .
The quantity c(σ, σ ′ ) is the rate at which the configuration σ is transformed into σ ′ when the system is in the state σ. We are usually interested in local dynamics which modify at most one spin at a time. We call this type of dynamics Glauber type dynamics. For a configuration σ and a vertex x, we denote by σ x the configuration obtained from σ by flipping the spin at the site x. So we require that c(σ, σ ′ ) vanishes when σ and σ ′ differ in more than one spin and the quantity c(σ, σ x ), sometimes denoted by c(x, σ), is the rate at which the spin at the site x flips when the system is in the state σ. The transition rates c(•, •) are defined on {-1, 1} V × {-1, 1} V and have thus the following properties: for any

σ ∈ {-1, +1} V and x ∈ V , c(σ, σ ′ ) = 0 if |{x ∈ V, σ(x) = σ ′ (x)}| ≥ 2, c(σ, σ x ) > 0 for any x ∈ V, (8)
c(σ, σ) = -y∈V c(σ, σ y ).

We say that the transition rates c(•, •) satisfy the detailed balance condition with respect to the Ising measure µ β if

∀ σ ∈ {-1, 1} V ∀ x ∈ V µ β (σ)c(σ, σ x ) = µ β (σ x )c(σ x , σ) . (9)
When this detailed balance condition holds, the associated dynamics is reversible with respect to the Ising measure µ β . Several rates satisfy the conditions (8) and the detailed balance condition [START_REF] Steif | A survey of dynamical percolation[END_REF]. These dynamics and their fundamental properties are presented in the paper of Schonmann [START_REF] Roberto H Schonmann | Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region[END_REF]. In the same paper appears the graphical construction of these dynamics, which provides a natural and intuitive picture of the associated Markov processes. Let us present some classical choices. Recalling the definition of the Ising Hamiltonian (6), we define further, for σ ∈ {-1, +1} V and x ∈ V ,

∆ x H(σ) = H(σ x ) -H(σ) . (10)
For dynamics which modify at most one spin at a time, we set c(x, σ) = c(σ, σ x ) .

Here are some possible choices for the rates:

The Metropolis dynamics:

c(x, σ) = exp -β max ∆ x H(σ), 0 .
The Heat bath dynamics:

c(x, σ) = 1 1 + exp β max ∆ x H(σ), 0)
.

Unnamed dynamics:

c(x, σ) = exp - β 2 ∆ x H(σ) .
These three choices satisfy the detailed balance condition ( 9) and correspond to one spin flip dynamics. We call them Glauber type dynamics. Let us try to express the rates with the help of the functions δ σ on the edges. Using the identity (5), we have, for any x ∈ V ,

∆ x H(σ) = y∈V :{x,y}∈E σ(x)σ(y) = e∈Ex 2δ σ (e) -1
where the set E x is the set of the edges e ∈ E having x as endvertex, i.e.,

E x = e = x, y ∈ E : y ∈ V .
For the unnamed dynamics, we obtain

c(x, σ) = exp -β e∈Ex δ σ (e) + β 2 |E x | .
On the lattice Z d , we have |E x | = 2d for any x ∈ Z d . Up to a constant multiplicative factor, the previous rates are equal to

c(x, σ) = exp -β e∈Ex δ σ (e) . (11)
We will mostly use this choice when trying to build a dynamical coupling between the Ising model and the FK model.

FK dynamics

In this section, we define the counterpart of a Glauber type dynamics for the FK model. We are interested in dynamics which modify at most one edge at a time. We build a Markov process on percolation configurations, which is defined through its infinitesimal generator L (see [START_REF] Thomas | Interacting particle systems[END_REF]). This generator L acts on functions g of the percolation configuration and it is of the following form. For any function g : {0, 1} E → IR, we have

∀η ∈ {0, 1} E Lg(η) = e∈E c(η, η e )(g(η e ) -g(η)) ,
where η e is the configuration obtained from η by changing the state of the edge e and the quantity c(η, η e ) is the rate at which the edge e changes its state when the system is in the state η. The transition rates c(•, •) are defined on {0, 1} E × {0, 1} E and have the following properties: for any η ∈ {0, 1} E and e ∈ E,

c(η, η ′ ) = 0 if |{e ∈ E, η(x) = η ′ (e)}| ≥ 2, c(η, η e ) > 0 for any e ∈ E, (12)
c(η, η) = - e∈E c(η, η e ).
We say that the transition rates c(•, •) satisfy the detailed balance condition with respect to the FK measure φ p,2 if

∀η ∈ {0, 1} E ∀ e ∈ E φ p,2 (η)c(η, η e ) = φ p,2 (η e )c(η e , η) . ( 13
)
When this detailed balance condition holds, the associated dynamics is reversible with respect to the FK measure φ p,2 . Several rates satisfy the conditions (12) and the detailed balance condition (13). A natural choice is the following. For an edge e = x, y ∈ E and a configuration η ∈ {0, 1} E , we set γ η (e) = 1 if the endpoints x, y of e are connected by a path of open edges in η which does not use the edge e itself, and we set γ η (e) = 0 otherwise. We define then c(e, η) One can check that these rates satisfy (12) and (13).

:= c(η, η e ) =        1 -p if η(e) = 1 , p if η(e) =

Our dream process

The main purpose of this paper is to construct a Markov process on the space {0, 1} E × {-1, +1} V whose marginal on the spins is a Glauber dynamics and which is reversible with respect to the coupling measure IP . Suppose that (η t , σ t ) t≥0 is such a Markov process and let q((η, σ), (η ′ , σ ′ )) be its transition rates. These transition rates satisfy the following usual conditions:

(η, σ) = (η ′ , σ ′ ) =⇒ q((η, σ), (η ′ , σ ′ )) ≥ 0, (14)
and for any (η, σ) ∈ {0,

1} E × {-1, +1} V , (η ′ ,σ ′ )∈{0,1} E ×{-1,+1} V q((η, σ), (η ′ , σ ′ )) = 0 . ( 15
)
The reversibility property with respect to the coupling measure IP is equivalent to the following detailed balance equation:

∀(η, σ), (η ′ , σ ′ ) ∈ {0, 1} E × {-1, 1} V IP (η, σ) q((η, σ), (η ′ , σ ′ )) = IP (η ′ , σ ′ ) q((η ′ , σ ′ ), (η, σ)). ( 16
)
In general, the marginals of a Markov process are not themselves Markov processes. Now, we know from Ball and Yeo [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF] (see theorem 10.4 of the appendix) that the second marginal (σ t ) t≥0 of the dynamics is a Markov jump process with rates c(σ, σ ′ ) if and only if for any η ∈ {0, 1} E , σ, σ ′ ∈ {-1, 1} V , the sum η ′ ∈{0,1} E q((η, σ), (η ′ , σ ′ )) does not depend on η.

We have then

∀η ∈ {0, 1} E ∀σ, σ ′ ∈ {-1, 1} V η ′ ∈{0,1} E q((η, σ), (η ′ , σ ′ )) = c(σ, σ ′ ) . ( 17
)
Suppose that this is the case. The second marginal (σ t ) t≥0 will then be the Glauber dynamics with the transition rates (11) as soon as the sum in ( 17) is equal, up to some positive constant, to the transitions rates given in (11). More precisely, we should have, up to some positive multiplicative constant,

c(σ, σ ′ ) =      (1 -p) |{e∈Ex, δσ(e)=1}| if σ ′ = σ x , -x∈V (1 -p) |{e∈Ex, δσ(e)=1}| if σ ′ = σ , 0 otherwise, (18)
where E x denotes the set of the edges e in E having x as endvertex. Ideally, we would also wish that the first marginal (η t ) t≥0 is a Markov jump process with rates c(η, η ′ ). This will be the case if and only if, for any

η ∈ {0, 1} E , η ′ ∈ {0, 1} E , σ ∈ {-1, 1} V , the sum σ ′ ∈{-1,1} V q((η, σ), (η ′ , σ ′ ))
does not depend on σ. We would have then

∀η, η ′ ∈ {0, 1} E ∀σ ∈ {-1, 1} V σ ′ ∈{-1,1} V q((η, σ), (η ′ , σ ′ )) = c(η, η ′ ) . (19)
So, to sum up, we dream of constructing a Markov process on {0, 1} E × {-1, 1} V with transition rates satisfying ( 14), ( 15), ( 16), ( 17), ( 18) and ( 19), that is a Markov jump process (η t , σ t ) t≥0 which is reversible with respect to the IP measure, whose marginal on the spins is a Glauber dynamics, and whose marginal on the edges is Markovian.

One change at a time

We present here our first try. We build a simple dynamics which updates at most one site or one edge at each time, which is reversible with respect to the coupling measure IP and whose marginal on the spins has the same rates as a Glauber dynamics, although it is not Markovian. Recall that, for x ∈ V , E x is the set of the edges e in E having x as endvertex, i.e.,

E x = e = x, y ∈ E : y ∈ V .
For σ ∈ {-1, 1} V and x ∈ V , σ x is the element of {-1, 1} V obtained from σ by reversing the spin at x, i.e.,

σ x (x) = -σ(x) , ∀ y ∈ V \ {x} σ x (y) = σ(y) .
For η ∈ {0, 1} E and e ∈ E, we denote by η e the element of {0, 1} E obtained from η by changing the value of η(e), i.e.,

η e (e) = 1 -η(e) , ∀f ∈ E \ { e } η ′ (f ) = η(f ) .
Before introducing the transition rates of the one change dynamics, we give a condition that the transition rates have to fulfill in order to have the properties announced at the beginning of the section.

Lemma 6.1. Let (η t , σ t ) t≥0 be a Markov jump process, which updates at most one site or one edge at each time and which is reversible with respect to the coupling measure IP.

Let us denote by q((η, σ), (η ′ , σ ′ )) the transition rates of (η t , σ t ) t≥0 . Let (η, σ) be a pair of compatible configurations. If there exists x ∈ V such that q((η, σ), (η, σ x )) = 0, then all the edges of E x are closed in η.

Proof. The detailed balance equation ( 16) yields that, for any η ∈ {0, 1}, σ ∈ {-1, 1} V and any x ∈ V ,

IP (η, σ)q((η, σ), (η, σ x )) = IP (η, σ x )q((η, σ x ), (η, σ)),
which gives, thanks to (1),

1I (η,σ)∈C q((η, σ), (η, σ x )) = 1I (η,σ x )∈C q((η, σ x ), (η, σ)).
Since (η, σ) are supposed to be compatible, the last equality implies that q((η, σ), (η, σ x )) = q((η, σ x ), (η, σ)) e∈Ex 1I η(e)≤1-δσ (e) .

Consequently, if q((η, σ), (η, σ x )) = 0 then necessarily η(e) ≤ 1 -δ σ (e) for any e ∈ E x . Since the configurations η and σ are compatible, necessarily η(e) = 0 for any e ∈ E x .

We define next the transition rates of our dynamics.

Definition 6.2. Let (η, σ), (η ′ , σ ′ ) be two elements of {0, 1} E × {-1, 1} V .
We consider several cases:

• If η ′ = η and there exists x ∈ V such that σ ′ = σ x , η(e) = 0 for any e ∈ E x , then we define c((η, σ), (η ′ , σ ′ )) = 1 .
• If σ ′ = σ and there exists e ∈ E such that η ′ = η e , then we set

c((η, σ), (η ′ , σ ′ )) = p1 I η(e)=0 δ σ (e) + (1 -p)1 I η(e)=1 . • Otherwise, if (η, σ) = (η ′ , σ ′ ), then we set c((η, σ), (η ′ , σ ′ )) = 0. • Finally, if (η, σ) = (η ′ , σ ′ ), then we set c((η, σ), (η, σ)) = - (η ′ ,σ ′ ), (η ′ ,σ ′ ) =(η,σ) c((η, σ), (η ′ , σ ′ )) .
Proposition 6.3. The transition rates introduced in definition 6.2 satisfy the detailed balance equation (16).

Proof. We check first that, for any (η, σ) in {0, 1} E × {-1, 1} V and e ∈ E,

IP (η, σ)c((η, σ), (η e , σ)) = IP (η e , σ)c((η e , σ), (η, σ)) (20) Let (η, σ) in {0, 1} E × {-1, 1}
V and e ∈ E be fixed. We have, since η(e) = 1 -η e (e), We also have, since η e = η on E \ {e},

f ∈E\{e} p1I η(f )=1 δ σ (f ) + (1 -p)1I η(f )=0 = f ∈E\{e} p1I η e (f )=1 δ σ (f ) + (1 -p)1I η e (f )=0 . ( 22
)
Noting that

IP (η, σ) = 1 Z p1I η(e)=1 δ σ (e) + (1 -p)1I η(e)=0 f ∈E\{e} p1I η(f )=1 δ σ (f ) + (1 -p)1I η(f )=0 ,
we get (20) by multiplying each side of ( 22) by (21). We now prove that, for any (η, σ) in {0, 1} E × {-1, 1} V and any x ∈ V ,

IP (η, σ)c((η, σ), (η, σ x )) = IP (η, σ x )c((η, σ x ), (η, σ)). (23)
Noting that c((η, σ), (η,

σ x )) = e∈Ex 1I η(e)=0 ,
and that δ σ (e) = δ σ x (e) for any e ∈ E \ E x , we deduce that

c((η, σ), (η, σ x ))IP (η, σ) = = 1 Z e∈Ex 1I η(e)=0 e∈E p1I η(e)=1 δ σ (e) + (1 -p)1I η(e)=0 = 1 Z e∈Ex ((1 -p)1I η(e)=0 )   e∈E\Ex p1I η(e)=1 δ σ (e) + (1 -p)1I η(e)=0   = 1 Z e∈Ex ((1 -p)1I η(e)=0 )   e∈E\Ex p1I η(e)=1 δ σ x (e) + (1 -p)1I η(e)=0   = 1 Z e∈Ex 1I η(e)=0 e∈E p1I η(e)=1 δ σ x (e) + (1 -p)1I η(e)=0 = c((η, σ x ), (η, σ))IP (η, σ x ).
This finishes the proof of condition (23). Since the transition rates c((η, σ), (η ′ , σ ′ )) given in definition 6.2 allow to perform at most one change at a time, the reversibility condition reduces to the two conditions (20) and ( 23), so the proof of Proposition 6.3 is complete.

The following proposition shows that, if the initial condition (η 0 , σ 0 ) is distributed as IP , then the infinitesimal behavior of the marginal on the spins of the process (η t , σ t ) t≥0 , whose rates are introduced in definition 6.2, is the same as the transition rates given in (18). Proposition 6.4. Let (η t , σ t ) t≥0 be a Markov jump process with the transition rates introduced in definition 6.2. Suppose that the pair (η 0 , σ 0 ) is distributed according to IP . Then, for any x ∈ V , for any σ ∈ {-1, 1} V and any s ≥ 0,

lim t→0 1 t IP(σ t+s = σ x |σ s = σ) = (1 -p) |{e∈Ex, δσ(e)=1}| . (24)
Proof. We have, for any σ ∈ {-1, 1} V , x ∈ V and s ≥ 0,

IP(σ t+s = σ x |σ s = σ) = η∈{0,1} E η ′ ∈{0,1} E IP(σ t+s = σ x , η t+s = η ′ , η s = η|σ s = σ) = η∈{0,1} E η ′ ∈{0,1} E IP(σ t+s = σ x , η t+s = η ′ |σ s = σ, η s = η)IP(η s = η|σ s = σ). (25)
Since (η t , σ t ) t≥0 is a Markov jump process with the transition rates c((η, σ), (η ′ , σ ′ )) introduced in definition 6.2, then lim

t→0 1 t IP(σ t+s = σ x , η t+s = η ′ |σ s = σ, η s = η) = c((η, σ), (η ′ , σ x )) = 1I η ′ =η e∈Ex 1I η(e)=0 . (26)
Hence we get, combining (25) and ( 26), for any s ≥ 0,

lim t→0 1 t IP(σ t+s = σ x |σ s = σ) = η∈{0,1} E e∈Ex 1I η(e)=0 IP(η s = η|σ s = σ).
The measure IP is, by proposition 6.3, a reversible and a stationary measure for the process (η t , σ t ) t≥0 . Hence, if (η 0 , σ 0 ) is distributed as IP , then, for any s ≥ 0, (η s , σ s ) is also distributed as IP . Consequently, under the hypothesis of proposition 6.4, we have

IP(η s = η|σ s = σ) = IP(η s = η, σ s = σ) IP(σ s = σ) = IP (η, σ) η ′ ∈{0,1} E IP (η ′ , σ)
.

We conclude that lim

t→0 1 t IP(σ t+s = σ x |σ s = σ) = η∈{0,1} E , η≡0 on Ex IP (η, σ) η ′ ∈{0,1} E IP (η ′ , σ) . ( 27
)
We compute these sums, starting with the very definition of the coupling measure IP . We have

η∈{0,1} E , η≡0 on Ex IP (η, σ) = 1 Z (1 -p) |{ e∈E:δσ(e)=0 }| (1 -p) |{ e∈Ex:δσ(e)=1 }| . ( 28
)
The sum in the denominator has already been computed in equation [START_REF] Grimmett | The stochastic random-cluster process and the uniqueness of random-cluster measures[END_REF]. Equations ( 3), ( 27) and (28) together yield the desired result (24).

We consider next the marginal on the edges and we obtain a similar result.

Proposition 6.5. Let (η t , σ t ) t≥0 be a Markov jump process with the transition rates introduced in definition 6.2. Suppose that the pair (η 0 , σ 0 ) is distributed according to IP . Then, for any e ∈ E, for any η ∈ {0, 1} E and any s ≥ 0,

lim t→0 1 t IP(η t+s = η e |η s = η) = (1 -p)1 I η(e)=1 + p1 I η(e)=0, γη(e)=1 + p 2 1 I η(e)=0, γη (e)=0 . (29)
Proof. We have, for any η ∈ {0, 1} E , e ∈ E and s ≥ 0,

IP(η t+s = η e |η s = η) = σ∈{-1,1} V σ ′ ∈{-1,1} V IP(σ t+s = σ ′ , η t+s = η e , σ s = σ|η s = η) = σ∈{-1,1} V σ ′ ∈{-1,1} V IP(σ t+s = σ ′ , η t+s = η e |σ s = σ, η s = η)IP(σ s = σ|η s = η). ( 30
)
Since (η t , σ t ) t≥0 is a Markov jump process with the transition rates c((η, σ), (η ′ , σ ′ )) introduced in definition 6.2, then lim The measure IP is, by proposition 6.3, a reversible and a stationary measure for the process (η t , σ t ) t≥0 . Hence, if (η 0 , σ 0 ) is distributed as IP , then, for any s ≥ 0, (η s , σ s ) is also distributed as IP . Consequently, under the hypothesis of proposition 6.5, we have

t→0 1 t IP(σ t+s = σ ′ , η t+s = η e |σ s = σ, η s = η) = c((η, σ), (η e , σ ′ )) = 1I σ ′ =σ p1I η(e)=0 δ σ (e) + (1 -p)1I η(e)=1 . (31)
IP(σ s = σ|η s = η) = IP(η s = η, σ s = σ) IP(η s = η) = IP (η, σ) σ ′ ∈{-1,1} V IP (η, σ ′ )
.

We conclude that lim

t→0 1 t IP(η t+s = η e |η s = η) = σ∈{-1,1} V p1I η(e)=0 δ σ (e) + (1 -p)1I η(e)=1 IP (η, σ) σ ′ ∈{-1,1} V IP (η, σ ′ )
.

(32) We compute these sums, starting with the very definition of the coupling measure IP . We have

σ∈{-1,1} V p1I η(e)=0 δ σ (e) + (1 -p)1I η(e)=1 IP (η, σ) = p1I η(e)=0, γη(e)=1 + p 2 1I η(e)=0, γη(e)=0 + (1 -p)1I η(e)=1 Φ p,2 (η) . ( 33
)
The sum in the denominator has already been computed in equation ( 2), it is equal to Φ p,2 (η). Equations ( 2), (32) and (33) together yield the desired result (29).

Although the infinitesimal behavior of the marginal on the spins of (η t , σ t ) t≥0 is the same as for the Glauber dynamics with the transition rates given in (18), the process (σ t ) t≥0 does not evolve according to a Glauber dynamics, because it is not a Markovian process.

Similarly, the marginal on the edges is not a Markovian process. We summarize these results in the following theorem.

Theorem 6.6. We suppose that the set of edges E is not empty. Let (η t , σ t ) t≥0 be a Markov jump process with the transition rates introduced in definition 6.2. Then (η t , σ t ) t≥0 is reversible with respect to the coupling measure IP. However its two marginal processes (η t ) t≥0 and (σ t ) t≥0 are not Markovian jump processes.

Proof. The reversibility property has been proved in proposition 6.3. Yet the sum

η ′ ∈{0,1} E c((η, σ), (η ′ , σ x ))
depends on η for some σ ∈ {-1, 1} V and x ∈ V . In fact, we have, for any σ ∈ {-1, 1} V and x ∈ V ,

η ′ ∈{0,1} E c((η, σ), (η ′ , σ x )) = c((η, σ), (η, σ x )) = e∈Ex 1I η(e)=0 .
Similarly, the sum

σ ′ ∈{-1,1} V c((η, σ), (η e , σ ′ ))
depends on σ for some η ∈ {0, 1} E and e ∈ E. In fact, we have, for any η ∈ {0, 1} E and e ∈ E,

σ ′ ∈{-1,1} V c((η, σ), (η e , σ ′ )) = c((η, σ), (η e , σ)) = p1I η(e)=0 δ σ (e) + (1 -p)1I η(e)=1 .
The proof of Theorem 6.6 is complete thanks to Theorem 3.1 in Ball and Yeo [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF] (see theorem 10.4 in the appendix).

Theorem 6.6 shows that the Markov jump process with the transition rates given in definition 6.2 does not fulfill our dream, which was to build a Markov jump process on {0, 1} E × {-1, 1} V , reversible with respect to the IP measure, with a Glauber dynamics as the marginal on the spins and such that the marginal on the edges is Markovian. The following theorem shows that there is no hope to realize this dream with a Markov jump process (η t , σ t ) t≥0 which updates at most one site or one edge at each time.

Theorem 6.7. Let (η t , σ t ) t≥0 be a Markov jump process which updates at most one site or one edge at each time and whose marginal on the spins is a Markov jump process with transition rates satisfying [START_REF] Roberto H Schonmann | Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region[END_REF]. Then (η t , σ t ) t≥0 cannot be reversible with respect to the coupling measure IP .

Proof. Let (η t , σ t ) t≥0 be a Markov jump process satisfying the hypothesis of theorem 6.7 and let q((η, σ), (η ′ , σ ′ )) be its transition rates. Since this Markov process updates at most one site or one edge at each time, then

η ′ ∈{0,1} E q((η, σ), (η ′ , σ x )) = q((η, σ), (η, σ x )). ( 34
)
Since the marginal on the spins of (η t , σ t ) t≥0 is a Markov process with transition rates c(σ, σ ′ ) satisfying ( 8), Theorem 3.1 in Ball and Yeo [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF] gives that

∀η ∈ {0, 1} E η ′ ∈{0,1} E q((η, σ), (η ′ , σ x ))) = c(σ, σ x ) .
Hence, for any η ∈ {0, 1} E , any σ ∈ {-1, 1} V and any x ∈ V q((η, σ), (η, σ x )) = c(σ, σ x ).

Suppose now that the detailed balance equation ( 16) is satisfied. We deduce then from (8) and lemma 6.1 that necessarily η(e) = 0 for e ∈ E x . We conclude that the detailed balance equation ( 16) is not satisfied for every η ∈ {0, 1} E and every σ ∈ {-1, 1} V . The proof of Theorem 6.7 is complete.

Theorem 3.2 in [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF] (see theorem 10.5 in the appendix) implies that, for a process like the above one, in which at most one marginal process moves at each time step, the marginal processes are Markovian if and only if they are independent. If that were the case, then the equilibrium measure would be a product measure. Therefore it is impossible to realize our dream with a process which changes only one edge or one spin at a time. In the next section, we construct a process which can change simultaneously one spin and its incident edges.

One site and the incident edges

We build here a dynamics which updates at most one site and its incident edges at each time, which is reversible with respect to the coupling measure IP and whose marginal on the spins is a Glauber dynamics. For x ∈ V , recall that E x is the set of the edges e in E having x as endvertex, i.e.,

E x = {e ∈ E, e = x, y , y ∈ V },
and define

C x = (η, σ) ∈ {0, 1} E × {-1, 1} V : η(e) ≤ δ σ (e)
for any e ∈ E x .

The following definition gives the transition rates of the dynamics.

Definition 7.1. Let (η, σ), (η ′ , σ ′ ) be two elements of {0, 1} E × {-1, 1} V . We consider several cases:

• If there exists x ∈ V such that σ ′ = σ x and η ′ (e) = η(e) for any e ∈ E \ E x , then we define c((η, σ), (η ′ , σ ′ )) = (1 -p) |{e∈Ex, η ′ (e)=0}| p |{e∈Ex, η ′ (e)=1}| 1 I (η ′ ,σ ′ )∈Cx .

• Otherwise, if (η, σ) = (η ′ , σ ′ ), then we set c((η, σ), (η ′ , σ ′ )) = 0.

• Finally, if (η, σ) = (η ′ , σ ′ ), then we set c((η, σ), (η, σ)) = -

(η ′ ,σ ′ ), (η ′ ,σ ′ ) =(η,σ) c((η, σ), (η ′ , σ ′ )) .
Proposition 7.2. The transition rates introduced in definition 7.1 satisfy the detailed balance equation ( 16).

Proof. Let (η, σ) and (η ′ , σ ′ ) be fixed in {0, 1} E × {-1, 1} V . We have to prove (16) only in the case where there exists x ∈ V for which

σ ′ = σ x , ∀e ∈ E \ E x η ′ (e) = η(e) , (35)
because in all the other cases, the detailed balance equation ( 16) is trivially satisfied. Let us fix x ∈ V . Suppose first that (η, σ) and (η ′ , σ ′ ) are two elements of C x satisfying (35). We get then, for e ∈ E x ,

1I η(e)=1 δ σ (e) = 1I η(e)=1 , 1I η ′ (e)=1 δ σ x (e) = 1I η ′ (e)=1 , therefore e∈Ex p1I η(e)=1 δ σ (e) + (1 -p)1I η(e)=0 c((η, σ), (η ′ , σ x )) = p |{e∈Ex, η(e)=1}| (1 -p) |{e∈Ex, η(e)=0}| (1 -p) |{e∈Ex, η ′ (e)=0}| p |{e∈Ex, η ′ (e)=1}| = e∈Ex p1I η ′ (e)=1 δ σ x (e) + (1 -p)1I η ′ (e)=0 c((η ′ , σ x ), (η, σ)).

Now we multiply each member of the last equality by

e∈E\Ex p1I η(e)=1 δ σ (e) + (1 -p)1I η(e)=0 = e∈E\Ex p1I η ′ (e)=1 δ σ x (e) + (1 -p)1I η ′ (e)=0
and we obtain the detailed balance equation ( 16). Suppose now that (η, σ) ∈ C x and (η ′ , σ x ) / ∈ C x . We have then, by the definition of the transition rates,

IP (η, σ)c((η, σ), (η ′ , σ x )) = 0 .
Moreover IP (η ′ , σ x ) = 0 because (η ′ , σ x ) / ∈ C, therefore IP (η ′ , σ x )c((η ′ , σ x ), (η, σ)) = 0. So here again equation ( 16) is satisfied. Finally suppose that (η, σ) / ∈ C x . Then IP (η, σ) = 0 and so IP (η, σ)c((η, σ), (η ′ , σ x )) = 0 which is equal to IP (η ′ , σ x )c((η ′ , σ x ), (η, σ)), since the transition rates c((η ′ , σ x ), (η, σ)) vanish as soon as (η, σ) / ∈ C x . We conclude that the detailed balance equation ( 16) is always satisfied.

We compute next the marginal dynamics on the spins. Proposition 7.3. For any η in {0, 1} E and any σ, σ ′ in {-1, 1} V , the sum

η ′ ∈{0,1} E c((η, σ), (η ′ , σ ′ ))
does not depend on η and it is equal to

c(σ, σ ′ ) =      (1 -p) |{e∈Ex, δσ(e)=1}| if σ ′ = σ x , -x∈V (1 -p) |{e∈Ex, δσ(e)=1}| if σ ′ = σ , 0 otherwise . Proof. Let η ∈ {0, 1} E be fixed. If |{y ∈ V, σ(y) = σ ′ (y)}| ≥ 2 then, by definition 7.1, η ′ ∈{0,1} E c((η, σ), (η ′ , σ ′ )) = 0. ( 36
)
Suppose now that there exists x ∈ V such that σ ′ = σ x . Then, by definition 7.1,

η ′ ∈{0,1} E c((η, σ), (η ′ , σ ′ )) = η ′ ∈{0,1} E , η ′ ≡η on E\Ex (1 -p) |{e∈Ex, η ′ (e)=0}| p |{e∈Ex, η ′ (e)=1}| 1I (η ′ ,σ x )∈Cx .
Our next task is to calculate the last sum. To this end, we use the same proof as for (3) (recall that IP (η, σ) is also given by ( 1)). We get

η ′ ∈{0,1} E c((η, σ), (η ′ , σ x )) = (1 -p) |{e∈Ex, δσ(e)=1}| . (37)
Finally, we have to calculate η ′ ∈{0,1} E c((η, σ), (η ′ , σ)). We have, from definition 7.1,

η ′ ∈{0,1} E c((η, σ),(η ′ , σ)) = c((η, σ), (η, σ)) = - (η ′ ,σ ′ ), (η ′ ,σ ′ ) =(η,σ) c((η, σ), (η ′ , σ ′ )) = - η ′ ∈{0,1} E x∈V c((η, σ), (η ′ , σ x )) = - x∈V η ′ ∈{0,1} E c((η, σ), (η ′ , σ x )) = - x∈V (1 -p) |{e∈Ex, δσ(e)=1}| . (38)
Equalities ( 36), ( 37) and (38) yield the statement of proposition 7.3.

We compute next the marginal dynamics on the edges.

Proposition 7.4. Let η and η ′ be two different elements of {0, 1} E .

• If there exists e = x, y ∈ E such that η ′ (e) = η(e) and η ′ ≡ η on E \ {e}, then

σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) = z∈{x,y} (1 -p) |{e∈Ez, η ′ (e)=0}| p |{e∈Ez, η ′ (e)=1}| 1 I (η ′ ,σ z )∈Cz .
• If there exists x ∈ V such that η ′ ≡ η on E \ E x and the configurations η, η ′ differ in more than one edge, then

σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) = (1 -p) |{e∈Ex, η ′ (e)=0}| p |{e∈Ex, η ′ (e)=1}| 1 I (η ′ ,σ x )∈Cx . • If for any x ∈ V , η ′ = η on E \ E x , then σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) = 0 .
Proof. Let η and η ′ be two fixed different elements of {0, 1} E . If for any x ∈ V , η and η ′ are different on E \ E x then we get, by definition 7.1,

∀ σ ′ ∈ {-1, 1} V c((η, σ), (η ′ , σ ′ )) = 0 .
Consequently,

σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) = 0. ( 39
)
Suppose now that there exists x ∈ V such that η ≡ η ′ on E \ E x . Let us define the set

V = y ∈ V, η ≡ η ′ on E \ E y .
We have

σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) = y∈V c((η, σ), (η ′ , σ y )). ( 40
) Let us set E = f ∈ E, η(f ) = η ′ (f ) .
By hypothesis, this set is not empty and it is a subset of E x because η = η ′ while η ≡ η ′ on E \ E x . Suppose first that E contains only one element, say E = {e}. Necessarily one endvertex of e is x. Let y be the other endvertex of e, so that e = x, y . We have then V = {x, y}. Indeed, if there exists a vertex z in V such that

z = x and η ≡ η ′ on E \ E z ,
then, since e ∈ E x and η(e) = η ′ (e), we have also e ∈ E z , whence e = x, z and z = y. Equation ( 40) can now be rewritten as

σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) = c((η, σ), (η ′ , σ x )) + c((η, σ), (η ′ , σ y )) . ( 41
)
Suppose that E contains more than one element. We claim that, in this case, the set V is reduced to {x}. Indeed, let e 1 and e 2 be two different elements of E. Let y ∈ V. Since η ≡ η ′ on E \ E y , necessarily both e 1 and e 2 belong to E y . We know also that both e 1 and e 2 belong to E x . Consequently e 1 ∈ E x ∩ E y and e 2 ∈ E x ∩ E y . It follows that y = x since e 1 and e 2 are different. From (40), we deduce that, whenever E contains more than one element,

σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) = c((η, σ), (η ′ , σ x )) . ( 42
)
Definition 7.1 and the three cases of equations ( 39), ( 41) and ( 42) complete the proof of Proposition 7.4.

From propositions 7.2, 7.3 and 7.4, we deduce the following theorem.

Theorem 7.5. Let (η t , σ t ) t≥0 be a Markov jump process with the transition rates introduced in definition 7.1. Then (η t , σ t ) t≥0 is reversible with respect to the coupling measure IP. Its second marginal process (σ t ) t≥0 is a Markov process evolving according to a Glauber dynamics while its first marginal (η t ) t≥0 is a non-Markovian jump process.

Proof. The reversibility property is deduced from Proposition 7.2. Proposition 7.3 shows that the sum

η ′ ∈{0,1} E c((η, σ), (η ′ , σ ′ ))
does not depend on η for any σ, σ ′ ∈ {-1, 1} V , moreover this sum is equal to the transition rates given in (18). The presence of the indicator function 1I (η ′ ,σ x )∈Cx in the formula of proposition 7.4 shows that the sum σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) depends on σ for some η, η ′ ∈ {0, 1} E . The proof of Theorem 7.5 is complete thanks to Ball and Yeo [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF].

So this second dynamics does not fulfill our dream.

The dream is not feasible

A Markov process on {0, 1} E × {-1, 1} V is said to be a Markovian coupling if both its marginals are Markov processes. In this section, we will work with the following additional hypothesis on the graph (V, E).

Hypothesis 8.1. We suppose that there exists a vertex x ∈ V which belongs to at least four edges. We suppose that there exists a spin configuration σ such that δ σ (e) = 0 for any e ∈ E.

For instance, when the graph is bipartite, the spin configuration σ can be obtained by setting minuses on one part of the graph and pluses on the other part. Theorem 8.2 below shows that, under hypothesis 8.1, we cannot build a Markovian coupling which is reversible with respect to the IP measure and whose marginal on the spins is a one spin flip Markov jump process, i.e., with transition rates satisfying [START_REF] Roberto H Schonmann | Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region[END_REF].

Theorem 8.2. Suppose that hypothesis 8.1 holds. Let (η t , σ t ) t≥0 be a Markovian jump process with rates q((η, σ), (η ′ , σ ′ )) satisfying ( 14), (15) and the detailed balance equation (16). If the second marginal (σ t ) t≥0 is a Markov jump process with transition rates satisfying [START_REF] Roberto H Schonmann | Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region[END_REF], then the first marginal (η t ) t≥0 is a continuous-time non-Markovian jump process.

Proof. Let (η t , σ t ) t≥0 be a Markovian jump process with transition rates q((η, σ), (η ′ , σ ′ )) satisfying ( 14), (15) and the detailed balance equation ( 16). Suppose also that the first marginal (η t ) t≥0 is a Markov jump process with transition rates c(η, η ′ ). Suppose also that the second marginal (σ t ) t≥0 is a Markov jump process with transition rates c(σ, σ ′ ) satisfying [START_REF] Roberto H Schonmann | Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region[END_REF]. From Ball and Yeo [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF] (see theorem 10.4 of the appendix), we have

∀η, η ′ ∈ {0, 1} E ∀σ ∈ {-1, 1} V σ ′ ∈{-1,1} V q((η, σ), (η ′ , σ ′ )) = c(η, η ′ ) , (43)
∀σ, σ ′ ∈ {-1, 1} V ∀η ∈ {0, 1} E η ′ ∈{0,1} E q((η, σ), (η ′ , σ ′ )) = c(σ, σ ′ ) . (44)
Let σ be a configuration as in hypothesis 8.1 and let x be a vertex of E belonging to at least four distinct edges. Let η be the configuration where all the edges are closed and let η x be the configuration where the edges exiting from x are opened, while all the other edges are closed. From formula (44) applied to η, σ and σ x , we have

η ′ ∈{0,1} E q((η x , σ x ), (η ′ , σ)) = c(σ x , σ) .
The only configuration η ′ compatible with σ is η, thus the above identity reduces to q((η x , σ x ), (η, σ)) = c(σ x , σ) .

The conditions [START_REF] Roberto H Schonmann | Slow droplet-driven relaxation of stochastic Ising models in the vicinity of the phase coexistence region[END_REF] implies that c(σ x , σ) is positive, therefore q((η x , σ x ), (η, σ)) is positive. Since we assumed that the dynamics is reversible, then the rate q((η, σ), (η x , σ x )) is also positive. From formula (43) applied to η, η x and σ, we have

σ ′ ∈{-1,1} V q((η, σ), (η x , σ ′ )) = c(η, η x ) ,
therefore the rate c(η, η x ) is positive. Let now σ be a spin configuration in which the spins of the vertices connected to x contain at least two negative spins and at least two positive spins. This is possible because we assumed that x is connected to at least four distinct vertices. Applying again formula (43), this time to η, η x and σ, we have

σ ′ ∈{-1,1} V q((η, σ), (η x , σ ′ )) = c(η, η x ) .
Since the rate c(η, η x ) is positive, then there exists a spin configuration σ ′ such that q((η, σ), (η x , σ ′ )) > 0 .

In the configuration η x , all the edges emanating from x are opened, hence the neighbours of x are connected. Since the configuration σ ′ has to be compatible with η x , then all the neighbours of x have the same spin in σ ′ , hence the configuration σ ′ has to differ from σ in at least two vertices. From the conditions (8), we should therefore have that c( σ, σ ′ ) = 0. Yet formula (44) applied to η, σ and σ x yields

c( σ, σ ′ ) = η ′ ∈{0,1} E q((η, σ), (η ′ , σ ′ )) ≥ q((η, σ), (η x , σ ′ )) > 0 ,
which is contradictory.

Notice that hypothesis 8.1 holds for a cubic box on the d-dimensional lattice when d ≥ 2.

Thus our dream process is not realizable in the lattice Z d for d ≥ 2.

9 One edge and one incident cluster

We drop here the requirement that the dynamics is local. We build a dynamics which updates at each time at most one edge and the vertices belonging to the open cluster of one of the endpoints of the edge. This dynamics will be reversible with respect to the coupling measure IP and its marginal on the edges is the FK dynamics described in section 4. For x ∈ V , recall that E x is the set of the edges e in E having x as endvertex, i.e., E x = {e ∈ E, e = x, y , y ∈ V } .

The following definition gives the transition rates of the dynamics.

Definition 9.1. Let (η, σ), (η ′ , σ ′ ) be two elements of {0, 1} E × {-1, 1} V . We consider several cases:

• If σ ′ = σ and there exists e ∈ E such that η ′ = η e and γ η (e) = 1, then we set c((η, σ), (η ′ , σ ′ )) = (1 -p)1 I η(e)=1 + p1 I η(e)=0 1 I (η,σ)∈C .

• If σ ′ = σ and there exists e ∈ E such that η ′ = η e , γ η (e) = 0 and δ σ (e) = 1, then we set

c((η, σ), (η ′ , σ ′ )) = 1 2 (1 -p)1 I η(e)=1 + p1 I η(e)=0 1 I (η,σ)∈C .
• If there exists x ∈ V and e ∈ E x such that η ′ = η e , γ η (e) = 0, η(e) = δ σ (e) and σ ′ is given by

∀y ∈ V σ ′ (y) =    σ(y) if y is not connected to x in η \ {e} , -σ(y) if y is connected to x in η \ {e} , then we set c((η, σ), (η ′ , σ ′ )) = 1 4 (1 -p)1 I η(e)=1 1 I (η,σ)∈C + p1 I η(e)=0 1 I (η ′ ,σ ′ )∈C . • Finally, if (η, σ) = (η ′ , σ ′ ), then we set c((η, σ), (η, σ)) = - (η ′ ,σ ′ ), (η ′ ,σ ′ ) =(η,σ) c((η, σ), (η ′ , σ ′ )) .
We first check that the dynamics associated to these rates is reversible with respect to the measure IP .

Proposition 9.2. The transition rates introduced in definition 9.1 satisfy the detailed balance equation ( 16).

Proof. Let (η, σ) and (η ′ , σ ′ ) be fixed in {0, 1} E × {-1, 1} V . We have to prove (16). We consider several cases, as in definition 9.1. Suppose first that σ ′ = σ and there exists e ∈ E such that η ′ = η e and γ η (e) = 1. We have then

IP (η, σ)c((η, σ), (η ′ , σ)) = 1 Z   f ∈E p1I η(f )=1 δ σ (f ) + (1 -p)1I η(f )=0   (1 -p)1I η(e)=1 + p1I η(e)=0 1I (η,σ)∈C = 1 Z   f ∈E\{e} p1I η(f )=1 + (1 -p)1I η(f )=0   (1 -p)p 1I (η,σ)∈C .
Since η ′ = η e and γ η (e) = 1, then (η, σ) ∈ C if and only if (η ′ , σ) ∈ C. Also, the product in the last formula is the same for η and η ′ , thus we obtain the detailed balance equation ( 16). Suppose next that σ ′ = σ and there exists e ∈ E such that η ′ = η e , γ η (e) = 0 and δ σ (e) = 1. We have then

IP (η, σ)c((η, σ), (η ′ , σ)) = 1 Z   f ∈E p1I η(f )=1 δ σ (f ) + (1 -p)1I η(f )=0   1 2 (1 -p)1I η(e)=1 + p1I η(e)=0 1I (η,σ)∈C = 1 Z   f ∈E\{e} p1I η(f )=1 + (1 -p)1I η(f )=0   1 2 (1 -p)p 1I (η,σ)∈C .
Since η ′ = η e and δ σ (e) = 1, then (η, σ) ∈ C if and only if (η ′ , σ) ∈ C. Also, the product in the last formula is the same for η and η ′ , thus we obtain the detailed balance equation (16). Suppose finally that there exists x ∈ V and e ∈ E x such that η ′ = η e , γ η (e) = 0, η(e) = δ σ (e) and σ ′ is obtained from σ by reversing all the spins of the sites which are connected to x by an open path in η \ {e}. We have then

IP (η, σ)c((η, σ), (η ′ , σ ′ )) = 1 Z   f ∈E p1I η(f )=1 δ σ (f ) + (1 -p)1I η(f )=0   1 4 (1 -p)1I η(e)=1 1I (η,σ)∈C + p1I η(e)=0 1I (η ′ ,σ ′ )∈C = 1 Z   f ∈E\{e} p1I η(f )=1 + (1 -p)1I η(f )=0   1 4 (1-p)p 1I η(e)=1 1I (η,σ)∈C +1I η(e)=0 1I (η ′ ,σ ′ )∈C .
To remove the symbol δ σ (f ) in the last line, we have used the fact that, if η(e) = 0 and (η ′ , σ ′ ) ∈ C, then (η, σ) ∈ C. Since we have

1I η(e)=1 1I (η,σ)∈C + 1I η(e)=0 1I (η ′ ,σ ′ )∈C = 1I η ′ (e)=1 1I (η ′ ,σ ′ )∈C + 1I η ′ (e)=0 1I (η,σ)∈C ,
then we can conclude that

IP (η, σ)c((η, σ), (η ′ , σ ′ )) = IP (η ′ , σ ′ )c((η ′ , σ ′ ), (η, σ))
and the detailed balance equation holds also in this case. We conclude that the detailed balance equation ( 16) is always satisfied.

We compute next the marginal dynamics on the edges. • If γ η (e) = 0 and η(e) = 0, then

σ ′ ∈{-1,1} V c((η, σ), (η e , σ ′ )) = p 2 .
• If γ η (e) = 0 and η(e) = 1, then

σ ′ ∈{-1,1} V c((η, σ), (η e , σ ′ )) = 1 -p .
Proof. Let η, e, σ be as in the statement of the proposition. We use the expression for the rates given in the definition 9.1. If γ η (e) = 1, then c((η, σ), (η e , σ ′ )) is null unless σ ′ = σ. Thus the sum reduces to c((η, σ), (η e , σ)), which is equal to p if η(e) = 0 and to 1 -p if η(e) = 1. Suppose next that γ η (e) = 0 and η(e) = 0. We consider two further cases. If δ σ (e) = 1, then the sum reduces again to c((η, σ), (η e , σ)), which in this case is equal to p/2. If δ σ (e) = 0, then the sum contains two terms, corresponding to the two spin configurations σ ′ obtained from σ by reversing all the spins of the sites which are connected to one extremity of e by an open path in η \ {e}. These two rates c((η, σ), (η e , σ ′ )) are both equal to p/4, and their sum is again equal to p/2. Suppose finally that γ η (e) = 0 and η(e) = 1. The sum contains three terms, one corresponding to σ ′ = σ and two corresponding to the two spin configurations σ ′ obtained from σ by reversing all the spins of the sites which are connected to one extremity of e by an open path in η \ {e}. The rate c((η, σ), (η e , σ)) is equal to (1 -p)/2. The two other rates c((η, σ), (η e , σ ′ )) are equal to (1 -p)/4. In total, the sum is equal to 1 -p.

The marginal dynamics on the spins is quite complicated, but it is not a Markov process.

Let us consider an edge configuration η and a spin configuration σ such that there exist x ∈ V and e ∈ E x such that γ η (e) = η(e) = δ σ (e) = 0. Let σ ′ be the spin configuration defined by

∀y ∈ V σ ′ (y) =    σ(y) if y is not connected to x in η \ {e} , -σ(y) if y is connected to x in η \ {e} .
We have then

η ′ ∈{0,1} E c((η, σ), (η ′ , σ ′ )) = c((η, σ), (η e , σ ′ )) = p 4 .
However the configuration σ ′ depends on η. Indeed, if we change the status of some edges f in E x \ {e}, we will change the set of the vertices which are connected to x in η \ {e}, and we will obtain an edge configuration η for which

η ′ ∈{0,1} E c((η, σ), (η ′ , σ ′ )) = c((η, σ), (η e , σ ′ )) = 0 .
Thus the above sum depends on the configuration η and the marginal process on the spin configurations is not a Markov process. This remark, together with propositions 9.2 and 9.3, yield the following theorem.

Theorem 9.4. Let (η t , σ t ) t≥0 be a Markov jump process on C with the transition rates introduced in definition 7.1. Then (η t , σ t ) t≥0 is reversible with respect to the coupling measure IP. Its first marginal (η t ) t≥0 is a Markov process evolving according to the FK dynamics while its second marginal process (σ t ) t≥0 is a non-Markovian jump process.

Proof. The reversibility property is deduced from Proposition 9.2. Proposition 9.3 shows that the sum σ ′ ∈{-1,1} V c((η, σ), (η ′ , σ ′ )) does not depend on σ for any η, η ′ ∈ {0, 1} E , moreover this sum is equal to the transition rates given in (18). The proof of Theorem 9.4 is complete thanks to the remark before the theorem and the result of Ball and Yeo [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF].

So this third dynamics does not fulfill our dream, although it is not even local.

10 One edge and one spin A notable inconvenient of the previous dynamics is that it might reverse simultaneously all the spins associated to an open cluster of the edge configuration. We introduce here a slight modification of this dynamics to ensure that at most one spin is changed at a time. For x ∈ V , recall that E x is the set of the edges e in E having x as endvertex, i.e.,

E x = {e ∈ E, e = x, y , y ∈ V } .
The following definition gives the transition rates of the dynamics.

Definition 10.1. Let (η, σ), (η ′ , σ ′ ) be two elements of {0, 1} E × {-1, 1} V . We consider several cases:

• If σ ′ = σ and there exists e ∈ E such that η ′ = η e and γ η (e) = 1, then we set c((η, σ), (η ′ , σ ′ )) = (1 -p)1 I η(e)=1 + p1 I η(e)=0 1 I (η,σ)∈C .

• σ ′ = σ and there exists e ∈ E such that η ′ = η e , γ η (e) = 0 and δ σ (e) = 1, then we set

c((η, σ), (η ′ , σ ′ )) = 1 2 (1 -p)1 I η(e)=1 + p1 I η(e)=0 1 I (η,σ)∈C .
• If there exists x ∈ V and e ∈ E x such that σ ′ = σ x , η ′ = η e , γ η (e) = 0, η(e) = δ σ (e) and η(f

) = 0 for f ∈ E x \ {e}, then we set c((η, σ), (η ′ , σ ′ )) = 1 4 (1 -p)1 I η(e)=1 1 I (η,σ)∈C + p1 I η(e)=0 1 I (η ′ ,σ ′ )∈C . • Otherwise, if (η, σ) = (η ′ , σ ′ ), then we set c((η, σ), (η ′ , σ ′ )) = 0. • Finally, if (η, σ) = (η ′ , σ ′ ), then we set c((η, σ), (η, σ)) = - (η ′ ,σ ′ ), (η ′ ,σ ′ ) =(η,σ) c((η, σ), (η ′ , σ ′ )) .
The difference compared to the rates of definition 9.1 is that, in definition 10.1, we allow to reverse the spins of a cluster only when it is reduced to a single vertex. This dynamics possesses the same properties as the dynamics of the previous section, namely, it is reversible with respect to the coupling measure IP, its first marginal process on the edges is a Markov process while its second marginal process on the spins is a non-Markovian jump process. The marginal dynamics on the edges differs from the FK dynamics in the following way. An edge can be opened between two different clusters only if one the two clusters is reduced to a single vertex. More precisely, the corresponding rates are the following. For x ∈ V and e ∈ E x , for any configuration η,

c(η, η e ) =        1 -p if η(e) = 1 , p if η(e) = 0 and γ η (e) = 1 , p/2 if η(f ) = 0 for f ∈ E x .
Funnily enough, this fourth dynamics can be seen as a contraction of the first dynamics, which could make only one change at a time. The following proposition shows that, if the initial condition (η 0 , σ 0 ) is distributed as IP , then the infinitesimal behavior of the marginal on the spins of the process (η t , σ t ) t≥0 , whose rates are introduced in definition 10.1, is quite similar to the transition rates of the Glauber dynamics as given in (18). The difference lies in the factor |E |. On the cubic lattice Z d , the factor |E x | is equal to 2d and it is independent of x.

Proposition 10.2. Let (η t , σ t ) t≥0 be a Markov jump process with the transition rates introduced in definition 10.1. Suppose that the pair (η 0 , σ 0 ) is distributed according to IP . Then, for any x ∈ V , for any σ ∈ {-1, 1} V and any s ≥ 0, lim The measure IP is, by proposition 6.3, a reversible and a stationary measure for the process (η t , σ t ) t≥0 . Hence, if (η 0 , σ 0 ) is distributed as IP , then, for any s ≥ 0, (η s , σ s ) is also distributed as IP . Consequently, under the hypothesis of proposition 10. q((η, σ), (η ′ , σ ′ )).

(53

)
Our goal is to present some conditions on the transition rates q((η, σ), (η ′ , σ ′ )) under which (σ t ) t≥0 or (η t ) t≥0 are Markov processes. This is a lumpability problem discussed in Ball and Yao (1993). We first recall the following definition of lumpability.

Definition 10.3. Let Ω be a countable set, {Ω 1 , • • • , Ω r } be a partition of Ω and h the function from Ω to {1, • • • , r} defined by h(x) = j if x ∈ Ω j . An homogeneous Markov chain (X t ) t≥0 with state space Ω is lumpable with respect to the partition Ω 1 , • • • , Ω r if (h(X t )) t≥0 is an homogeneous Markov chain for every initial distribution y 0 on {1, • • • , r} and its transition rates do not depend on the choice of the initial distribution y 0 .

Let us take the following partition of Ω:

Ω = σ∈{-1,1} V Ω σ , Ω σ = (η, σ) ∈ Ω : η ∈ {0, 1} E .
In this case, the function h is the projection from Ω to {-1, 1} V defined by h((η, σ)) = σ.

Our purpose is to obtain necessary and sufficient conditions under which the Markov process ((η t , σ t )) t≥0 is lumpable with respect to the partition (Ω σ ) σ∈{-1,1} V .

It is well known (see for instance Theorem 3.1 in Ball and Yao (1993)1 ) that a Markov process is lumpable with respect to the partition (Ω j ) 1≤j≤r if and only if there exist positive real numbers λ i,j , 1 ≤ i, j ≤ r, such that ∀i = j , ∀ x ∈ Ω i , y∈Ω j q x,y = λ i,j .

In order to simplify the reading of this paper, we adapt this criterion of lumpability to the case of the partition (Ω σ ) σ∈{-1,1} V and the Markov process ((η t , σ t )) t≥0 . We summarize it in the following theorem.

Theorem 10.4. Let ((η t , σ t )) t≥0 be a continuous-time Markov process with finite state space {0, 1} E × {-1, 1} V and transition rates q((η, σ), (η ′ , σ ′ )). Then the following statements are equivalent:

(i) For any σ = σ ′ , there exist positive numbers c(σ, σ ′ ) such that ∀ η ∈ {0, 1} E , η ′ ∈{0,1} E q((η, σ), (η ′ , σ ′ )) = c(σ, σ ′ ) .

(ii) The spin marginal (σ t ) t≥0 of the Markov process ((η t , σ t )) t≥0 is a Markov process for every initial distribution σ 0 on {-1, 1} V and its transition rates c(σ, σ ′ ) do not depend on the choice of the initial distribution of σ 0 .

  0 and γ η (e) = 1 , p/2 if η(e) = 0 and γ η (e) = 0 .

  p1I η(e)=1 δ σ (e) + (1 -p)1I η(e)=0 × p1I η(e)=0 δ σ (e) + (1 -p)1I η(e)=1 = p1I η e (e)=1 δ σ (e) + (1 -p)1I η e (e)=0 × p1I η e (e)=0 δ σ (e) + (1 -p)1I η e (e)=1 . (21)

Combining ( 30 1 t

 301 ) and (31), we get, for any s ≥ 0, lim t→0 IP(η t+s = η e |η s = η) = σ∈{-1,1} V p1I η(e)=0 δ σ (e) + (1 -p)1I η(e)=1 IP(σ s = σ|η s = η).

Proposition 9 . 3 .

 93 Let η be an element of {0, 1} E and let e ∈ E. Let also σ be an element of {-1, 1} V such that (η, σ) ∈ C. • If γ η (e) = 1, then σ ′ ∈{-1,1} V c((η, σ), (η e , σ ′ )) = p1 I η(e)=0 + (1 -p)1 I η(e)=1 .

t→0 1 t 1 t 1 t

 111 IP(σ t+s = σ x |σ s = σ) = 1 4 |E x |p(1 -p) |{ e∈Ex:δσ(e)=1 }| .(45)Proof. We have, for any σ ∈ {-1, 1} V , x ∈ V and s ≥ 0,IP(σ t+s = σ x |σ s = σ) = η∈{0,1} E η ′ ∈{0,1} E IP(σ t+s = σ x , η t+s = η ′ , η s = η|σ s = σ) = η∈{0,1} E η ′ ∈{0,1} E IP(σ t+s = σ x , η t+s = η ′ |σ s = σ, η s = η)IP(η s = η|σ s = σ).(46)Recall that (η t , σ t ) t≥0 is a Markov jump process with transition rates c((η, σ), (η ′ , σ ′ )), thuslim t→0 IP(σ t+s = σ x , η t+s = η ′ |σ s = σ, η s = η) = c((η, σ), (η ′ , σ x )) . (47)Using the expression of the rates c((η, σ), (η ′ , σ ′ )) given in definition 10.1, together with formulas (46) and (47), we obtain that, for any s ≥ 0, lim t→0 IP(σ t+s = σ x |σ s = σ) = η∈{0,1} E e∈Ex p)1I η(e)=1 1I (η,σ)∈C + p1I η(e)=0 1I (η e ,σ x )∈C IP(η s = η|σ s = σ) . (48)

  2, we haveIP(η s = η|σ s = σ) = IP(η s = η, σ s = σ) IP(σ s = σ) = IP (η, σ) η ′ ∈{0,1} E IP (η ′ , σ) . (49)Equation (3) yields thatη ′ ∈{0,1} E IP (η ′ , σ) = 1 Z (1 -p) |{ e∈E:δσ(e)=0 }| .(50)and for (η, σ) = (η ′ , σ ′ ), q((η, σ), (η, σ)) = -(η ′ ,σ ′ ): (η,σ) =(η ′ ,σ ′ )

Ball and Yao's result is stated under their condition

2.2 which is satisfied in the actual context.

Reporting (49) and (50) in (48), we conclude that lim

We compute these sums as follows. We have, for e ∈ E x ,

Reporting the values of these sums in formula (51), we obtain

Appendix

Let (V, E) be a finite graph and let (X t ) t≥0 = ((η t , σ t )) t≥0 be a time-homogeneous continuous-time Markov process with state space Ω = {0, 1} E × {-1, 1} V and with infinitesimal generator Q defined as follows. For any function f defined on Ω with values in R, we have

where q((η, σ), (η ′ , σ ′ )) are the transition rates defined by, for (η, σ) = (η ′ , σ ′ ),

The analog of Theorem 10.4 holds for the edge marginal process of ((η t , σ t )) t≥0 . For a proof, we refer to Theorem 3.1 in Ball and Yeo [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF]. We also recall Theorem 3.2 of Ball and Yeo [START_REF] Ball | Lumpability and marginalisability for continuous-time Markov chains[END_REF], that we adapt to our context.

Theorem 10.5. Suppose that the transition rates q((η, σ), (η ′ , σ ′ )) of the Markov process (η t , σ t ) t≥0 satisfy q((η, σ), (η ′ , σ ′ )) = 0 =⇒ (η, σ) = (η ′ , σ ′ ) or (η = η ′ and σ = σ ′ ) or (η = η ′ and σ = σ ′ ).

Then (η t ) t≥0 and (σ t ) t≥0 are both Markov processes if and only if they are mutually independent.


Agathe Jaillon
email: agathe.jaillon@isae-supaero.fr

Julien Jumel
email: julien.jumel@u-bordeaux.fr

Frédéric Lachaud
email: frederic.lachaud@isae-supaero.fr

Éric Paroissien
email: eric.paroissien@isae-supaero.fr

Jordi Renart
email: jordi.renart@udg.edu

Adhesive thickness influence on a structural methacrylate adhesive behavior

Introduction

Structural adhesive bonding importance has been growing steadily in the last decades as transport sector's current problematic is to there products' reduce mass. In addition, compared to riveting and welding, adhesive bonding offers better properties when joining dissimilar materials such as metals and composites which are a pair used more and more frequently. However, adhesive bonding suffers greatly from a lack of confidence from industries as validating the bond quality need either destructive testing or long and costly nondestructive testing. Both these solutions can hardly be implemented at an industrial level. Nevertheless, with the implementation of robotics it is possible to automate and control the entire bonding processes. In this context a collaborative project called S3PAC (Système de Supervision et de Simulation de la Production d'Assemblage par Collage) has been launched in order to offer a fully supervised and automated industrial bonding process. Each bonding steps from the cleaning of the adherends to the raw adhesive laying and the final thickness of the structure is controlled while monitoring curing time and temperature. Moreover, to assess the quality of the structure and to validate that its mechanical behavior is compatible with the part's specifications, the bonded joints characteristics are implemented in a numerical simulation. One of this project purposes is to model a fracture behavior under complex loading of the SAF30MIB adhesive a methacrylate structural adhesive manufactured by AEC Polymers/Bostik. In this work, the adhesive bond thickness influence will be investigated for pure modes and mixed-mode loadings. The experimental results in mode I will be compared to 2D plain stress finite element (FE) simulations and to the 1Dmacro-element (ME) technique [1]. Experimental characterization of the methacrylate adhesive is carried out through the realization of tensile test on bulk adhesive, double cantilever beam (DCB) and ARCAN tests [2]. The bonded assembly tests were performed for two adhesive thicknesses 0.2mm and 1mm.

Experimentation

Experimental tests presented in this part were carried out using SAF30MIB adhesive a methacrylate adhesive manufactured by AEC Polymers/Bostik. It is a thermoplastic adhesive that polymerize at room temperature in less than 20 minutes. When testing bonded specimen, adherends are aluminum 2024 that has a measured Young's modulus and Poisson's ratio of respectively E=70GPa and υ=0.33.

a) Tensile test on Bulk adhesive

Bulk specimens were manufactured in a Teflon mold. According to the NF-EN ISO 527-2 standard, dog bone specimens with a length of 150mm, a width of 10mm and a height of 4mm were manufactured. The methacrylate adhesive polymerizes at room temperature but it is sensitive to oxygen and the methacrylate solvent tends to dissipate. As a consequence, once the mold imprints were filled, the adhesive overload was removed with a spatula and the imprints were recovered with a polypropylene film. The adhesive was left to cure at room temperature for 24 hours. Specimens have then been tested in monotonous tensile under a controlled displacement rate of 5mm/min on an Instron 100kN tensile machine. Displacement and strain field were obtained by 3D DIC (Digital image correlation) with two sets of Pike 505 B cameras.

b) ARCAN tests

Arcan specimens were manufactured using a bonding tool designed to be tailored to the adherends geometry (Figure 1a andb). Prior to bonding, the adherends surfaces were degreased with isopropanol. Adhesive bond thickness is assured with a metal calibration gauge for 1mm bond thickness and two layers of Teflon film for the 0.2mm bond thickness. Raw adhesive was applied on adherends screwed on the bottom half of the tool. The other half was then positioned on top of the latter. The excess of adhesive was removed with a plastic rod in order to reduce edge effects [2]. Specimens are left to polymerize for 24 hours. Specimens were embedded in the modified arcan which is divided into two moon shape parts (figure 1c). Monotonous tests were performed on a Zwick 10kN machine under a controlled displacement rate of 0.1mm/min for 0.2mm-thick specimen and 0.2mm/min for 1mm-thick specimens. For each bond thickness two specimens were tested in tension, two specimens were tested in shear and two specimens were tested in tensile/shear. c) DCB tests DCB specimens were manufactured using tailored bonding too. The adherends were 195mm long, 15mm wide and 10mm thick. These dimensions were chosen in order to prevent plastic deformation in the adherends. Prior to bonding, adherends were degreased with isopropanol. Specimens' thickness was assured with calibrated Teflon films. For bond thicknesses of 0.2mm and 1mm, respectively two layers and 10 layers of Teflon film were used. Initial crack length was set to 35mm.for every specimen. DCB test are carried out on a Zwick 10kN machine at a controlled displacement rate of 2mm/min for both adhesive bonds thicknesses. Displacement and rotation of the adherends were determined using DIC with two reflex cameras Canon EOS 750D. Both cameras took 1 picture every two seconds. Cameras are positioned on each side of the specimen in order to have images of the whole specimen and zoomed in on the initial crack.

Modeling a) Arcan Model

A 2D plain stress FE model has been realized in order to investigate the stress concentrations in the adhesive layer and to validate that the experimental adherends geometry used is offering a homogenous stress distribution. The arcan specimen geometry has been simplified in order to reduce computation time and the plan strain hypothesis has been chosen. For tensile loading, 2 symmetry conditions have permitted the modelling of a quarter of specimen. For shear and tensile/shear loading, the whole specimen has been considered. Boundary conditions are applied on the middle line of the adhesive. Stresses and strains in the adhesive were investigated along the overlap at the contact between adhesive and adherend. For simplification purposes, a kinematic bonding has been set between the adhesive layer nodes and the adherends'. The adhesive behavior is supposed to be perfectly elastic. In order to validate the specimen geometry, the model has been implemented for 2 adhesive layer thicknesses, 2 adhesive edge shape and for 3 adherends geometry (figure3): a simple model without beaks which is supposed to have the highest singularities, a model with straight edges beaks and one with round beaks. The adhesive geometry is either straight or lightly curved. Mesh is set to be the finest close to the high stress regions which are close to the edge. For both adhesive thicknesses the elements size is of 10μmx10μm. 3a). The model is implemented with plan strain hypothesis. Adherends are modeled with a linear elastic behavior and adhesive behavior has been implemented with a cohesive zone model (CZM). A classical bilinear traction-separation law has been chosen as a first approximation [3]. The traction separation law is defined by the initial adhesive modulus (Yt), the tensile stress at crack propagation (Smax) and the fracture energy (GIc). The parameters used are listed in table 1. Mesh is at its finest close to the initial crack with a size of 0.2mmx0.2mm. Loading is modeled by two reference points rigidly linked to the upper and lower adherends. The upper reference node is the one driving the load and the lower reference node is clamped.

Results and Discussion

a) Bulk tests results Bulk tests experimental results have stressed out that the adhesive has a nonlinear behavior with an almost perfectly perfect plastic region. These tensile tests have also permitted the calculation of the Young's tensile Modulus which is of 610 MPa and the Poisson ratio υ=0.37.

b) Arcan finite element stress analysis

Simulations of the ARCAN specimen under tensile loading have been carried out in order to choose the geometry presenting the most homogenous stress distribution. The goal was also to decrease the stress singularities between adhesive and adherends close to the edge. Like Cognard et al [2], it was showed that a curved adhesive at the edge enables the reduction of the stresses. But, it is the adherends' geometry that has the most impact on adhesive stress peaks at the interface. Indeed, straight geometry generates high stresses in the adhesive whereas both beaks geometry enables a drastic reduction of stress singularities.

c) Arcan experimental results

A Matlab script has been developed for the analysis of the Arcan test results. It uses the displacement data calculated by DIC correlated to determine the real tangential and normal displacement. Indeed, the experimental set up tends to rotate during the tests. This rotation is taken into account and permits the correction of the displacement for each image. The analysis of the 3 load cases have permitted to define the adhesive failure envelope where it can be seen that the adhesive bond thickness has little influence on the stresses at failure for shear and tensile-shear loadings (figure 4a). But for tensile loadings, a 1mm-bond thickness reduces the stress at failure by 20%. However, as it appears on figure 4b, the adhesive bond thickness has a large influence on the adhesive deformation when loaded in pure shear.

d) DCB experimental results

DCB experimental results showed an influence of the bond thickness on the load necessary to initiate the crack propagation, and the fracture energy, but the initial slope of the load-displacement curve is not affected by the variation of thickness. Numerical simulations are in good agreement for 0.2mm-bond thickness. However, for 1mm-bond the initial adhesive modulus and tensile stress at crack propagation determined for 0.2mm-bonds are not applicable for 1mm-bonds, as the FE model overestimates the load and both models underestimate the opening at the beginning of the crack propagation. Moreover, both numeric simulations overestimate the elastic modulus (figure 5).

Conclusions

The experimental and numerical analysis of the adhesive layer thickness has stressed out that bond thickness has an impact on the bonded assembly mechanical properties. Numerical simulation has permitted to validate the Arcan specimen geometry used experimentally. Moreover, numerical simulations of DCB tests highlights that the cohesive parameters determined for a 0.2mm-bonds are not assuring a good fit when used for 1mm-bond thickness.

Figure 1 .

 1 Figure 1. a) ARCAN test specimen; b) Experimental device to mold ARCAN specimens; c) Experimental ARCAN setup

Figure 2 .

 2 Figure 2. a) Quarter Arcan Model b) beak shapes studied c) Adhesive layer shapes studied b) DCB model A 2D FE model of a DCB specimen has been developed to compare to experimental data. The model dimensions are the same than the experimental ones (figure3a). The model is implemented with plan strain hypothesis. Adherends are modeled with a linear elastic behavior and adhesive behavior has been implemented with a cohesive zone model (CZM). A classical bilinear traction-separation law has been chosen as a first approximation [3]. The traction separation law is defined by the initial adhesive modulus (Yt), the tensile stress at crack propagation (Smax) and the fracture energy (GIc). The parameters used are listed in table1. Mesh is at its finest close to the initial crack with a size of 0.2mmx0.2mm. Loading is modeled by two reference points rigidly linked to the upper and lower adherends. The upper reference node is the one driving the load and the lower reference node is clamped.

  Figure 3. a) DCB FE model; b) ME model.

Figure 4

 4 Figure 4. a) Stress at failure envelope; b) load-deformation curve for shear loading.

Figure 5 .

 5 Figure 5. Comparison between experimental results and numerical prediction in terms of applied load versus opening at loading point for 2 thicknesses

Table 1 .

 1 Cohesive zone model properties.

	Bond thickness	Yt	Smax	GIc
	(mm)	(MPa)	(MPa)	(J/mm 3 )
	0.2	610	10	2
	1	610	10	4.3
	A 1D-beam model based on ME technique is developed. It
	is based on simplified hypotheses to model the joint behav-
	ior and uses the minimization of the potential energy for

solving. Both the adhesive layer and the adherends are gathered in 1 4-node element. Nonlinear adhesive material can be supported

[1]

. Boundary conditions are different than the FE model as the upper and lower adherends' nodes at the extremity of the specimen are blocked in the

Da Silva, L. F. & Campilho, R. D. Advances in numerical modelling of adhesive joints, 2012, Springer (a) (b) (a) (b) Beam element 1st ME ith ME

Acknowledgements

As part of the collaborative project S3PAC (FUI 21), this work was co-funded by BPI France, the Occitanie region and the Nouvelle Aquitaine region.


Emeric Guéguen
email: emeric.gueguen@insa-rennes.fr

Matthieu Crussière
email: matthieu.crussiere@insa-rennes.fr

Jean-Franc ¸ois Hélard
email: jean-francois.helard@insa-rennes.fr

Combination of OFDM and Spread Spectrum for High Data Rate UWB: optimization of the spreading length

Keywords: UWB, MB-OFDM, SS-MC-MA, spreading length optimization, self-interference

In this paper we investigate a new waveform based on Spread Spectrum Multi-Carrier Multiple Access (SS-MC-MA) for Ultra Wide Band (UWB) systems obtained by the combination of Orthogonal Frequency Division Multiplexing (OFDM) and Code Division Multiple Access (CDMA). This system is a great solution to combat frequency selectivity and narrowband interferers, and to manage the coexistence of several users and piconets. It also brings an interesting degree of freedom since the spreading codes of SS-MC-MA allow to optimize jointly the assignment of the number of used codes and the coding rates. We proposed methods to limit the Self-Interference (SI) between the spreading sequences by a judicious positioning of the subcarriers which carry spread data, and a selection of the spreading codes. The effect of the spreading code length is studied and shows that the code length has to be increased when the code rate gets higher to optimize the system performance. Through simulations it is demonstrated that the new UWB system outperforms the well known Multi-Band OFDM Alliance (MBOA) one.

I. INTRODUCTION

Since the Federal Communications Commission (FCC) opened the frequency range from 3.1 GHz to 10.6 GHz for unlicensed operation of Ultra Wide Band (UWB) in 2002 [START_REF]First report and order, revision of part 15 of the commission's rules regarding ultra-wideband transmission systems[END_REF], UWB has become a very attractive technology to create short range wireless communications with high data rates in a congested frequency sprectrum. As free spectrum access systems, UWB systems have to be designed not to disturb already existing narrowband systems such as Wireless Local Area Network (WLAN) 802.11a for example.

Since 2003, the standardization process of the Institute of Electrical and Electronics Engineers (IEEE) 802.15.3a task group, which aims at defining a high data rate physical layer for Wireless Personal Area Networks (WPAN), has seen the emergence and the confrontation of two main propositions. The first one is based on a pulse radio approach using a Direct Sequence Code Division Multiple Access (DS-CDMA) ternary codes and is supported by the UWB Forum. The second one follows a multi-carrier multi-band approach based on Orthogonal Frequency Division Multiplexing (OFDM) and is proposed by the Multi-Band OFDM Alliance (MBOA) consortium. The MBOA solution, supported by the main actors of the general public and component industries, seems to be a competitive solution by it's efficiency to cope with channel selectivity. The aim of this paper is to propose a new system for UWB applications combining the multi-band OFDM approach with spread spectrum (SS). More precisely, we investigate the use of the so-called Spread Spectrum Multi-Carrier Multiple Access (SS-MC-MA) system for UWB applications. Particularly, it is highlighted that under a spreading length optimization, SS-MC-MA offers for the future WPAN good performance and great flexibility for the resource allocation between users of a same piconet. In that sense, section II briefly presents a critical analysis of the MBOA solution and points out the advantages of combining SS with multi-band OFDM in the UWB context. In section III, the principle of the new waveform for UWB is presented and a description of the related system is introduced as an evolution of the MBOA solution. Optimization parameters for the system are proposed to improve the performance of the proposed system in section IV. In section V, the results obtained with different spreading code length are presented and discussed. Finally, section VI concludes this paper.

II. THE OFDM AND CDMA COMBINATION INTEREST

The MBOA consortium proposes to divide the available band into 14 sub-bands of 528 Mhz each, as illustrated in Fig. 1. The MBOA solution is described in detailed in [START_REF] Batra | Multi-Band OFDM physical layer proposal for IEEE 802.15 task group 3a[END_REF] with the main parameters.

The MBOA solution offers some advantages for high data rate UWB applications, among which the signal robustness 1-4244-0521-1/07/$20.00 ©2007 IEEE against the channel selectivity and the efficient exploitation of the signal energy received within the prefix duration. The main asset of the MBOA solution, as compared with the competitive impulsive radio solution, lies in its multi-carrier component which brings to the system a well recognized robustness to deep fades. The impulsive solution can actually hardly make use of all of the received energy, the number of the RAKE fingers being compulsorily limited for complexity reasons. However, the MBOA solution is relatively limited in a multiuser and multi-piconet context. Particularly, when the only three first sub-bands of the first mode are considered, conflicts immediately appear when a fourth user is added within a piconet, whereas scenarios going up to 6 simultaneous users have to be considered in practice.

To go beyond this limitation, recent studies have proposed to add a CDMA component to the MBOA solution in order to improve the system robustness or the resource sharing between several users [START_REF] Park | Performance of UWB DS-CDMA/OFDM/MC-CDMA system[END_REF]. This spreading component essentially allows to organize the access of several users to a common resource. Taking into account the UWB channel characteristics, frequency selectivity and slow time variations in an indoor environment, the spreading sequences are generally applied along the frequency axis, leading to the so-called Multi Carrier-CDMA (MC-CDMA) system. With MC-CDMA, the symbols of all users are transmitted by all the subcarriers as depicted in Fig. 2, the spreading code length L c being lower or equal to the subcarrier number N p of the OFDM multiplex. Compared to the "traditional" MBOA solution, and beyond a greater facility in the resource sharing, the MC-CDMA system also presents a better robustness against channel frequency selectivity and improves the UWB signal robustness against narrowband interferers ([3], [START_REF] Schmidt | Ultra wideband transmission based on MC-CDMA[END_REF]). This last point is fundamental for uncontrolled access to the spectral resource like UWB. In [START_REF] Park | Performance of UWB DS-CDMA/OFDM/MC-CDMA system[END_REF] however, authors suggest to use an MC-CDMA signal with a bandwidth B w = 1.58 GHz, equivalent to 3 sub-bands of the MBOA signal, which leads to an highly increase of the sampling frequency of the analog-to-digital conversion.

In order to fully benefit from the advantages of the SS component, we have proposed in [START_REF] Guéguen | Crussière Combination of OFDM and CDMA for high data rate UWB[END_REF] an SS-MC-MA waveform, which is new for UWB applications and offers better performance and more flexibility in the resource management than MBOA as detailed in the sequel.

III. THE SS-MC-MA BASED UWB SYSTEM

A. SS-MC-MA principle

The SS-MC-MA scheme, illustrated in Fig. 2, consists in assigning to each user a specific set of subcarriers according to a Frequency Division Multiple Access (FDMA) approach. Code dimension L c can then be exploited for an adaptive resource optimization and sharing (modulation type, data rate, . . . ). Spreading in the frequency domain yields a diversity gain and, like with MC-CDMA, improves the signal robustness against narrowband interferers. With an SS-MC-MA signal, symbols are transmitted simultaneously on a specific subset of subcarriers by the same user and undergo the same distortions. Self-interference (SI) which then replaces the Multiple Access Interference (MAI) obtained with MC-CDMA signals, can be easily compensated for using a single-user detection with only one complex coefficient per subcarrier.

B. System studied

The proposed system basically consists in an evolution of the MBOA system as depicted in Fig. 3. In this figure, the MBOA transmission chain is presented in continuous lines and the functions that are added to obtain the SS-MC-MA waveform are displayed using dashed lines. These functions are mainly the Hadamard Transform ("Fast Hadamard Transform": FHT) at the transmitter and the inverse transform (IFHT) at the receiver. In addition, Minimum Mean Square Error (MMSE) single user detection is applied. The spreading operation turns out to be simple, so the MBOA chain complexity is not significantly increased by the addition of the spreading component.

The main parameters of the SS-MC-MA system are listed in Table I. Walsh-Hadamard orthogonal spreading codes are used in this study to limit the SI and several spreading codes lengh L c can be used. To compare performances of the SS-MC-MA system with different L c , the number of useful subcarriers for MBOA is reduced from 100 to 96 for each OFDM symbol. This means that 4 additional guard subcarriers are added compared to the MBOA solution.

The SS-MC-MA signal waveform generated at the output of the IFFT expresses:

S(t) = +∞ i=-∞ NST 2 n=-NST 2 P l=1 D l (i)c l,m × Π(t -iT ZP ) e j2πn∆F(t-iTZP) (1)
where ∆ F , N ST and T ZP represent the subcarriers spacing, the total number of used subcarriers and the spacing between two consecutive OFDM symbols, respectively (Table I). Π(t) is a rectangular window defined by:

Π(t) = 1 0 t T FFT 0 T FFT t T FFT + T ZP . ( 2
)
C l = [c l,1 ... c l,m ... c l,L ]
is the l th Walsh-Hadamard orthogonal spreading code and D l (i) represents the P complex symbols, belonging to a QPSK constellation and which are transmitted by a block of L c subcarriers considered during the OFDM symbol i. P then represents the load, and is namely equal to L c in the full load case and to L c /2 in the half load case. m denotes a chip of the l th spreading code and is varying from

1 to L c , with m = n mod(L c ).
The MMSE detection coefficients used to compensate for the channel effects write:

g n,i = h * n,i |h n,i | 2 + 1 γn,i (3)
where h n,i and γ n,i represent the complex channel response and the signal to noise ratio for subcarrier n of symbol i, respectively.

C. SS-MC-MA advantages

Let us consider the case of the SS-MC-MA system working in the mode 1 of the MBOA standard (Fig. 1) as it is proposed in most of the studies.

1) Case of three or less users:

The SS-MC-MA system allows the allocation of a 528 MHz sub-band for each user. This system offers the same performance and advantages as MC-CDMA, but needs a more simple channel estimation process in reception. In fact, with SS-MC-MA a given subcarrier is distorted by only one channel, the one of the user associated with this subcarrier. At the contrary, with an MC-CDMA system, each subcarrier is corrupted by different channels of different users, which increases considerably channel estimation complexity. In that case, each user has to estimate the response of many channels all over the total available bandwidth.

2) Case of more than three users: The code dimension could be exploited in SS-MC-MA to share a same 528 MHz sub-band between 2 or even 3 users if necessary. In that case, the generated signal within a given block corresponds to an MC-CDMA signal, but with a limited number of users per block (2 or even 3). Thus, it is possible to have 6 or more users in the first mode with SS-MC-MA compared to the MBOA solution for which conflicts appear from 4 users.

More generally, in a multi-piconet context, the possibility of easily modifying the number of spreading codes assigned to a given user in a given piconet, allows the SS-MC-MA scheme to offer a more flexible and efficient dynamic resource sharing than the MBOA solution does.

IV. SPREADING COMPONENT HANDLING

A. Spreading block assignment strategy

In SS-MC-MA systems, two different frequency allocation schemes can actually be considered to form the spreading blocks.

The first frequency allocation scheme is a "the standard block interleaving scheme" which consists in interleaving the subcarriers assigned to one user so that the chips of the spread symbol belonging to that user are regularly distributed across the whole bandwidth. Thus, maximum frequency diversity is made available at the receiver.

The second scheme is called "adjacent subcarrier scheme" as it gathers the chips of one user's spread symbol on neighboring subcarriers. In contrast to the first scheme, the adjacent subcarrier scheme provides a weaker exploitation of the available frequency diversity. However, the correlation between channel coefficients of adjacent subcarriers is more important, and consequently the channel variance σ 2 h is smaller. It is explained in [START_REF] Cariou | A simple and efficient channel estimation for MIMO OFDM code division multiplexing uplink systems[END_REF] that the variance of the SI σ 2 SI is proportional to σ 2 h over a specific subset of subcarriers. So, by using adjacent subcarriers, the channel variance is limited and so is the SI.

In this paper, the "adjacent subcarrier scheme" has been chosen in order to limit the SI. In this case, the frequency diversity is jointly exploited by the spreading component and the channel coding combined with binary interleaving.

B. Optimization of the spreading code length

The spreading code length L c has a direct influence on the SS-MC-MA system performance. The longer the spreading codes are, the more the system takes advantage of the frequency diversity. Moreover, the system flexibility increases with L c by the possibility to obtain a broad choice of data rates. However, the subcarriers of one spreading block undergo a stronger distortion due to the channel selectivity, which induces an increase of the SI. Using shorter spreading codes length reduces the SI to the detriment of a weaker channel diversity exploitation. In that case, for low coding rates, the channel decoder is expected to compensate for the diversity loss. In other words, the channel coding combined with binary interleaving should allow to recover the frequency diversity. At the contrary, for high coding rates, i.e. for coding rates R that tend to 1, the system tends to a non-coded system, so the using of longer spreading code may be preferable in order to collect enough diversity.

C. Selection of the spreading codes

In presence of multipath channels, the orthogonality between spreading sequences is broken and not completely restored by the MMSE detector, then a residual SI term remains. The analytic expression of the SI power associated to the data j for the case of a synchronous SS-MC-MA transmission is given by:

σ 2 SI,j = (P -1)R j (0)L c α + P m=1 m =j                            2R j (1) Lc-1 n=1 w (j,m) n w (j,m) n+1 βj,m + 2R j (2) Lc-2 n=1 w (j,m) n w (j,m) n+2 γj,m + . . . 2R j (L c -1)w (j,m) 1 w (j,m) Lc                            (4)
where R j is the autocorrelation defined as R j (pq) = E[a p,j .a q,j ]. a n,j = h n,j .g n,j is the coefficient affecting subcarrier n after equalization, with h n,j and g n,j the channel and equalization coefficients respectively. w j,m n = c n,j .c n,m represents the product between the chip elements of the spreading sequences used by data j and m at the subcarrier n, and P L c is the number of active codes.

An optimized spreading code assignment is proposed in [START_REF] Mottier | A spreading sequences allocation procedure for MC-CDMA transmission systems[END_REF] to minimize the SI. Judicious subsets of P spreading sequences, whose minimal number of transitions (+1/ -1) among each possible product vector W (j,m) = (w

(j,m) 1 , w (j,m) 2 , . . . , w (j,m) Lc
) is maximum, are selected. Indeed, each product vector W (j,m) can have between 0 and L c -1 transitions. Then, depending on the set of selected spreading sequences, the set of corresponding product vectors has a given minimum which can be different from the minimum of another set. The selected spreading sequences subset is the one whose minimum vectors product is maximal compare to the minimum of the other subsets. In this case, the sum over m of negative terms β j,m in (4) decreases, which reduces the SI due to large positive value α. W (j,m) has to be understood, here, as a measure of the ability to reduce interference between data j and m. With this criterion, the largest degradation among two symbols could be minimized.

V. SYSTEM PERFORMANCE

This section presents results obtained with different spreading code lengths L c and the performance of the SS-MC-MA system for a given L c .

A. UWB channel modeling

The channel model used is the one adopted by the IEEE 802.15.3a channel modeling sub-committee for the evaluation of UWB physical layer proposals [START_REF] Foerster | Channel modeling sub-committee report final[END_REF]. This model results from Saleh-Valenzuela model for indoor application [START_REF] Saleh | A statistical model for indoor multipath propagation[END_REF]. This ray based model takes into account clusters phenomena highlighted during channel measurements. Mathematically, the impulse response of the multipath model is given by:

h k (t) = X k Z k z=0 B k b=0 α k (z, b)δ (t -T k (z) -τ k (z, b)) (5)
where X k is the log-normal shadowing of the k th channel realization, T k (z) is the delay of cluster z, and α k (z, b) and τ k (z, b) represent the gain and the delay of multipath b within cluster z respectively. The mean excess delay τ m and the root mean square delay spread τ rms for the 4 channel models CMi are presented in detail in [START_REF] Foerster | Channel modeling sub-committee report final[END_REF].

Frames of 150 OFDM symbols are used in simulations, and one different channel realization is applied on each new frame. The 100 different realizations are used for each CMi. In this paper, performances are estimated for UWB channel model CM1 which is in Line Of Sight (LOS) configuration.

B. The spreading code length effects

The objective is to find the best compromise between the spreading code length L c and the coding rate R. So, we present the system performance obtained versus L c : for a given R and a given load P . Optimized SS-MC-MA systems are simulated for a given E b /N 0 . Three coding rates R = [1/3, 1/2, 3/4] and three loads P = [L c /4, L c /2, L c ] are considered.

Results are exhibited for channel CM1 in Fig. 4. They show that L c and R have a strong influence on the SS-MC-MA system performance. For low coding rates, the curves tendency shows that it is better to use short code lengths (Fig. 4(a)) and reciprocally, that the more the coding rate is increased, the longer the code length should be (Fig. 4(b) and Fig. 4(c)). Indeed, the use of short spreading code lengths allows to minimize the SI and, combined with a low coding rate, it allows to benefit from the channel diversity. At the contrary, when the coding rates increase, the decoder is less able to exploit the channel diversity and longer spreading codes are necessary to compensate for this weakness. Note that for high coding rates, the system tends to a non-coded system. So, the observed behavior is consistent with the conclusions already drawn in [START_REF] Stephan | Adaptive multi-carrier spread-spectrum with dynamic time-frequency codes for UWB applications[END_REF] which shows that the performance of the SS-MC-MA system without channel coding are optimal with L c set to its maximal value to benefit from the maximum of diversity.

C. Performance of the SS-MC-MA system with L c = 16

In Fig. 5, L c = 16 was chosen to assess the performance of the SS-MC-MA system since it seems to be a good compromise for the coding rates used, according to the previous results. The joint assignment of the number P of spreading codes and the coding rate R provides different data rates. Table II introduces the code number/coding rate pairs that lead to SS-MC-MA data rates values that are very close of the ones of the MBOA solution. Fig. 5 exhibits the results obtained in the ideal case of perfect channel estimation for the data rates of Table II.

Compared to MBOA, the gain in term of required E b /N 0 to obtain a BER equal to 10 -4 with SS-MC-MA on channel CM1 varies from 0.5 to 1 dB. The best gain is obtained for low coding rate, while the performances of the two systems tend to be close when R tends to one.

VI. CONCLUSION

In this paper we have presented some methods to optimize the performance of an UWB system based on the SS-MC-MA waveform. In particular, we have selected an adjacent subcarrier scheme and introduced a code selection approach to reduce the SI. In addition, the spreading code length effects have been studied versus the coding rate in order to find the best compromise to exploit at best the channel diversity without degrading the performance by introducing too much SI. We can conclude that an optimal working of the SS-MC-MA system consists in increasing the spreading factor when the coding rate is getting higher.

To complete these results, we have plotted the performance of the SS-MC-MA system for a given spreading code length. We have hereby shown that the optimized new system outper- forms MBOA, which tends to prove that the new waveform is more adapted to face frequency selectivity. Eventually, we have improved the MBOA system by simply adding an Hadamard transformation function.

Fig. 1 .

 1 Fig. 1. Channels distribution for MBOA solution.

Fig. 2 .

 2 Fig. 2. Data distribution of different users for MC-CDMA and SS-MC-MA systems.

4 Fig. 4 .

 44 Fig. 4. SS-MC-MA system performance versus the spreading code length Lc for channel model CM1.

Fig. 5 .

 5 Fig. 5. Performance of the SS-MC-MA system for channel model CM1.

TABLE II POSSIBLE

 II DATA RATES WITH SS-MC-MA

	Data rate Modulation	Coding rate	Lc	Load	Coded bit
	(Mbit/s)		(R)		(P)	per symbol
	51.2	QPSK	1/3	1 6	4	4 8
	76.7	QPSK	1/3	1 6	6	7 2
	115.1	QPSK	1/3	16	9	108
	153.6	QPSK	1/3	16	12	144
	192	QPSK	1/2	16	10	120
	307	QPSK	1/2	16	16	192
	409	QPSK	2/3	16	16	192
	460	QPSK	3/4	16	16	192

ACKNOWLEDGMENT

The authors would like to thank France Télécom R&D/RESA/BWA which supports this study within the contract 461 365 82.