Pierre Nolin

Critical Exponents of Planar Gradient Percolation

Keywords:

published or not. The documents may come

Introduction

We study some aspects of "gradient percolation". This is a model of inhomogeneous site percolation where the probability for each site to be occupied varies along some fixed direction, for instance the y-axis in the plane. It has been introduced by physicists (see [START_REF] Sapoval | The fractal nature of a diffusion front and the relation to percolation[END_REF][START_REF] Desolneux | Self-organized percolation power laws with and without fractal geometry in the etching of random solids, in Fractal Geometry and Applications : A Jubilee of Benoit Mandelbrot[END_REF]) to modelize phenomena like diffusion or chemical etching, for example to describe the interface created by welding two pieces of metal. They argued that this is one of the cases in which one can actually "see" some aspects of critical percolation in real life, without privileging any value of the parameter a priori (notion of selfcriticality).

Since we will be concentrating on the triangular planar lattice anyway, let us briefly describe the model in this particular case, even if it makes sense for other lattices and dimensions.

Recall that if one colors each cell of a honeycomb lattice independently in black or white with respective probability p and 1 -p, then when p > 1/2 there is an infinite connected component of black cells, and when p < 1/2 there is an infinite connected component of white cells. The number 1/2 is called the critical probability of this homogeneous percolation model : the particular value p = 1/2 gives rise to critical percolation. This model has recently received a lot of attention, which has led among others to the computation of the "critical exponents" (see [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF]).

Suppose now that a large integer N is given. We consider an inhomogeneous percolation model, where each cell z is colored in black and white independently, but with a probability that depends on z. More precisely, a cell z with y-coordinate equal to y(z) ∈ [0, N ] will be colored in black with probability p(z) = y/N (when y < 0, we take p(z) = 0 and when y > N , we take p(z) = 1). It is then easy to see that there almost surely is a (unique) infinite black connected component (that contains the half space {y > 1}) and a unique white connected component (containing the halfspace {y < 0}). Furthermore, the upper boundary of this white cluster and the lower boundary of the black cluster coincide. This separating curve is called the "percolation front".

Intuitively, it is quite clear what happens when N is large. The percolation front will tend to be localized near the line {y/N = 1/2}. Furthermore, since at this level, the percolation is close to critical, the fine structure of the percolation front will be described in terms of critical percolation and its critical exponents.

Let us now briefly describe the main results of the present paper. Suppose that we restrict ourselves to a strip of height N and of length ℓ N . Then, as N → ∞ (provided ℓ N goes to infinity too -not too fast but not too slowly -for instance ℓ N = N is OK for what follows), one can with high probability still define "the" percolation front separating the two "giant components". We call R N the front, and T N its length (i.e. number of steps). We shall see that for each positive δ, when N → ∞, and with high probability:

• The front will remain in the strip of width N 4/7+δ near the line {y = N/2}.

• The front will not remain in the strip of width N 4/7-δ near the line {y = N/2}.

• The expected length t N = E[T N ] of the front satisfies N 3/7-δ ℓ N ≤ t N ≤ N 3/7+δ ℓ N .

• T N is close to its expected value, i.e. T N /t N is close to 1.

The proofs of these results build on the following mathematical results and ideas : Kesten's hyperscaling relations [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF], Smirnov's conformal invariance result [START_REF] Smirnov | Critical percolation in the plane : conformal invariance, Cardy's formula, scaling limits[END_REF], and the computation of the critical exponents for SLE 6 by Lawler, Schramm and Werner [START_REF] Lawler | Values of Brownian intersection exponents II : Plane exponents[END_REF]. More precisely, we shall use directly Figure 1: A simulation for N = 50 and ℓ N = 100 results of the latter two papers, and develop further some ideas introduced in the first one. In particular, Smirnov and Werner showed in [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF] how to derive the critical exponents for percolation from those for SLE 6 , so we shall use directly their computations, and properties of SLE 6 will not be needed.

These results had been conjectured by Sapoval et al. in [START_REF] Sapoval | The fractal nature of a diffusion front and the relation to percolation[END_REF][START_REF] Desolneux | Self-organized percolation power laws with and without fractal geometry in the etching of random solids, in Fractal Geometry and Applications : A Jubilee of Benoit Mandelbrot[END_REF] and also raised in the percolation survey paper by Beffara and Sidoravicius in the Encyclopedia for mathematical physics [START_REF] Beffara | Percolation theory, to appear in Encyclopedia of Mathematical Physics[END_REF].

In fact, it is rather easy to simulate gradient percolation. It has turned out to be an efficient practical tool to obtain numerical estimates for the critical probability of percolation on various lattices (by using the mean height of the front in order to approximate p c , see e.g. [START_REF] Rosso | Determination of percolation probability from the use of a concentration gradient[END_REF][START_REF] Ziff | The efficient determination of the percolation threshold by a frontier-generating walk in a gradient[END_REF]), for instance the square lattice, and it has also been one of the first ways to get numerical evidence for values of the critical exponents of standard percolation (that then supported the conjectures based on Coulomb gas and conformal field theory).

Let us stress the fact that the anisotropy of this model yields that despite the fact that the front converges to a straight line in the fine mesh limit (i.e. a curve of dimension one), its length for a lattice approximation of mesh-size δ (in a rhombus) behaves roughly like δ -3/7 (i.e. it has δ -10/7 steps). Here, the critical exponents do not correspond directly to a fractal dimension for the limiting object.

Preliminaries for (standard) percolation

We recall in this section some known facts concerning percolation (not gradient percolation) that we will use later on.

Setting

The setting in this paper will be site percolation in two dimensions on the triangular lattice. We will represent it as usual as a random (black or white) coloring of the faces of the hexagonal lattice. It has been proved by Harry Kesten [START_REF] Kesten | The critical probability of bond percolation on the square lattice equals 1/2[END_REF] and it is by now a classical fact that for this model, the critical probability is p c = 1/2 and that there is almost surely no infinite cluster when the percolation parameter is taken to be 1/2. The reason why we focus here on this lattice is that it is (at present) the only one for which conformal invariance in the scaling limit has been proved (Smirnov [24]). Conformal invariance combined with the study of SLE 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] entails the exact value of the so-called "critical exponents" that will be instrumental in our considerations. For some other lattices, some inequalities have been proved, that should imply weaker but nevertheless interesting statements, but we will not develop this aspect in the present paper.

The percolation parameter will be denoted by p : each site is occupied (or black) with probability p, and vacant (white) otherwise, independently of each other. The corresponding probability measure on the set of configurations will be referred to as P p , and E p will be the expectation. We will use oblique coordinates, with the origin in 0 and the basis given by 1 and e iπ/3 (in complex notation). The parallelogram of corners a j + b k e iπ/3 (j, k = 1, 2) will thus be denoted by [a 1 , a 2 ] × [b 1 , b 2 ], and we will often use

S(n) = [-n, n] × [-n
, n] and refer to it as the "box of size n". We will denote its boundary by ∂S(n).

For two positive functions f and g, f ≍ g means that there exist two positive and finite constants C 1 and C 2 such that C 1 g ≤ f ≤ C 2 g (so that their ratio is bounded away from 0 and +∞), and f ≈ g means that log f /log g → 1 (when p → 1/2 or when n → ∞, which will be clear from the context).

We now recall some relevant results on critical percolation that we will use in the present paper.

S(n)

Figure 2: The triangular lattice, its associated basis and S(n)

Arm exponents

We first briefly recall some facts concerning critical exponents for the existence of a certain number of "arms". These exponents describe the asymptotic behavior of the probability of certain exceptional events :

Let us consider a fixed integer j ≥ 2. For each positive integers m ≤ n, define the event A j (m, n) that there exist j disjoint monochromatic paths from ∂S(m) to ∂S(n) that are not all of the same color (each path is either completely black/occupied or completely white/vacant, and there is at least one vacant path, and one occupied path). As 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], we could also prescribe the cyclic order of the paths without changing the results that we state below.

Combining the property of conformal invariance in the scaling limit (see [START_REF] Smirnov | Critical percolation in the plane : conformal invariance, Cardy's formula, scaling limits[END_REF][START_REF] Camia | The full scaling limit of two-dimensional critical percolation[END_REF]) with the study of SLE made by Lawler, Schramm and Werner [START_REF] Lawler | Values of Brownian intersection exponents II : Plane exponents[END_REF], it has been proved that : Proposition 1 ( [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF]). For all fixed j ≥ 2, m ≥ j,

P 1/2 (A j (m, n)) ≈ n -(j 2 -1)/12
(1)

when n → ∞.

In fact, we will use this result only for j = 2, 3 and 4. Let us just remark that to derive these three exponents, it might be possible to bypass the use of the rather involved results of Camia and Newman [START_REF] Camia | The full scaling limit of two-dimensional critical percolation[END_REF] as these are exponents related to outer boundaries of clusters, so that "universality"-based ideas might just be enough.

The value of the related "one-arm" exponent is 5/48 and has been derived in [START_REF] Lawler | One-arm exponent for critical 2D percolation[END_REF], but we shall not need it in the present paper.

Behavior near criticality

In the seminal paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF], Harry Kesten showed how the behavior of percolation at its critical point was related to the asymptotic behavior of percolation near its critical point, and derived the so-called hyperscaling relations that link some of the previous arm exponents to other critical exponents describing the behavior of connectivity probabilities near p = p c .

A key idea in his article is to use a certain "characteristic length" L(p) defined in terms of crossing probabilities (sometimes referred to as "spongecrossing probabilities"). This idea (or adaptations of it) was also used in papers concerning finite-size scaling, e.g. [START_REF] Chayes | The low-temperature behavior of disordered magnets[END_REF][START_REF] Chayes | Finite-size scaling and correlation lengths for disordered systems[END_REF][START_REF] Borgs | Uniform boundedness of critical crossing probabilities implies hyperscaling[END_REF][START_REF] Borgs | The birth of the infinite cluster : finite-size scaling in percolation[END_REF].

Let us introduce some more notation. We will denote by

C H ([a 1 , a 2 ] × [b 1 , b 2 ]) (resp. C V ([a 1 , a 2 ] × [b 1 , b 2 ])) the event that there exists a horizontal (resp. vertical) occupied crossing of the parallelogram [a 1 , a 2 ] × [b 1 , b 2 ],
and by C * H , C * V the same events with vacant crossings. So, we have for example

P p (C H ([a 1 , a 2 ] × [b 1 , b 2 ])) = 1 -P p (C * V ([a 1 , a 2 ] × [b 1 , b 2 ])) (2)
Consider now rhombi [0, n] × [0, n]. At p = 1/2, P p (C H ([0, n] × [0, n])) = 1/2. When p < 1/2 (sub-critical regime), this probability tends to 0 when n goes to infinity, and it tends to 1 when p > 1/2 (super-critical regime).

We define a quantity that will roughly measure the scale up to which these crossing probabilities remain bounded away from 0 and 1. For each fixed ǫ 0 > 0, we define

L(p, ǫ 0 ) = min{n s.t. P p (C H ([0, n] × [0, n])) ≤ ǫ 0 } when p < 1/2 min{n s.t. P p (C * H ([0, n] × [0, n])) ≤ ǫ 0 } when p > 1/2 (3)
If we use the Russo-Seymour-Welsh theory (see e.g. [START_REF] Grimmett | Percolation[END_REF][START_REF] Kesten | Percolation theory for mathematicians[END_REF]), we see that for each k ≥ 1, there exists some δ k > 0 (depending only on ǫ 0 ) such that

∀N ≤ L(p), P p (C H ([0, kN ] × [0, N ])) ≥ δ k (4)
For symmetry reasons, this bound is also valid for horizontal vacant crossings. These estimates for crossing probabilities are then the basic building blocks on which many further considerations are built. One of the main results of Kesten's paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF] is for instance the following one-arm probability estimate :

P p [0 ∂S(n)] ≍ P 1/2 [0 ∂S(n))] (5)
for all n ≤ L(p) (uniformly in p). In Section 4.1 of the present paper, we will derive the analogous result for "two-arm probabilities". This result is basically saying that when n is not larger than L(p), things can be compared to critical percolation. On the other hand, the definition of L(p) shows that when n > L(p), the picture starts to look like super/subcritical percolation. For instance:

Lemma 2 (exponential decay with respect to n/L(p)). If ǫ 0 has been chosen sufficiently small, there exists a constant C > 0 such that for all n, all p < 1/2,

P p (C H ([0, n] × [0, n])) ≤ Ce -n/L(p) (6)
Variants of this result are implicitly used or mentioned in Kesten's paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF] or in other papers on finite-size scaling. We now give its proof, as it just takes a couple of lines and clarifies things.

Proof. Observe first that for all integer n,

P p (C H ([0, 2n] × [0, 4n])) ≤ C ′ [P p (C H ([0, n] × [0, 2n]))] 2 (7)
with (i = 0, 1, j = 0 . . . 2). Indeed, consider a horizontal crossing of the big parallelogram : we can extract from it two pieces, one between its extremity on the left side and its first intersection with the vertical median x = n, and in the same way another one starting from the right side. These two subpaths both cross one of the sub-parallelograms "in the easy way" : as they are disjoint by construction, the claim follows by using the BK inequality ( [START_REF] Van Den Berg | Inequalities with applications to percolation and reliability[END_REF][START_REF] Grimmett | Percolation[END_REF]).

C ′ =
We then obtain by iterating :

C ′ P p (C H ([0, 2 k L(p)] × [0, 2 k+1 L(p)])) ≤ (C ′ ǫ 1 ) 2 k (8)
as soon as

ǫ 1 ≥ P p (C H ([0, L(p)] × [0, 2L(p)])).
Recall that by definition,

P p (C H ([0, L(p)] × [0, L(p)])) ≤ ǫ 0 .
Consequently, the RSW theorem entails that for all fixed ǫ 1 > 0, if we take ǫ 0 sufficiently small, we get automatically (and independently of p) that We now choose

P p (C H ([0, L(p)] × [0, 2L(p)])) ≤ ǫ 1 (9)
ǫ 1 = 1/(e 2 C ′ ). For each integer n ≥ L(p), we can define k = k(n) such that 2 k ≤ n/L(p) < 2 k+1
, and then,

P p (C H ([0, n] × [0, n])) ≤ P p (C H ([0, 2 k L(p)] × [0, 2 k+1 L(p)])) ≤ e -2 k+1 /C ′ ≤ e × e -n/L(p) ,
which is also valid for n < L(p), thanks to the extra factor e.

Combining the results of Kesten's paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF] (in particular remark following Lemma 8, p. 144-145) with the derivation of the arm estimates of Proposition 1 (actually only the 4-arm exponent, counting points which are pivotal for the existence of a crossing) leads to the following :

Proposition 3 ([14, 25]). When p → 1/2, L(p) ≈ |p -1/2| -4/3 (10)
Remark 4. It has been shown in [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF] that for any fixed ǫ 1 and ǫ 2 with

0 < ǫ 1 , ǫ 2 ≤ ǫ 0 , L(p, ǫ 1 ) ≍ L(p, ǫ 2 ). ( 11
)
Thus, the particular choice of ǫ 0 is not really important here (as soon as it is sufficiently small).

3 Localization of the front

Framework of gradient percolation

We will now define the model itself and fix some notation. The starting point is still site percolation on the triangular lattice. We first consider a strip S N of finite width 2N (we assume it to be even for simplicity), centered around the x-axis, such that the y-coordinate varies between -N and N . This strip may for the moment be unbounded in one or both directions, and we will denote its length by ℓ N .

In this strip, we assume the parameter to decrease linearly according to y, that is we choose it to be

p(y) = 1/2 -y/2N (12)
With this choice, all the sites on the bottom edge B N will be occupied (p = 1), all the sites on the top edge T N vacant (p = 0). The corresponding probability measure will be denoted by P.

When we perform such a percolation, two opposite regions appear. At the bottom of S N , the parameter is close to 1, we are in a super-critical region and most occupied sites are connected to the bottom edge. On the contrary, we observe on the top a big cluster of vacant sites. The characteristic phenomenon of this model is the existence of a "front", a continuous line touching in the same time the occupied sites connected (by an occupied path) to the bottom of the strip, and the vacant sites connected to the top (by a vacant path). Definition 5. A front will be any interface γ N (in the sense of a curve formed by sides of the lattice hexagons) between an occupied (horizontal) crossing and a vacant (horizontal) crossing.

When the length ℓ N of the strip is finite, there is not necessarily a unique front. For instance, there is a positive probability to observe two horizontal crossings, one vacant and one occupied above it. If the strip is infinite, by independence of the different columns there exists almost surely a column on which all the sites, except the highest one, are occupied. In that case, the front is unique and the result of an exploration path (starting from the top of this random column).

However, for practical purposes, it will be more natural to consider finite length strips. The argument above shows that on an infinite strip, exceptional events occur (the front touches the top and the bottom of the strip), and we may expect them not to happen when the length ℓ N is not too large compared to N . We now use the strip

N

S N = [0, ℓ N ] × [-N, N ].
We will often consider sub-strips of this big strip. For the sake of simplicity, the strip [0, ℓ N ] × [-⌊N α ⌋, ⌊N α ⌋] will be written [±N α ].

Localization

We study now the convergence of the front when it is unique. Temporarily, we will consider, instead of "the" front, the highest horizontal crossing, and we will denote it by R N . Note that the sites just above R N form a vacant horizontal crossing of S N , so that the upper boundary ρ N of R N is a front. We shall see a little bit later that it is indeed the front (i.e. the only interface) with high probability.

We a priori expect it to be close to the line {y = 0} corresponding to the sites where p is critical. A hand-waiving argument goes as follows : if we are at a distance approximately N 4/7 from the line, say for instance above it, the corresponding parameter will be about 1/2 -N -3/7 and the associated characteristic length of order N 4/7 too (by using [START_REF] Chayes | The correlation length for the high-density phase of Bernoulli percolation[END_REF]). Hence, the strip on which everything looks like critical percolation has width of order N 4/7 , and outside of this strip, one is in the super-or sub-critical regime. This explains • For all δ > 0, there exists a δ ′ > 0 such that for all sufficiently large N and all ℓ N ≥ N 4/7 ,

P(R N ⊆ [±N 4/7-δ ]) ≤ e -N δ ′ (13)
• For all δ > 0, for all γ ≥ 1, there exists a δ ′ > 0 such that for all sufficiently large N , and all N 4/7 ≤ ℓ N ≤ N γ ,

P(R N [±N 4/7+δ ]) ≤ e -N δ ′ (14)
Proof. Let us first derive the statement concerning P(R N ⊆ [±N 4/7-δ ]).

For that purpose, consider disjoint rhombi of the form

[i, i + 2N 4/7-δ ] × [-N 4/7-δ , N 4/7-δ ] (i = 0, 2N 4/7-δ + 1, 4N 4/7-δ + 2 . . .)
We can take at least ℓ N /3N 4/7-δ such rhombi, and each of them possesses (independently of the other ones) a vertical vacant crossing with probability larger than

P 1/2+N -3/7-δ /2 C * H ([0, 2N 4/7-δ ] 2 ) .
But (using [START_REF] Chayes | The correlation length for the high-density phase of Bernoulli percolation[END_REF], the critical exponent for L)

L(1/2 + N -3/7-δ /2) ≈ (N -3/7-δ /2) -4/3 ≈ N 4/7+4δ/3 , hence L(1/2 + N -3/7-δ /2) ≫ 2N 4/7-δ ,
so that there exists a vertical vacant crossing with probability larger than ǫ 0 . By independence, a "block" entailing that R N [±N 4/7-δ ], will occur with probability larger than

1 -(1 -ǫ 0 ) ℓ N /3N 4/7-δ ,
which proves the claim as ℓ N ≥ N 4/7 by assumption.

Let us now turn our attention to the quantity P(R N [±N 4/7+δ ]). Assume that R N is not entirely contained in the strip [±N 4/7+δ ].

If at some point it is for example above this strip, we face the following alternative :

• Either R N is in the strip [±N 4/7+δ
/2] at some point. In that case, consider the following rhombi, located between the lines y = N 4/7+δ /2 and y

= N 4/7+δ : [i, i+N 4/7+δ /2]×[N 4/7+δ /2, N 4/7+δ ] (i = 0, 1, 2 . . .).
It is easy to see that R N will have to cross vertically or horizontally one of them.

There are at most ℓ N such rhombi, and they are in a zone for which p ≤ 1/2 -N -3/7+δ /4, a crossing thus occurs with probability less than

ℓ N P 1/2-N -3/7+δ /4 (C H ([0, N 4/7+δ /2] 2 ) ∪ C V ([0, N 4/7+δ /2] 2 )) ≤ 2ℓ N P 1/2-N -3/7+δ /4 (C H ([0, N 4/7+δ /2] 2 )) ≤ 2Cℓ N e -N 4/7+δ /2L(1/2-N -3/7+δ /4)
by using Lemma 2 (of sub-exponential decay).

But (by [START_REF] Chayes | The correlation length for the high-density phase of Bernoulli percolation[END_REF])

L(1/2 -N -3/7+δ /4) ≈ (N -3/7+δ /4) -4/3 ≈ N 4/7-4δ/3
, and ℓ N does not grow too fast, so that the considered case has a probability tending to 0 sub-exponentially fast.

N N / 2

4/7+δ 4/7+δ , where the percolation parameter remains lower than 1/2 -N -3/7+δ /4. In that case, R N will cross vertically or horizontally one of the rhombi (forming a "column") [0, N 4/7+δ /2]×[j+N 4/7+δ /2, j+N 4/7+δ ] (j = 0, 1, 2 . . .).

There are at most N such rhombi. Once again, Lemma 2 entails that a crossing occurs with probability less than 2CN e -N 4/7+δ /2L(1/2-N -3/7+δ /4)

As before, L(1/2 -N -3/7+δ /4) ≈ N 4/7-4δ/3 , so that the obtained probability tends to 0 sub-exponentially fast.

If R N is below the strip at some point, the reasoning is the same : consider instead the vacant crossing bordering R N . Hence, the final probability just has to be multiplied by 2.

Uniqueness of the front

In the previous subsection, we have focused on the highest horizontal crossing R N . Clearly, the results remain valid if we consider the lowest horizontal vacant crossing R * N instead. Recall that R N is bordered above by a horizontal vacant crossing, so that its upper boundary ρ N is a front. Similarly, R * N is bordered below by an occupied crossing, and its lower boundary ρ * N is a front too. Note that ρ * N is always below ρ N . It is easy to see that uniqueness of the front amounts to checking whether ρ N and ρ * N coincide. It is also equivalent to verifying that R N is connected to the bottom B N by an occupied path (or that R * N is connected to the top T N by a vacant path). We are now going to prove that this indeed occurs with a very large probability.

Note that if one starts an exploration process from the top-left corner of S N , one discovers the top-most occupied crossing without discovering the status of the sites below it. This RSW-type observation will be essential in our proof.

Proposition 7 (Uniqueness). Assume that ℓ N ≥ N 4/7+δ for some δ > 0. Then, there is a δ ′ > 0 (depending on δ) such that for all sufficiently large N ,

P(ρ N = ρ * N ) ≥ 1 -e -N δ ′ (15)
Proof. As announced, we will work with R N . Our goal is to show that the probability for R N not to be connected to B N is very small. For that purpose, we first divide the strip S N into disjoint sub-strips (S i N ) as follows. For ǫ := δ/4, we choose N 3ǫ /6 disjoint sub-strips of length 3N 4/7+ǫ (not necessarily covering entirely S N ) of the type

S i N = [n i N , n i N + 3N 4/7+ǫ ] × [-N, N ] (i = 0, . . . , N 3ǫ /6 -1).
Consider one of these strips S i N . R N crosses it horizontally, and remains "below" its highest horizontal crossing, that we denote by r i N . Consequently, it will be sufficient to show that one of the r i N 's is connected to B N in S i N . We now fix an i, and we try to find a lower bound for the probability that r i N is connected to the bottom of S i N by an occupied path that stays in that sub-strip. Let us suppose for notational convenience that i = 0 and n i N = 0. Note first that with probability at least 1/2, there exists a vacant top-tobottom crossing of the rhombus [N 4/7+ǫ , 2N 4/7+ǫ ] × [0, N 4/7+ǫ ] (percolation is sub-critical in this region), so that a lowest point z on r i N in the middle part [N 4/7+ǫ , 2N 4/7+ǫ ] × [-N, N ] of the strip lies below the x-axis with probability at least 1/2 + o(1) (the localization result (Theorem 6) tells us that r i N remains below the height N 4/7+ǫ with high probability). Now, if we have explored this highest crossing of S 0 N "from above", we have not yet discovered the status of the sites below it, so that we can apply

N -N

4/7+ε 4/7+ε the FKG inequality for events involving only the state of these remaining sites. We are now going to show that in the case where r 0 N passes below the x-axis, the conditional probability that it is connected to the bottom part of the sub-strip by an occupied crossing is bounded from below by a quantity of order N -2ǫ .

A way to prove this goes as follows. Let us first choose z, and define the annulus S(z, 2N 4/7-ǫ ) \ S(z, N 4/7-ǫ ) around z. Since it is contained in the region where p ≥ 1/2 -2N -3/7-ǫ , and since the characteristic length corresponding to this value of the parameter is of order

L(1/2 -2N -3/7-ǫ ) ≈ N 4/7+4ǫ/3 ,
there is a probability at least δ 4 4 (this is the constant coming from RSWtheory) to observe an occupied circuit in this annulus.

We now want to connect this circuit to the bottom boundary of the substrip. Note that the part of the circuit that is below r 0 N together with r 0 N contain an occupied circuit around the segment

I = z + [-N 4/7-ǫ , N 4/7-ǫ ] × {-N 4/7-ǫ }.
We need the following simple lemma for critical percolation:

Lemma 8. Consider the rhombus [-N 4/7+ǫ , N 4/7+ǫ ]×[-2N 4/7+ǫ
, 0] and the sub-interval I N = [-N 4/7-ǫ , N 4/7-ǫ ] × {0} on its top edge. Then the event

C I N V ([-N 4/7+ǫ , N 4/7+ǫ ] × [-2N 4/7+ǫ
, 0]) that there exists a vertical occupied crossing connecting I N to the bottom edge has a probability at least

P 1/2 C I N V ([-N 4/7+ǫ , N 4/7+ǫ ] × [-2N 4/7+ǫ , 0]) ≥ C N 2ǫ (16)
for some universal constant C (depending neither on N nor on ǫ).

Proof. Consider the parallelogram [0, N 4/7+ǫ ] × [-2N 4/7+ǫ , 0], and cover its top edge by less than N 2ǫ intervals I j N = [n ′j N -N 4/7-ǫ , n ′j N + N 4/7-ǫ ] × {0} of length 2N 4/7-ǫ . We know from the RSW theorem that there exists a vertical occupied crossing with probability at least δ 2 > 0, so that

δ 2 ≤ j P 1/2 C I j N V ([0, N 4/7+ǫ ] × [-2N 4/7+ǫ , 0]) But for each j, P 1/2 C I j N V ([0, N 4/7+ǫ ] × [-2N 4/7+ǫ , 0]) ≤ P 1/2 C I j N V ([n ′j N -N 4/7+ǫ , n ′j N + N 4/7+ǫ ] × [-2N 4/7+ǫ , 0]) = P 1/2 C I N V ([-N 4/7+ǫ , N 4/7+ǫ ] × [-2N 4/7+ǫ , 0])
by translation invariance. Hence

δ 2 ≤ N 2ǫ × P 1/2 C I N V ([-N 4/7+ǫ , N 4/7+ǫ ] × [-2N 4/7+ǫ , 0]) ,
which completes the proof. Note that a repeated application of the RSW theorem (log(N 2ǫ ) times) would have given a lower bound of the type N -κǫ that would have been also sufficient for our purpose here.

Putting the pieces together, with the help also of the FKG inequality, we get that for each i ≤ N 3ǫ /6 -1, on the event that R N remains localized, the probability that it is connected to the bottom part of the strip in the sub-strip S i N is bounded from below by C ′ /N 2ǫ independently for each i. The proposition then follows readily : indeed, R N is connected to B N with probability at least

1 -(1 -C ′ N -2ǫ ) N 3ǫ /6 ≥ 1 -e -N ǫ ′ ( 17
)
for some positive ǫ ′ . The previous results suggest to make some restrictions about the length ℓ N of the strip. In the following, we will thus assume that ℓ N = o(N γ ) for some γ ≥ 1 (to ensure convergence), and that there exists a δ > 0 such that ℓ N ≥ N 4/7+δ (to ensure uniqueness). This second hypothesis entails that the event corresponding to uniqueness has a probability tending to 1 sub-exponentially fast, so that we can restrict to it.

From now on, we will simply refer to the front and denote it by F N , which means that we will implicitly neglect the error term in the estimates that we derive. In particular, the front will be exactly the set of edges from which two arms can be drawn, one occupied to the bottom B N , and one vacant to the top T N .

Length of the front

We would like now to study the length T N of the front i.e. its number of edges. The preceding remark shows that we will need a two-arm probability estimate for that purpose. Unless otherwise stated, the expression "two arms" refers to two arms of opposite color.

We will often have to count edges rather than sites. Since we are mostly interested in rough estimates, we will then just use the fact that number of edges and number of corresponding sites are comparable (i.e. up to a multiplicative factor of 6). To simplify notations, it will be convenient to associate to each edge e one of its two neighboring sites x e , which we do arbitrarily and once for all.

Two-arm estimates

We are now going to derive the analog of (5) in the case of two arms and for non-constant p. This lemma will enable us to estimate the probability of having two arms from an edge e in the "critical strip". The goal is to show roughly that

P p [A 2 (2, n)] ≍ P 1/2 [A 2 (2, n)] (n ≤ L(p))
In fact, for our purpose, we will have to consider, instead of P p , product measures P ′ with associated parameters p ′ (v) which may depend on the site v but remain between p and 1 -p (we will simply say that P ′ is "between p and 1 -p"). The present situation is a little more complicated than for the one-arm estimate, because of the lack of monotonicity (A 2 events correspond to the combination of one path of each type, so that they are neither increasing nor decreasing).

For a parallelogram R containing 0 in its interior, we denote by Γ 2 (0, R) the event that there exist an occupied path r 1 and a vacant path r 2 from the set ∂0 of vertices neighboring 0 to the boundary ∂R of R (this is the analog to Γ(0, R) in Kesten's paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF], that corresponds to the existence of 4 arms alternatively occupied and vacant). Let us now state and prove the result: Lemma 10. Uniformly in p, P between p and 1 -p, n ≤ L(p), we have

P [Γ 2 (0, S(n))] ≍ P 1/2 [Γ 2 (0, S(n))] (18)
Proof. This proof is an adaptation of the proof of Theorem 1 in Kesten's paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]. We first recall some estimates contained in this paper, and then we adapt the original proof for 1 arm to the case of 2 arms. Note that Lemma 8 in [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF] is the "four-arm" version of this two-arm result. Let us first introduce the events that we use throughout the proof. All of them are exact analogs of events defined in Kesten's paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]. We consider a parallelogram R such that 0 ∈ R.

• In the case of 1 arm, we do not lose much (a factor 4) by imposing the extremity to be on a specified edge of R, at least if 0 is not too close to one of the edges. Here we impose the occupied arm to arrive on the bottom edge of R, and the vacant arm on the top edge. This event is denoted by Ω 2 (0, R).

• This event is a bit difficult to extend, so we may want to add furthermore "security strips". This leads to the definition of the event ∆ 2 ,

S(2 ) S(2 ) k-1 k A(1,k) A(2,k)
Figure 9: Definition of the event ∆ 2 analog of the event ∆ (in the case of 4 arms). Albeit more restrictive, this event has a probability that remains comparable to Ω 2 (0, R).

We consider thus the two horizontal strips

A(1, k) := [-2 k-1 , 2 k-1 ] × [-2 k , -2 k-1 ] A(2, k) := [-2 k-1 , 2 k-1 ] × [2 k-1 , 2 k ]
and we define for a site v in S(2 k-1 ),

∆ 2 (v, S(2 k )) = {Γ 2 (v, S(2 k )) occurs, with r i ∩(S(2 k )\ S(2 k-1 )) ⊆ A(i, k
) and there exist two horizontal crossings, one occupied of A(1, k) and one vacant of A(2, k)}.

• We define similarly for a parallelogram R ′ contained in the interior of S(2 k ):

Γ2 (S(2 k ), R ′ ) =
{There exist an occupied path r 1 and a vacant path r 2 from ∂R ′ to the bottom and top edges respectively of S(2 k ), which are (with the exception of their extremities on

∂R ′ ) contained in S(2 k ) \ R ′ }.
and for a site v in S(2 k-1 ), j ≤ k -2, the following strips (centered on v):

These two events are respectively increasing and decreasing, and in that case, d dt Pt (A 2 ∩ B 2 ) can be expressed as (see Lemma 1 in [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]):

v∈S(2 k-3 )
dp(v, t) dt Pt (v is pivotal for A 2 , but not for B 2 , and B 2 occurs)

-Pt (v is pivotal for B 2 , but not for A 2 , and A 2 occurs)

A vertex v ∈ S(2 k-3 ), v = 0, is pivotal for A 2 iff there exist two paths both containing v such that 1. Their sites, except for v, are respectively all occupied and all vacant.

2. The first path connects ∂0 to the bottom side of S(2 k ).

3. The second path separates 0 from the bottom side of S(2 k ).

In the case of 1 arm, Kesten used a vacant loop around the origin, here the separating path can have its extremities on the boundary. This will not change the computations, since we still have 4 arms locally, that we will sum in the same way.

Actually, we must also assume that k ≥ 7 and put apart the vertices which are too close to the origin, for instance those in S [START_REF] Lawler | Values of Brownian intersection exponents II : Plane exponents[END_REF]. For these sites, Pt (v is pivotal for A 2 , but not for B 2 , and B 2 occurs) ≤ Pt ( Γ2 (S(2 k ), S( 16)))

≤ C 4 Pt (Ω 2 (0, S(2 k )))
for some universal constant C 4 .

We now associate to each site v ∈ S(2 k-3 ) \ S( 16) a parallelogram R(v) such that 0 / ∈ R(v), so that we make appear four arms locally. We choose them to present the following property : if j is such that 2 j+1 < d(v, 0) ≤ 2 j+2 , then R(v) is included in S(2 j+3 ) \ S(2 j ). This will imply that Pt (v is pivotal for A 2 , but not for B 2 , and B 2 occurs)

≤ Pt Γ 2 (0, S(2 j )) ∩ Γ(v, R(v)) ∩ Γ2 (S(2 k ), S(2 j+3 )) = Pt [Γ 2 (0, S(2 j ))] Pt [Γ(v, R(v))] Pt [ Γ2 (S(2 k ), S(2 j+3 ))]
by independence (these events depending on sites in disjoint sets).

We can take the parallelograms R(v) like in Kesten's paper (see [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF] p. 117). As noticed in this paper, the precise choice is not really important.

On one hand, we have to ensure that the four arms are not too small. For that purpose, the distances between v and each of the sides of R(v) must be comparable to the distance between 0 and v. On the other hand, if the announced property is satisfied, we will be in position to join the paths outside R(v), that are respectively between 0 and ∂S(2 j ), and between ∂S(2 j+3 ) and ∂S(2 k ). For the sake of completeness, let us just recall briefly how Kesten chooses these R(v).

Consider v = (v 1 , v 2 ) / ∈ S(16). If |v 1 | ≤ |v 2 | ≤ 2 k-3 and 16 ≤ 2 j+1 < v 2 ≤ 2 j+2
, take l 1 and l 2 such that l 1 2 j-2 < v 1 ≤ (l 1 + 1)2 j-2 and l 2 2 j-2 < v 2 ≤ (l 2 + 1)2 j-2 , and define

R(v) = [(l 1 -2)2 j-2 , (l 1 + 2)2 j-2 ] × [l 2 2 j-2 -2 j , l 2 2 j-2 + 2 j ]. If v 2 < 0, take the image of R((v 1 , -v 2 )
) by the symmetry with respect to the x-axis. Finally, if |v 2 | < |v 1 |, simply exchange the roles of the first and second coordinates. We can easily check that these parallelograms possess by construction the desired property. Now, by joining the two terms Pt [Γ 2 (0, S(2 j ))] and Pt [ Γ2 (S(2 k ), S(2 j+3 ))], we can make appear Pt [Ω 2 (0, S(2 k ))] : indeed, we get easily from the Russo-Seymour-Welsh theorem (see Lemma 6 in [14])

δ 2 16 Pt [∆ 2 (0, S(2 j ))] Pt [ ∆2 (S(2 k ), S(2 j+3 ))] ≤ Pt [∆ 2 (0, S(2 k ))]
as, by assumption, 2 k ≤ L(p) and j ≤ k -5 (here a slight generalization of the FKG inequality is needed (see for instance Lemma 3 in [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]), invoking zones where "everything is monotonous"). We obtain then from

Pt [∆ 2 (0, S(2 j ))] ≥ C -1 2 Pt [Γ 2 (0, S(2 j ))]
(which follows from (20)), and

Pt [ ∆2 (S(2 k ), S(2 j+3 ))] ≥ C -1 3 Pt [ Γ2 (S(2 k ), S(2 j+3 ))] (using (21)), that Pt [Γ 2 (0, S(2 j ))] Pt [ Γ2 (S(2 k ), S(2 j+3 ))] ≤ C 5 Pt [∆ 2 (0, S(2 k ))] ≤ C 5 Pt [Ω 2 (0, S(2 k ))]
These two paths can then be extended, so that they go out of the strip [±N 4/7+ǫ ] : Lemma 8 (contained in the proof of uniqueness) implies that this can be done with probability at least

(CN -2ǫ ) 2 = C ′ N -4ǫ (24)
On the other hand, since we stay in the strip [±3N 4/7-ǫ ], of associated characteristic length L(1/2 ± 3N -3/7-ǫ ) ≈ N 4/7+4ǫ/3 , we get that P(∆ 2 (x e , S(x e , 2 j ))) ≍ P(Γ 2 (x e , S(x e , 2 j )))

≍ P 1/2 (Γ 2 (x e , S(x e , 2 j ))) ≥ P 1/2 (Γ 2 (0, S(2N 4/7-ǫ ))),
and the 2-arm exponent implies that

P 1/2 (Γ 2 (0, S(2N 4/7-ǫ ))) ≈ (2N 4/7-ǫ ) -1/4 ≫ N -1/7 (25)
We can thus construct two arms going out of the strip [±N 4/7+ǫ ] with probability at least N -1/7-4ǫ (for N large enough).

In that case, e has a high probability to be connected to the top and to the bottom of S N . Indeed, the front would go out of the strip [±N 4/7+ǫ ] otherwise, which occurs with a probability less that ǫ N , for some ǫ N (independent of e) tending to 0 sub-exponentially fast. The conclusion follows, by summing the lower bound over all edges e in the strip [2N 4/7+ǫ , ℓ N -2N 4/7+ǫ ] × [±N 4/7-ǫ ] : for N large enough,

e⊆S N P(e ∈ F N ) ≥ (2N 4/7-ǫ )(ℓ N -4N 4/7+ǫ ) × (N -1/7-4ǫ -ǫ N ) ≥ N 3/7-δ ℓ N 4.3 Convergence in L 2
Here everything happens in the strip [±N 4/7 ], and it is possible to determine whether or not an edge e is on the front on a vicinity of size N 4/7 (with probability very close to 1). Distant points will thus be almost completely decorrelated, and there is a phenomenon of "averaging". More precisely, we get the following bound on Var[T N ]: Proposition 12. For each δ > 0, we have for N sufficiently large:

Var[T N ] ≤ N 11/7+δ ℓ N (26)
By combining this result to the estimates on E[T N ] of the previous subsection, we immediately get that if ℓ N ≥ N 5/7+δ for some δ > 0, then

Var[T N ] = o(E[T N ] 2 ) ( 27
)
and consequently

Theorem 13. If for some δ > 0, ℓ N ≥ N 5/7+δ , then

T N E[T N ] -→ 1 in L 2 , when N → ∞ (28)
Remark 14. On each vertical line, amongst the (about) N 4/7 edges which lay in the critical strip, approximately N 3/7 of them will be on the front, that is a fraction 1/N 1/7 . If we take ℓ N to be equal to N , we get E[T N ] ∼ N 10/7 , and this exponent can indeed be observed numerically.

Proof. For an edge e, the event "e ∈ F N " will be denoted by F e . Using the expression

T N = e I Fe (29)
we get Var

[T N ] = e,f [P(F e ∩ F f ) -P(F e )P(F f )] (30)
First, notice that we can restrict the summation to the edges e, f ⊆ [±N 4/7+δ ] : like before, the remaining term tends to 0 sub-exponentially fast.

The idea is to replace F e by an event Fe which depends only on sites in a box around e of size N 4/7+δ , and such that P(F e ∆ Fe ) → 0 sub-exponentially fast (uniformly in e). To construct such an event, we simply invoke the result of uniqueness, which implies that we can take with an error term ǫ N tending to 0 sub-exponentially fast. Now, for an edge e, there are at most 6(2N 4/7+δ + 1) 2 edges f for which the two associated boxes intersect, and in that case, P( Fe ∩ Ff ) -P( Fe )P( Ff ) ≤ P( Fe ) ≤ N -1/7+δ for N large enough, uniformly in e (by using the 2-arm exponent in the box S(x e , N 4/7-2δ )). We deduce from it

Outer boundary

To further describe F N , we can also consider its outer boundary. Actually, two curves arise : the "upper" outer boundary, and the "lower" outer boundary. We denote respectively by U + N and U - N the lengths of these curves. These quantities can be computed in a similar way as the length T N of the front.

We now have to count the number of 3-arm points (one occupied and two vacant arms for U + N , two occupied and one vacant for U - N ). There is no extra difficulty to adapt Kesten's result : being pivotal still leads to 4 arms locally, and we get the same summation, that can be conducted identically. We have thus the following estimate on E[U ± N ] : Proposition 16. For all δ > 0, we have for N sufficiently large:

N 4/21-δ ℓ N ≤ E[U ± N ] ≤ N 4/21+δ ℓ N (31)
The upper bound can be treated in the same way, but for the lower bound, we have to be a little more careful : we have to ensure that we can extend the two arms of the same type into two disjoint macroscopic arms. It can be done by using the fact that we can impose an occupied crossing in a N 4/7+ǫ × N 4/7+ǫ rhombus to arrive in a "corner" of size N 4/7-ǫ with probability at least N -κǫ for some constant κ (for instance applying the RSW result log N 2ǫ times).

The decorrelation arguments that we used for the length of the front can also be applied to the outer boundary, entailing convergence in L 2 if ℓ N is not too small. Proposition 17. For each δ > 0, we have for N sufficiently large:

Var[U ± N ] ≤ N 4/3+δ ℓ N (32)
Hence, if for some δ > 0, ℓ N ≥ N 20/21+δ , then

U ± N E[U ± N ] -→ 1 in L 2 , when N → ∞ (33)
Note that when ℓ N = N , this result implies that U ± N is of order N 25/21 with high probability.

Open questions

We have used here the logarithmic equivalence of Proposition 1, that has been proved in [START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF]. It is in fact conjectured that this equivalence holds up to multiplicative constants. For instance, the stronger assumption P 1/2 (A 4 n ) ≍ n -5/4 would lead to L(p) ≍ |p -1/2| -4/3 , which would make it possible to derive a sharper description of the front (in most constructions, only one application of the RSW theorem would be sufficient).

One of the major issues would be to describe (if it exists) the scaling limit of the front, once properly renormalized. We could first argue as follows : in each of the sub-strips [±N 4/7-δ ], everything "looks like criticality", so the front will look (locally) the same as the border of a cluster in critical percolation, i.e. a SLE 6 .

However, we do not know at present how the front "bounces", that is to say what happens when we are at a distance approximately N 4/7 from the critical line. In fact, the front could be linked to the kind of objects F. Camia, L. Fontes and C. Newman have recently tried to construct for percolation near criticality, when we take the parameter to be p = 1/2 + λδ 3/4 , with a λ possibly inhomogeneous.

  10 2 some universal constant. It suffices for that (see figure) to divide the parallelogram [0, 2n] × [0, 4n] into 4 horizontal sub-parallelograms [0, 2n]×[in, (i+1)n] (i = 0 . . . 3) and 6 vertical ones [in, (i+1)n]×[jn, (j+2)n]

Figure 3 :

 3 Figure 3: A horizontal crossing of the parallelogram [0, 2n] × [0, 4n]

Figure 4 :

 4 Figure 4: The strip where the percolation parameter p varies

Figure 5 :

 5 Figure 5: Localization of the front (2N = 500, ℓ N = 1000)

Figure 6 :

 6 Figure 6: R N crosses horizontally or vertically one of the small rhombi

Figure 7 :

 7 Figure 7: Construction of an arm going out of [±N 4/7+ǫ ]

Figure 8 :

 8 Figure 8: Existence of an arm from I N to the bottom side

Fe := { 2

 2 arms e top and bottom sides of ∂ S(x e , N 4/7+δ ) ∩ S N } We have thus Var[T N ] = e f P( Fe ∩ Ff ) -P( Fe )P( Ff ) + ǫ N

Var[T N ] ≤ e [ 6 (

 e6 2N 4/7+δ + 1) 2 × N -1/7+δ ] + ǫ N and finallyVar[T N ] ≤ [6(2N 4/7+δ + 1)(ℓ N + 1)] × [6(2N 4/7+δ + 1) 2 N -1/7+δ ] + ǫ N ≪ N 11/7+5δ ℓ N Remark 15.Recall that for ordinary percolation, it is still an open problem to show that the typical length of an interface is comparable to its expected length (see open question 2 in[START_REF] Smirnov | Critical exponents for two-dimensional percolation[END_REF]). Here, things are easier because localization of the front implies a faster decorrelation.

Acknowledgements

The author would like to thank warmly his supervisor W. Werner for suggesting this problem to him and his guidance. He also thanks J. van den Berg and F. Camia for useful discussions and remarks.

and then ∆2 (S(2 k ), S(v, 2 j )) = { Γ2 (S(2 k ), S(v, 2 j )) occurs, with associated paths r i satisfying r i ∩ (S(v, 2 j+1 ) \ S(v, 2 j )) ⊆ Ã(i, j), and there exist two horizontal crossings, one occupied of Ã(1, j) and one vacant of Ã(2, j)}.

Let us now state the estimates that we will use. We will not prove them, since these are exact analogs for 2 arms of results stated for 4 arms in Kesten's paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF], except that we generalize the condition "between p and 1/2" to "between p and 1 -p". This generalization can be done as the only tool used in the proofs is the Russo-Seymour-Welsh theorem (we also implicitly use the symmetry about 1/2, which implies that L(p) = L(1-p)).

In the following, P denotes any product measure "between p and 1 -p".

(1) Extension of ∆ 2 : There exists a constant C 1 < ∞ such that

for all p, P and 2 k ≤ L(p).

(2) Comparison of Γ 2 and ∆ 2 (analog of Lemma 4 in [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]) : There exists a constant C 2 < ∞ such that

for all p, P and 2 k ≤ L(p).

(3) Comparison of Γ2 and ∆2 (analog of Lemma 5 in [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF]) : There exists a constant C 3 < ∞ such that

for all p, P , 2 k ≤ L(p) and j ≤ k -2.

These prerequisites being recalled, we are now able to turn to the proof itself. For that purpose, we will have to adapt Kesten's original proof. Recall that we consider a parameter p, a measure P between p and 1 -p and an integer n ≤ L(p). The parameters of P will be denoted by p(v).

1st step : We first notice that we can replace Γ 2 (0, S(n)) by Ω 2 (0, S(2 k )), with k such that 2 k ≤ n < 2 k+1 . Indeed, the previous estimates entail

and

2nd step : We would like now to use Russo's formula, but for technical reasons which will become clear in the next step, this will work nicely only for the points that are not too close to the boundary. We therefore first have to see how the change of p(v) for the points that are close to the boundary affects the probabilities that we investigate. More precisely, we start with

). The resulting measure P is between p and 1 -p : so,

thanks to [START_REF] Sapoval | The fractal nature of a diffusion front and the relation to percolation[END_REF] and [START_REF] Desolneux | Self-organized percolation power laws with and without fractal geometry in the etching of random solids, in Fractal Geometry and Applications : A Jubilee of Benoit Mandelbrot[END_REF], and similarly P (Ω 2 (0, S(2 k ))) ≍ P (∆ 2 (0, S(2 k-3 ))), which allows us to conclude, since P 1/2 (∆ 2 (0, S(2 k-3 ))) = P (∆ 2 (0, S(2 k-3 ))) by definition.

3rd step : In order to handle the sites v located in S(2 k-3 ), we will apply a generalization of Russo's formula to the family of measures ( Pt

which corresponds to a linear interpolation between 1/2 p(v) in S(2 k-3 ). The event Ω 2 (0, S(2 k )) can be written as the intersection of A 2 (0, S(2 k )) = {There exists an occupied path from ∂0 to the bottom side of S(2 k )} and B 2 (0, S(2 k )) = {There exists a vacant path from ∂0 to the top side of S(2 k )}.

Hence, Pt (v is pivotal for A 2 , but not for B 2 , and B 2 occurs)

For symmetry reasons, the same is true if we invert A 2 and B 2 . Consequently, dividing by Pt [Ω 2 (0, S(2 k ))] will make appear its logarithmic derivative in the left-hand side of Russo's formula : 16) 16)

dp(v, t) dt

Finally, the first term that we obtain (the sum encoding the existence of 4 arms locally) is exactly the same as in Kesten's paper [START_REF] Kesten | Scaling relations for 2D-percolation[END_REF] (end of the proof of Theorem 1 p. 140), and its integral between 0 and 1 is bounded by some universal constant C 6 : 16)

The desired conclusion then follows.

Expected length of the front

With this result established, we are now able to study properties of the front in the "critical strip". Roughly speaking, only the sites in [±N 4/7 ] must be taken into account, and each of these sites has a probability approximately (N 4/7 ) -1/4 = N -1/7 (the two-arm exponent being equal to -1/4) to be on the front. Starting from this idea, we will prove the following estimate on the expectation of T N :

Proposition 11. For all δ > 0, we have for N sufficiently large:

Proof. Throughout the proof, we will use the fact that

We consider first the upper bound. Take ǫ = δ/4 : we have

and it follows from localization and the fact that

), that the first term tends to 0 sub-exponentially fast. We can thus restrict the summation to the vertices in the strip [±N ≈ (N 4/7-2ǫ ) -1/4 (by using the 2-arm exponent)

Now we just have to sum this inequality to get the desired result : for N large enough, e⊆[±N 4/7+ǫ ] P(e ∈ F N ) ≤ 6(2N 4/7+ǫ + 1)(ℓ N + 1) × N -1/7+ǫ

≤ N 3/7+δ ℓ N Let us turn now to the lower bound. We restrict to the edges e in the strip [2N 4/7+ǫ , ℓ N -2N 4/7+ǫ ] × [±N 4/7-ǫ ], with ǫ = δ/6. For such edges, we would like to estimate the probability of having two arms, one occupied to B N and one vacant to T N , we will so use the event ∆ 2 rather than Γ 2 . Indeed, take j such that N 4/7-ǫ < 2 j ≤ 2N 4/7-ǫ : the probability of having two arms, one occupied to the bottom of ∂S(x e , 2 j ), and one vacant to the top, is at least P(∆ 2 (x e , S(x e , 2 j )))


Enrique Wulff Barreiro
email: enrique.wulff@icman.csic.es

Owners profile: the molecular origin of human cancer in history*

Keywords: Oncogenetics, Cancer, History of biochemistry, Co-citation analysis, Point mutation, Sentiment analysis

From 1977 to 1980, the search for genes capable of inducing genotype and phenotype alteration was successfully introduced by DNA-mediated gene transfer experiences. The next step was the purification of the first mammalian gene by genetic means, in 1980. And by 1981, the first cloning of a human oncogene took place. Indeed, when the epochal discovery was made, in 1982, that the functional activation of a human oncogene was caused by a single point mutation, the significance of the presence of Spanish scientists in this pioneering work acquired historical perspective. Evidence that the interaction of a carcinogen with a ras proto-oncogene creates the event that initiates the tumor formation was found to be a major achievement. And a group of Spanish researchers, around the biochemist Severo Ochoa, played a main role in this decoding of the initial human carcinogenesis processes. But in spite they solved the mystery of unlocking the truth on the molecular nature of human oncogenes, their eponymous vindication in relationship with the first steps of oncogenetics was not strong enough. To explore this symbolic reality, a sentiment analysis was made of the point or points where the top stories and events were co-referenced in the documents. Also these data provide an illustration of the emergence of oncogenetics taken on the scale supplied by co-citation analysis to understand the history of science.

Introduction

The elucidation of the nature of the oncogene, its real function that would be conserved through evolution, needed screening protocols to ferret out these cancer genes from tumors. If the induction of cancer through the transformation of cells by viruses dates back from the early sixties, like the discovery of the genes (ras genes) transforming rat sarcoma viruses, it was not until 1977 that the methods of gene transfer were successful by using total vertebrate and mammalian DNA [START_REF] Pellicer | Gene purification by transfection methods[END_REF]. Àngel Pellicer had achieved this historical transfer of the thymidine kinase (TK) gene from herpes virus into a cell line of mouse, and Mariano Barbacid figured out that a mutation of ras was responsible for the tumor. Thanks to the development of gene transfer technologies, particularly since the advent of recombinant DNA technology, could be fitted to describe the ups and downs in the pulse of oncogene and mutation, in particular when Barbacid and Pellicer were preceded in the 1989 Nobel Prize for Medicine or Physiology by other scientists from the US who also worked on oncogenes.

In this contribution a widely used technique, the genetic transformation of eukaryotic, is historically approached as a most central matter in the onset of oncogenetics. A strategy that is used to generate a reversal one which ultimately sought to make sense of the genes and proteins responsible for cancerous transformation and yielding more effective therapies. Relying on which Pellicer's work with transfection was communicated to Manuel Perucho and Barbacid in summer 1978, and they realized that animal genes could be transferred from cell to cell and expressed with ease. This Spanish people in the early eighties oncogenes wars -from the developer of the main technique (Pellicer) to the couple of workers (Perucho and Barbacid) who for the first time identified the genetic responsibility for the transformation and cloned the human transforming gene -were some of the kindest and most caring people in the biochemical community in the US. To disparage their efforts those who have followed the 1989 Nobel track involved in heated discussions their significant accomplishments. For the task of the historian of science and technology willing to measure this doubtless influential tradition in incipient oncogenetics, taking the grand sweep in statistical historiography is good both in this particular science and for this particular country.

The Spanish team with which Barbacid raised the question that caused him to establish the molecular base of human neoplasia comprised E. Santos, D. Martín Zanca, and V. Notario. The group of spanish researchers that worked with A. Pellicer, involved in this decoding of the human carcinogenesis processes, was composed by I. Guerrero, V. Corces, P. Calzada and A. Villasante. M. Perucho engaged in a group which included as Spanish collaborators C. Almoguera, C. Lama and J. Jordano, focused on the characterization of human and murine cellular oncogenes and provided a method to detect mutations in the genes transcripts.

In 1989, the Nobel Prize in Physiology or Medicine was awarded to the discovery that oncogenes in animal tumor viruses are derived from cell genes (proto-oncogenes) [START_REF] Newmark | Nobel for oncogenes[END_REF]. Ultimately, by the fact that normal cells contain genes that may cause cancer if they are altered [START_REF] Garfield | The 1989 Nobel Prize in Physiology or Medicine is awarded to J. Michael Bishop and Harold E. Varmus for their contribution to cancer research[END_REF]. The Nobel Committee credited the article [START_REF] Stehelin | Oncogènes: de modèles simples vers des modèles plus sophistiqués[END_REF] published in Nature, where the prize winners reported on the presence of DNA closely linked to the src oncogene of the avian sarcoma virus as part of the genetic material of the chicken cells [START_REF] Newmark | Nobel for oncogenes[END_REF]. In one of his rare references to this work, M. Barbacid suggested that this intelligence of the cellular origin of some viral oncogenes, warned scientists about the possibility that there were also oncogenes associated to neoplasms induced by non viral agents [START_REF] Barbacid | ras oncogenes: their role in neoplasia[END_REF]. M. [START_REF] Barbacid | ras oncogenes: their role in neoplasia[END_REF]A. Pellicer, in 1983 and1984, found the role played by the ras oncogene activation in human neoplasia and documented it in various carcinogen-induced animal tumour model systems. The first tested chemically induced ras oncogenes activation, through N-nitroso-N-methylurea, of mammary carcinoma in rats [SUKUMAR et al., 1983]. The second obtained a similar result by gamma radiation in brain and thyme lymphoma, also in rats [GUERRERO et al., 1984]. The two used ras oncogene activated by carcinogens with the same mutations than those present in human tumours. And Barbacid received in 1988, the considered by the scientific community as the highest prize in cancer research, the Josef Steiner Foundation (University of Berne) Award for providing evidence that the interaction of a carcinogen with a rasproto-oncogene creates the event that initiates tumor formation. While Barbacid found a MNU-specific glycine to aspartic acid mutation at the H-ras gene in rat mammary tumors, Pellicer identified a cytosine to adenine transversion as the activating mutation in N-ras associated to hematologic malignancies. Both authors discovered mutations as responsible for the malignant activation at the exonic sequence. The modest fact that a single mutation could induce the development of cancer, the fact that ras oncogenes were not a 'laboratory artefact', took the scientific community by surprise, and led oncogenetists to a heated discussion on its sufficiency or necessity for cancer.

The personal perspective of a Nobel Prize laureate ensures the closest attention about a small segment of a very big chase. Severo Ochoa's description, specially in the United States, as the biochemist of the biochemists and the ideal of what the true scientist should be [SRINIVASAN et al., 1979], highlights his heritage and identity as a key decoder of the genetic code, because this project turned out to depend critically on biochemical tools such as his enzyme (polynucleotide phosphorylase, with which he produced ribonucleic acids and obtained 1959 Nobel Prize in Physiology or Medicine). Working in the New York University School of Medicine until 1974, he trained specifically to Spain bounded biochemists, in a sufficient number indicative of a scientific school. S.Ochoa presented some of the contributions by M. Barbacid, and M. Perucho to the National Academy of Sciences. And at the same time he knew their doctoral dissertations advisors D. Vázquez, M.L. Salas and J. Salas. The act of working with a Nobel Prize has brought about, along the history of the award, a greater amount of prizes and a largest number of young people between those who received it [START_REF] Zuckermann | Nobel laureates in science : patterns of productivity, collaboration, and authorship[END_REF]. The US government owns the royalties and license uses of these three Spanish scientists. But the 1989 Nobel Prize winners were a second literature that found its way to the award by making use, in terms of scientific policy, of practices own to the management of transfer of scientific results.

So the pioneering nature of these Spanish researchers work, in a highly competitive research race, was exposed to the science stratification that existed in the US at the beginning of the eighties, to the fragmentation awaiting any innovative idea, when the conditions for its eponymic vindication are not enough strong. And the guarantee that was involved with S. Ochoa presenting the works was not effective. After the 1989 Nobel award, the authority of the scientist M. Barbacid fell upon the laureates (since the attribution of the Prize both have been successively directors of the National Institutes of Health). In the US, the mention to Mariano Barbacid acquired the unfair nature of a memorial practice; referred by the official speeches and festivities [START_REF] Marx | Oncogenes reach a milestone[END_REF]. The awarded research program imported the discoveries of the program that never climbed the podium, and sets the diffusion patterns for the scientific activity of M. Barbacid. The 89 Nobel winners anticipated the Spanish scientists.

Methodology

A question on which is the universe of resources that we have taken to portray the results of earlier studies in oncogenetics, is convened by the gauged ensemble of publications. Right in this sense, there is any publications taken to get answers to the scientific community on the emergence of oncogenetics from virus work, the study of eukaryotic transfection, and the somatic mutation theory of cancer (see appendix). It can be thought along these lines that, those scientists who participated in the oncogenes races of the earlier eighties underlined their visibility and independence, so striving to unlock their contributions to the best point of acknowledgement. There are definitely many things that can be done with these regular instances research colleagues express their influences towards each other. So that the methods employed to screen for data to draw models to illustrate the ideas about oncogenes could be useful in solving the problems necessary to justify this shifting, and developing area in its early times.

Typically there are two types of assets that can be recognized as sensible when following the development of a scientific specialty. One type is focused on catching the intellectual problems and processes on which a kind of cross-over indicators can be built. And the other one is based in attention as recorded by citations, basically figuring out whether a scientist has more citations than normally expected for a given situation. And the reason why it is interesting is that certain scientists results have been in use more than others. There is here a large correlation to size, so your different teams get more citations than some more outside investigators out there. And the way the possibilities of almost all these tools can be applied, very much comply with your imagination really.

We look at enhancing, can you say, reversal strategies based on time distribution of ideas as usual, we have looked at the concept of oncogene amplification during tumorigenesis, we have looked at how fundamental effects are between top events. There are a lot of different things that can be done here. By using co-reference analysis that considers the family matrix, timing and relevance of the couple of concepts considered as a pair, top-events defining the relationship between the co-referenced pair of ideas and, portraying the link to the underlying top story to summarize the relationship.

In terms of earnings per author in scientific projects in their community, it would be definitely interesting to see how a scientist is portrayed into the scientific papers for a series of years and to see how that modify how the scientific community reacts when the actual learning is announced. Co-citation activity plot, looking at analyst reading moving shifts in intellectual focus, is particularly adequate in this sense. You may be able to use news in some way to position yourself before intellectual changes occurred. Showing how a scientific interest grows, a model runs into trouble or important predictions are disconfirmed by experiment. So in the order of coming to identify when there is a likely scientific event rightly coming out, you may look at what kind of patterns of contents tends to trigger any action from investigators and, then you may be able to position yourself before hand. The positive substantial skew with which co-citations really correlates with historical processes, is definitely true when you look at the question of time. Here the relationship is about to reach the most bottom of the 1973 new transfection protocol by Graham and van der Eb's calculations and it scrutinize the scientific proportions of 1981 and 1982 keystones. By looking into the macro-space we have piloted a methodology to relate a global index, like the co-reference indicator, with a more traditional input-output connectivity, like co-citation analysis.

Intellectual problems and processes

We take a technical approach to very fundamental research problems. However, research on ras genes is justified because they play a fundamental role in basic cellular functions due to their phylogenetic origin, in the early eighties the focus was on their role in neoplastic development. Thus we look for cross-overs between the first time a human-tumor-derived cell transformed mouse cells and, the surprise that single point mutations were behind oncogenic ability to induce tumors. The purpose here is not simply to show the chronology within which these discoveries occurred but rather to indicate the pattern in which they occurred, since that pattern can indicate (roughly) the several stages to which the intellectual problem has developed. These phases correspond roughly to the processes where the oncogenes had been detected in human and other animal tumors through transfection, and the point mutation work at the twelfth codon presumably affecting the ability of the protein to contribute to human tumors.

In each of the stages (see Figures 1 &2), a cross-over from below and above translates into a bullish view, and a cross-over from above and below translates into a bearish view. A cross-over is considered as driven by co-references, in particular, the co-reference indicator was created as a simple rate of journals to time, applying a span of time sufficient to satisfy the given purpose of the paper examined. A simple strategy is considered in which we are in the knowledge space of the working literature all the time. High density of useful material is entered like signals to generate a significant history. It must be emphasized that for some of these given intellectual problems, positions were overlapped and we have hold them. In terms of this analytical scheme a fixed time horizon is applied and all grouped papers are created equally.

The tide of excitement swept up in the currents of history, in the competitive races for the oncogene of the early eighties. The characterization of human and murine cellular oncogenes was a race, and the search for the function of ras was definitely a race.

Intellectual Problem: Study of the isolation of a human oncogene constituted Manuel Perucho (a Cold Spring Harbor scientist) central intellectual problem until the autumn of 1982. The response of biochemists to the fact that cellular genes (proto-oncogenes) may become activated by somatic mutations that mimic the changes imposed upon these loci during endogenous retroviral transduction, was to show in the transfection system that methylated and unmethylated DNAs were equally well accepted by the recipient cell. By that it meant the fortunate rediscovery of ras genes and, in the summer of 1981, gene transfer assays revealed for the first time that a defined genetic element was responsible for the transformation of NIH-3T3 cells. Based on this, after looking at Figure 1, it turns out that the work performed was science as good as what was going on at Bethesda, where Barbacid had created a new Laboratory of Cellular and Molecular Biology at National Cancer Institute (NCI). Both teams were in a race now; they wanted to clone the bladder oncogene.

Intellectual Problem: Since Mariano Barbacid had branched out from virology, he was backed by the right immunological approach to generate transforming viruses, from several mammalian species, which transformed activities were stably associated with oncogenes picked up by these retroviruses from their genomes. This homology between ras genes of rat sarcoma viruses and activated oncogenes contained in human tumors showed for the first time the bridge between the pioneering work carried out with tumor viruses during the 1960s and 1970s and the realms of those working on human cancer [START_REF] Morange | From the regulatory vision of cancer to the oncogene paradigm, 1975-1985[END_REF]. We will return to the nature of this bridge later on, across the standard rodent models provided by Àngel Pellicer (Figure 2). With the evidence that the ras oncogene was some sort of a common denominator for cancer, scientists had to determine how in the evolution of cancer the ras gene degenerated from a healthy, vital cellular gene into a miserable oncogene where the mutation is dominant. Taking the scientific community by surprise a single ras gene proved able to induce significant transformations in NIH-3T3 able to confer on them the ability to induce tumors in nude mutations. Mammalian ras genes acquired their properties to induce malignancy by single point mutations in codons 12 [SULLIVAN et al., 1983].

To get a better understanding of the nature of the relationships between these two intellectual problems, we consider them both when the target was the first three ras genes identified in the mammalian genome as well as when the related single point mutations were targeted. Both problems are required to have a high related mention in the history of oncogenetics. To see if we are actually able to identify any impact on the course of events, we turn to the plotting of the co-references against their return relationships between July 1978 and October 1986 (Figure 1). With 21 top stories and 167 co-references, and a Hit Ratio of 11, on average the strategy generates 6.2 sized-groups. In processing this information Perucho is hypothesized the key driver of the race (he stands out for 45.5% of the return on event; Barbacid (40.7%), Pellicer (13.7%)).

To understand the relationships in greater detail we can either consult the top events or either we consider the cross-overs between the two programs in the co-references staging. Comparing the co-references spikes, looking at the first top story back in December 1981, we find that this first spike claim the identification of the first member of the ras gene family. Otherwise we do observe that the first common assay for the detection of mutant ras alleles (based on the ability of these alleles to transform the mouse NIH/3T3 cell line, which meant the rediscovery of ras genes), uttered in scientific circles in December 1981, had a common impact in the co-reference scenario from the point of view of both programs. On the existence of transforming genes in human tumors, its clonage as an oncogene (H-ras-1) was carried out by the same team and published in April 1982. A couple of months later, in light of this same T24 bladder oncogene (homologous with Harvey sarcoma virus ras gene (H-ras)) discovery, the result has come out of that row from the retrovirologists side. This positive performance drives a particularly visible cross-over between equally faster indicators from both approaches. The bearish trend recorded in the characterization of oncogenes research front cross the bullish trend faced by the oncogenes and the genetics of human cancer research front. We take a look again, to highlight the fundamental homology between the T24 human bladder oncogene and the ras transforming genes coming from the BALB-and Harvey-murine sarcoma viruses. The event is made by correlating the biochemical program with the oncogenetics.

Within mutation research getting access to using data from the high-expressing human bladder carcinoma cells to support the point mutation case, at the which cloning things were quite down for the second time, was a result that can be clearly appreciated by observing the second cross-over. Then things begin to hit up again with the identification in human lung and pancreas human carcinomas of a common oncogene which shares sequences with the onc gene of the Kirsten strain of murine sarcoma virus. Also it was obtained that about 15% of the human tumors contain activated oncogenes (December 1982). And with the human neuroblastoma cell line transforming gene identified and found to be designated N-ras (January 1983). To support the case a few months later N-ras gene was cloned and sequenced (April 1983), which was the second decision in favor of characterization's group, followed by a falling-out, an indicative intent for appealing results from the oncogenetical program searched to consolidate Severo Ochoa decisive recognition (August 1983). A fifth cross-over has been added in the framework of the identification of oncogenes in a NMU induced point mutation [START_REF] Sukumar | Induction of mammary carcinomas in rats by nitrosomethylurea involves malignant activation of H-ras-1 locus by single point mutations[END_REF]).

An actual attraction followed along a month later, until both programs claimed for the January 1984 result, by the NYU pathologist Àngel Pellicer, relative to the identity of activated oncogenes in a carcinogen-induced animal tumor model.

Then things went up again this time holding a position, along this same month of January 1984, by a second hit currently reassuring the scientific community after the identification of a mutated kras oncogene in a tumor biopsy from a lung cancer patient. Two quick mentioned events refer the synthesis of large quantities of p21 in E. coli (Sep. 1984, 01) and Ngene identified in animal models (Sep. 1984, 07). There is a noise" in a particular event for September 14, 1984, novelty provides information about another cross-over. The success of the NYU pathologist rodent models appears again, as the evidence that gamma rays could be the initiating carcinogen able to activate c-K-ras oncogenes through a somatic mutation and to induce lymphomas in mice. Two months later the activation of human c-K-ras makes again correlate both research programs [START_REF] Yamamoto | Activation of a human c-K-ras oncogene[END_REF].

May 1985 shows NMU activates H-ras-1 and October that gene amplification is an alternative pathway for oncogene activation. And finally by November 1985, Ochoa was able to settle the case by his support to a clinically oriented point mutation detection method, then expression of the positive association between ras genes and mitogenic factors. This event tailors specifically the next cross-over. Little before that ras was found activated by mitogenic factors with loss of the N-ras allele in a mouse thymic lymphoma (Dec. 85). Then amplification, a consistent sign of a more dangerous type of malignant transformation, was proved to be a reversible process that depends on the relative abundance of oncogene sequences, as can be observed across July 1986. Chromatin promoter region, in c-K-ras, was described by the characterization program (Sept. 86). At last, it is time to appreciate the building of a new research area, as indicated by the ras oncogene mimic neuronal differentiation event (Oct. 1986).

Over the monthly periods considered the number of co-references stories between the two research fronts was ranging between 75 and 87 respectively, consulting the top stories we were able to summarize the relationships in only fifteen cutting lines. From the quantitative point of view incorporating co-referencing allows adequate products picking up, increasing cross-axis correlations, and even the research direction. The top event must provide useful information to understand the relationships and whether to expect positive or negative spill-over effects. The case symbol top events detected over time from the respected research programs included anything from methodological issues, to basic discoveries and to actual, veridic and follow-up acknowledged results; and finally a settlement by Ochoa. Hence, in this case it would have been possible to successfully having model the characterization of cellular oncogenes versus human cancer & viral transforming genes programs as a part of an automated analysis process by simply using the top events together with spikes and using co-referencing.

Let's try to look at another simpler example. In this case we look at news co-referencing between the murine lymphoma system to study the mutated oncogenes and carcinogenic agents (molecular oncology animal models developed by Pellicer) and the activation of distinct oncogenic pathways in the induction of mammary carcinomas by single point mutations achieved by Barbacid. As can be observed that has proved to be a lot of action over time, in fact 25 top stories and 376 coreferences between 1978 and 1986, but just three cross-overs. Hit Ratio of 18, on average, the strategy generates 15,32references-sized groups (Figure 2). We hypothesize that Barbacid did lead the investigations (he gave recognition to 47% of the events; Perucho (32%), Pellicer (21%)).

From this Figure 2 information around scientifically linked programs correlations delivers an impressive combination of size and performance, detecting a contagion effect extended to all the events across the temporal chain. All pairs contribute positively to performance. Also, visible at a glance, is that the steep diagonals soars steadily upwards and downwards to the same scientific keystones, specially since September, 1984. And it is probably fair to state that these two researchers would be considered related at least from the summer of 1978.The first time the two programs were co-referenced was when Pellicer shared with Barbacid a landmark experiment on the detection of single copy genes after transfection. Then first release came out that an activated ras gene was found in a human tumor in December 1981. Watching out for the first ras gene cloned (H-ras) allows us to check the T24 human bladder carcinoma oncogene April 1982 result. To understand the relationship between oncogene cloning and identification, we observe the agreement of co-referencing for May-December 1982, including the point mutation responsibility in activation finding by May 1982. There is quite a lot ranging between H-ras molecular cloning from human bladder cells to the thrilling original identification of the human trk oncogene. This last contribution was published in December 1982, and for many years it was the sole example of a TK (tyrosine kinase) oncogene activated in a solid human tumor, been the target of the first cross-over, the characterization of the transforming gene of neuroblastoma cells (N-ras). Ochoa communicated the spontaneous acquisition of transforming properties by a human proto-oncogene, mediated by a single point mutation, and occurred at the same nucleotide. This approach was published in August 1983.

On the date when the scientific community was astonished by the fact that a single injection of a carcinogenic agent conferred its extraordinary power to the oncogenic ras gene (December 1983), a powerful model in the study of leukemogenesis takes all its relevancy, pushing down the co-references by January 1984. In this last case, the event showed a clear increase of the transformed cells following K-ras gene activation induced by radiation. So the early stages of carcinogenesis study in in vivo systems, was associated to oncogenes identified in animal models. Also in January 1984, a comparative analysis of the mutation responsible for the activation of the human c-K-ras oncogene in two human lung tumors indicated that point mutations in different tumors are not identical. In February 1984, an activated oncogene found in a cancer patient improved on its own the supply of evidence to reassure the scientific community that ras oncogenes were not a 'laboratory artefact'. After that, we do see an event communicated by Ochoa to the National Academy of Sciences of the US. Top-story was driven by the achievement of the synthesis of large quantities of both normal and transforming p21 proteins. And it resulted from the need of significant amounts of purified p21 proteins to unveil the biochemical role that ras gene products play in normal cellular growth and differentiation, as well as in malignant transformation (September 1, 1984). Looking into the next data we arrive to a common recognition of the N-ras genes (from a thymic lymphoma chemically induced) cloning and sequencing event (September 7, 1984). These two last results are identified with the best scores [START_REF] Yamamoto | Activation of a human c-K-ras oncogene[END_REF]. From there the more close the relationship between both programs, the more time analysis is driven by the same information across the competitive landscape. In fact, for September 14, 1984, we get a signal looking at gamma rays as activators of c-K-ras oncogenes by somatic mutations. And as an expression of the relative common importance of the reciprocal usages of both programs, the following result refers to the activation of a human c-Kras oncogene.

Let's take a look at the next examples. Primarily for May 1985, concerning direct proof for the generation of tumors in mice involving genes, a series of experiments revealed that 85 percent of lab rats injected with a mutagenic chemical contracted mammary carcinoma. Whenever we travel from spring to winter 1985, we are presented with October 1985 mechanism, corresponding to the evidence of the malignant transformation and ras gene amplification relationship, that increased amount of normal proto-oncogene proteins might alter the basic regulatory controls of cell proliferation. A detecting point mutation method was revealed by November 1985. And December 1985, it was time to observe the loss of the normal allele in mouse thymic tumors that contained oncogenic versions of ras genes. We do appreciate the relation between the oncogenetics program and research front code 84-4046, an edge supported by quantitative as well as qualitative alterations in mammalian ras genes concerning the allele loss oncogenic dominance. Besides increasing the number of signals moving beyond strict cross-over strategy, the result showed promise, previously unformulated strategies based on the suggestion that the expression of a minimum level of mutant ras gene products may be critical in tumorigenesis. What increased the likelihood of reversibility of the malignant transformation caused by mutant human ras oncogenes (July 1986). Looking at the performing graphics (Figure 2), we get an information issue as mentioned previously.

Finally, after considering the last cross-over around the chromatin structure of the promoter region (once promoter mutation theory offered an alternative to point mutation), we turn out to the other side of the equation, cellular differentiation, because ras genes can have other effects besides the induction of uncontrolled proliferation, activating the pathways of differentiation. In October 1986, the human cancer research front clearly perceived as a substantial experiment the study on the effects of ras genes on the cells in the neuroendocrine tumor of the medulla of the adrenal glands. These cells can differentiate into neuron-like cells if infected with ras-containing retroviruses or microinjected with oncogenic p21 ras proteins. So demonstrating that the introduction of a ras oncogene was able to arrest growth and mimic the steps of neuronal differentiation. This induction of terminal differentiation in neural cells implied ras oncogenes in a signal transduction pathway commonly used by NGF, a strong result with consequences on the research of the receptors for nerve growth-factor and other neurotrophins.

As we have seen using co-reference data makes easy to get notify of new or reciprocal relationships and, could be available input for a case history management process as to this a more timely, measured and correlation tool.

Emergence of oncogenetics

A modification of the ras protein itself, due to a point mutation in the coding sequence of the gene, was placed by the authors of its discovery in the general framework of cancer studies [START_REF] Morange | From the regulatory vision of cancer to the oncogene paradigm, 1975-1985[END_REF]. The outstanding accomplishment (of 1982) was the introduction of oncogenetics as a new period within the history of human genetics, where important intellectual changes occurred on a scale much smaller than a year's span [START_REF] Sullivan | Understanding rapid theoretical change in particle physics: a monthby-month co-citation analysis[END_REF]. Co-citation studies have been proved to be a powerful tool sensitive to very rapid shifts in intellectual focus, having used it to ran into this area, we now provide a sequence of the events (Figure 3).

The period from 1973 to 1982 is ideal because it codifies the first set of successful assays in the domain of oncogenicity. Two Dutch scientists, Frank Graham and Alex Van der Eb developed a new technique to introduce foreign DNA into mammalian cells as a means of studying gene functions [START_REF] Malumbres | RAS oncogenes: the first 30 years[END_REF]. The Dutch scientists had transfected DNA into cultured human kidney cells. First they diluted DNA of low complexity (an adenovirus that causes respiratory diseases in humans) in isotonic saline containing phosphate at low concentration, and after adding calcium chloride, the formation of a calcium phosphate precipìtate resulted. Once the investigators put these crystals onto cultured human kidney cells, they incubated the cells for a couple of weeks. Clearly, after that, the foreign genes integrated into the chromosomes of the kidney cells [START_REF] Angier | Natural obsessions : striving to unlock the deepest secrets of the cancer cell[END_REF]. Circumventing the dependence on pathogens calcium phosphate had permitted transfection of foreign genes into the host DNA of cultured cells. Put another way, transfection has had the same effect on cultured cells as infection.

However, until 1977, the reference to transfection must be restricted as the methods of gene transfer were successful only when this DNA of low complexity was used [START_REF] Pellicer | Gene purification by transfection methods[END_REF]. Figure 3 shows the status of Graham and Van Eb technique, which we can distinguish in the left upper region, with a commonality of usage (estimated by number of connections) similar to the main discovery in the field, the single point mutation mechanism of oncogenic activation, represented in the right corner (Reddy). The basic concern of this research, beyond understanding the function of ras, was to understand the genetics of human cancer. This more purely genetic concern and one which is more a user of recombinant -DNA technology than a source of that technology, is represented in the right region. All the events and findings of the past in this area, co-exist at a single time in the past, have been placed together at the same moment, in 1982.

Turning first to the left region, mostly a blow-up of the 1981 picture (Figure 3), which we may roughly identify with the characterization of human and murine cellular oncogenes research front, this crucial instrumentality is represented by the hill labelled "Southern E, 1975". A simple method, in that it works with diced DNA transfered to lots of dry paper, Southern blot, the electrophoretic transfer of proteins from polyacrylamide gels to nitrocellular sheets, said also the frontiers within which emerged in time the new research specialty of oncogenetics. This technique was ubiquitous in all the biological laboratories in the context of forming DNA:RNA hybrids and, its need investigated the possibilities of development of genetic engineering. In 1975 Edward Southern, a Scottish biologist, published the recombinant DNA work that he had deviced and with which the biochemists involved in the early eighties oncogene races spend many hours preparing various types of blots.

The middle and rather central stage of this region is occupied by Perucho's key paper. The practical concern here is the newly resurrected ras gene model, that determined for the first time the detection of mutant ras alleles (elements called H-, K-and N-ras, implicated in a wide range of human tumours). The most initial assay it was based on the ability of these alleles to transform the mouse NIH/3T3 cell line. In the chronicle of oncogenetics, evidence was offered of a first sign of transformation efficiency using DNAs from different human-tumor-derived cell lines. The cell lines analyzed which gave rise to foci containing the transfected oncogenes were SK-CO-1 in the case of colon carcinoma, and Calu-1 and SK-LU-1 in lung carcinoma; on the charge of DNA derived from a neuroblastoma SK-N-SH line, another member of the ras genes family was identified, later characterized as the N-ras.

In Figure 3 we see that it was Mike Murray, getting a larger link with the central node, who supplied evidence for the same colon carcinoma oncogene (this time using SW480 cell line; also he distinguished a transforming gene associated with a leukemic cell line, HL60). He made a difference, but it was certainly not a stellar accomplishment. On the left, the triangularity shaped group of concepts concerns also this accurate measure of the transforming potential of the altered ras genes. And, in fact, it treated all the similar results been reported previously, either by Chiaho Shih or by Theodore G. Krontiris and Geoffrey Cooper. Together with Pulciani (in the cluster of Barbacid, but on the right inmediate link with this region), these scientists were the inadvertent rediscoverers of ras genes when, using gene transfer assays, they established the existence of dominant oncogenes in humans. Shih's paper in Nature analyzed DNA isolated from mouse fibroblasts transformed by human tumor DNA, and Krontiris and Cooper also had transfected DNA, from human bladder carcinoma cells (although they were published in PNAS, which has a far less circulation (and reputation) than Nature). Shih's initial transfection success had been met with scepticism, hence, in effect, the caveat was solved when he showed transforming activity in NIH-3T3 cells (a mouse line created by G.J. Todaro) infected with DNA from humar tumour cell lines. The remaining concept in this region stand for stage-specific transforming genes. Once Cooper leave Weinberg's labo (easy come, easy go), he identified stage-specific transforming genes in MCF-4 mammary carcinoma cells (tx-4). As expressed by the work of Lane, at the left end of the region. It turned out that these genes were specific for the differentiating stage of the tumor cell in which they were identified. These observations were driven from both human and mouse tumors; as such, for instance, tx-4 is present in both human MCF-7 mammary carcinoma cells and in several rodent mammary carcinomas.

From this fairly coherent region a bridge connects to the right side via Channing Der, a postdoctoral fellow in Geoffrey Cooper's laboratory. The ras connection arouse the interest of biologists through his experiments. He used retroviral oncogene probes to hybridize DNA isolated from NIH-3T3 cells that had been transformed with various human tumour DNAs. But, above all, he showed that transfected oncogene were identical to the retrovirally encoded oncogenes. The most straightforward interpretation of his findings was that retroviral oncogenes were derived from cellular genes [START_REF] Vecchio | Oncogenes of DNA and RNA tumor viruses and the origin of cellular oncogenes[END_REF]. The special properties of both Harvey (Ha-MSV) and Kirsten (Ki-MSV) murine sarcoma viruses, according to which they were actually triple recombinants between the replication competent (helper) retrovirus that conferred on them infectious properties, rat cellular sequences of ancestral retroviral origin known as 30S RNA and rat cellular genes (H-ras in Ha-MSV and K-ras in Ki-MSV) [START_REF] Malumbres | RAS oncogenes: the first 30 years[END_REF], were exploited in the 1981 transduction work associated with Ellis in the right part. So 1982 Der's positive answer to the question of whether or not human oncogenes were related to the cellular homologues of the retroviral oncogenes was based on the description of the molecular clones of viral DNAs previous 1981 description by Ellis.

But the linkage between the regions is one of historical as well as operational dependence. Investigators to establish the functional side of the oncogene, started from the spring of 1982 results, when Goldfarb (he had been a previous member of Weinberg's team), Pulciani and Shih reported on the molecular cloning of a human transforming gene from the T24 and bladder carcinoma cell lines. This substructure, that do group in a rather diffuse triangle at the middle of the Figure 3 center, evolved into the trapezoidal map, at the right of the right region, associated with the ability of a single point mutation to lead to the carcinoma. As a result that led to a heated debate in the oncogene community [START_REF] Duesberg | Activated proto-onc genes : sufficient or necessary for cancer?[END_REF].

But it was not until the autumn of 1982 that the nucleotide sequences of the Harvey-and Kirsten-sarcoma viruses oncogenes (isolated from NIH-3T3 cells) were published. That was in fact the nature of the bridge between regions left and right via Der (Figure 3). A fairly coherent sub-group to the right of center illustrates the relationship between three indepent reports (Der, Parada and Santos) having found similar results. All these three were able to generate an information relative to cells transformed with DNA isolated from EJ bladder and lung carcinoma cells having additional sequences that hybridized to probes specific for the v-h-ras and the v-k-ras oncogenes, respectively. As a consequence of these results, taken together, and before getting into the last section of this graph, the historical data available suggested some questions. Was it possible that nobody thought to hybridize a blot containing DNA from transfected NIH-3T3 cells with a virus ras probe until those days? Another question that comes is if perhaps researchers wanted human cancer to be distinct from virus-induced tumors of rodent? That nobody thought that those oncogenes transduced by retroviruses could be mutated in human cancer prepared the surprise. A single point mutation was responsible for the malignant activation of ras oncogenes. As it was established that p21 is produced constitutively in all human tissue, revealing an important role of ras genes in normal cell growth. And ras protein is known to need guanine triphosphate, or GTP, to be active, being the role of p21 to transduce molecular signals to the cell nucleus. The simplicity of a single amino acid change in the p21 protein product was too much to take. As a primary event point mutations must occur all the time, every day in every cell, but people do not develop cancer until old age. It was not clear how a single mutation in a membrane-based protein made GTP enzymatic activity so greatly reduced, made ras do in humans enough to provoke the biochemical modifications occuring in cancer cells. Mutations are a major activating mechanism for the ras family genes, mainly in codons 12, 13 and 61, resulting in their conversion from protooncogenes to activated oncogenes. As shown vertically, in a roughly linear fashion, two results associated to the mechanism of activation to the right region. This connection is concerned with that human transforming genes were mutant ras genes, and was announced in winter of 1982.

At the right corner on our Figure 3 (Tabin, Reddy), the limited constraint with which ras genes acquire transforming properties, is in contradiction to the prediction that located the alteration in the gene promoter. But the frantic pace of research made a revelation in response to the question on having previously thought that those oncogenes that were transduced by retroviruses could be mutated in human cancer.

Conclusions

The frequent detection of ras-related oncogenes activated by single point mutations is a genetic change detected in about 15% of all human tumors. But there are people thinking they were looked for by seeking a cure for cancer in our time. In the search for basic cancer therapies the biochemical and biological properties of ras proteins shows that oncogenic mutants are the target of immunotherapies searching to induce immune responses against them in patients. There is a way for the history and pathogenesis of cancer to benefit from basic research on oncogenes in terms of prevention, diagnosis and therapy of human cancer.

In order to figure out whether a treatable strategy was available, one having had to precisely timing the connection between human oncogenes, its mechanism of activation, and the ras proteins, watching out for those unexpected events surrounding the identification and subsequent characterization of oncogenes in human tumors, as well as the discovery of the point mutation, helps in mapping factors whose regulatory activity converges. Hence, it is possible to find powerful moves validating the pioneering work carried out with tumor viruses during the 1960s and 1970s. Members of the Ras superfamily of small GTP-binding proteins have been implicated in the regulation of diverse biological functions, including tumor progression. Activating mutations in Ras genes are commonly found in a variety of human tumors. As well as a molecular diagnosis to identify those oncogenes that exert their transforming properties by mechanisms involving gene amplification was put at disposal by Ochoa in 1985. The thirty-three anniversary of the identification of the first human oncogene, makes the area of research into the three closely related proteins, H-, K-, and N-RAS more accountable to virology than ever before.

Figure 1 .

 1 Figure 1. Plotting the Co-References between the between the spring of 1982 results on the existence of transforming genes in human tumor cells.

Figure 2 .

 2 Figure 2. Plotting the Co-References between the ras oncogenes and carcinogen-induced animal tumor models.

Figure 3 .

 3 Figure 3. Oncogenetics Co-Citation activity plot for 1973-1982.

Appendix

These are the 25 research articles selected for studying this case history. The specifications consist of: number of citations (until September 21, 2014); articles communicated by Severo Ochoa; together with the research fronts papers from which the co-references have been plotted.