Margherita Disertori
email: margherita.disertori@iam.uni-bonn.de

Christophe Sabot
email: sabot@math.univ-lyon1.fr

Pierre Tarres
email: tarres@maths.ox.ac.uk

Pierre Tarrès

Transience of edge-reinforced random walk

come

1. Introduction 1.1. Setting and main result. Let (Ω, F , P) be a probability space. Let G = (V, E, ∼) be a nonoriented connected locally finite graph without loops. Let (a e ) e∈E and (w e ) e∈E two sequences of positive weights associated to each edge e ∈ E.

Let (X n ) n∈N be a random process that takes values in V , and let F n = σ(X 0 , . . . , X n ) be the filtration of its past. For any e ∈ E, n ∈ N ∪ {∞}, let (1.1) Z n (e) = a e + n k=1

1I {{X k-1 ,X k }=e} be the number of crosses of e up to time n plus the initial weight a e . Then (X n ) n∈N is called Edge Reinforced Random Walk (ERRW) with starting point i 0 ∈ V and weights (a e ) e∈E , if X 0 = i 0 and, for all n ∈ N,

(1.2) P(X n+1 = j | F n ) = ½ {j∼Xn } Z n ({X n , j}) k∼Xn Z n ({X n , k})
.

On the other hand, let (Y t ) t 0 be a continuous-time process on V , starting at time 0 at some vertex i 0 ∈ V . Then (Y t ) t 0 is called a Vertex-Reinforced Jump Process (VRJP) with starting point i 0 and weights (w e ) e∈E if Y 0 = i 0 and if Y t = i then, conditionally on (Y s , s t), the process jumps to a neighbour j of i at rate w {i,j} L j (t), where

L j (t) := 1 + t 0 1I {Ys=j} ds.
The Edge Reinforced Random Walk was introduced in 1986 by Diaconis [START_REF] Coppersmith | Random walks with reinforcement[END_REF]; the Vertex-Reinforced Jump Process was proposed by Werner in 2000, and initially studied by Davis and Volkov [START_REF] Davis | Continuous time vertex-reinforced jump processes[END_REF][START_REF] Davis | Vertex-reinforced jump processes on trees and finite graphs[END_REF]; for more details on these models and related questions, see [START_REF] Sabot | Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model[END_REF] for instance.

The aim of this paper is to prove transience of the edge-reinforced random walk (ERRW) for large a e > 0, i.e. small reinforcement.

2000 Mathematics Subject Classification. primary 60K37, 60K35, secondary 81T25, 81T60. This work was partly supported by the LABEX project MILYON and ANR project MEMEMO 2. Theorem 1. On Z d , d 3, there exists α c (d) > 0 such that, if a e > α c (d) for all e ∈ E, then the ERRW with weights (a e ) e∈E is transient a.s.

Recall that almost-sure positive recurrence of ERRW and VRJP for large reinforcement (i.e. if a e < αc , resp. w e < wc for all e ∈ E, for some αc , wc > 0), and transience of VRJP for small reinforcement (i.e. if w e > w c for some w c > 0) were proved by Sabot and Tarrès in [START_REF] Sabot | Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model[END_REF], using localisation/delocalisation results of Disertori, Spencer and Zirnbauer [START_REF] Disertori | Anderson localization for a supersymmetric sigma model[END_REF][START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]. Another proof of recurrence of the ERRW and VRJP was proposed, shortly afterwards, by Angel, Crawford and Kozma [START_REF] Angel | Localization for linearly edge reinforced random walks[END_REF].

The proof of Theorem 1 follows from estimates on the fluctuation of a field (U i ) associated to the limiting behaviour of the reinforced-random walk. Let us first recall two earlier results from Sabot and Tarrès [START_REF] Sabot | Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model[END_REF].

Theorem 2 (Sabot and Tarrès [START_REF] Sabot | Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model[END_REF]). On any locally finite graph G, the ERRW (X n ) n 0 is equal in law to the discrete time process associated with a VRJP in random independent conductances W e ∼ Gamma(a e , 1 ).

The next result concerns VRJP (Y t ) t 0 on a finite graph G, with |V | = N, given fixed weights (w e ) e∈E ; let P V RJP i 0 be its law, starting from i 0 ∈ V .

Proposition 1 (Sabot and Tarrès [START_REF] Sabot | Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model[END_REF]). Suppose that G is finite and set N = |V |. For all i ∈ V , the following limits exist P V RJP i 0 a.s.

U i = lim t→∞ (log L i (t) -log L i 0 (t)) .
Theorem 3 (Sabot and Tarrès [START_REF] Sabot | Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model[END_REF]). (i) Under P V RJP i 0 , (U i ) i∈V has the following distribution on H 0 = {(u i ) ∈ R V : u i 0 = 0} dρ w,Λn (u) = 1 (2π) (N -1)/2 e -j∈V u j e -H(w,u) D[m(w, u)] j∈V \{i 0 } du j (1.3) where H(w, u) = 2 {i,j}∈E w i,j (cosh(u i -u j ) -1)

and D(w, u) is any diagonal minor of the N × N matrix m(w, u) with coefficients

m i,j =    w i,j e u i +u j if i = j -k∈V w i,k e u i +u k if i = j
(ii) Let C be the following positive continuous additive functional of X:

C(s) = i∈V L 2 i (s) -1,
and let Z t = Y C -1 (t) . Then, conditionally on (U i ) i∈V , Z t is a Markov jump process starting from i 0 , with jump rate from i to j 1 2 w i,j e U j -U i .

In particular, the discrete time process associated with (Y s ) s 0 is a mixture of reversible Markov chains with conductances w i,j e U i +U j .

Remark 1. The diagonal minors of the matrix m(w, u) are all equal since the sum on any line or column of the coefficients of the matrix are null. By the matrix-tree theorem, if we let T be the set of spanning trees of (V, E, ∼), then D[m(w, u)] = T ∈T {i,j}∈T w {i,j} e u i +u j .

Notation and convention. In the following we fix d 3.

A sequence σ = (x 0 , . . . , x n ) is a path from x to y in Z d if x 0 = x, x n = y and x i+1 ∼ x i for all i = 1, . . . , n.

Let

Λ n = {i ∈ Z d , |i| ∞ n} be the ball centred at 0 with radius n, and let

∂Λ n = {i ∈ Z d , |i|∞ = n}
be its boundary. We denote by E the set of edges in Z d and by E n the set of edges contained in the hypercube Λ n . We denote by Ẽn the associated set of directed edges. Let (a ij ) i,j∈Z d ,i∼j be the family of initial positive weights for the ERRW and

a := inf e∈E a e .
In the proofs, we will denote by Cst(a 1 , a 2 , . . . , a p ) a positive constant depending only on a 1 , a 2 , . . . a p , and by Cst a universal positive constant.

Let P 0 (resp. P Λn 0 ) be the law of the ERRW on Z d (resp. on Λ n ). Theorem 3 ensures that on every finite volume Λ n the ERRW is a mixture of reversible Markov chains with random conductances (W U e ) e∈En where W U ij = W ij e U i +U j and the law for the random variables (W, U) has joint distribution on R En + × H 0 given by dρ Λn (w, u) = dρ w,Λn (u) e∈En e -we w ae-1 e Γ(a e ) dw e , where dρ w,Λn (u) was defined in (1.3). Let E Λn 0 the average with respect to the joint law for (W, U) (mixing measure). Note that we cannot define this average on an infinite volume, since we do not know if the limiting measure exists. Denote by • Λn the corresponding marginal in U.

Warning. We use a capital letter W, U • • • to denote a random variable and a smallcase letter w, u, . . . to denote a particular realization of the variable. The same is true for any function of such variables. In some cases though we do not state the argument explicitely, to avoid heavy notations. It should become clear from the context when the corresponding argument has to be regarded as a random variable.

The proof of Theorem 1 will follow from the following result.

c (m, d) > 0 such that, if a a c (m, d) then, for all n ∈ N, x, y ∈ Λ n , cosh m (U x -U y ) Λn 2.
The proof of Theorem 4 is the purpose of the rest of this paper, and adapts the argument of Disertori, Spencer and Zirnbauer [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF], which implied in particular transience of VRJP with large conductances (w e ). But before, let us show how Theorem 4 implies Theorem 1.

Proof of Theorem 1. Given (w, u) ∈ R E + × H 0 , denote by P w u 0 the law of the Markov chain in conductances w u i,j = w ij e u i +u j starting from 0. Let H ∂Λn be the first hitting time of the boundary ∂Λ n and H0 be the first return time to the point 0.

We seek to estimate P 0 (H ∂Λn < H0 ) the probability the ERRW hits the boundary of Λ n before coming back to 0. Since this event depends only on the history of the walk inside the hypercube Λ n , we have

(1.4) P 0 (H ∂Λn < H0 ) = P Λn 0 (H ∂Λn < H0 ) = E Λn 0 P W U 0 (H ∂Λn < H0 )
where in the last term we used the last part of Theorem 3. Now P w u 0 (H ∂Λn < H0 ) is the probability for a Markov chain inside Λ n with conductances w u to hit the boundary before coming back to 0. This probability can be related to the effective resistance between 0 and ∂Λ n (see for instance Chapter 2 of [START_REF] Lyons | Probability on Trees and Networks[END_REF])

R(0, ∂Λ n , w u ) = inf θ: Ẽn→R e∈ Ẽn θ(e) 2 w u e
where the infimum is taken on unit flows θ from 0 to ∂Λ n : θ is defined on the set of directed edges Ẽn and must satisfy θ((i, j)) = -θ((j, i)) and j∼v θ((v, j)) = δ v,0 for all v ∈ Λ n . Denote by R(0, ∂Λ n ) the effective resistance between 0 and ∂Λ n for conductances 1.

Classically we have

w u 0 R(0, ∂Λ n , w u ) = 1 P w u
0 (H ∂Λn < H0 ) with w u 0 = j∼0 w u 0,j . Using (1.4) and Jensen's inequality 1

P 0 (H ∂Λn < H0 ) = 1 E Λn 0 P W U 0 (H ∂Λn < H0 ) E Λn 0 1 P W U 0 (H ∂Λn < H0 ) = E Λn 0 W U 0 R(0, ∂Λ n , W U ) . (1.5)
We will show below that, if min e∈E a e = a > max{a c (3, d), 3}, then

E Λn 0 W U 0 R(0, ∂Λ n , W U ) Cst(a, d)R(0, ∂Λ n ) (1.6)
This will enable us to conclude: since lim sup R(0, ∂Λ n ) < ∞ for all d 3, (1.5) and (1.6) imply that P 0 ( H0 = ∞) > 0. ✷ Proof of (1.6). Let θ be the unit flow from 0 to ∂Λ n which minimizes the L 2 norm. Then

R(0, ∂Λ n , w u ) {i,j}∈En 1 w u i,j θ 2 (i, j), and R(0, ∂Λ n ) = {i,j}∈En θ 2 (i, j).
Now, if a > max{ã c (3, d), 3} then, using Theorem 4,

E Λn 0 W U 0 W U i,j = l∼0 E Λn 0 W 0,l W i,j e U 0 -U i e U l -U j l∼0 E Λn 0 W 3 0,l W 3 i,j E Λn 0 e 3(U 0 -U i ) E Λn 0 e 3(U l -U j ) 1/3 = l∼0 E Λn 0 W 3 0,l E Λn 0 W -3 i,j e 3(U 0 -U i ) Λn e 3(U l -U j ) Λn 1/3 Cst(a, d),
where we also used the fact that W i,j are independent Gamma distributed random variables. ✷ 1.2. Organization of the paper. Section 2 introduces to the main lines of the proof, the geometric objects that are needed (diamonds and deformed diamonds), and state some Ward identities. Section 3 provides some estimates on the probability of existence of good points i (from which there are no large deviations on B ij at small scales) in certain boxes. A key step in our proof is to study the law of U after integration over the conductances W e , in other words to focus on the marginal . Λ . Note that it is one of several possible approaches to the question of recurrence/transience for ERRW. Indeed, one could instead focus on the analysis of the law of the random conductances W ij e U i +U j given by Coppersmith-Diaconis formula, or directly study ERRW from its tree of discovery using the existence of a limiting environment, as done in [START_REF] Angel | Localization for linearly edge reinforced random walks[END_REF], or possibly conclude from a.s. results conditionally on the conductances W e , e ∈ E.

In fact we add a Gaussian Free Field variable S with conductances W U ij before integrating over W , as in the first step of [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF], since the corresponding joint law again has more transparent symmetries and is better suited for the subsequent analysis. More precisely, let

B xy := cosh(U x -U y ) + 1 2 e Ux+Uy (S x -S y ) 2
for all x, y ∈ V (not necessarily neighbours).

Then (W, U, S) has the following distribution on

R E + × H 0 × H 0 (2.1) 1 (2π) (N-1)/2 e -j∈Λ u j e -i,j∼i W ij (B ij -1) D[m(W, u)] k =i 0 du k ds k e∈E e -we w ae-1 e Γ(a e ) dw e
The marginal of this law in (U, S) after integration over W , which we still call . Λ or . by a slight abuse of notation, is given in the following Proposition 2.

Proposition 2. The joint variables (U, S) have density on

H 0 × H 0 (2.2) µ a,V (u, s) = 1 (2π) (N -1)/2 e 1 B ae e e -j∈Λ u j D[M V (u, s)],
where

D[M V (u, s)] is any diagonal minor of the N × N matrix M V (u, s) defined by M i,j =    c i,j , i = j -k c i,k , i = j and c i,j := a i,j e u i +u j B i,j , ∀, i ∼ j
Note that we slightly abuse notation here, in that B i,j is considered alternatively as function of (u, s) and (U, S).

Proof. Integrating the density in (2.1) with respect to the random independent conductances W e ∼ Gamma(a e , 1 ) we obtain e -j u j (2π) (N-1)/2 e e -we w ae-1 e dw e Γ(a e ) e -j∼i w ij (B ij -1) D(w, u)

= e -j u j (2π) (N-1)/2
T i∼j∈T e u i +u j

(2π) (N-1)/2 T i∼j∈T a ij e u i +u j B ij = µ a,V (u, s)
where in the second line we expand the determinant as a sum over spanning trees and In the proof of delocalization in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF], the first step was a set of relations (Ward identities) generated by internal symmetries. Their analogue is stated in Lemmas 1 and proved in Section 5, and involves a term of effective resistance D x,y depending on the variable U, defined as follows.

I
Definition 1. Let D x,y be the effective resistance between x and y for the conductances c x,y i,j := c i,j B x,y e -ux-uy = a i,j e u i +u j -ux-uy B x,y B i,j .

Lemma 1. For all m a/4 then, for all x, y ∈ Λ (not necessarily neighbours),

B m x,y (1 -mD x,y ) = 1.
One of the consequences of Lemma 1 is that, if the effective resistance D xy is small, then we can deduce a good bound on B m x,y . However, there is positive probability that this resistance D xy might be large, so that it would be more useful to show an inequality restricted to the event that D xy is small. Although it is not possible to derive such an identity directly, we can define events χ xy on which such identities hold, and so that D xy is small on χ xy . These events will depend on geometric objects defined in the next Section 2.2, called diamonds and deformed diamonds.

Diamonds and deformed diamonds.

Definition 2. Let l = 0 be a vector in R d and x ∈ Z d . We denote by C l

x the cone with base x, direction l and angle π/4

C l x = {z ∈ R d , ∠(xz, l) π 4 } = {z ∈ R d , (z -x) • l √ 2 2 |z -x||l|}
where ∠(xz, l) is the angle between the vector xz and the vector l.

If x and y are in Z d , we call Diamond the set

C y-x x ∩ C x-y y ∩ Z d .
to which we add a few points close to x and close to y so that the set is connected in Z d . We denote this set by R x,y .

Remark 2. By the expression " a few points" above we mean that we add some extra points to C y-x

x ∩ C x-y y ∩ Z d at a bounded distance from x, y so that the resulting set becomes connected in the lattice. The distance at which the points can be added is bounded by a constant depending only on the dimension. Of course, all the estimates below will be independent of the choice of these points. We will repeat this operation several times hereafter, without extra explanation.

In the course of the inductive argument, some deformed diamonds appear, they are formed of the intersection of two cones with smaller angles than for diamonds. For l ∈ R d , l = 0, and x ∈ R d we set

Cl x = {z ∈ R d , ∠(xz, l) π 16 }. Definition 3. A deformed diamond is a set of the following form Cl x ∩ Cx-y y ∩ Z d ,
(plus a few points close to x and to y so that the set is connected in

Z d , see Remark 2) where x ∈ Z d , l ∈ R d , l = 0 and y ∈ Z d is a point such that y ∈ Cl x .
We also denote a deformed diamond by R x,y (and it will always be clear in the text wether R x,y is a diamond or deformed diamond). Definition 4. It will be useful to write each (exact or deformed) diamond as a non disjoint union of two sets

R x x,y = {z ∈ R x,y , |z -x| f x |y -x|}, R y x,y = {z ∈ R x,y , |z -y| f y |y -x|}, where the pair (f x , f y ) is such that 1/5 f x 1, 1/5 f y 1 and f x + f y 1 + 1
5 . This condition on (f x , f y ), and the fact that the maximal angle in deformed diamonds is π/16, indeed ensures that that R x,y = R x x,y ∪ R y x,y . Note that the choice of (f x , f y ) is not unique. Here is the main proposition of the section. It ensure that under the condition χ xy , there is a uniform bound on the effective resistance D N x,y . Proposition 3. Fix α ∈ (0, 1/4) and b > 0. There exists a constant C = Cst(d, α, b) such that for any deformed diamond R x,y ⊆ Λ, if χ xy is satisfied, then

χ ij = ½ {B ij b|i-j| α } .
D N x,y C/a,
where a = inf(a i,j ).

Remark 3. Note that the constant C is independent of the precise shape of the deformed diamond.

Proposition 3 is proved in Section 6. It partly relies on the following two Lemmas 2 and 3, which will also be useful in other parts of the proof.

Lemma 2 (lemma 2 of [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]). For all x, y, z ∈ Λ,

B xz 2B xy B yz . Proof. Elementary computation. Lemma 3. For all i, j ∈ Z d , i ∼ j, x, y ∈ Z d , we have (2.3) (c xy ij /a) -1 16(B iz ) 2 (B jz ) 2 . for both z = x and z = y.
In particular, let c = b -4 /64, β = 4α. Assume that χ xy holds, then the electrical network with conductances

(γ i,j ) i,j∈Λ, i∼j = (c xy ij /a) i,j∈Λ, i∼j , satisfies γ i,j c|i -z| -β for all i, j ∈ R z xy , i ∼ j, for z = x, y,.
The proof of Lemma 3 partly relies on the following two Lemmas, which will also be useful in other parts of the proof.

Proof of Lemma 3. First note that e ±(ux-uy) B xy e ±(ux-uy) cosh(u x -u y ) 1/2, so that (2.4) e u i +u j -ux-uy B x,y 1 2 max[e u i +u j -2ux , e u i +u j -2uy ].

Now

(2.5)

e u i +u j -2ux = e u i -ux e u j -ux 1 4 (cosh(u i -u x ) cosh(u j -u x )) -1 1 4 (B ix B jx ) -1 .
On the other hand, note that

(2.6) B ij 2B ix B jx .
Inequalities

B m i x i y i (1 -m i D N x i y i ) 1,

and

(2.8)

n i=1 B m i x i y i χ x i y i (1 -m i D N x i y i ) 1.
Inequality (2.8) implies, by Lemma 3, that if, additionally, the regions R x i y i , i = 1, . . . , n are deformed diamonds and m i < a/C for all i, then (2.9)

n j=1 B m j x j y j χ x j y j n j=1 (1 -m j C/a),
where C is the constant considered in Proposition 3.

Lemma 1 is proved in Section 5. The rest of the proof is similar to the argument in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF], which consists in deducing upper bounds of n j=1 B m

x j y j for some fixed m from these "protected" estimates, through an induction on the maximal length |x i -y i | on the protected estimates, i = 1 . . . n, using Chebyshev inequalities in order to deal with the "unprotected" parts of the estimates. We summarise the argument in Section 4.

Estimates on good points

Definition 7. A point x ∈ Λ is called n-good if B x,y b|x -y| α ,
for all y ∈ Λ with distance 1 |x -y| 4 n from x.

Given z ∈ Λ, let R n (z) (denoted by R n when there is no ambiguity) the cube with side 4 n and barycenter z. We denote by χ c

Rn the indicator of the event that there is no n-good point in R n .

In this section we present an estimate on the event χ c

Rn , more precisely we bound from above the indicator function by a sum of terms involving B i,j b|j-i| α for points i, j at distance at most 4 n . This will be used in the main inductive argument in Section 4. It follows lemma 9 of [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF], but, for convenience of the reader, we give the proof which is only a few lines long.

If R n is a hypercube of side 4 n then it is the disjoint union of 4 d sub-hypercubes of side 4 n-1 . We can select the 2 d corner subcubes that we denote

(R i n , i = 1, . . . , 2 d ) so that d(R i n , R j n ) > 2 × 4 n-1 for i = j.
Repeating this procedure hierarchically we can construct a family of cubes (R v n ) with side 4 n-k for v running on the set {1, . . . , 2 d } k . We consider the natural structure of rooted (2 d + 1)-regular tree of the set

R = {root} ∪ ∪ n k=1 {1, . . . , 2 d } k ,
where "root" is the root of the tree, corresponding the cube R n (= R root n ). We denote by d v the depth of a point v ∈ R with d root = 0 and

d v = k if v ∈ {1, . . . , 2 d } k .
We denote by T n the set of connected subtrees T of R containing the root and which have the property that any vertex x ∈ T have either 2d or 0 descendant in T . We denote by L T the set of leaves of such a tree T ∈ T n . (Remark that the set {L T , T ∈ T n } is also the set of maximal totally unordered subsets of R for the natural "genealogical order" on R.)

For an element v ∈ R with k = d v we set S c R v n = x∈R v n , y∈Λ 4 n-k-1 <|x-y|<4 n-k χ c x,y x∈R v n , y∈Λ 4 n-k-1 <|x-y|<4 n-k B m
x,y b m |x -y| αm Lemma 5. (lemma 9 of [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]) With the notations above we have

χ c Rn T ∈Tn v∈L T S c R v n .
Proof. If there is no n-good point in R n then either there is no (n -1)-good point in any of the subcubes {R i n , i = 1, . . . , 2 d } or else there exists at least one pair (x, y) ∈ R n × Λ with 4 n-1 < |x -y| < 4 n and B x,y > b|x -y| α . This gives the first level inequality

χ c Rn S c Rn + 2d i=1 χ c R i n-1
.

Then the proof follows by induction on the integer n. For n = 0, R 0 (z) is the singleton z and χ c R 0 = S c R 0 = 0 which initializes the induction. If lemma 5 is valid at level n -1, then, obviously, the previous inequality implies it at level n. Our goal in this section is to prove the following theorem.

Theorem 5. Let m = a 1/8 , let ρ = 1/2 and assume a a 0 for some constant a 0 1. For all n 1 , n 2 and n 3 0, let R x i y i , i = 1, . . . n 1 , R p j q j , j = 1, . . . n 2 , and R r k s k , k = 1, . . . n 3 be respectively subsets of class 1, 2 and 3. Then we have

n 1 i=1 B m x i y i n 2 j=1 B 3m p j q j χ p j q j n 3 k=1 B 3m r k s k 2 n 1 (1 + ρ) n 2 2 n 3 .
The proof is by induction on max 1 i n 1 i -y i |. Let (H) ℓ be the following statement: Theorem 5 holds if max

1 i n 1 |x i -y i | ℓ.
The first step in the induction is the case n 1 = 0, proved in Section 4.1.

Let us now show that (H) ℓ-1 implies (H) ℓ . We do the proof only in the case where n 1 = 1, the general case being only notationally more involved. It remains to upper bound R(x, y) . Now, R(x, y) being an expansion over "bad" points z (i.e. sites z such that χ c xz holds), we expand it into four terms. We prove in Section 4.2 that R 1 (x, y) 1/16 if b Cst(a) and αm d (Case 1 in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]).

(ii) Second, fix a constant M, which will only depend on the dimension d. For a site z far from x (i.e. with |z -x| > a Then v c x,y,z = 1 if there is a large scale "bad" event originating from a point near z. We will show in Section 4.3 that the corresponding term R 2 (x, y)

1/16 if if b Cst(a) and αm 10d (Case 2a in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]).

(iii) Third, we consider the case where v x,y,z holds, i.e. there is no large scale "bad" event near z and, furthermore, there is exists a point g with |g -z| |z -x| 1/2 that is good up to scale |z -x| 1/2 .

More precisely, given

i ∈ Λ, R > 0, let G(i, R) = h: |i-h| R χ ih ;
then G(i, R) = 1 iff i is "good" up to distance R. Recall the similar Definition 7 that a site x ∈ Λ is called n-good if χ xy = 1, for all y with |y -x| 4 n .

Let

g x,y,z = max

g: |g-z| |z-x| 1/2 G(g, M|z -x| 1/2 ).
Then g x,y,z = 1 iff we can find a site g in the ball of radius |z -x| 1/2 centered at z, which is good up to distance M|z -x| 1/2 . Now, if v x,y,z = g x,y,z = 1, then we can find a deformed diamond from x to g close to z such that χ x,g = 1, so that we can apply the induction hypothesis. If we let

R 3 (x, y) = z∈ Rx xy : |z-x|>a 1/4 B m xy χ c xz v x,y,z g x,y,z j: |j-x|<|z-x| χ xj , then R 3 (x, y)
1/16 if a Cst and αm 3d (Case 2b in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]): this is proved in Section 4.4.

(iv) Fourth, if g x,y,z = 0 and M Cst(d) then there is no good point up to distance M|z -x| 1/2 in the hypercube of side length 4M|z -x| 1/2 . This implies that χ c R n(x,z) (z) 4.1. Proof of Theorem 5 in the case n 1 = 0. The proof is similar to the one of lemma 8 in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]. First, note that the case n 3 = 0 follows from Lemma 4, for a Cst(C).

Let us do the proof in the case n 2 = 0 and n 3 = 1, the argument for the general case being only notationally more involved.

Let δ > 0. For all p, q ∈ Λ, p ∼ q, let ξ pq = 1I {Bpq 1+δ} . Let us first deal with the first term in the right-hand side of (4.3): B pq 1 + δ implies 0 (U p -U q ) 2 /2 cosh(U p -U q ) -1 δ.

Using

Choose δ > 0 such that a 1/4 √ 2δ = 1, i.e. δ = a -1/2 /2.

Let z = x, y, and assume {p,q},p,q∈Rxy ξ pq holds. Then, for all j ∈ R xy , (4.4)

|U z -U j | a 1/4 √ 2δ = 1.
Subsequently, for all p, q ∈ R xy , p ∼ q, e -2 2 e 2Uz (S p -S q ) 2 e Up+Uq 2 (S p -S q ) 2 B pq -1 δ,

which implies e Uz |S p -S q | √ 2δe. Using again our choice of δ, we deduce that, again if Let us now deal with the second term in the right-hand side of (4.3): fix p, q ∈ R xy , p ∼ q, and use that, by Markov inequality,

j ∈ R xy , e Uz |S z -S j | a 1/4 √ 2δe = e, so that
ξ pq B pq 1 + a a/2
.

Let (x 0 , . . . , x n ) be a path of minimal distance from x to y inside R xy , which does not go through the edge {p, q}. By repeated application of Lemma 2,

2B xy 0 j n-1 2B x j x j+1 .
Therefore, letting ℓ = |x -y|, we have

B 3m xy ξ c pq 2 3m(ℓ-1) (1 + δ) a/2 B a/2 pq 0 j n-1 B 3m x j x j+1 2 3m(ℓ-1) (1 + δ) a/2 2 1 - 3m a -ℓ
exp (3mℓ -aδ/3) exp 3a 3/8 -a 1/2 /6 . (4.7)

In the second inequality, we use (2.7) and note that, if r ∼ s, then D N rs = a rs a. In the third inequality we assume a Cst and, for x ∈ [0, 1/2], 1 -x e -x and (1 + x) -1 e -2x/3 . Finally, we use δ = a -1/2 /2, m = a 1/8 and ℓ a 1/4 (since R xy is of Class 3) in the last inequality.

In summary, (4.3), (4.6) and (4.7) together imply, if a Cst(C), In order to apply the induction assumption, we would need to construct a deformed diamond R xz and diamond R zy which do not intersect within R xy . This is not true in general, but we can add an intermediate point a ∈ R xy such that R xz , R za and R xz are respectively one deformed diamond and two diamonds within R xy and disjoint, except at endpoints (see Figure 1, and Lemma 12 [START_REF] Angel | Localization for linearly edge reinforced random walks[END_REF] in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF] for more details). Now, using again Lemma 2,

B 3m xy 3/2 + Cst(d)ℓ d exp 3a 3/8 -a 1/2 /6 2 if a Cst(d).
B m xy χ c xz 2 2m b -2m |z -x| -2αm B 3m xz B m za B m
ay . Therefore, using the induction assumption,

R 1 (x, y) z∈ Rx xy : |z-x| a 1/4 4 m b -2m |z -x| -2αm 4(1 + ρ) 4 b 2 m Cst a 1/4 r=1 r d-1-2αm 1/16
if b Cst(a) and αm d. As above, we use

(4.9) χ c jk B m jk b -m |j -k| -αm .
In order to apply the induction assumption, we need to expand B 2m jk B m xy into a product of terms arising from disjoint diamonds within R xy . It is an easy geometric result to show that, under our assumptions on z, j and k, we can choose four intermediate points a i ∈ R xy , so that R xa 1 , R a i a i+1 (i = 1, . . . , 3) and R a 4 y are diamonds with disjoint interiors which do not overlap with the diamond R jk (see Figure 2, and Lemma 12 [START_REF] Coppersmith | Random walks with reinforcement[END_REF] in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF] for more details). Now, using (4.9) and Lemma 2,

B m xy χ c jk 2 4m B m xa 1 3 i=1 B m a i a i+1 B m a 4 y B m jk b -m |j -k| -αm ,
which implies, using the induction assumption,

B m xy χ c jk 2 4m 2 6 b -m |z -x| -αm/2 .
Now, for each z ∈ Rx xy , there are of the order of |z -x| d+d/2 pairs (j, k) satisfying (4.8). Therefore On the other hand, if χ c xz occurs then, using B gz b|z -g| α b|z -x| α/2 and Lemma 2, we deduce that

2B xg B xz /B gz b|z -x| α /(b|z -x| α/2 ) = |z -x| α/2 . Hence B m xy χ c xz χ xg 2 m B m xg B m gy χ c xz χ xg 2 3m (B 3m xg χ xg )B m gy |z -x| -αm .
As in the proof of the case i = 1, we introduce an intermediate point a ∈ R xy such that R ga and R ay are diamonds, disjoint from each other and from R xg , except at endpoints (see Figure 1, and Lemma 12 [START_REF] Angel | Localization for linearly edge reinforced random walks[END_REF] in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF] for more details).

Therefore B m xy χ c xz χ xg 2 3m (1 + ρ)2 2 |z -x| -αm .
There are less than |z -x| d/2 choices for g, so that

R 3 (x, y) 2 3m r>a 1/4 r d-1+d/2-αm 1/16
if αm 3d and a Cst.

4.5. Proof of R 4 (x, y) 1/16. If a Cst, then R n(x,z) (z) is inside R xy ; recall that its side length is of the order of |z -x| 1/2 << |y -x|.
As in the proof of the case i = 2, we can choose four intermediate points a i ∈ R xy , so that R xa 1 , R a i a i+1 (i = 1, . . . , 3) and R a 4 y are diamonds with disjoint interiors which do not overlap with the hypercube R n(x,z) (z) (see Figure 2, and Lemma 12 [START_REF] Coppersmith | Random walks with reinforcement[END_REF] in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF] for more details). Now, using Lemma 5,

B m xy χ c R n(x,z) (z) 2 4m B m xa 1 3 i=1 B m a i a i+1 B m a 4 y T ∈Tn xv ∈Rv, yv∈Λ,v∈L T 4 n-kv -1 <|xv -yv | 4 n-kv v∈L T B m xvyv b m |x v -y v | αm .
This implies, letting n v = n -d v and using the induction assumption,

B m xy χ c R n(x,z) (z) 2 4m 2 6 T ∈Tn v∈L T (4 nv ) d (4 nv 2) d b m 4 (nv-1)αm = 2 4m+6 I n(x,z)
where, for all n 0,

I n = T ∈Tn v∈L T γ2 -ζnv , with γ = (4 α /b) m 2 d , ζ = 2(αm -2d).
It follows from the structure of the trees T n that

I n = γ2 -ζn + (I n-1 ) 2 d , I 0 = γ.
Assume αm 4d and b Cst(α, d), so that γ 1/4 and ζ αm. Then we deduce by elementary induction that I n 2 -αmn for all n 1. Note that 4 n(x,z) 2 M |z -x|. In summary,

R 4 (x, y) 2 4m 2 -αmM/2 Cst r>a 1/4 r d-1-αm/4 1/16
if αm 8d and a Cst.

Proof of Ward inequalities: Lemmas 1 and 4

We start this section with two lemmas, that will be useful in the subsequent proofs. The first is an elementary lemma which expresses the equivalent resistance on a conductance network as a quadratic form and relates the quantity D x,y to the corresponding term in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]. Lemma 6. Let (Λ, E) be a finite connected graph and (c e ) e∈E a conductance network on E. We set c i = j∼i c i,j . Let i 0 ∈ Λ be a fixed vertex and M be the matrix given by

(M i,j ) =    c i,j , i = j -c i , i = j
(which is the Matrix of the generator of the Markov process with jump rates (c i,j )). Let N be the restriction of M to Λ \ {i 0 }. Denote by G the Λ × Λ symmetric matrix defined by G(i 0 , y) = G(y, i 0 ) = 0 for any y and G(x, y) = -N -1

x,y if x, y = i 0 . If D x,y is the equivalent resistance between x and y, then

D x,y = G(x, x) -2G(x, y) + G(y, y) =< (δ x -δ y ), G(δ x -δ y ) > .
Remark 4. In comparison with [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF], it means that the term G x,y which appears in formula (5.4) in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF] is the equivalent resistance between x and y with conductances c x,y i,j = e -tx-ty B x,y β i,j e t i +t j .

Proof. We first interpret probabilistically the matrix G. Let G i 0 (x, y) be the Markov chain with transition probabilities p i,j = c i,j c i and killed at its first entrance hitting time of i 0

G i 0 (x, y) = E x   H i 0 k=0 ½ X k =y   ,
where

H i 0 = inf{k 0, X k = i 0 }. Then, clearly, G(x, y) = 1 c x G i 0 (x, y),
Then exercise 2.61 in chapter 2 of [START_REF] Lyons | Probability on Trees and Networks[END_REF] yields the result.

The next lemma ensures that the joint density µ a,V (u, s) given in (2.2) has bounded "moments" up to a certain order. for any choice of m 1 , . . . m n such that m j a/2 for all j = 1, .., n.

Proof. By expression (2.2), we have

n j=1 B m j e j = 1 (2π) (N-1)/2 e 1 B āe e D[M a,V (u, s)]e -j∈Λ u j k =i 0 du k ds k ,
where we set āe j = a e j -m j for j = 1, ..n and āe = a e for all other edges. Note that āe j a e j /2 since m j a/2 for all j. Expanding the minor as a sum over spanning trees we deduce where we have used the bound a e j 2(a e j -m j ) so we can replace a e with ā in the determinant.

Proof of lemma 1. For more readability we provide an elementary derivation of the Ward identity which does not involve fermionic integral, even though it could be deduced from the more general proof of lemma 4. Consider the graph (V, Ẽ) where we add an extra edge ẽ = {x, y} to E (possibly creating a double edge). We put a weight a ẽ = -m on this edge. Denote by μa,V (u, s) the corresponding density, and by Ma,V the corresponding matrix in (2.1). Using the expression (2.2), we deduce ,s) is a probability measure on Ẽ. The bound above holds for any m a/4, hence μa,V (u, s) is integrable. Moreover it is an analytic function in the parameters a e , and therefore (5.1) μa,V (u, s)

k =i 0 du k ds k = 1.
Now let N(u, s) be the restriction of M V (u, s) to the subset V \ {i 0 }, and let Ñ(u, s) be the corresponding matrix for the new graph. Expanding the determinant with respect to the extra term coming from the new edge ẽ we deduce, letting N = N(u, s),

det( Ñ(u, s)) = det(N) -(-m) e ux+uy B x,y (½ x =i 0 det(N) x,x -2½ x =i 0 ,y =i 0 det(N) x,y + ½ y =i 0 det(N) y,y ) = det(N) 1 -m e ux+uy B x,y < (δ x -δ y ), G(δ x -δ y ) > ,
where in the previous expressions, det(N(u, s)) x,y is the minor where we remove line x and column y, and where in the last line G is the matrix defined in Lemma 6, with the conductances c i,j defined in Proposition 2. Using lemma 6, we deduce

det( Ñ(u, s)) = det(N(u, s))(1 -mD x,y ). Therefore 1 = μa,V (u, s) k =i 0 du k ds k = B m x,y (1 -mD x,y )µ a,V (u, s) k =i 0 du k ds k .
Proof of Lemma 4. In [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF], the protected Ward estimates are a consequence of Berezin identity stated in appendix C, proposition 2 of [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]. The starting point is to write the determinant term D[M(W, u)] as a fermionic integral (cf e.g. [START_REF] Disertori | Modèles de matrices aléatoires et supersymétrie[END_REF][START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]) with new pairs of anti commuting variables (ψ i , ψ i ). This leads to

(5.2) µ a,V (u, s, ψ, ψ) = e∈E 1 B ae e e -e∈E ae Be (Se-Be) e -j∈Λ u j , where S i,j = B ij + e u i +u j (ψ i -ψ j )(ψ i -ψ j )
is the same supersymmetric expression introduced in [START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF],

u i 0 = s i 0 = 0 and ψ i 0 = ψ i 0 = 0. Then dµ a,V (u, s) = dµ a,V (u, s, ψ, ψ) k =i 0 dψ k dψ k .
From the mathematical point of view, the fermionic integral should be understood as an interior product with respect to the variables (ψ k , ψ k ), cf e.g. [START_REF] Disertori | Modèles de matrices aléatoires et supersymétrie[END_REF]. Since the fermionic variables are antisymmetric, we have (S e -B e )

(s ij ) ij ], then (5.4) f [(S ij ) ij ]dµ a,V (u, s, ψ, ψ) = f (0), if f [(s ij ) ij ]
is integrable with respect to dµ a,V (u, s, ψ, ψ) (which means that after integration with respect to the fermionic variables it is integrable in the usual sense).

Remark that if f is a polynomial (of degree bounded by a), then it is a direct consequence of the fact that the measure dµ a,V (u) has integral one, which can be proved either by supersymmetric or probabilistic arguments (as we did in (5.1) above). Indeed let (x i , y i ), i = 1, . . . , n be n pairs of points such that R x i y i are regions whose interiors are disjoint. As in the proof of lemma 1 above let Ẽ = E ∪ (∪ n i=1 {x i , y i }) the graph obtained by adding the edges {x i , y i }. We assign to each new edge {x i , y i } the conductance a x i ,y i = -m i . Let Ma,V be the corresponding matrix. Then

n i=1 S m i x i y i = 1 (2π) (N-1)/2   e∈ Ẽ 1 B ae e   D[ Ma,V (u, s)]e -j∈Λ u j k =i 0 du k ds k ,
In order to show that the density is integrable, we proceed as in the proof of Lemma 1. This time we have to choose n simple paths γ i , i = 1, ..n in E connecting x i to y i , which do not intersect; this is possible, since the regions R x i y i have disjoint interiors. Then

n i=1 S m i x i y i = 1
Now by the same arguments as in [7, lemma 7] we have

D[ Ma,V (u, s)] D[M a,V (u, s)] n j=1 (1 -m j D N x j y j ).
This completes the proof of (2.7). Finally to prove the "protected" Ward estimate (2.7) we proceed exactly as in [7, lemma 6]. The main idea is to approximate the characteristic function χ(x) by a sequence of smooth decreasing functions χ δ (x). Now, by symmetry, S m xy χ δ (S xy ) = 1 ∀x, y.

Integration over the fermionic variables yields

S m xy χ δ (S xy ) = B m xy χ δ (B xy )e m+ χ ′ (Bxy ) χ(Bxy ) (Sxy-Bxy) = 1 (2π) (N-1)/2 e∈ 1 B ae e B m xy χ δ (B xy )D[ Ma,V (u, s)]e -j∈Λ u j k =i 0 du k ds k ,
where Ma,V is defined on the graph Ẽ = E ∪ (x, y) and the edge (x, y) has conductance

a xy = -m + B xy -χ ′ (B xy ) χ(B xy ) e ux+uy > -m, since χ ′ < 0. Hence D[ Ma,V (u, s)] > D[ Mā,V (u, s)] = D[M a,V (u, s)](1 -mD xy )
where āxy = -m and āe = a e for all other e. Finally

B m xy χ δ (B xy )(1 -mD xy ) 1.
The proof of the general statement follows from a combination of this argument with the ideas used for the estimate (2.7).

Proof of Proposition 3

Denote by E x,y = {{i, j}, i ∈ R x,y , j ∈ R x,y , i ∼ j} the set of non-directed edges in R x,y . We denote by Ẽx,y the associated set of directed edges. By construction, aD N x,y is the effective conductance between x and y of the network with edges E x,y and conductances (γ i,j ) {i,j}∈Ex,y . Let F x,y be the set of unit flows from x to y with support in Ẽx,y : precisely, θ ∈ F x,y if θ is a function θ : Ẽx,y → R which is antisymetric (i.e. θ(i, j) = -θ(j, i)) and such that div(θ) = δ x -δ y , where div : R x,y → R is the function div(θ)(i) = j∈Rx,y, j∼i θ(i, j).

Recall that (see for instance [START_REF] Lyons | Probability on Trees and Networks[END_REF], Chapter 2) aD N

x,y = inf θ∈Fx,y {i,j}∈Ex,y 1 γ i,j (θ(i, j)) 2 . (6.1)

The strategy is now to construct explicitly a flow θ such that under the condition χ x,y , the energy (6.1) is bounded by a constant depending only on d, α and b. This flow will be constructed as an integral of flows associated to sufficiently spread paths.

Remind that a deformed diamond is a set of the following form and p(z) = x + r(z)(y -x) the projection of z on the line (x, y).

For h ∈ (0, 1) we denote Rx x,y = {i ∈ R x,y , r(i) h}, Ry x,y = {i ∈ R x,y , r(i) h}. From the assumption on f x , f y , there exists h ∈ [1/10, 9/10] such that Rx

x,y ⊆ R x x,y and Ry

x,y ⊆ R y

x,y

1 . We fix now such a h ∈ [1/10, 9/10]. We set

∆ h = {z ∈ R d , r(z) = h} ∩ Cl x ∩ Cx-y y .
If is clear from the construction that

|∆ h | Cst(d)|x -y| d-1 ,
where |∆ h | is the surface of ∆ h (note that Cst(d) does not depend on the value of h ∈ [1/10, 9/10]).

To any path σ = (x 0 = x, . . . , x n = y) from x to y we can associate the unit flow from x to y defined by

θ σ = n i=1 ½ (x i-1 ,x i ) -½ (x i ,x i-1 ) .
For u ∈ ∆ h , let L u be the union of segments

L u = [x, u] ∪ [u, y].
Clearly L u ⊆ Cl

x ∩ Cx-y y by convexity. There is a constant Cst(d), such that for any u ∈ ∆ h , we can find a simple path σ u in R x,y from x to y such that for all k = 0, . . . , |σ u | dist(L u , σ u (k)) Cst(d), (6.2) and we define θ as

θ = 1 |∆ h | ∆ h θ σu du,
which is a unit flow from x to y.

The path σ u can visit a vertex i only if dist(i, L u ) c 0 = Cst(d). This implies that, for all i, j: j∼i

|θ(i, j)| 1 |∆ h | ∆ h ½ {dist(i,Lu) c 0 } du. Now if r(i) ∈ [0, h], then ∆ h ½ {dist(i,Lu) c 0 } du Cst(d) h r(i) d-1
1 Indeed, if i ∈ R x,y , the angle ∠(z, i), (x, y) π/8 for z = x, y. Hence, if i ∈ Rx x,y , then |i -x| h cos(π/8) |x -y|. Hence Rx x,y ⊆ R x x,y as soon as h f x cos(π/8). But cos(π/8) 0.92 and f x 1/5 so that f x cos(π/8) 0.18. Similarly Ry x,y ⊆ R y x,y as soon as 1 -h f y cos(π/8), since f y cos(π/8) 0.18. Using f x + f y > 1 + 1/5 it implies that (f x + f y ) cos(π/8) > 1 and we can find h ∈ [1/10, 9/10] such that Rx

x,y ⊆ R x x,y and Ry x,y ⊆ R y x,y .

Hence, if r(i) ∈ [0, h] Now, under the condition χ x,y , we know that if i ∈ Rx

x,y ⊆ R x x,y , then γ i,j c|i -x| -β and if i ∈ Ry

x,y ⊆ R y x,y , then γ i,j c|i -y| -β with β = 4α < 1. This implies

Theorem 4 .

 4 Fix d 3. For all m > 0, there exists a

  e (a e ) = dw e Γ(a e ) e -weBe w ae-

2. 3 .

 3 Estimates on effective conductances. Definition 5. Given a diamond or deformed diamond R xy ⊆ Λ, let D N xy be the effective resistance of the electrical network with the same conductances c x,y i,j in R xy as in Definition 1, and Neumann boundary conditions on ∂R xy . Definition 6. Fix b > 1 and α 0. Given i, j ∈ Λ, let

  Given a deformed diamond R xy ⊆ Λ and the two corresponding regions R x xy and R y xy , let χ xy = j∈R x xy χ xj j∈R y xy χ yj .

4 .Definition 8 .

 48 Inductive argument Fix b > 1 and α ∈ (0, 1/4), which will be chosen later; let C = Cst(d, α, b) be the constant considered in Proposition 3. The sets R xy in our induction are classified as follows: • Class 1: diamonds R xy , with |x -y| > a 1/4 ; • Class 2: deformed diamonds R xy , with |x -y| > a 1/4 ; • Class 3: deformed diamonds R xy , with |x -y| a 1/4 .

  Assume (H) ℓ-1 . Let x, y ∈ Λ be such that |x -y| = ℓ. Let Rxy be the deformed diamond between x and y introduced in Definition 3, with l = y -x; let Rx xy and Ry xy be its two parts in Definition 4, respectively from x and y, withf x = f y = 1/(2 cos(π/16)). |j-x|<|z-x| χ xj .Then it follows from the expansion of the partition of the unity u xy + R(x, y) + R(y, x) . The first term in the right-hand side of (4.1) can be upper bounded, by (2.9) in Lemma 4: if a Cst(C) (recall m = a 1/8 ), then (4.2) B m xy u xy (1 -mC/a) -1 1 + ρ 3/2.

( i )

 i First, the sites z close x, i.e. with |z -x| a 1/4 : R 1 (x, y) = z∈ Rx xy : |z-x| a 1/4 B m xy χ c xz j: |j-x|<|z-x| χ xj .

(4. 5 ) e U j +Uz 2 (e 3 2 .

 522 S j -S z ) 2 e 2 e 2Uz (S j -S z ) 2 Inequalities (4.4)-(4.5) together imply that B zj cosh(1) + e 3 /2 = Cst. Therefore χ xy holds with b = Cst and α = 0 implies (4.6) B 3m xy {p,q},p,q∈Rxy ξ pq (1 -3mC/a) -1 3/2, assuming a Cst(C) (recall m = a 1/8 ).

4. 2 .m xy χ c xz 2 m B m xz B m zy χ c xz 2 m

 22 Proof of R 1 (x, y) 1/16. Let z ∈ Rx xy be such that |z -x| a 1/4 . Using χ c xz B 2m xz b -2m |z -x| -2αm and Lemma 2, we obtain B b -2m |z -x| -2αm B 3m xz B m zy .

Figure 1 .Figure 2 .

 12 Figure 1. (Figure 5 in [7]) We add one intermediate point a. The two angles θ 1 and θ 2 are greater than π/8.

R 2 ( 4 . 4 .

 244 Cst(a) and αm 10d. Proof of R 3 (x, y) 1/16. Let z ∈ Rx xy be such that |z -x| > a 1/4 . If v x,y,z = g x,y,z = 1, then there exists g with |g -z| |z -x| 1/2 such that, for all h ∈ R xy with |g -h| |z -x|/5, χ gh holds. Let R xg be the deformed diamond in Definition 3, with l = y -x, and choose f x = (|z -x| -1)/|g -x| = 1 -O(a -1/8 ), f y = 1/5 in Definition 4. Then χ xg holds.

Lemma 7 .

 7 Let (x i , y i ), i = 1, . . . n be n pairs of nearest neighbor points and let e 1 , . . . e n be the corresponding undirected edges in E. Then

c e e -j∈Λ u j k =i 0 du k ds k ,

 0 ā,V (u, s)]e -j∈Λ u j k =i 0 du k ds k = 2 n dµ ā,V (u, s) = 2 n

  xy D[ M|a|,V ] where |a| xy = m and |a e | = a e > 0 for all e ∈ E. Now let γ a simple path in E connecting x to y. Then, by Lemma 2, the same argument as in the proof Lemma 7 above, |μ a,V (u, s)| 2 m(|γ|-1) e∈γ B 2m e μ|a|,V (u, s) 2 m(|γ|-1) 2 |γ| μ | a|,V (u, s) where | a| e = |a| e -2m for all e ∈ γ and | a| e = |a| e otherwise. By Proposition 2, dμ | a|,V (u

  few points close to x and to y so that the set is connected inZ d ) where x ∈ Z d , l ∈ R d , l = 0 and y ∈ Z d is a point such that y ∈ Cl x For z ∈ R d we set r(z) = (z -x) • (y -x) |y -x| 2 ,

1 =

 1 Cst(d) |i -x| d-1 . Similarly we have if r(i) ∈ [h, 1]

  j: j∼i |θ(i, j)| Cst(d) |i -y| d-1 .



  x,y , i =x c -1 |i -x| β-2(d-1) + i∈ Ry x,y , i =y c -1 |i -y| β-2(d-1)  For k ∈ N, |{i ∈ R x,y , k |i -x| < k + 1}| Cst(d)k d-1 , and similarly for x replaced by y. k β-(d-1) Cst(d, α, b). since β = 4α < 1 and d 3.

  holds, where n(x, z) is the part of M log |z -x|/(2 log 4). Then, if we let R 4 (x, y) =

		B m xy χ c xz v x,y,z χ c R n(x,z) (z)	χ xj ,
		z∈ Rx xy : |z-x|>a 1/4		j: |j-x|<|z-x|
	we apply the "good points" expansion obtained in Section 3 to show in Section 4.5 that
	1/16 if a Cst, b Cst(α, d) and αm 4d (Case 2c in [7]). In summary, R 4 (x, y)
			4
		R(x, y)	i=1	R i (x, y) ,
	and R i (x, y) i = 1, 2, 3 and 4.	1/16 is proved in Sections 4.2, 4.3, 4.4 and 4.5, respectively in the cases

  If f (s ij ) is a smooth function of the variable (s i,j ), we understand f (S i,j ) as the function obtained by Taylor expansion of the fermionic variables (ψ i , ψ i , ψ j , ψ j ); this expansion is finite since the fermionic variables are antisymmetric. Hence, formally f (S i,j ) is function with values in the exterior algebra constructed from (ψ i , ψ i , ψ j , ψ j ). The same can be generalized to f [(S ij ) i,j∈V ×V ]. The Berezin identity (see for instance proposition 2 in[START_REF] Disertori | Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model[END_REF]) implies that for any smooth function f [

					2 = 0, and
	Then	e -ae Be (Se-Be) = 1 -	a e B e	(S e -B e ) =	1 Be (S e -B e ) 1 + ae
	(5.3)	µ a,V (u, s, ψ, ψ) =	e∈E	1 e S ae	e -j∈Λ u j


Thavamaran Kanesan
email: thavamaran.kanesan@ieee.org

Son Thai Le

Damien Roque

Andrew D Ellis

Non-rectangular perfect reconstruction pulse shaping based ICI reduction in CO-OFDM

Keywords:

In this paper, we propose to increase residual carrier frequency offset tolerance based on short perfect reconstruction pulse shaping for coherent optical-orthogonal frequency division multiplexing. The proposed method suppresses the residual carrier frequency offset induced penalty at the receiver, without requiring any additional overhead and exhaustive signal processing. The Q-factor improvement contributed by the proposed method is 1.6 dB and 1.8 dB for time-frequency localization maximization and out-of-band energy minimization pulse shapes, respectively. Finally, the transmission span gain under the influence of residual carrier frequency offset is ~62% with out-of-band energy minimization pulse shape.

Introduction

In the past years, coherent optical-orthogonal frequency division multiplexing (CO-OFDM) has been an attractive technology in high speed communication, due to its inherent advantages of higher resilience towards chromatic dispersion and polarization mode dispersion, flexibility in accessing individual subcarriers in multi-user environment, and a simplified equalization scheme [START_REF] Benyuan | Recent progress in transmission fibers for capacity beyond 100-Tbit/s[END_REF]. However, apart from the disadvantage of high peak-toaverage ratio (PAPR), the CO-OFDM is very sensitive towards carrier frequency offset (CFO) [START_REF] Jian | Advantage of optical fast OFDM over OFDM in residual frequency offset compensation[END_REF], which will effectively introduce inter-carrier interference (ICI).

In CO-OFDM transmission demonstration, CFO estimation and compensation was carried out in [START_REF] Simin | Multiband real-time coherent optical OFDM reception up to 110 Gb/s with 600-km transmission[END_REF][START_REF] Zhou | A simple and efficient frequency offset estimation algorithm for high-speed coherent optical OFDM systems[END_REF][START_REF] Ju | An efficient and frequency-offset-tolerant channel estimation and synchronization method for PDM CO-OFDM transmission[END_REF]. However, in high speed CO-OFDM systems, residual uncompensated CFO always exists with Gaussian distribution [START_REF] Jian | Advantage of optical fast OFDM over OFDM in residual frequency offset compensation[END_REF]. It is crucial to minimize the residual CFO to not only avoid ICI, but also relaxes the complexity of the initial CFO estimator and enhances the receiver robustness against optical local oscillator instability [START_REF] Jian | Advantage of optical fast OFDM over OFDM in residual frequency offset compensation[END_REF][START_REF] Barbieri | OFDM versus single-carrier transmission for 100 Gbps optical communication[END_REF]. However, the algorithms for ICI mitigation are computationally demanding, and the implementation of such algorithm is highly not feasible due to the high speed nature of optical communications [START_REF] Barbieri | OFDM versus single-carrier transmission for 100 Gbps optical communication[END_REF]. For an example, some real-time CO-OFDM transmission demonstrations were not able to include even the first stage CFO compensation due to the lack of resources in field programmable gate array (FPGA) [START_REF] Kaneda | Real-Time 2.5 GS/s coherent optical receiver for 53.3-Gb/s sub-banded OFDM[END_REF][START_REF] Simin | Demonstration of 12.1-Gb/s single-band real-time coherent optical OFDM reception[END_REF], unless a state-of-the-art FPGA with high capacity was utilized as the CO-OFDM receiver as demonstrated in [START_REF] Simin | Multiband real-time coherent optical OFDM reception up to 110 Gb/s with 600-km transmission[END_REF]. An efficient approach in mitigating the impact of residual CFO is to use pulse shaping techniques with good time and frequency localization properties. Recently Li et al [START_REF] Li | Reduction of intercarrier interference based on window shaping in OFDM RoF systems[END_REF] introduced pulse shaping with root raised cosine (RRC) in optical OFDM, where RRC has demonstrated some improvements in comparison to rectangular pulse shape for receiver operating under the influence of residual CFO.

In this paper, we are introducing short perfect reconstruction (orthogonal) pulse shapes in CO-OFDM, namely time-frequency localization maximization (TFL) and out-of-band energy minimization (OBE), and presenting an opportunity of taking a new approach in providing higher residual CFO tolerance with pure discrete pulse shapes. We will introduce RRC for comparison with TFL and OBE. The major advantages of the proposed short pulse shapes (TFL and OBE) are that: A) it does not require additional overhead beyond adding cyclic prefix, and B) in the case of hardware implementation, the proposed method only requires additional multipliers at the transmitter and receiver, simply because TFL and OBE are short length filters essentially designed in discrete domain allowing short length block processing. However, RRC is an analogue pulse shape with infinite impulse response that matches the Nyquist criterion. In the case of implementation, truncation is required for RRC to achieve short pulse shape; consequently the discrete finite impulse response of the RRC pulse shape would not allow perfect reconstruction and will not meet the Nyquist criterion.

Principal of proposed pulse shapes and impact of residual CFO

Pulse shaping allows the signal to spread in the time-frequency plane. Such technique is important in the case of time-frequency selective channels [START_REF] Kozek | On the eigenstructure of underspread WSSUS channels[END_REF]. Since the pulse shapes are usually longer than each data block, the underlying transceiver requires polyphase decomposition and advanced equalization schemes. To maintain a minimum complexity, short perfect reconstruction pulse shapes will be used, which are shorter or equal than each data block [START_REF] Roque | Performances of weighted cyclic prefix OFDM with low-complexity equalization[END_REF].

Input-output relationship

Information is carried by a sequence of complex symbols denoted {c m,n } (m,n) , where m is the subcarrier index, n is the block index. The symbols are assumed to be independent and identically distributed. Each c m,n is placed in the time-frequency plane at coordinates (m/M, nN), where M is the number of subcarriers and N represents the number of sample per subchannel symbol period. The transmitted signal can be written as a function of discrete time k:

, , ( , ) [ ] [ ], Z. m n m n m n s k c k k g = Î å ( 1
)
where Z is the integer set, γ m,n [k] is a time-frequency shifted version of the prototype pulse shape γ[k] defined as:

2 , 2 1 [ ] [ ] , [ ] Z m j k M m n k k nN e k . M p g g g = - Î , 2 ] Z , 2 k . ] Z , 2 (2)
where 2 (Z) 2 (Z) is the set of square summable sequences. A necessary condition for perfect reconstruction of the transmitted symbols is the linear independence of the vectors {γ m,n }. This requires N/M ³ 1, but orthogonality and completeness are not mandatory [START_REF] Kozek | Nonorthogonal pulseshapes for multicarrier communications in doubly dispersive channels[END_REF].

Let x and y be two sequences of 2 (Z) 2 (Z) , we define the inner product

, 2 1 [ ] [ ] , [ ] Z m j k M m n k k nN e k . M p g g g = - Î , 2 1 [ ] [ ] , [ , 2 , 2 m j k j k 2 m j k j k k k nN 1 [ ] [ ] [ ] [ ] , 2 , 2 , 2 , 2 , 2 1 k . ] Z ] Z , 2 , 2 , 2 j k j k g , 2 , 2 [ ] [ ] , [ , [ , 2 , 2 [ ] [ ] [ ] ] Z ] Z , 2 , 2 , 2 ] Z , 2 k . ] Z , 2 k . ] Z , 2 (3)
In the presence of an ideal channel, such that r[k] = s[k], the received signal is projected on the dual family and if we let (m,n) ¹ (p,q), where the qth received symbol on the pth subchannel can be written as:

, , , , , , , , ( , , )
, , , p q p q p q p q p q m n p q m n m n p q c s c c

g g = [ ] [ ] k k [ ] [ ] [ ] [ ] g g [ ] [ ] [ ] [ ] [ ] [ ] ,
a perfect reconstruction system is denoted as orthogonal. In order to perform an N-size block processing, we restrict our analysis on short filters, that is to say [ ] [ ] 0 k k

g g = = [ ] [ ] k k [ ] [ ] [ ] [ ] g g [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] , if k > N -1 or k < 0.
This particular condition allows an N-size block processing. The resulting transmission scheme can be efficiently realized thanks to the use of fast algorithms [START_REF] Roque | Modulations multiporteuses WCP-OFDM: evaluation des performances en environment radiomobile[END_REF]. Thus, this generalizes the conventional OFDM transceiver by allowing non-rectangular pulse shapes while preserving a low complexity, as shown in Fig. 2 (detailed system description will be given in the Section 3). If we denote x[k], 0 £ k £ M -1, entering the cyclic prefix Insertion block, and its output as

y[k], 0 £ k £ N -1, then we have y[k] = x[2M -N + k] for 0 £ k £ N -M -1 and y[k] = x[N -M + k] for N -M £ k £ N -1. The resulting N samples are weighted by [ ], 0 1 k k N g £ £ -.
At the receiver side, the dual operation is performed in the filtering block to ensure perfect reconstruction of the transmitted symbols, that is to say ,

, m n m n c c = c c c c c c if r[k] = s[k].
In particular, cyclic prefix Folding block is defined by the following: if we denote x[k], 0 £ k £ N -1 as entering cyclic prefix Folding block, and its output is

y[k], 0 £ k £ M -1, then we have y[k] = x[N -M + k] for 0 £ k £ 2M -N -1 and y[k] = x[k -(2M -N)] + x[k -(M -N)] for 2M -N £ k £ M -1.
If the received signal r[k] is equal to the transmitted signal s[k], then the complex symbols {c m,n } can be exactly reconstructed provided that the following perfect reconstruction conditions are fulfilled:

[ ] [ ] [ ] [ ] 1, for 0 1, [ ] [ ] 1, for 1. k k k M k M k N M k k N M k M g g g g g g • + + + = £ £ -- • = -£ £ - [ ] 1, fo k k k M k M [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1, fo [ ] [ ] [ ] [ [ ] [ ] [ ] [ ] [ ] [ ] ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] k k [ ] 1, fo [ ] g g g g [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1, fo [ ] [ ] 1, [ ] [ ] [ ] [ ] [ ] [ ] ( 5
)
Through this relation, one may recover the expression of rectangular pulse shape used for cyclic prefix based OFDM: if [ ] 1 k g = for 0 £ k £ N -1 and 0 otherwise and perfect reconstruction is satisfied when [ ] 1

k g = [ ] 1 k [ ] [ ] g for N -M £ k £ N -1 and 0 otherwise.

CO-OFDM in the presence of residual CFO

As specified in Section 1, CO-OFDM is very sensitive towards residual CFO; assuming perfect frame synchronization and no sampling frequency offset, a simple relation can be established by the following expression:

[ ] ( )[ ] [ ] [ ] [ ] j k r k Hs k z k s k e z k pf = + = + ( )[ ] [ ] ( )[ ] [ ] ( )[ ] )[ ] [ ] ( )[ ] ] ( )[ ] [ ] (6)
where H H is the channel model, f is the residual CFO, and z[k] is a complex white Gaussian noise sequence. Taking into account ( 1), ( 4), and ( 6), we derive the expression of the qth estimated symbol on the pth subcarrier:

2 , , , , 2 [( )/ ] , ( , ) , [ ] [ ] [ ] [ ] 1 [ ] [ ] . j k p q p q p q p q k Z k Z j m p M k m n p q k Z m n
Using the hypothesis of short length filters, we notice

that [ ] [ ] 0 k nN k qN g g - - = [ ] [ ] [ ] [ ] N k qN [ ] [ [ ] [ ] [ ] [ ] [
] for n ¹ q, so that there is no interference between OFDM symbols. Consequently, we can focus on a single OFDM symbols estimation and omit n and q indices: ,

1 1 2 [( )/ ] , , 0
1: TFL pulse shape exhibits better localization in time and frequency than OBE. The difference between TFL and OBE is that TFL has higher sidelobe energy and quick energy decay; on the other hand OBE has lower sidelobe energy and slow energy decay. RRC shows good sidelobe energy, however it introduces high inter-symbol interference.

1 1 1 [ ] [ ] p p m p M M j m p M k p m p p p p m m p p m k Z m m p s i A c c k k e z c A c A z M M M f p f f f g g - - - + = Î = ¹ = + = + + å å å 1 1 1 1 j m p M 2 [( ) 2 [( ) k k e [ ] [ ] 2 [( ) 2 [( ) 1 1 2 [( ) 2 [( ) 1 M M c c c c 1 M M M M c c c c å c c c c
+ z c A f A [ ] p m k Z A f g g [ ] [ ] m k Z m k Z å m , , 1: m m p p , , , , m m p 1: i = ¹ 1: m m p m m p 1: å , , , , (8)
where s p is the useful part of the signal, i p is ICI term, and z p is the filtered noise. The useful part of the signal should be maximized and the rest should be minimized in terms of mean power. First of all, in presence of a small residual CFO such that f < 1/M, with overlapped frequency domain signals, ICI mitigation suggests using frequency localized pulse shapes. On the other hand, the presence of additive white Gaussian noise justifies the use of orthogonal signaling (i.e. matched filtering) [START_REF] Proakis | Digital Communications[END_REF].

Perfect reconstruction filters

In this section, we will describe the two families of prototype filters that are used herein over time-frequency dispersive channel, namely rectangular pulse shape and short perfect reconstruction filters.

Biorthogonal rectangular pulse shapes

The cyclic prefix OFDM is known as a rectangular multicarrier modulation with worst frequency localization compared to any non-rectangular shaped OFDM signals [START_REF] Farhang-Boroujeny | OFDM versus filter bank multicarrier[END_REF][START_REF] Strohmer | Optimal OFDM design for time-frequency dispersive channels[END_REF]. Therefore we choose rectangular pulse shape as a benchmark in our work. The CO-OFDM biorthogonal rectangular pulse shapes can be defined as:

RECT RECT 1, if 0 1, [ ] 0, otherwise, 1, if 1, [ ] 0, otherwise. k N k N -M k N k g g £ £ - ì = í î £ £ - ì = í î RECT [ ] k [ ] ] g = (9)
The first N -M coefficients of γ RECT form the cyclic prefix in order to mitigate inter-symbol interference introduced by time-dispersive channels. At the receiver side, the cyclic prefix is removed by RECT g RECT g

. The time and frequency response of the rectangular pulse shape is shown in Figs. 1(a) and 1(b), respectively.

Orthogonal perfect reconstruction filters

The prototype filters described in [START_REF] Pinchon | Closed-form expressions of optimal short PR FMT prototype filters[END_REF] are derived from two optimization criteria: TFL and OBE. Using ∆ = N -M, closed-form expressions are given for both filters:

TFL (2 1) sin if 0 1, 4 1 if 1, [ ] (2( ) 1) sin if 1, 4 0 otherwise. k k k M k k M k N p g p ì + ae ö £ £ D - ç ÷ ï D è ø ï ï D £ £ - = í D -+ ae ö ï £ £ - ç ÷ ï D è ø ï î (10) OBE (2 1) cos if 0 1, 2 1 if 1, [ ] (2 1) cos if 1, 2 0 otherwise. k a b k k M k k a b M k N g ì + ae ö + £ £ D - ç ÷ ï D è ø ï ï D £ £ - = í + ae ö ï + £ £ - ç ÷ ï D è ø ï î ae ö 1) ì + (2 (2 (2 (2 ç ÷ ç ÷ 1) ae ö 1) ç ÷ ç ÷ ì + a b a b a b a b a b a b a b a b è ø 2 ç ÷ 2 a b a b a b D ç ÷ ç ÷ ì + (2 (2 (2 (2 a b ç ÷ ç ÷ (2 a b ae ö (2 a b ì + (2 (2 (2 (2 a b ç ÷ 2 a b ç ÷ ç ÷ ae ö (2 1) (2 (2 ç ÷ ç ÷ 1) ae ö 1) ç ÷ ç ÷ ae ö a b a b a b a b a b a b a b è ø 2 ç ÷ 2 a b a b a b D ç ÷ ç ÷ ae ö (2 (2 a b ç ÷ ç ÷ (2 a b ae ö (2 (2 (2 a b ç ÷ 2 a b ç ÷ ç ÷ (11)
where a a and b b are two constants given in [17] and depends on the ratio of M / ∆. The time- frequency analysis of the prototype pulse shapes are shown in Fig. 1, and plotted against rectangular pulse as a reference. From Fig. 1, TFL pulse shape exhibits better localization in time and frequency than OBE. In principal, both OBE and TFL are short length, perfect reconstruction (orthogonal) and have symmetric impulse response. The difference between these pulse shapes is that TFL has higher sidelobe power (lower than rectangular pulse shape) and quick power decay; on the other hand OBE has lower sidelobe power and slow power decay.

Since RRC pulse shape is investigated in this paper, OBE filter may be directly compared with RRC because they share the same optimization criterion. But, it is important to understand that the RRC pulse shape is a continuous time function, hence it has to be truncated in order to fit an N coefficients impulse response. Such a truncation prevents the perfect reconstruction and yield inter-symbol interference [START_REF] Siohan | Analysis and design of OFDM/OQAM systems based on filterbank theory[END_REF]. A comparison between TFL and RRC is not equitable because they are not optimized from the same criterion, where the former is optimized jointly in time and frequency resulting in minimization of second order moment with the constraint of N non-zero coefficients. The RRC pulse shape is shown in Fig. 1.

System overview

In order to evaluate the proposed method of non-rectangular perfect reconstruction pulse shapes, an end-to-end extensive simulation is carried out with 40 Gb/s quadrature phase shift keying (QPSK) CO-OFDM system. The simulation system is constructed in MATLAB TM , as shown in Fig. 2. As illustrated in Fig. 2, the serial data stream is converted to M = 512 parallel structured data prior to being mapped into gray coded QPSK modulation format. The first symbol is used for channel estimation, and 8 pilot symbols are placed in an interval of every 64 subcarriers for phase noise estimation. The QPSK data is then transformed into OFDM signal by performing IFFT of size 1024 operation, apart from the data subcarriers, the rest are padded with zeros for oversampling purpose. The duration of OFDM symbol is 40.1 ns. The OFDM signal is then appended by cyclic prefix at the rate of 25% to combat inter-symbol interference in long haul transmission. The extended OFDM symbol is then weighted with various pulse shapes, including rectangular, RRC with α of 0.1, 0.2, and 1, and the newly introduced TFL and OBE.

After the insertion of training symbol, the total transmitting symbols are 200. The electrical OFDM signal is optically modulated by IQ modulator, and transmitted over 400 km to 3600 km of single mode fiber (SMF), with loss coefficient: 0.2 dB/km, PMD coefficient: 0.1 ps/km 1/2 , CD coefficient: 16 ps/nm.km, and optical fiber nonlinearity coefficient: 1.22 W -1 km -1 . The optical fiber loss is compensated for every 80 km span by utilizing erbiumdoped fiber amplifier (EDFA) with 16 dB gain and noise figure of 6 dB. The laser phase noise is modeled using the Wiener-Levy process, expressed as the variance σ 2 = 2πvt, where v is the combined laser linewidth and t is the time difference between two samples [START_REF] Randel | Analysis of RF-pilot-based phase noise compensation for coherent optical OFDM systems[END_REF]. In this simulation, the amplified spontaneous emission (ASE) noise is added relative to each fiber span (80 km). At the receiver, the coherently detected signal is first frame synchronized and sampling frequency offset is not induced to purely study the mitigation of ICI due to residual CFO. To achieve optimum pulse shaping, a receiver dual filtering with the same shape as the transmitted is utilized for weighting. Rest of the receiver process is exactly the reverse of the transmitter except for the channel and phase noise estimations and compensations.

Results and discussions

Table 1 and Fig. 3 present the back-to-back and optical launch power analysis, respectively, and the signal quality is quantified through the usage of Q-factor based on direct error counting. The CO-OFDM transmission with rectangular pulse shape without any CFO is the benchmark, in other word it is the ideal received signal without ICI. Then, we deliberately introduce a residual CFO of 2 MHz and investigate the additional tolerance that can be provided by the proposed pulse shapes. By utilizing a rectangular pulse shape and 2 MHz residual CFO, the Q-factor at the optical launch power of -5 dBm deteriorated from 12.9 dB to 8.9 dB, which clearly shown the criticalness of ICI and resulted in dropping below the 7% forward error correction (FEC) limit of 9.1 dB [START_REF] Sano | 30 x 100-Gb/s all-optical OFDM transmission over 1300 km SMF with 10 ROADM nodes[END_REF]. In the case of RRC pulse shape, the Q-factor at -6 dBm optical launch power for α of 0.1 and 0.2 is 9.3 dB and 9.1 dB, respectively, unveiling the optimum α as 0.1 and consequently agreeing with [START_REF] Li | Reduction of intercarrier interference based on window shaping in OFDM RoF systems[END_REF]. However, increasing the α to 1 introduces excessive intersymbol interference, resulting in very worsen Q-factor, where at -8 dBm optical launch power, the Q-factor is 7.3 dB. One important aspect that needs to be addressed here is that the optimum optical launch power point decreases with respect to the increasing α. This is due to the proportional increase of PAPR relative to the increase in α. Now we have shown the impact of applying RRC and confirmed the optimum α, although applying RRC resulted in Qfactor above the FEC limit, but the improvement is rather insignificant. Henceforth, the investigation on RRC will be dropped from rest of the paper. It is clear from Fig. 3 that both TFL and OBE pulse shapes achieved the highest Q-factor under residual CFO of 2 MHz. At the optical launch power of -6 dBm, the Q-factor for TFL and OBE is 10.5 dB and 10.7 dB, respectively. The Q-factor improvement with respect to the rectangular pulse shape is 1.6 dB and 1.8 dB for TFL and OBE, respectively. It is clear that OBE provides better performance compared to TFL, although the difference is small. The superiority of OBE over TFL is due to the lower energy of sidelobes, as shown in Fig. 1(b). The overall improvement of both OBE and TFL pulse shapes over rectangular pulse shape is due to the improvement of frequency localization that results in lower energy sidelobes (see Fig. 1), hence under the influence of residual CFO, ICI occurs and lower energy sidelobes minimizes the impact. Similar to RRC, both TFL and OBE demonstrate a shift in the optimum optical launch power region due to the increase in PAPR. However, neither of these pulse shapes affect the peak power of OFDM signal. The pulse shaping function trims the symbol power over a period, as shown in Fig. 1, resulting in decreased average power compared to rectangular pulse shape. In other word, rectangular pulse shape retains the shape of the symbol over its period. Therefore, reduction in average power from pulse shaping results in increase of PAPR that shifts the optimum launch power region. The paper on PAPR evolution relative to pulse shaping by Roque et al can be referred for further information [START_REF] Roque | Weighted cyclic prefix OFDM: PAPR analysis and performances comparison with DFT-precoding[END_REF]. Since 2 MHz residual CFO was purely chosen under arbitrary condition, it is important to understand the credibility of the proposed pulse shapes. Therefore, we have conducted a detailed investigation of the performances of rectangular, TFL, and OBE pulse shapes over residual CFO(x), 0 £ x £ 4 MHz; for the transmission span of 1600 km, as shown in Fig. 4. At CFO of 0, all three pulse shapes exhibit almost similar performance. For residual CFO > 0.5 MHz, rectangular pulse shape performs worse than both TFL and OBE with continuous degradation in Q-factor with a very sharp slope, effectively demonstrating the rectangular pulse shape's sensitivity towards ICI. Since TFL pulse shape's side lobe is lower than rectangular pulse shape and higher than OBE, it provides an optimum compensation for ICI up to 1 MHz. However, OBE closely follows the trend of TFL up to 1 MHz. Beyond 1 MHz, due to OBE's much lower power based sidelobes, it exhibits better resilience towards increasing ICI compared to TFL and rectangular.

This investigation has revealed the possibilities of using TFL and OBE under selected conditions based on a CO-OFDM receiver performance. For an example, if the residual CFO is < 1 MHz and > 1 MHz, TFL and OBE can be employed, respectively. However, it is important to note that in a real field deployed system, over a span of time, the residual CFO will increase, although not indefinitely. Therefore, it is advisable to utilize OBE as it performs similar to rectangular and TFL pulse shapes for residual CFOs of < 0.5 MHz and < 1 MHz, respectively, and demonstrates exemplary performance for residual CFO beyond 1 MHz. The limit of CFO for OBE is approximately 4 MHz for 1600 km at the specified FEC limit. Beyond this limit, the transmission distance has to be proportionally shortened, or more robust equalization schemes have to be utilized, and such optimization is beyond the scope of this paper.

Linewidth is a major factor that contributes to phase noise and as well as the quality of a specific laser. In order to determine if the quality of an optical LO at the receiver of CO-OFDM can be flexible in terms of quality, next phase of the investigation will be covering the impact of linewidth broadening on the proposed pulse shapes under the condition of no CFO. The rational for no CFO is to purely study the effect of phase noise. Figure 5 illustrates the performance of rectangular, TFL and OBE pulse shapes under no CFO, at 1600 km transmission and expanding combined laser linewidth that proportionally increases the phase noise. It is clear that additional phase noise induced by higher linewidth lasers have very similar impact on all three pulse shapes under no CFO. Hence, it is clear that the proposed pulse shapes behaves closely to the rectangular pulse shape and can be employed for ideal receivers, although ideal receivers is highly unlikely to occur in a practical system. Since it is well known that physical hardware, both optical and electrical will always introduce offsets and distortions, Fig. 4 has already shown the usefulness of the proposed pulse shapes.

Finally, CO-OFDM mostly drawn great interest in the area of long-haul optical communication systems [START_REF] Shieh | 107 Gb/s coherent optical OFDM transmission over 1000-km SSMF fiber using orthogonal band multiplexing[END_REF]. Therefore, in the first framework of these new pulse shapes, it is important to stress the proposed system and investigate its potential transmission ability. Figure 6 presents the transmission investigation for rectangular, TFL, and OBE pulse shapes from 400 km to 3600 km spans. The residual CFO is fixed at 2 MHz and a combined linewidth of 200 kHz. Taking 7% FEC as the first benchmark, rectangular pulse shape is capable of achieving this reference point for slightly less than 1600 km transmission. However, TFL and OBE pulse shapes achieved Q-factor improvements of 1.6 and 1.8 dB, respectively at the same transmission point. In the case of TFL and OBE pulse shapes at 7% FEC point, the achievable distances are 2400 km and 2600 km, respectively. It is clear that by employing OBE pulse shape, the improvement of transmission span is 1000 km with respect to rectangular pulse shaped system, which effectively translates into a ~62% gain. Long haul system can also be designed by taking 20% soft decision (SD)-FEC as the benchmark that results in Q-factor of 6.75 dB [START_REF] Dayou | Transmission of 115×100G PDM-8QAM-OFDM channels with 4bits/s/Hz spectral efficiency over 10,181km[END_REF]. In the case of 20% SD-FEC, rectangular pulse shape is still able to reach a 3100 km transmission distance. For TFL pulse shape, the 20% SD-FEC meets at the span of 3500 km and OBE effectively has not reached the threshold of 20% SD-FEC even up to 3600 km transmission, thus showing the credibility of the proposed pulse shapes.

Conclusion

A simple residual CFO compensation method was proposed herein, based on short nonrectangular perfect reconstruction pulse shapes for CO-OFDM. Employing both OBE and TFL pulse shapes will significantly reduce the receiver complexity compared to employing exhaustive algorithms. The investigation reveals that TFL performs better for residual CFO < 1 MHz and vice versa for OBE. However, employing OBE for CFO < 1 MHz does not introduce significant penalty, and since the optical LO for CO-OFDM will drift over time, OBE would be the ideal pulse shape in real-time condition. It is also shown in the paper that the proposed pulse shapes is independent of laser phase noise, which is not the case for conventional residual CFO compensating algorithms. Finally, by employing OBE, a transmission span of 3600 km could be achieved without hitting the 20% SD-FEC limit.

  norm \\x\\ = ‹x, x› 1/2 . At the receiver side, the dual family of {γ m,n } is written as { , m n g g } and we have:

2

 2

  this expression that perfect reconstruction of the transmitted symbols needs biorthogonality, namely , m,n) and (p,q); δ represents the Kronecker delta. If [ ] [ ] k k

2

 2

Fig. 1 .

 1 Fig.1. Rectangular (black), TFL (blue), OBE (red) and RRC (green, α = 1) depiction of (a) impulse response and (b) frequency response. Rectangular has very high energy sidelobes. TFL pulse shape exhibits better localization in time and frequency than OBE. The difference between TFL and OBE is that TFL has higher sidelobe energy and quick energy decay; on the other hand OBE has lower sidelobe energy and slow energy decay. RRC shows good sidelobe energy, however it introduces high inter-symbol interference.

Fig. 2 .

 2 Fig. 2. System diagram of CO-OFDM with proposed pulse shaping and receiver dual filtering. Acronyms-S/P: serial-to-parallel, IFFT: inverse fast Fourier transform, CP: cyclic prefix, P/S: parallel-to-serial, TS: training symbol, DAC: digital-to-analogue converter, I: in-phase, Q: quadrature, OLO: optical local oscillator, ADC: analogue-to-digital converter, and FFT: fast Fourier transform.

Fig. 3 .

 3 Fig.3. Optical launch power investigation with no CFO for rectangular pulse shape, and 2 MHz residual CFO across rectangular, TFL, OBE and RRC. α is 0.1, 0.2, and 1 for RRC pulse shape. The transmission distance is 1600 km.

Fig. 4 .

 4 Fig. 4. Impact of utilizing rectangular, TFL and OBE pulse shapes under varying residual CFOs at transmission distance of 1600 km.

Fig. 5 .

 5 Fig. 5. Rectangular, TFL and OBE pulse shapes behavior under continuous broadening of combined linewidth with no residual CFO, and at transmission distance of 1600 km.

Fig. 6 .

 6 Fig. 6. Transmission investigation of Rectangular, TFL and OBE pulse shapes with 2 MHz residual CFO.

Table 1 . Back-to-back Q-factor of CO-OFDM with various pulse shapes at 2 MHz residual CFO

 1

			Q-factor
					RRC α =	RRC α =	RRC α =
	Residual CFO	Rectangular	TFL	OBE	0.1	0.2	1
	2 MHz	17.4 dB	18.7 dB	19.6 dB	17.7 dB	17 dB	11.4 dB

Acknowledgments

This work was partly funded by the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement (FOX-C, n° 318415), EPSRC (Grant EP/J017582/1-UNLOC), and Aston Institute of Photonic Technologies for supporting the publication fee.


Athanase Ndihokubwayo
email: athanand@yahoo.com

Compressive and Flexural Strengths for Considerable Volume Fly-Ash Concrete

Keywords: Considerable Volume of Fly Ash Concrete, Compressive Strength, Flexural Strength

The use of considerable volume of fly ash concrete in replacement of cement is economically and environmentally benefic because the fly ash is cheaper and produces fewer emissions of greenhouse gases than the conventional Portland cement. By using 30; 40 and 50 percent of fly ash in cementitious materials, after comparing different results of samples tested, the 28 days high-performance fly ash concrete resisted to the compressive loading forces respectively equal to 30.9MPa; 29.7MPa and 26.2 MPa while for flexural loading forces, the high-performance fly ash concrete resisted respectively 5.1 MPa; 4.0MPa and 3. 9 MPa. These results were found after an experimental study done in the concrete laboratory and may inspire the sustainable construction especially for infrastructures with no high loading forces.

Introduction

The high-volume fly ash concrete (HVFAC) system has emerged as a future powerful concrete system that would be far more durable and resource-efficient than those made of conventional Portland cement concrete. As the Portland cement industry is responsible for approximately 7percent of the world's carbon dioxide emission, Malhotra et al. [START_REF] Malhotra | High-Performance[END_REF] proposed that the fly ash content must be above 50 percent by mass of total cementitious for making a concrete for sustainable development They also proposed that for concrete mixture with specified 28 days compressive strength of 30 MPa or higher, slumps >150mm, and water-to-cementitious materials ratio of the order of 0.30, the use of highrange superplasticizers is mandatory. For concrete mixture with slumps less than 150mm and 28 days compressive strength of less than 30 MPa, high fly ash concrete mixture with a water-to-cementitious material ratio of the order of 0.4 may be used without superplasticizers. The Confederation of Indian Industry; centre of excellence for sustainable development [START_REF]Confederation of Indian Industry[END_REF] proposed that for obtaining a superior performance of HVFAC, the water-to-cementitious ratio should be bellow 0.4 and preferably ≤ 0.35 .The use of superplasticizer is most of the time essential.

Vargas and José A. [START_REF] Vargas | A Designer's View of Fly Ash Concrete[END_REF] defined a HVFAC as a concrete in which at least 50 percent by weight or mass of the cementitious material content is the fly ash. The applications (such as foundations) where strengths of 21 to 28 MPa are acceptable and high early strength is not required may be produced without special admixtures. Compared to the conventional Portland concrete, the HVFAC has some qualities: lower cost, very high environmental benefits (reduced CO 2 ), easier flowability and workability, negligible bleeding, quicker finishing, higher ultimate strength up to 90 days, higher crack resistance, very higher resistance to penetration of chloride ions after 3 months and very higher electrical resistivity after 3 months.

About the volume the emissions of greenhouse gases which can be reduced, Ehrenberg and Geisler [START_REF] Ehrenberg | Ecological Properties of Blast-furnace cement-part of life cycle[END_REF] wrote that the replacement of 1 ton of Portland cement by the fly ash reduces the overall CO 2 emissions by approximately 1 ton. Langley et al. [START_REF] Langley | Practical uses for High-Volume Fly Ash Concrete[END_REF] specified that another benefit of using fly ash in concrete is that fly ash makes beautiful architectural concrete because its light in color and its extreme workability ensures smoother finishes. In addition, fly ash in plaster virtually eliminates defects like crazing, map cracking, drying shrinkage cracks, grinning, expansion and popping.

With the aim to contribute to the study of the compressive and flexural loading resistance for a considerable volume fly ash concrete, a total of 144 samples were made in the concrete laboratory (72 for compressive test and 72 for flexural test).

Materials

The materials used were composed by the cementitious, aggregates, superplasticizer and pure water. The chemical properties of the cementitious used are shown in the

Materials Proportioning

The methods used for making the fly ash concrete were based on materials proportioning system according to the percent of the cementitious contents. The Table 3-1 and the Table 3-2 show respectively our proposition for materials proportioning and the volume of materials. For compressive samples the moulds size is 150mm cubic while for flexural samples the size is 100 mm x 100 mm x 400 mm.

The concrete -mixer used could mix at once concrete whose volume is the total volume of 6 samples for compressive test and 6 samples for flexural test .

The slump test was done for each fresh concrete and the measure was ranged between 120 and 180mm. For making the concrete more compact and homogeneous, a vibrating table was used.

Testing Procedures a) Compressive test

Compressive strength is the capacity of a material to withstand axially directed pushing forces. When the limit of compressive strength is reached, materials are crushed. The cubic concrete specimen is placed on the platform of compression testing machine without any packing between the cube and the steel platens of the testing machine. The load is then applied on a smooth surface on the cube steadily and uniform starting from zero at a rate of 35 N/mm 2 /min till the cube fails.

b) Flexural Test

The flexural test is one measure of the tensile strength of concrete. It is a measure of an unreinforced concrete to resist failure in bending. The specimen is placed symmetrically on the load-applying and support blocks. In the present test, the distance between the support-block axis and the extremity of the specimen is 50 mm. The load is then rapidly applied until approximately 50 percent of the breaking load has been reached.

Figure1. Compressive Test

Test Results and Discussion

For 28 days fly ash concrete, with water-to-cementitious ratio equal to 0.39 and gravel-to-sand ratio equal to 2.12, by using superplasticizer product, the experimental results showed that the high-performance compressive strengths of the fly ash concrete were 30.9 MPa; 29.7MPa and 26.2 MPa respectively for 30, 40 and 50 percent of fly ash in cementitious materials.

For flexural test results, the high-performance of 28 days fly ash concrete were 5.1 MPa, 4 MPa and 3.9 MPa respectively for 30; 40 and 50 percent of fly ash in cementitious materials. The water-to-cementitious ratio and the gravel-to-sand ratio were 0.46 and 2.12 for 30 percent while they were 0.39 and 2.12 for 40 and 50 percent of fly ash in cementitious materials.

The superplasticizer used played a big part in concrete strengthening comparatively to the concrete samples made without the product.

By analyzing the charts (Figure 5-1 and Figure 5-2), the allures of charts 7 days and 28 days are not exactly proportional; either for compressive or flexural test results; this because the climatic and storage conditions varied and changed the parameters of concrete hardening.

Conclusions

In addition to the environmental protection benefits by reducing the volume of the greenhouse gases emissions, the use of fly ash in the construction domain contributes also to the reduction of the cementitious cost because the fly ash is cheaper compared to the cement. For developing countries, the economic factor is very important especially for designing concrete structures where; in many cases; structure loading forces are not higher. The present study may inspire the actors in the construction domain by choosing appropriate percentage of the fly ash to use in cementitious materials according to the pre-established limits of structure loading.

Figure 2 .

 2 Figure 2. Flexural Test. For Figure.4-2: (1): Bed of testing machine; (2): Load -applying support blocks; (3): Head of testing machine; (4): Loading plate; (5): Steel ball; (6):Rectangular specimen

Figure 3 .

 3 Figure 3. Compressive Result Charts

Figure 4 .

 4 Figure 4. Flexural Result Charts

Table 2 -

 2 1.The specific surface of the cement used is 310 m 2 /kg while the fly ash fineness (80µm residue on sieve) is 24 percent. The mechanical properties of aggregates used are shown below in

	Some chemical properties for the superplasticizer used:
	Na2SO4:10.2%; Solid Content: 50%; Chlorine ion: 0.01%;
	PH: 6.5-8.5; Fluidity: E250 mm; Density: 1.08g/cm 3

Table 1 .

 1 Cementitious chemical properties

	Spec.	Cementitious (%) Cement (42. 5) Fly ash (F)
	SiO2	21.47	50.0
	Al2O3	5.81	32.85
	FeO3	4.04	4.21
	CaO	59.64	4.91
	MgO	3.24	1.58
	Na2O	-	1.42
	K2O	-	1.35
	LOI	2.44	3.22

Table 2 .

 2 Aggregate properties

	Item	Gravel	Sand
	D max (mm)	9.52	4.75
	Specific gravity	2.64	2.65
	Absorption (%)	1.3	1.85
	Unit weight (kg/m3)	1522	1761
	Passing #50 sieve (%)	0.0	8.6
	Fineness modulus	6.32	3.24
	Compressive strength(MPa)	93.55	-
	Modulus of elasticity(Gpa)	51.04	-

Table 3 .

 3 Materials proportion

	FA and C (%)	Ref.	W/C	G/S
		1	0.51	1.94
	30	2	0.46	2.12
	70	3	0.46	2.12
		4	0.39	2.12
		5	0.51	1.94
	40	6	0.46	2.12
	60	7	0.46	2.12
		8	0.39	2.12
		9	0.46	2.12
	50	10	0.51	2.12
	50	11	0.46	2.12
		12	0.39	2.12

Table 4 .

 4 Volume

	of materials

ACKNOWLEDGEMENTS

The author would like to sincerely thank all those who contributed to the laboratory facilities for making and testing the fly ash samples.

Compressive Test Results