Vincent Beffara
email: vincent.beffara@ens-lyon.fr

Pierre Nolin
email: nolin@cims.nyu.edu

ON MONOCHROMATIC ARM EXPONENTS FOR 2D CRITICAL PERCOLATION

Keywords: Percolation, critical exponent, scaling limit. 1286

We investigate the so-called monochromatic arm exponents for critical percolation in two dimensions. These exponents, describing the probability of observing j disjoint macroscopic paths, are shown to exist and to form a different family from the (now well understood) polychromatic exponents. More specifically, our main result is that the monochromatic j -arm exponent is strictly between the polychromatic j -arm and (j + 1)-arm exponents.

Introduction.

Percolation is one of the most-studied discrete models in statistical physics. The usual setup is that of bond percolation on the square lattice Z 2 , where each bond is open (resp., closed) with probability p ∈ (0, 1) (resp., 1p), independently of the others. This model exhibits a phase transition at a critical point p c ∈ (0, 1) (in this particular case, p c = 1/2): for p < p c , almost surely all connected components are finite, while for p > p c there exists a unique infinite component with density θ(p) > 0. Site percolation is defined in a similar fashion, the difference being that the vertices are open or closed, instead of the edges; one can then see it as a random coloring of the lattice, and use the terms black and white in place of open and closed.

The behavior of percolation away from the critical point is well understood; however it is only recently that precise results have been obtained at and near criticality. For critical site percolation on the regular triangular lattice, the proof of conformal invariance in the scaling limit was obtained by Smirnov [START_REF] Smirnov | Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits[END_REF], and SLE processes, as introduced by Schramm [START_REF] Schramm | Scaling limits of loop-erased random walks and uniform spanning trees[END_REF] and further studied by Lawler, Schramm and Werner [START_REF] Lawler | Values of Brownian intersection exponents I: Half-plane exponents[END_REF][START_REF] Lawler | Values of Brownian intersection exponents II: Plane exponents[END_REF], provide an explicit description of the interfaces (in the scaling limit) in terms of SLE [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF] (see, e.g., [START_REF] Werner | Lectures on two-dimensional critical percolation[END_REF]).

This description allows for the derivation of the so-called polychromatic arm exponents [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF], which describe the probability of observing connections across annuli of large modulus by disjoint connected paths of specified colors (with at least one arm of each color), and also the derivation of the one-arm exponent [START_REF] Lawler | One-arm exponent for critical 2D percolation[END_REF]. Combined with Kesten's scaling relations [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF], these exponents then provide the existence and the values of most of the other critical exponents, like, for example, the exponent β = 5/36 associated with the density of the infinite cluster, as p ↓ p c , θ(p) = (pp c ) 5/36+o (1) .

On the other hand, very little is known concerning the monochromatic arm exponents (i.e., with all the connections of the same color-see below for a formal definition) with more than one arm. Here, the SLE approach does not seem to work, and, correspondingly, there is no universally established conjecture for the values of those exponents. One notable exception, however, is the two-arm monochromatic exponent, for which an interpretation in terms of SLE( 6) is proposed at the end of [START_REF] Lawler | One-arm exponent for critical 2D percolation[END_REF]-but again no explicit value has been computed. That particular exponent is actually of physical interest: known as the backbone exponent, it describes the "skeleton" of a percolation cluster. Even the existence of these exponents is not clear, as there does not seem to be any direct sub-additivity argument.

In this paper, we prove that the monochromatic exponents do exist, and investigate how they are related to the polychromatic exponents. We show that they have different values than their polychromatic counterparts. As an illustration, our result implies that the backbone of a typical large percolation cluster at criticality is much "thinner" than its boundary.

Background.

2.1. The setting. We restrict ourselves here to site percolation on the triangular lattice, at criticality (p = p c = 1/2). Recall that it can be obtained by coloring the faces of the honeycomb lattice randomly, each cell being black or white with probability 1/2 independently of the others. In the following, we denote by P = P 1/2 the corresponding probability measure on the set of configurations. Let us mention, however, that many of the results of combinatorial nature based on Russo-Seymour-Welsh-type estimates should also hold for bond percolation on Z 2 , due to the self-duality property of this lattice.

Let S n denote the ball of radius n in the triangular lattice (i.e., the intersection of the triangular lattice with the Euclidean disc of radius n, though the specifics of the definition are of little relevance), seen as a set of vertices. We will denote by ∂ i S n (resp., ∂ e S n ) its internal (resp., external) boundary, that is, the set of vertices in (resp., outside) S n that have at least one neighbor outside (resp., in) S n , and, for n < N, by S n,N := S N \ S n the annulus of radii n and N . To describe critical and near-critical percolation, certain exceptional events play a central role: the arm events, referring to the existence of a number of crossings ("arms") of S n,N , the color of each crossing (black or white) being prescribed. DEFINITION 1. Let j ≥ 1 be an integer and σ = (σ 1 , . . . , σ j ) be a sequence of colors (black or white). For any two positive integers n < N, a (j, σ )-arm configuration in the annulus S n,N is the data of j disjoint monochromatic, nonselfintersecting paths (r i ) 1≤i≤j -the arms-connecting the inner boundary ∂ e S n and the outer boundary ∂ i S N of the annulus, ordered counterclockwise in a cyclic way, where the color of the arm r i is given by σ i . We denote by

A j,σ (n, N ) := ∂ e S n j,σ ∂ i S N (2.1)
the corresponding event. It depends only on the state of the vertices in S n,N .

We will write down color sequences by abbreviating colors, using B and W for black and white, respectively. To avoid the obvious combinatorial obstructions, we will also use the notation n 0 = n 0 (j ) for the smallest integer such that j arms can possibly arrive on ∂ e S n 0 [n 0 (j ) is of the order of j ] and only consider annuli of inner radius larger than n 0 . This restriction will be done implicitly in what follows.

The so-called color exchange trick (noticed in [START_REF] Aizenman | Path crossing exponents and the external perimeter in 2D percolation[END_REF][START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF]) shows that for a fixed number j of arms, prescribing the color sequence σ changes the probability only by at most a constant factor, as long as both colors are present in σ (because an interface is needed to proceed). The asymptotic behavior of that probability can be described precisely using SLE [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]: it is possible to prove the existence of the (polychromatic) arm exponents and to derive their values [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF], which had been predicted in the physics literature (see, e.g., [START_REF] Aizenman | Path crossing exponents and the external perimeter in 2D percolation[END_REF] and the references therein). THEOREM 2. Fix j ≥ 2. Then for any color sequence σ containing both colors,

P(A j,σ (n, N)) = N -α j +o(1) (2.2)
as N → ∞ for any fixed n ≥ n 0 (j ), where α j = (j 2 -1)/12.

The value of the exponent for j = 1 (corresponding to the probability of observing one arm crossing the annulus) has also been established [START_REF] Lawler | One-arm exponent for critical 2D percolation[END_REF] and it is equal to 5/48 (oddly enough formally corresponding to j = 3/2 in the above formula).

For future reference, let us mention the following facts about critical percolation that we will use.

1.

A priori bound for arm configurations: there exist constants C, ε > 0 such that for all n < N,

P(A 1,B (n, N)) = P(A 1,W (n, N)) ≤ C n N ε . (2.3)
For all j ≥ 1, there exist constants c j , β j > 0 such that for all n < N,

P(A j,B...BB (n, N)) ≥ c j n N β j . (2.4)
2. Quasi-multiplicativity property: For any j ≥ 1 and any sequence σ , there exist constants C 1 , C 2 > 0 such that for all n 1 < n 2 < n 3 ,

C 1 P(A j,σ (n 1 , n 2 ))P(A j,σ (n 2 , n 3 )) ≤ P(A j,σ (n 1 , n 3 )) ≤ C 2 P(A j,σ (n 1 , n 2 ))P(A j,σ (n 2 , n 3 )).
The first of these two properties actually relies on the so-called Russo-Seymour-Welsh (RSW) lower bounds, that we will use extensively in various situations: roughly speaking, these bounds state that the probability of crossing a given shape of fixed aspect ratio is bounded below independently of the scale. For instance, the probability of crossing a 3n × n rectangle in its longer direction is bounded below, uniformly as n → ∞. We refer the reader to [START_REF] Grimmett | Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften[END_REF] for more details.

In the second one, the independence of arm events in disjoint annuli implies that one can actually take C 2 = 1, which we will do from now on. The lower bound is obtained using a so-called separation lemma, as first proved by Kesten [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF], Lemma 6 (technically, he does the proof in the case of four arms, but the argument extends easily to the general case). The monochromatic case is rather easier, as is follows directly from RSW estimates and Harris's lemma.

A correlation inequality.

For two increasing events, the probability of their disjoint occurrence can be bounded below by the classic van den Berg-Kesten (BK) inequality [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF]; Reimer's inequality, conjectured in [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF] and proved in [START_REF] Reimer | Proof of the van den Berg-Kesten conjecture[END_REF], extends it to the case of arbitrary events. A key ingredient in our proof will be a not-that-classic correlation inequality which is an intermediate step in the proof of Reimer's inequality. Note that actually, in the simpler case of increasing events (which is the only one we need here), this inequality was obtained earlier by Talagrand [START_REF] Talagrand | Some remarks on the Berg-Kesten inequality[END_REF]; but we choose to follow the more established name of the general result even for this particular case.

Instead of using the terminology in Reimer's original paper [START_REF] Reimer | Proof of the van den Berg-Kesten conjecture[END_REF] we follow the rephrasing (with more "probabilistic" notation) of his proof in the review paper [START_REF] Borgs | The van den Berg-Kesten-Reimer inequality: A review[END_REF]. Consider an integer n, and = {0, 1} n . For any configuration ω ∈ and any set of indices S ⊆ {1, . . . , n}, we introduce the cylinder

[ω] S := { ω : ∀i ∈ S, ωi = ω i },
and more generally for any X ⊆ , any S : X → P({1, . . . , n}),

[X] S := ω∈X [ω] S(ω) .
For any two A, B ⊆ , we denote by A • B the disjoint occurrence of A and B (the notation A B is also often used to denote this event)

A • B := ω : for some S(ω) ⊆ {1, . . . , n}, [ω] S ⊆ A and [ω] S c ⊆ B .
Recall that Reimer's inequality states that P(A • B) ≤ P(A)P(B). (2.5) We also denote by ω = 1ω the configuration obtained by "flipping" every bit of the configuration ω ∈ , and if X ⊆ , we define X := { ω : ω ∈ X}. We are now in a position to state the correlation inequality that will be a key ingredient in the following: THEOREM 3 ([10], Theorem 1.2). For any A, B ⊆ , we have

|A • B| ≤ |A ∩ B| = | Ā ∩ B|. (2.6)
For the sake of completeness, let us just mention that this inequality is not stated explicitly in that form in [START_REF] Reimer | Proof of the van den Berg-Kesten conjecture[END_REF]. It can be deduced from Theorem 1.2 by applying it to the flock of butterflies B = {(ω, f (ω)) : ω ∈ A • B}, where f (ω) coincides with ω exactly in the coordinates in S(ω). Here, S(ω) ⊆ {1, . . . , n} is the subset of indices associated with A and B by the definition of disjoint occurrence, that is, so as to satisfy Equivalently, it can be obtained from Lemma 4.1 in [START_REF] Borgs | The van den Berg-Kesten-Reimer inequality: A review[END_REF] by taking X = A•B, and

S : X → P({1, . . . , n}) associated with A • B (so that [X] S ⊆ A and [X] S c ⊆ B).

Statement of the results.

In this paper, we will be interested in the asymptotic behavior of the probability of the event A j,σ (n 0 (j ), N) as N → ∞ for a monochromatic σ , say σ = B . . . B, so that A j,σ simply refers to the existence of j disjoint black arms. Our first result shows that this probability follows a power law, as in the case of a polychromatic choice of σ . THEOREM 4. For any j ≥ 2, there exists an exponent α j > 0 such that

P(A j,B...B (n, N)) = N -α j +o(1) (2.7)
as N → ∞ for any fixed n ≥ n 0 (j ).

These exponents α j are known as the monochromatic arm exponents, and it is natural to try to relate them to the previously mentioned polychromatic exponents α j .

Consider any j ≥ 2; we start with a few easy remarks. On the one hand, one can apply Harris's lemma (more often referred to in the statistical mechanics community as the FKG inequality, which is its generalization to certain nonproduct measures; we will follow that convention in what follows): it implies that

P(A j +1,B...BW (n 0 , N)) = P A j,B...B (n 0 , N) ∩ A 1,W (n 0 , N) ≤ P(A j,B...B (n 0 , N))P(A 1,W (n 0 , N)),
and by using item 1 above, we get that, for some constants C, ε > 0,

P(A j +1,B...BW (n 0 , N)) ≤ CN -ε P(A j,B...B (n 0 , N)) (2.8)
or, in other words, that α j < α j +1 . On the other hand, inequality (2.6) directly implies that

P(A j,B...BB (n 0 , N)) = P A j -1,B...B (n 0 , N) • A 1,B (n 0 , N) ≤ P A j -1,B...B (n 0 , N) ∩ A 1,W (n 0 , N) = P(A j,B...BW (n 0 , N)),
hence α j ≥ α j . We will actually prove the following, stronger result. THEOREM 5. For any j ≥ 2, we have

α j < α j < α j +1 . (2.9)
The monochromatic exponents α j thus form a family of exponents different from the polychromatic exponents.

We would like to stress the fact that the case of half-plane exponents (or more generally, boundary exponents in any planar domain) is considerably different: indeed, whenever a boundary is present, the color-exchange trick implies that the probability of observing j arms of prescribed colors is exactly the same for all color prescriptions, whether mono-or poly-chromatic. In particular there is no difference between the monochromatic and polychromatic boundary exponents. (For the reader's peace of mind, he can notice that the presence of the boundary provides for a canonical choice of a leftmost arm; the lack of which is precisely the core idea of the proof of our main result in the whole plane.)

We will first prove the inequality α j < α j (which is the main statement in the above theorem, the other strict inequality being the simple consequence of the FKG inequality we mentioned earlier), since its proof only requires combinatorial arguments, and postpone the proof of the existence of the exponents to the end of the paper.

In order not to refer to the α j 's, we adopt the following equivalent formulation of the inequality: what we formally prove is that, for any j ≥ 2, there exists ε > 0 such that for any N large enough,

P(A j,B...BB (n 0 , N)) ≤ N -ε P(A j,B...BW (n 0 , N)). (2.10)
The proof of this inequality only relies on discrete features such as self-duality and RSW-type estimates, and hence it is possible that our proof could be extended to the case of bond percolation on Z 2 , where the existence of the exponents, which strongly relies on the knowledge of the scaling limit, is still unproved; however, the statement of duality in that case is different enough (the color exchange trick for instance has no exact counterpart) that some of our arguments do not seem to extend directly.

The set of winding angles.

3.1. Strict inequalities between the exponents. Our proof is based on an energy versus entropy consideration. The difference between the monochromatic and the polychromatic j -arm exponents can be written in terms of the expected number of "really different" choices of j arms out of a percolation configuration with j arms: for a polychromatic configuration, this number is equal to 1, whereas for a monochromatic configuration, it grows at least like a positive power of the modulus, and the ratio between these two numbers behaves exactly like (N/n) α j -α j because, for fixed disjoint arms (r 1 , . . . , r j ) with respective lengths ( 1 , . . . , j ), the probability that they are present in the configuration with a prescribed coloring does not depend on that coloring (it is equal to 2

-( 1 +•••+ j ) ).
More precisely, but still roughly speaking, the proof relies on the following observation: given a configuration where j black arms are present, there are many ways to choose them, since by RSW there is a positive density of circuits around the origin [allowing "surgery" on the arms (see Figure 1)], while if we consider a configuration with arms of both colors, then there is essentially only one way to select them. Of course the geometry of an arm is quite intricate and many local modifications, on every scale, are always possible, both in the monochromatic and polychromatic setups; what we mean here is that this choice is unique from a macroscopic point of view. To formalize this intuition, we thus have to find a way of distinguishing two macroscopic choices of arms, and for this we will use the set of winding angles associated with a configuration. DEFINITION 6. For any configuration of arms, one can choose a continuous determination of the argument along one of the arms; we call winding angle of the arm (or simply angle for short) the overall (algebraic) variation of the argument along that arm.

FIG. 1. To a given monochromatic configuration correspond many different "macroscopic" ways to choose the arms, contrary to the polychromatic case. The left-hand picture shows the sort of realization in which many macroscopically different 5-arm configurations can be found-the actual topology needed is a little bit more involved; see Figure 3.

Clearly, the winding angles of the arms corresponding to a given (j, σ )-arm configuration differ by at most 2π . However, for the same percolation configuration, there might exist many different choices of a (j, σ )-arm configuration, corresponding to different winding angles: we denote by I j,σ (n, N ) the set of all the winding angles which can be obtained from such a configuration; we omit the subscript from the notation whenever j and σ are clear from the context (notice though that whenever there are enough arms for two different arm events with the same j to be realized simultaneously, the corresponding sets of winding angles are essentially the same). For the sake of completeness, we also declare I j,σ (n, N ) to be empty if the configuration does not contain j arms of the prescribed colors.

We will actually rather use Īj,σ (n, N ), the set of angles obtained by "completing" I j,σ (n, N )

Īj,σ (n, N ) := α∈I j,σ (n,N) (α -π, α + π].
It is an easy remark that in the polychromatic case (σ nonconstant), we have for any α ∈ I j,σ (n, N )

I j,σ (n, N ) ⊆ (α -2π, α + 2π)
(because two arms of different colors cannot cross), so that Īj,σ (n, N ) is an interval of length at most 4π . In the monochromatic case (σ constant), no such bound applies [and actually it is not obvious that Īj,σ (n, N ) is an interval; this is proved as Proposition 7 below].

In the case of a polychromatic arm configuration, considering successive annuli of a given modulus as independent, one would expect a central limit theorem to hold on the angles, or at least fluctuations of order √ log N . On the other hand, for a monochromatic configuration, performing surgery using circuits in successive annuli should imply that every time one multiplies the outer radius by a constant, the expected largest available angle would increase by a constant, so that one would guess that, by a careful choice of arms, the total angle can be made of order ± log N .

Fix ε > 0, and let A ε 1,B (resp., A ε 1,W , resp., A ε j,σ ) be the event that there exists a black arm (resp., a white arm, resp., j arms with colors given by σ ) with angle larger than ε log N between radii n 0 and N . Applying inequality (2.6) with A = A j -1,B...B and B = A ε 1,B , if the above intuition was correct, this would imply

P(A j,B...BB ) P(A • B) ≤ P(A j -1,B...B ∩ A ε 1,W ) = P(A ε j,B...BW
), and we could expect

P(A ε j,B...BW ) ≤ N -ε P(A j,B...BW )
by a large-deviation principle. However, proving this LDP seems to be difficult, and we propose here an alternative proof that relies on the same ideas, but bypasses some of the difficulties.

PROOF OF THEOREM 5. We shall in fact prove (2.10). As noted earlier, assuming Theorem 4 (which will be proved in Section 4), Theorem 5 follows immediately.

Step 1. First, note that it suffices to prove that the ratio

P(A j,B...BB (n, N)) P(A j,B...BW (n, N))
can be made arbitrarily small as N/n → ∞, uniformly in n: indeed, assuming that this is the case, then for any δ > 0, there exists R > 0 such that this ratio is less than δ as soon as N/n ≥ R. Then, as a direct consequence of the quasi-multiplicativity property (item 2 above), we have

P(A j,B...BB (n, R k n)) ≤ C k-1 2 P(A j,B...BB (n, Rn)) • • • P(A j,B...BB (R k-1 n, R k n)) ≤ C k-1 2 δ k P(A j,B...BW (n, Rn)) • • • P(A j,B...BW (R k-1 n, R k n)) ≤ C k-1 2 δ k (C -1 1 ) k-1 P(A j,B...BW (n, R k n)), and for δ = 1/(2C 2 C -1 1 ) this gives P(A j,B...BB (n, R k n)) ≤ 2 -k P(A j,B...BW (n, R k n)), (3.1)
which immediately implies that for some C, ε > 0,

P(A j,B...BB (n, N)) ≤ C N n -ε P(A j,B...BW (n, N)).
In particular, applying this for n = n 0 (and N large enough) leads to the inequality that we need.

Step 2. The key step of the proof is as follows. Given a configuration with j arms in an annulus of large modulus, we use RSW-type estimates to prove the existence of a large number of disjoint sub-annuli of it, in each of which one can find black paths topologically equivalent to those in Figure 2 (in the case j = 2) or its reflection. Every time this configuration appears, one has the possibility to replace the original arms (in solid lines on the figure) with modified, and still disjoint, arms, obtained by using one of the dashed spirals in each of them. The new arms then land at the same points on the outer circle, but with a winding angle differing by 2π . This allows us to show that, with high probability, the set of angles Ī (n, N) contains an interval of length at least ε log(N/n), for some ε > 0 (which can be written in terms of the RSW estimates). We now proceed to make the construction in detail.

FIG. 2. When they encounter this configuration, the arms (here in solid lines) can be modified, detouring via the dashed lines, to make an extra turn.

Let j ≥ 2, and let m be a positive integer. Define a j -spiral between radii m and 4m as the configuration pictured in Figure 3. More precisely, a j -spiral is the union of 4 families of j black paths in a percolation configuration, namely:

• j disjoint rays between radii m and 4m; FIG. 3. Generalization of Figure 2 in the case of j ≥ 3 arms. The additional circuits (in solid lines) are needed to apply Menger's theorem; the circles of radii m and 4m (resp., 2m and 3m) are drawn in heavy (resp., dotted) lines, the spiraling paths in dashed lines and the active points are marked with a black square.

• j disjoint "spiraling paths" contained in the annulus S 2m,3m , each connecting two points of one of the rays and making one additional turn around the origin; • j disjoint circuits around the origin, contained in the annulus S m,2m ; • j disjoint circuits around the origin, contained in the annulus S 3m,4m . RSW-type estimates directly show that, uniformly as m → ∞, the probability of observing a j -spiral between radii m and 4m is bounded below by a positive constant (depending only on j ). In addition, with each such spiral we associate two families of j active points: for each of the j rays, oriented starting at radius m, its last intersection with the circle of radius 2m is called an inner active point, and its first intersection with the circle of radius 3m after it, an outer active point; in particular, the section of the ray between its two active points remains within the annulus S 2m,3m .

The same RSW arguments show that with positive probability, such a j -spiral actually satisfies a few additional properties (which we will consider part of the definition from now on): the different pieces remain well separated whenever they are not forced to intersect by topological constraints; whenever two pieces intersect, they can do so multiple times, but all intersection points remain close to each other, and it is easy to check that this is not a problem in the proof. In fact, the only thing that might cause trouble is that the rays may cross each of the circles of radii 2m and 3m several times; one cannot use RSW to avoid it, but it is taken care of by our choice of the active points.

The presence of j -spirals in disjoint annuli are independent events, each with positive probability, so that, for some ε > 0, the probability of the event E (ε) j (n, N) of having at least ε log(N/n) + 1 disjoint j -spirals between radii n and N goes to 1 as N/n goes to infinity. The presence of j -spirals being an increasing event, the FKG inequality ensures the conditional probability of E (ε) j (n, N), given the existence of j black arms between radii n and N , still goes to 1 as N/n goes to infinity.

We now explain how to use j -spirals to perform surgery on black arms. Assume that between radii n and N , there are a certain number s ≥ 2 of disjoint spiral configurations; let (m i ) 1≤i≤s be the corresponding values of m. Assume in addition that there are j disjoint arms between radii n and N . The first remark is the following: for every i ∈ {1, . . . , s -1}, the event A j,B...BB (3m i , 2m i+1 ) is realized, and one can choose j disjoint arms (γ i,k ) 1≤k≤j in the annulus S 3m i ,2m i+1 accordingly. The main part of the argument then consists of proving that, within the union of all those arms together with the spirals, it is always possible to find 2 s-1 j -arm configurations between radii n and N , spanning winding angles in an interval of length 2(s -2)π .

Each of the constructed arms will consist of sections of two types, connecting at the active points (and disjoint otherwise, by our choice of the active points): the "movable" ones, formed out of the rays and spiraling parts of the j -spirals and contained in the union of the annuli S 2m i ,3m i ; and the "intermediate" ones, made from the arms γ i,k (which we have chosen to stay in the annuli S 3m i ,2m i+1 ) and from pieces of the j -spirals lying outside the active points. Within every j -spiral there are two choices for the corresponding movable section, which is how arms of very different winding angles will be produced. For that, all we need to prove is the existence of one family of j disjoint arms following this decomposition.

The key argument is as follows. Define 1 as the union of the (γ 1,k ) 1≤k≤j (which cross the annulus S 3m 1 ,2m 2 ), together with the parts of the j -spiral 1 (resp., 2 ) outside its outer active points (resp., inside its inner active points). It is easy to check that, whenever one marks (j -1) vertices on 1 , there still exists a path completely contained in 1 and avoiding the marked points, and connecting one of the outer active points of 1 to one of the inner active points of 2 . Indeed, marking (j -1) vertices leaves untouched at least one arm, one inner (resp., outer) circle and one ray of 1 (resp., 2 ). Menger's theorem (see [START_REF] Diestel | Graph Theory[END_REF]) then ensures that 1 contains j disjoint arms, each connecting one of the outer active points of 1 to one of the inner active points of 2 .

The same construction can be performed between each pair of successive jspirals, to produce the desired "intermediate" sections. An obvious variation of the construction can also be applied in the annuli S n,2m 1 and S 3m s ,N , leading to the following fact: whenever there are j arms between radii n and N , and the event E (ε) j (n, N ) is realized, the set Ī (n, N) contains an interval of length at least 2πε log(N/n) (obtained by playing with the appropriate movable sections), and this occurs with conditional probability going to 1 as N/n goes to infinity. Notice that the winding angles of the original j arms need not be within the interval we just constructed; however, this has no bearing on what follows.

Step 3. We now use the BK inequality to control the probability, given the presence of j black arms between radii n and N , that there is a choice of arms with a very large winding angle. In fact, the argument shows a little more: it is very unlikely, even without the conditioning, that there is a single arm with large winding.

Let R m,m be the intersection of the annulus S m,m with the cone C := {z ∈ C : |arg(z)| < π/10} (using the standard identification of the plane R 2 with the complex plane). We will consider two families of "rectangles": the R e k ,e k+2 (which we call the long ones), and the R e k ,e k+1 (the wide ones). It is easy to see that any curve connecting two points of the plane of arguments -π/10 and +π/10 while staying within the cone C has to cross at least one of these rectangles between two opposite sides. More precisely, it must either cross a long rectangle from one straight side to the other, or a wide rectangle from one curved side to the other. Now, assume that there exist j arms between radii n and N , and that their winding angle is at least equal to 2πK log(N/n) + 4π (where K is a positive constant which will be chosen later). Each of these arms has to cross the cone C at least (the integer part of) K log(N/n) times, so the configuration contains j K log(N/n) disjoint paths (at least), each one crossing one of the rectangles. On the other hand, the probability that a rectangle of a given shape is crossed by a path is bounded above by 1δ for some δ > 0, as provided by RSW estimates.

Combined with the BK inequality, this implies that the probability that there are j arms winding of at least an angle of 2πK log(N/n) + 4π is bounded above by

S := (l k ,l k ) k (1 -δ) l k +l k ,
where the sum is taken over all log(N/n)-tuples of (l k , l k ) having a sum equal to j K log(N/n). The number of such tuples is the same as the number of choices of 2 log(N/n) -1 disjoint elements out of (j K + 2) log(N/n) -1, so we obtain that

S ≤ (1 -δ) jK log(N/n) (j K + 2) log(N/n) -1 2 log(N/n) -1 .
It is then a straightforward application of Stirling's formula to obtain that

S ≤ C exp[(-cK + C log K) log(N/n)],
where the constants c and C do not depend on the value of K. Choosing K large enough, one then obtains that

S ≤ C n N β j +1
[where β j is the same as in (2.4)].

Hence, the conditional probability, given A j,B...BB (n, N ), that Ī (n, N) is contained in the interval of length 4πK log(N/n) + 8π centered at 0 goes to 1 as N/n goes to infinity. Dividing that interval into sub-intervals of length ε 2 log(N/n), and using the previous step, we get that for one of them, say i ε (n, N ),

P i ε (n, N ) ⊆ Ī (n, N)|A j,B...BB (n, N ) ≥ C ,
where C > 0 is a universal constant.

Step 4. We are now in a position to conclude. If we take α min such that

P A j,B...BW (n, N ) ∩ {α min ∈ Īj,B...BW (n, N )} is minimal among all α min ∈ i ε (n, N ) ∩ (4π Z), then P A j,B...BW (n, N ) ∩ {α min ∈ Īj,B...BW (n, N )} ≤ 4π ε/2 log(N/n) P(A j,B...BW (n, N))
since, as we noted earlier, whenever there are arms of different colors, 4πZ cannot contain more than one element of Ī (n, N). On the other hand, we know from the previous step that

P A j,B...BB (n, N ) ∩ {α min ∈ Īj,B...BB (n, N )} ≥ P A j,B...BB (n, N ) ∩ {i ε (n, N ) ⊆ Īj,B...BB (n, N )} ≥ C P(A j,B...BB (n, N)).

If we apply (2.6) to

≤ P A j,B...BW (n, N ) ∩ {α min ∈ Īj,B...BW (n, N )} ≤ 4π ε/2 log(N/n) P(A j,B...BW (n, N)),
which completes the proof.

3.2. The density of the set of winding angles. In this section, we further describe the set of winding angles I (n, N), which happened to be a key tool in the previous proof, in the monochromatic case. We prove that (conditionally on the existence of j disjoint black arms) Ī (n, N) is always an interval, as in the polychromatic case. For that, we use the following deterministic statement that I (n, N) does not have large "holes": PROPOSITION 7. Let j ≥ 1 and σ = B . . . BB of length j . Let α, α ∈ I j,σ (n, N ) with α < α ; then there exists a sequence (α i ) 0≤i≤r of elements of I j,σ (n, N ), satisfying the following two properties:

• α = α 0 < α 1 < • • • < α r = α ; • for every i ∈ {0, . . . , r -1}, α i+1 -α i < 2π .
This result directly implies that Ī (n, N) is an interval, and the construction of the previous subsection, creating extra turns (step 2 of the proof), gives a lower bound on the diameter of Ī (n, N): we hence get that for σ constant, there exists some ε > 0 (depending only on j ) such that Ī (n, N) is an interval of length at least ε log(N/n) with probability tending to 1 as N/n gets large.

The main step in the proof of the density result is the following topological lemma: LEMMA 8. Let j ≥ 1, and let γ 1 , . . . , γ j be j disjoint Jordan curves contained in the (closed) annulus {n ≤ |z| ≤ N}, ordered cyclically and each having its starting point on the circle of radius n and its endpoint on the circle of radius N . For each k ∈ {1, . . . , j}, let α k be the winding angle of γ k (as defined above) and let δ k be the ray [ne 2iπk/j , Ne 2iπk/j ]. Assume that, for each pair (k, k ), the intersection of γ k and δ k is finite. Then, provided all the α k are larger than 2π(1 + 2/j ), the union of all the paths γ k and δ k contains j disjoint paths δ1 , . . . , δj , all having winding angle 2π/j and sharing the same endpoints as the δ k .

In other words: starting from two collections of paths, if their angles differ enough, one can "correct" the one with the smaller angle in such a way as to make it turn a little bit more. PROOF OF LEMMA 8. We shall construct the paths δk explicitly. The first step is to reduce the situation to one of lower combinatorial complexity, namely to the case where the starting points of the γ k are separated by those of the δ k . For each k ≤ j , let τ k = inf{t : γ k (t) ∈ [ne iπ(2k-1)/j , Ne iπ(2k-1)/j ]} (which is always finite by our hypotheses), and let

:= j k=1 {γ k (t) : 0 ≤ t ≤ τ k }.
intersects each of the δ k finitely many times, so each of the δ k \ has finitely many connected components: let be the union of those components that do not intersect the circle of radius N , and let 0 := {n ≤ |z| ≤ N} \ ( ∪ ). Let be the connected component of 0 having the circle of radius N as a boundary component. is homeomorphic to an annulus, and for each k, the point γ k (τ k ) is on its boundary; by construction, the γ k (τ k ) are intertwined with the (remaining portions of the) rays of angles 2πk/j . We will perform our construction of the δk inside ; continuing them with the δ k outside then produces j disjoint paths satisfying the conditions we need.

Up to homeomorphism, we can now assume without loss of generality that for each k, γ k (0) = ne iπ(2k-1)/j . The only thing we lose in the above reduction is the assumption on the angles of the γ k ; but since it takes at most one turn for each of the γ k to reach the appropriate argument, we can still assume that the remaining angles are all larger than 4π/j . In particular, each of the γ k will cross the wedge between angles 2πk/j and 2π(k + 1)/j in the positive direction before hitting the circle of radius N .

For every k ≤ j , let θ k (t) be the continuous determination of the argument of γ k (t) satisfying θ k (0) = (2k -1)π/j , and let

T k := t > 0 : 2πk j < θ k (t) < 2π(k + 1) j and ˜ k = {γ k (t) : t ∈ T k }.
We now describe informally the construction of δk . Start from the point ne 2πik/j , and start following δ k outward, until the first intersection of δ k with ˜ k . Then, follow the corresponding connected component of ˜ k , until intersecting either δ k or δ k+1 ; follow that one outward until it intersects either ˜ k or the circle of radius N ; iterating the construction, one finally obtains a Jordan path joining ne 2πik/j to Ne 2πi(k+1)/j , and contained in the union of δ k , δ k+1 and ˜ k (see Figure 4). All that remains is to prove that the δk are indeed disjoint; by symmetry, it is enough to do so for δ1 and δ2 . Besides, because the γ k are themselves disjoint, any intersection point between δ1 and δ2 has to occur on δ 2 (at least in the case j > 2, but the case j = 2, where they could also intersect along δ 1 , again follows by symmetry).

The intersection of δ1 with δ 2 consists of a finite collection (I m ) of compact intervals; besides, the points of the intersection are visited by δ1 in order of increasing distance to the origin. Similarly, the intersection of δ2 with δ 2 consists of a finite collection (J l ) of compact intervals, which are also visited in order of increasing distance to the origin.

Suppose that I p and J p have a nonempty intersection, and let z 0 be the intersection point lying closest to the origin. Let p 0 and q 0 be such that z 0 ∈ I p 0 ∩J q 0 ; notice that z 0 is the endpoint closest to the origin of either I p 0 or J q 0 . According to the order in which γ 1 (resp., γ 2 ) visits the endpoints of I p 0 (resp., J q 0 ), this gives rise to eight possible configurations; it is straightforward in all cases to apply Jordan's theorem to prove that γ 1 and γ 2 then have to intersect, thus leading to a contradiction.

For the purpose of the proof of Proposition 7, we will need a slight variation of the lemma, where the hypothesis of finiteness of the intersections between paths is replaced with the assumption that the paths considered are all polygonal lines. The proof is exactly the same though, and does not even require any additional notation: whenever two paths, say γ k and δ k , coincide along a line segment, the definition of k amounts to considering some of the endpoints of this segment as intersections, which in other words is equivalent to shifting γ k by an infinitesimal amount toward the exterior of the wedge used to define T k in order to recover finiteness.

PROOF OF PROPOSITION 7. The previous lemma is stated with particular curves on which a surgery can be done, but it can obviously be applied to more general cases through a homeomorphism of the annulus. The general statement is then the following (roughly speaking): assuming the existence of two families of j arms with different enough winding angles, it is possible to produce a third family using the same endpoints as the first one but with a slightly larger winding angle.

We are now ready to prove Proposition 7. Consider a configuration in which one can find two families of crossings, say (λ k ) and (λ k ), in such a way that for every k, the difference between the winding angles of λ k and λ k is at least 2π . Let α 0 be the minimal angle in the first family, and apply the topological lemma with δ k = λ k and γ k = λ k : one obtains a new family of pairwise disjoint paths (λ 1 k ), which share the same family of endpoints as the (λ k ), the endpoint of λ 1 k being that of λ k+1 (with the obvious convention that j + 1 = 1).

One can then iterate the procedure, applying the topological lemma with this time δ k = λ 1 k , and still letting γ k = λ k ; one gets a new family (λ 2 k ) with the endpoints again shifted amongst the paths in the same direction. Continuing as long as the winding angle difference is at least 2π , this construction produces a sequence (λ i k ) of j -tuples of disjoint paths, the winding angles of which vary by less than 2π at each step. Besides, the construction ends in finitely many steps, for after j steps, each of the winding angles has increased by exactly 2π . This readily implies our claim. REMARK 9. Notice that, as early as the second step of the procedure, (λ n k ) and (λ k ) will always coincide on a positive fraction of their length, which is why we needed the above extension of the lemma.

Existence of the monochromatic exponents.

We now prove Theorem 4, stating the existence of the monochromatic exponents α j . For that, we use a rather common argument, presented, for example, in [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF]: since the quasi-multiplicativity property holds (item 2 above), it is actually enough to check that there exists a function f j (which will automatically be sub-multiplicative itself; one can take C 2 = 1 in the quasi-multiplicativity property) such that, for every R > 1, P(A j,B...BB (n, Rn)) → f j (R) (4.1) as n → ∞. Notice that RSW-type estimates provide both the fact that the left-hand term in bounded above and below by constants for fixed R as n → ∞, and a priori estimates on any (potentially subsequential) limit, of the form

R -1/ε j ≤ f j (R) ≤ R -ε j ,
where ε j depends only on j . By Menger's theorem (see, e.g., [START_REF] Diestel | Graph Theory[END_REF]), the complement of the event A j,B...BB (n, N) can be written as D j (n, N ) = {There exists a circuit in S n,N that surrounds ∂ i S n and contains at most j -1 black sites}. This makes it possible to express the event A j,B...BB (n, N ) in terms of the collection of all cluster interfaces (or "loops"): it is just the event that there does not exist a "necklace" of at most (j -1) loops, with white vertices on their inner boundary and black ones on their outer boundary, forming a chain around ∂ i S n and such that two consecutive loops are separated by only one black site.

Standard arguments show that the probability that two interfaces touch in the scaling limit is exactly the asymptotic probability that they "almost touch" (in the sense that they are separated by exactly one vertex) on discrete lattices; it is, for example, a simple consequence of the fact that the polychromatic six-arm exponent is strictly larger than 2, which in turn is a consequence of RSW-type estimates (the fact that the polychromatic five-arm exponent is equal to 2 being true on any lattice on which RSW holds, at least for colors BW BW W ).

What this means, is that to show convergence of the probability in (4.1), it is enough to know the probability of the corresponding continuous event. While we do not know the exact value of the limit, it is nevertheless easy to check that the event itself is measurable with respect to the full scaling limit of percolation, as constructed by Camia and Newman in [START_REF] Camia | Two-dimensional critical percolation: The full scaling limit[END_REF], and that is enough for our purpose. Notice that the measurability of the event in terms of the full scaling limit is ensured by the exploration procedure described in that paper: it is proved there that for every ε > 0, all loops of diameter at least ε (with the proper orientation) are discovered after finitely many steps of the exploration procedure. REMARK 10. Proving the existence of the exponents only requires the existence of the function f j , but it is easily seen that in fact the value of α j describes the power law decay of f j (R) as R goes to infinity. However, deriving the value of the exponent directly from the full scaling limit seems to be difficult, and we were not able to do it.

  [ω] S(ω) ⊆ A and [ω] S(ω) c ⊆ B for all ω ∈ A • B. For this particular B, we indeed have Red(B) ⊆ A and Yellow(B) ⊆ B.

  A = A j -1,B...B (n, N ) ∩ {α min ∈ Īj-1,B...B (n, N )} and B = A 1,B (n, N ), we obtain that C P(A j,B...BB (n, N)) ≤ P A j,B...BB (n, N ) ∩ {α min ∈ Īj,B...B (n, N )} = P(A • B) ≤ P(A ∩ B) = P A j,B...BW (n, N ) ∩ {α min ∈ Īj-1,B...B (n, N )}

FIG. 4 .

 4 FIG.[START_REF] Diestel | Graph Theory[END_REF]. The construction of the δk (in the case j = 5). The dotted lines are the paths γ k , and the heavy lines are the δk obtained at the end of the construction.

Acknowledgments. We are indebted to J. van den Berg for pointing out inequality (2.6) to us. We are also very grateful to W. Werner for many stimulating discussions, and to an anonymous referee for many insightful remarks about the first version of this paper. Part of this research was realized during a semester (resp., year) spent by Pierre Nolin (resp., Vincent Beffara) at Université de Genève, and both authors would like to thank the mathematics department there for its hospitality, in particular Stanislav Smirnov.

Grant OISE-07-30136. MSC2010 subject classifications. 82B43, 60K35.


Vincent Beffara

Cardy's formula on the triangular lattice, the easy way

We give a simple derivation of Cardy's formula for site-percolation in the triangular lattice, as proved by Smirnov; our main goal, besides simplifying the proof, is to pinpoint why the triangular lattice is so special ...

Introduction

The problem of determining the scaling limit of critical percolation in two dimensions is longstanding, and although there have been substantial progress in the understanding of said limit via the study of SLE 6 processes, convergence is known only in the case of critical site-percolation on the triangular lattice, for which the essential estimate, known as Cardy's formula, was proved by Smirnov in a much celebrated paper (see [?, ?]).

In several steps of Smirnov's proof, the specific structure of the triangular lattice is used in a seemingly essential way, and it is a very natural question to try and determine where exactly it is necessary to be in this specific case, and where the construction can be unified -and thus simplified ... The aim of this short note is to provide a mostly self-contained re-writing of Smirnov's proof, written in such a way as to emphasize the one key part of the proof where the precise geometry of the graph is used. As such, it does not contain any new result per se (except for the quite anecdotal Corollary 1), but we do believe that it will provide the reader with some insight into the model.

For general background on percolation, we refer the avid reader to the books of Grimmett [?] and Kesten [?]; and besides Smirnov's papers, related work on the scaling limit of percolation includes a paper of Camia and Newman [?], where they provide the full details about the convergence of percolation cluster boundaries to SLE processes, and a paper of Smirnov and Werner [?] where they derive the values of critical exponents for 2D percolation.

We will keep the same notations as in Smirnov's paper as much as possible; at various point we will be a bit terse or even a bit sloppy, since writing the proof in full would take many more pages, be tedious to read and probably not be more informative.

Acknowledgments Of course I would like to thank the organizers of the workshop, and especially Ilia Binder who convinced me to contribute to these proceedings. I would also like to mention that part of this construction was done independently by Tóth, who is the one who had the idea of replacing H in Smirnov's proof with the more symmetric version we use here.

1 Notations and setup

The graph

Let G δ be the hexagonal lattice, embedded as usual to preserve its rotational symmetry of order 3, and scaled by a factor δ > 0 (which we will refer to as the mesh of the lattice). The vertices of G δ all have degree 3. In all the following discussion, e will always stand for an oriented edge of G δ -which, by a slight abuse of notation, we will denote as e ∈ G δ .

Let G * δ be the dual graph of G δ , defined as follows: The set of vertices (resp. faces) of G * δ is in bijection with the set of faces (resp. vertices) of G δ , and there is an edge between two vertices of G * δ if and only if the two corresponding faces of G δ share an edge. It can be embedded in the plane by choosing each of its vertices to be the center of the corresponding regular hexagon in G δ , in which case it is exactly the usual triangular lattice, scaled by the same factor of δ. If e is an edge of G δ , we will denote by e * its dual edge, i.e. the unique edge of G * δ intersecting it, oriented in such a way that the frame (e, e * ) is direct.

For each vertex x of G δ , there are three oriented edges having x as their origin. Letting τ = e 2πi/3 , if e is one of these edges, we will denote the other two by τ.e and τ 2 .e. Seeing each edge as a complex number (the difference between the positions of its endpoint and its origin), this corresponds to complex multiplication; however, we prefer seeing it as an action of the group {1, τ, τ 2 } on the set of oriented edges of G δ , since this will make the constructions below easier. When needed we will also use the notation -e for the edge sharing the same endpoints as e but with opposite orientation.

Notice the following, fundamental identity: For each oriented edge e ∈ G δ , e * + τ (τ.e) * + τ 2 (τ 2 .e) * = 0.

(1)

Here, τ (τ.e) * means the product of two complex numbers, namely τ and the difference between the endpoint and the origin of (τ.e) * -we will stick to this convention, that a dot means the action of τ on edges while no dot means complex multiplication. This identity is actually trivial (using the fact that the action of τ on edges can itself be seen as complex multiplication, the sum above is equal to (1 + τ 2 + τ 4 )e * ), but as it will appear it is essentially the only place in the construction where we use the fact that G * δ is the usual, equilateral triangular lattice. It is to be noted that the above identity actually characterizes the triangular lattice; more precisely, the sum is 0 if and only if the triangle around the origin of e is equilateral. So, it seems that the triangular lattice is the only one (apart from trivial modifications of it) in which a fully combinatorial proof like follows is possible ...

The model

Let Ω be a smooth, bounded, simply connected domain in the complex plane. We will be interested in critical site-percolation on the intersection of Ω with G * δ . More specifically, the question we are interested in is the following. Let A, B, C and D be four points on the boundary of Ω, in that order. For every δ > 0, let Ω δ be the largest connected component (in terms of graph connectivity) of the intersection of Ω with G δ , and let Ω * δ be its dual graph. Ω δ should be seen as a discretization of Ω at scale δ. Let A δ , B δ , C δ and D δ be the vertices of Ω δ that are closest to A, B, C and D respectively.

The model we are considering is now critical site-percolation on Ω * δ , i.e., each vertex of Ω * δ is taken to be open with probability p c = 1/2, independently of all the others. Let C δ (Ω, A, B, C, D) be the event that there is an open crossing in Ω * δ , between the intervals A δ B δ and C δ D δ of its boundary. Standard Russo-Seymour-Welsh estimates (cf. [?] for instance) tells us that at criticality, the probability of C δ (Ω, A, B, C, D) is bounded away from both 0 and 1 as δ goes to 0; The main result we prove here is the following:

Theorem 1 (Smirnov) The probability of the event C δ (Ω, A, B, C, D) has a limit f (Ω, A, B, C, D) as δ goes to 0. Moreover, the limit is conformally invariant, in the following sense: If Φ is a conformal map from Ω to another simply connected domain Ω = Φ(Ω), and extends continuously to ∂Ω, then f (Ω, A, B, C, D) = f (Φ(Ω), Φ(A), Φ(B), Φ(C), Φ(D)).

As will appear naturally in the proof and was first pointed to by Carleson, f has a most simple expression in the case where Ω itself is an equilateral triangle with vertices A, B and C and D on the interval (CA): Then f (Ω, A, B, C, D) = |CD|/|CA|. By conformal invariance, this gives the value of f for every conformal rectangle.

The proof

General framework

The framework of the proof here is the same as in Smirnov's proof, but we describe it in some detail for sake of self-containedness. Recall that Ω δ is a discrete approximation of Ω; for every point z be a point of the Ω δ lying inside one of the triangles of Ω * δ , we will denote by E A,δ (z) the event that there exists a simple path of open vertices in Ω * δ , separating A δ and z from B δ and C δ -and E B,δ (z), E C,δ (z) similarly, with obvious circular permutations of the letters. Let

H A,δ (z) (resp. H B,δ (z), H C,δ (z)) be the probability of E A,δ (z) (resp. E B,δ (z), E C,δ (z)).
From now on, in many cases where this does not introduce confusion, we will omit the mention of δ in those notations.

From Russo-Seymour-Welsh theory, we obtain the following: There are two positive constants K E and ε E such that, for every δ > 0 and any two points z and z in Ω δ (up to the boundary of the domain),

H A,δ (z ) -H A,δ (z) K E z -z ε E (2)
and a similar bound for H B and H C . Hence, if we suitably extend these functions continuously to Ω, we obtain a family of uniformly Hölder maps from Ω to R. The family is then relatively compact with respect to uniform convergence, and it is hence possible to extract a subsequence (H A,δn , H B,δn , H C,δn ) n>0 , with δ n → 0, which converges uniformly to a triple of Hölder maps

(h A , h B , h C ) from Ω to [0, 1],
with the same exponent and norm as the H A 's.

If we can identify the triple (h A , h B , h C ) uniquely, we will obtain convergence of the family (H A,δ , H B,δ , H C,δ ) as δ goes to 0. Then, making the additional remark that E C,δ (D δ ) is the same event as C δ (Ω, A, B, C, D), this will conclude the proof of the first statement in Theorem 1.

The way we will do that is slightly different from Smirnov's, the difference being essentially in the determination of the boundary conditions (see Subsection 2.4). Introduce the following two functions:

H δ (z) := H A,δ (z) + τ H B,δ (z) + τ 2 H C,δ (z) S δ (z) = H A,δ (z) + H B,δ (z) + H C,δ (z).
(This is slightly different from Smirnov's H, but the idea is exactly the same and the version we use here is more symmetric.) They also form a family of uniformly Hölder maps, and the subsequence (H δn ) (resp. S δn ), with the same (δ n ) as were introduced previously, converges to

h := h A + τ h B + τ 2 h C , resp. s := h A + h B + h C .
The key step of the proof, to which the next two sections are devoted, is to prove that the functions h and s are holomorphic on Ω. Subsection 2.4 then concludes the proof. Notice though that since s is both holomorphic and real-valued, it has to be constant, and it is easy to see from boundary conditions that it is actually equal to 1:

Corollary 1 As the mesh of the discretization vanishes, the sum of the probabilities of E A,δ (z), E B,δ (z) and E C,δ (z) converges to 1, uniformly in z ∈ Ω.

The exact same proof works for any triangulation in which Russo-Seymour-Welsh holds, so this fact seems to be a fundamental property of critical two-dimensional percolation (and might be the key to understanding universality in this particular, limited case, though this is hardly even speculative). As of this time, no direct, combinatorial proof of the corollary seems to be known.

Discrete integration

The theorem best adapted in the present case to prove that h is indeed holomorphic is Morera's theorem, which we recall without proof.

Theorem 2 (Morera) Let Ω be a simply connected domain of the complex plane, and let f : Ω → C be a continuous function. Then f is holomorphic if, and only if, for every simple, closed, smooth curve γ contained in Ω, the integral of f along γ vanishes.

So, let γ be a simple, closed, smooth curve contained in Ω, and for every δ > 0 let γ δ be a discretization of γ contained in Ω δ , i.e. a finite chain (γ δ (k)) 0 k N δ of pairwise distinct vertices of G δ , ordered in the positive direction, such that for every index k, γ δ (k) and γ δ (k + 1) are nearest neighbors, and chosen in such a way that the Hausdorff distance between γ δ and δ goes to 0 with δ. Notice that N δ can be taken of order δ -1 , which we shall assume from now on.

Since the subsequence (δ n ) is chosen so that h δn converges uniformly to h, we have the following fact: As n goes to infinity,

I n (γ) := N δn k=0 H δn (γ δn (k)) + H δn (γ δn (k + 1)) 2 (γ δn (k + 1) -γ δn (k)) → γ h(z)dz. (3)
So, to prove that h is holomorphic, it is sufficient to show that I n (γ) goes to 0 as n goes to infinity.

The discrete curve γ δ surrounds a finite family of faces of G δ , which we shall denote by

• γ δ . Besides, for every (oriented) edge e = xy in G δ , define the following notations:

H δ (e) := H δ (x) + H δ (y) 2 , ∂ e H := H(y) -H(x)
(without dividing by the length of e). Last, if f is a face of G δ , let ∂f be its oriented boundary, seen as a set of oriented edges. With these notations, we get a shorter notation for I n (where e ∈ γ δ means that e is of the form γ δn (k)γ δn (k + 1) for some k) and the following identity:

I n (γ) = e∈γ δ eH δ (e) = f ∈ • γ δ e∈∂f eH δ (e). (4)
Indeed, in the last equality, each boundary term is obtained exactly once with the correct sign, and each interior term appears twice with opposite signs -for every edge e, H δ (-e) = H δ (e).

If f is a face of G δ , let c(f ) be its center (it is a vertex of G * δ ). The sum of eH(e) around f can be rewritten in the following fashion: If x 0 , x 1 , . . ., x |f | = x 0 are the vertices of f , we can "integrate by parts" (which is the same as re-indexing part of the sum -we do it explicitly this time, and shall do it quietly a few more times below) and obtain

e∈∂f eH δ (e) = |f |-1 k=0 (x k+1 -x k ) H(x k+1 ) + H(x k ) 2 = |f |-1 k=0 (x k+1 -c(f )) H(x k+1 ) + H(x k ) 2 - |f |-1 k=0 (x k -c(f )) H(x k+1 ) + H(x k ) 2 = |f |-1 k=0 (x k -c(f )) [H(x k+1 ) -H(x k )] + [H(x k ) -H(x k-1 )] 2 = |f |-1 k=0 x k + x k+1 2 -c(f ) [H(x k+1 ) -H(x k )].
Putting this back in the previous sum, notice that for x, y nearest neighbors in the interior of γ, the term H(y) -H(x) appears twice, once for each face of G δ adjacent to xy, and that the factors (x + y)/2 cancel between them, leaving only the difference between the centers of the faces, i.e. the dual edge of xy.

On the boundary, we obtain of the order of δ -1 terms, each being of order δ 1+ε H , so the contribution of the boundary terms goes to 0 with δ, and we finally obtain

I n (γ) = 1 2 e⊂γ δ e * ∂ e H + o(1), (5)
with the slight abuse of notation that e ⊂ γ δ means that e is an edge of Hδ lying in the interior of γ. We will keep this notation from now on.

For every edge e = xy of Ω δ let P A,δ (e) (and similarly P B and P C ) be the probability that E A,δ (y) is satisfied, but E A,δ (x) is not. Again by Russo-Seymour-Welsh, we get the following estimate, uniformly in δ and e: P A δ (e) K P δ ε P . (6)

Besides, we have for every edge e:

∂ e H A = P A (e) -P A (-e).

So, replacing H by its definition in (5), and re-indexing the sum to obtain each oriented edge in exactly one term, we get the following expression for the discrete contour integral:

I n (γ) =
e⊂γ δ e * P A (e) + τ P B (e) + τ 2 P C (e) + o(1).

(7)

Color-swapping

The main combinatorial tool that we will use in what follows, and in fact the only place where the fact that the model is critical percolation at p c = 1/2, is the following (stated here with our notations):

Proposition 1 (Smirnov) For every edge e of Ω δ , we have the following identities:

P A (e) = P B (τ.e) = P C (τ 2 .e).

We refer the reader to [?] for the (elementary, but very clever) proof of this result; said proof does not actually use the fact that we are considering the triangular lattice, or even the fact that 1/2 is the critical parameter: It extends to site-percolation with parameter 1/2 on any planar triangulation. Using this identity in (7), we obtain where the last step is obtained by re-indexing the sum one last time. Each term in the sum is then equal to 0, by Equation ( 1), so we get I n (γ) = o(1) which is exactly what we needed. It follows that the limit h is holomorphic on Ω.

I n (γ) =
Replacing H with S in the above computation, it is easy to check that one gets the sum, over the same family of oriented edges, of (e * + (τ.e) * + (τ 2 .e) * )P A (e); but the identity e * + (τ.e) * + (τ 2 .e) * = 0

is true in any lattice, as it is the sum of the oriented edges of the triangle around the origin of e. Notice how this does not use the exact details of the triangulation anymore, only Russo-Seymour-Welsh estimates, so it holds on any triangulation for which those hold, as announced.

Boundary conditions

The last step we have to perform, now that we know that h is holomorphic, is to identify enough boundary conditions to specify it uniquely. This can be done in a very simple way, as follows.

Let z be a point on the boundary of Ω δ lying between B and C. It is clear that h A (z) = 0, and it is true that h B (z) + h C (z) = 1 (either from Corollary 1, or by looking at the discrete exploration process started at A and aiming at the boundary interval [BC], for which H B (z) (resp. H C (z)) is the probability that it touches [zC] before (resp. after) [Bz]). Hence, h(z) lies on the interval [τ, τ 2 ] of the complex plane. Besides, h(B) = τ and h(C) = τ 2 , so h induces a continuous map from the boundary interval [BC] of Ω onto [τ, τ ]. By Russo-Seymour-Welsh yet again, h is one-to-one on this boundary interval.

Similarly, h induces a bijection between the boundary interval [AB] (resp.

[CA]) of Ω and the complex interval [1, τ ] (resp. [τ 2 , 1]), so putting the pieces together we see that h is a holomorphic map from Ω to C which extends continuously to Ω and induces a continuous bijection between ∂Ω and the boundary of the triangle with vertices at 1, τ and τ 2 .

From standard results of complex analysis ("principle of corresponding boundaries", cf. for instance Theorem 4.3 in [?]), this implies that h is actually a conformal map from Ω to the interior of the same triangle. But we know that h maps A (resp. B, C) to 1 (resp. τ , τ 2 ), and this determines it uniquely. In other words, there is only one possible limit for the triple (H A , H B , H C ) as δ goes to 0, which gives conformal invariance for free and concludes the proof of Theorem 1.

As a corollary of Theorem 1, we get a nice expression for h A : If Φ Ω,A,B,C is the conformal map from Ω to the triangle mapping A, B and C as previously (which means of course that Φ Ω,A,B,C = h) then H A,δ (z) → 2 (Φ Ω,A,B,C (z)) + 1 3 .

If in particular Ω is the equilateral triangle itself, then h is the identity map and we obtain Cardy's formula in Carleson's form: if D ∈ [CA] then

P pc ([AB] ↔ [CD]) = |CD| |AB| .

  e⊂γ δ e * P A (e) + τ P A (τ 2 .e) + τ 2 P A (τ.e) + o(1) = e⊂γ δ (e * + τ (τ.e) * + τ 2 (τ 2 .e) * )P A (e) + o(1),


Cristel Onésippe
email: cristel.onesippe@univ-ag.fr

Fernando Toro

Ketty Bilba
email: ketty.bilba@univ-ag.fr

Silvio Delvasto
email: delvasto@visionsatelite.com.co

Marie-Ange Arsène
email: maarsene@univ-ag.fr

Influence of Fibers Weight Fraction and Nature of Fibers on Thermal and Mechanical Properties of Vegetable Fibers / Cement Composites

Keywords: Vegetable fibers/cement composites, Thermal properties, Flexural strength, Alkali treatment, Heat treatment

Nowadays, in the purpose of energy saving and eco-friendly technologies and materials, increasing interest is accorded to natural vegetable fibers. Many studies have shown that vegetable fibers are an alternative to asbestos in fiber reinforced cement products manufacturated by the Hatscheck process. Few works have demonstrated the low thermal conductivity of such composite materials resulting from effect of the mixture fiber/matrix. The aim of this study is to produce low-cost building materials in the purpose of saving energy using heat-treated and chemically treated sugar cane bagasse and fique fibers as reinforcement in cement (OPC) based materials. Consequently, the influence of the type and weight of additives on the insulating thermal properties (conductivity) and on the mechanical properties (flexural strength) are prospected.

This study evidences that composites composed with pyrolyzed bagasse fibers have better thermal properties than others composites. The higher mechanical properties are noted for composites manufactured with alkali treated fibers.

INTRODUCTION

Composite materials incorporating vegetable natural fibers have known an increasingly interest during the past few decades [START_REF] Toledo Filho | Development of vegetable fibre-mortar composites of improved durability[END_REF][START_REF] Savastano | Brazilian waste fibres as reinforcement for cement-based composites[END_REF], Savastano et al. 2003[START_REF] Bilba | Sugar cane bagasse fibre reinforced cement composites. Part I. Influence of the botanical components of bagasseon the setting of bagasse/cement composites[END_REF][START_REF] Arsène | Chemically and thermally treated vegetable fibers for reinforcement of cement-based composites[END_REF][START_REF] Coutts | Wood Fibre Reinforced Cement and Concrete[END_REF]]. These environmentally friendly materials are low-cost [START_REF] Coutts | Wood Fibre Reinforced Cement and Concrete[END_REF]] and offer advantages such as reduction of electrical consumption by air conditioning. Moreover, the use of vegetable fibers in replacement of synthetic fibers is interesting in developing countries where the availability of tropical plants and agricultural wastes is important. Due to health reasons, since 1973, various regulations are applied to restrict and ban the use of asbestos in France [START_REF] Bilba | Influence of chemical treatment of vegetable fibers on insulating behavior of vegetable fibers/cement composites[END_REF]] and other countries. Many studies have shown that, regarding the mechanical behaviour, vegetable fibers are an interesting alternative to asbestos in fiber reinforced cement products manufactured by the Hatscheck process [START_REF] Savastano | Brazilian waste fibres as reinforcement for cement-based composites[END_REF], Savastano et al. 2003[START_REF] Savastano | Performance of low-cost vegetable fibrecement composites under weathering[END_REF]] although few works have demonstrated the low thermal conductivity of such vegetable fiber-Portland cement based composite materials [START_REF] Toledo Filho | Development of vegetable fibre-mortar composites of improved durability[END_REF][START_REF] Savastano | Performance of low-cost vegetable fibrecement composites under weathering[END_REF] resulting from effect of the mixture fiber/matrix.

The aim of this study is to produce low-cost insulating materials for building construction in order to obtain habitability and to save energy. Although thermal properties are really involved in applications of such materials, it is essential to evaluate if their mechanical properties are in good agreement with the civil engineering codes [START_REF] Blankenhorn | Effects of fiber surface treatments on mechanical properties of wood fiber-cement composites[END_REF]. Consequently, the influence of the type of treatment and amount of fibers on the mechanical and thermal properties are prospected.

MATERIALS AND METHODS

Samples were elaborated at Universidad del Valle (Cali, Colombia) using a process followed by [START_REF] Delvasto | IIBCC 10 th Int. Inorganic-Bonded Fiber Composites Conference[END_REF] and de Gutiérrez and al. [2005] while the thermal and mechanical studies were developed at Université des Antilles et de la Guyane (Guadeloupe, France). Two types of vegetable fibers were used: bagasse, solid residue of sugar cane after juice extraction (provided by Sugar Cane Factory in Guadeloupe) and Colombian plant fique (provided by a Colombian Fique sacks factory). They will be noted respectively "BAG" and "FIQ".

Two types of treatments were applied:

• Heat-treatment : pyrolysis under controlled atmosphere (N 2 flow, 2L/h) during 2 hours and at 200°C which is the optimal temperature to obtain retified fiber without formation of char [Bilba & Ouensanga 1996]. The pyrolyzed fibers will be named "BAGP". • Chemical treatment: attack by a 5% by mass alkaline solution of Ca(OH) 2 for bagasse and fique fibers. The alkali treated fibers will be named "BAGB" and "FIQB".

A Portland cement marketed as ASTM type I usually used in Colombia, a graded natural river sand, a limestone filler and water were used as the matrix. 93.5 % by weight of the limestone filler is Calcium carbonate (CaCO 3 ). The size distribution, determined by using a Malvern laser diffraction analyzer, of Portland cement gives an average granulometry of 15µm and the limestone filler has an average particle size about 1 µm.

According to these results, the Portland cement has a factory addition what is a calcareous material.

The sand is appropriated for use in cement based mortars, its apparent density is about 2605 kg/m 3 and its average particle size is approximately 0.6 mm.

Composites Preparation

The content of sand and calcium carbonate were kept constant at 50% and 30% by weight of cement respectively. The composites studied were elaborated with a matrix based on Portland cement. The proportions of the matrix compounds were in weight percent with respect to Portland cement (wrtc): sand 50 %, limestone powder 30 %, bentonite 1.5 and 3 %, cellulose pulp 4 %, silica fume 5 %, an aqueous copolymer dispersion of butyl acrylate and styrene from BASF (Acronal ® 296 D) 7.5 %. The amount of fibers was changed from 0 % to 3 % wrtc. Slurry of each mix contains a water/cement ratio of 0.9 (weight by weight) being prepared before its pouring on a casting bed that was subjected to vacuum to obtain a flat sheet. After the slurry dewatering process, pads (160 x 50 x 8 mm 3 ) cut from the fresh laminate were cured for 21 days at 100% of relative humidity and left to air dry in the laboratory for 7 days at room temperature.

About 19 mixes were elaborated: 3 samples for the 3 types of fibers for 2 fiber contents and 1 as control (without any fiber). The detail of the mixes is shown in [Table 1]. The mixes reinforced with fique alkali treated (FIQB) correspond to codes: M86, M90; The mixes reinforced with bagasse alkali treated (BAGB) correspond to codes: M71, M74. The mixes reinforced with pyrolyzed bagasse (BAGP) correspond to codes: M80, M83. The composites will be noted "CBAGP", "CBAGB" and "CFIQB".

Thermal conductivity testing

Thermal tests were conducted under controlled laboratory conditions (temperature ~ 20°C and relative humidity of 70-80%) on 112 days old specimen. The apparatus used was a thermal conductivimeter "CT-mètre" with a thermal probe commercialized by Controlab (Saint-Ouen, France). Six measurements were conducted for each composite with one hour interval between each measurement in order to evaluate the standard deviation of the results.

Flexural strength testing

Composites flexural strength was measured under three-point bend loading with a span of 50mm. The tests were conducted 112 days after specimen fabrication. The specimens were loaded continuously to failure at a speed rate of 0.5 mm/minute. The tests were carried out in an Instron (model 3367 -Guyancourt, France) testing machine equipped with a 500N load cell.

RESULTS AND DISCUSSION

In this paper, results of the tests on 112 days old specimen are presented. At this age, the principal hydration processes of cement have almost completely been done and the formulations of the composites are very stable [START_REF] Bilba | Influence of chemical treatment of vegetable fibers on insulating behavior of vegetable fibers/cement composites[END_REF], Taylor 1997]. The evolution of thermal conductivity according to the amount of fibers is reported in [Figure 1]. Dash lines correspond to control specimen.

The thermal conductivity values are of same order of magnitude whatever the type of fibers. Evolution of mechanical properties determined by three-point bending test is presented in [Figure 2].

Influence of weight fraction of fibers

By increasing the weight fraction of fibers, there is a progressive decrease of the thermal conductivity. This result is in good agreement with [START_REF] Asasutjarit | Development of coconut coir-based lightweight cement board[END_REF] results: when cement/fiber is higher, the thermal conductivity of a cement board increases. The weaker heat conductor materials are CBAGP because the more pronounced addition of fibers effect is for those composites; indeed all the thermal conductivity values of CBAGP are lower than the control specimen one. The effect of fiber content has been correlated with the treatment. At low fiber content (1.5% wrtc), strengthening effect is observed only for composites with pyrolyzed bagasse fibers. On the contrary, at 3% (wrtc) of fibers, the better mechanical properties are measured for composites elaborated with bagasse and fique fibers using alkaline treatment.

In case of thermally treated fibers, the flexural strength decreases with increasing fiber content. A strengthening effect of mechanical properties, with increasing fiber content, is noted for composites containing chemically treated fibers. However, at low content of chemically treated fiber (1.5 %), the composites flexural strength is lower than for the control specimen. This behaviour with 1.5% weight fraction of fibers points out the importance of fiber adhesion and porosity. Between 1.5% and 3% (wrtc) of fiber content, mechanical properties are supposed to follow a linear variation. By extrapolation, a fiber content of 2% is found. In these conditions, flexural strengths are closer for the two composites made of bagasse fibers and remain higher than the control specimen one.

These results are confirmed by former results [5] that evidence an optimal pyrolyzed bagasse fiber content of 2% (wrtc) for improved mechanical properties and an increase of mechanical strength thanks to the addition of pyrolyzed bagasse fibers or alkali treated fique fibers.

Effects of fiber treatment

As shown in [Figure 3], pyrolysis at 200°C significantly improves thermal properties of these composites because thermal conductivity values of CBAGP are lower than the control specimen one. The value of thermal conductivity is more than 20 % lesser for the CBAGP than for the commercial cellulose boards which report 0.68 W/K/m [START_REF] Asasutjarit | Development of coconut coir-based lightweight cement board[END_REF]].

The most important decrease of thermal conductivity values is noted for alkali treated fibers/cement composites. Nevertheless, up to 3% of alkali treated fibers, the values keep higher than for the control specimen. This means that the inclusion of pyrolyzed fibers limits the heat transfer. With 3% of pyrolyzed bagasse fibers, the composites do not offer greater mechanical properties because the strength of CBAGP is lower than the control specimen, whereas strengths of alkali treated fibers/cement composites are higher than the control specimen one. At 3% weight fraction of fibers, CBAGB and CFIQB behave better and seem to be well designed for improved mechanical properties of building materials.

Influence of nature of fibers

A comparison of mechanical and thermal properties of CFIQB and CBAGB allows an assessment of effect of botanical origin of fibers. According to Fig. 3, the less insulating materials are elaborated with alkali treated fique fibers as all thermal conductivity values for CFIQB are higher than the others.

Regarding the strength values (Fig. 2 and 3), its trend according to the weight fraction of fibers is similar for CBAGB and CFIQB but we note that CBAGB present higher flexural strength than CFIQB.

Alkali treatment dissolves the hemicellulose matrix of vegetable fibers and thus, increases surface roughness. However, it improves little or degrades slightly the vegetable fibers properties. While heat treatment (pyrolysis), which increases significantly fibers mechanical strength, increases in a less way the vegetable fibers roughness in comparison with alkali treatment [START_REF] Arsène | Chemically and thermally treated vegetable fibers for reinforcement of cement-based composites[END_REF]]. Thus, these two factors allow the interpretation of the flexural strength according to fiber weight fraction and nature of fibers.

CONCLUSIONS

Thermal conductivity of vegetable fibers/cement composites is an important parameter in the use of such materials in construction for saving energy. In this study, particular attention is given to the influence of weight fraction of fibers, fibers treatment and nature of fibers on thermal properties.

Composites elaborated with bagasse fibers are weaker heat conductor materials than those elaborated with fique fibers. Although for thermal insulating applications, one should prefer the weaker heat conductor that is to say CBAGP. But if requirements on mechanical strength are more severe, composites with alkali treated fibers have to be preferred.

Treatment effectiveness can be limited by roughness or fiber/matrix adhesion problems.

In order to evaluate the interest of a treatment or the nature of reinforcement, we have to compare the composites characteristics to control specimen (without any fiber) characteristics. In this study, the reference specimen contains aggregates and plasticizers and fiber content is limited to 3% (wt). In such conditions, improving effect of alkali treatment is not noted, at the contrary of results reported by [START_REF] Arsène | Chemically and thermally treated vegetable fibers for reinforcement of cement-based composites[END_REF] in the case of an ordinary Portland cement paste. This study allows the determination of an "optimal" weight fraction of fibers. Indeed, if one wants to find a good deal between great thermal and mechanical properties, a formulation within 2.25% of bagasse fibers appears to be optimal.

Composites with fique fibers present weaker mechanical properties than composites with bagasse fibers.

This study has to be supplemented by additional experiments on specimen containing higher fiber content. Indeed, it will be useful in order to determine the potentialities of use of these composites as insulating materials.

Figure 1 .Figure 2 .

 12 Figure 1. Evolution of thermal conductivity of 112 days old vegetable fibers/cement composites according to weight fraction of fibers.

Figure 3 .

 3 Figure 3. Comparison of thermal conductivity and flexural strengths of 112 days old vegetable fibers/cement composites according to weight fraction of fibers.

Table 1 :

 1 Proportions of the mixes elaborated.

	Composites	Fibres	Bentonite	Fibres	Acrylic	Silica	Pulp
		and	(% wrtc)	(% wrtc)	Polymer	Fume	(% wrtc)
		treatment			(% wrtc)	(% wrtc)
	Control (M 75)		3	0	7,5	5	4
	CFIQB (M90)	FIQB	3	1,5	7,5	5	4
	CBAGB (M74)	BAGB	3	1,5	7,5	5	4
	CBAGP (M83)	BAGP	3	1,5	7,5	5	4
	CFIQB (M86)	FIQB	3	3	7,5	5	4
	CBAGB (M71)	BAGB	3	3	7,5	5	4
	CBAGP (M80)	BAGP	3	3	7,5	5	4

T 11, Influence of Fibers Weight Fraction and Nature of Fibers on Thermal and Mechanical Properties of Vegetable Fibers / Cement Composites, Cristel Onésippe et. al.

ACKNOWLEDGMENTS

This work was supported by the ECOS-Nord Project "Vegetable fibers-cement composites" (Ref: C05P03). ECOS-Nord is also supported by MEN/ICETEX/COLCIENCIAS from the Colombian Government and by the French Government.