Élien Aur

Deya

Ivan Nourdin
email: inourdin@gmail.com

Invariance principles for homogeneous sums of free random variables

Keywords: central limit theorems, chaos, free Brownian motion, free probability, homogeneous sums, Lindeberg principle, universality, Wigner chaos

We extend, in the free probability framework, an invariance principle for multilinear homogeneous sums with low influences recently established by Mossel, O'Donnel and Oleszkiewicz in [Ann. of Math. (2)

Introduction and background

Motivation and main goal. Our starting point is the following weak version (which is enough for our purpose) of an invariance principle for multilinear homogeneous sums with low influences, recently established in [START_REF] Mossel | Noise stability of functions with low influences: Invariance and optimality[END_REF].

Theorem 1.1 (Mossel-O'Donnel-Oleszkiewicz). Let (Ω, F , P ) be a probability space (in the classical sense). Let X 1 , X 2 , . . . (resp., Y 1 , Y 2 , . . .) be a sequence of independent centered random variables with unit variance satisfying moreover

sup i≥1 E[|X i | r ] < ∞ resp., sup i≥1 E[|Y i | r ] < ∞ for all r ≥ 1.
Fix d ≥ 1, and consider a sequence of functions f N : {1, . . . , N } d → R satisfying the following two assumptions for each N and each i 1 , . . . , i d = 1, . . . , N :

(i) (full symmetry) f N (i 1 , . . . , i d ) = f N (i σ(1) , . . . , i σ(d) ) for all σ ∈ S d ;

(ii) (normalization) d! N j1,...,j d =1 f N (j 1 , . . . , j d ) 2 = 1. Also, set

Q N (x 1 , . . . , x N ) = N i1,...,i d =1 f N (i 1 , . . . , i d )x i1 • • • x i d (1)
and

Inf i (f N ) = N j2,...,j d =1
f N (i, j 2 , . . . , j d ) 2 , i = 1, . . . , N.

Then, for any integer m ≥ 1,

E[Q N (X 1 , . . . , X N ) m ] -E[Q N (Y 1 , . . . , Y N ) m ] = O(τ 1/2 N ), ( 2
)
where τ N = max 1≤i≤N Inf i (f N ).

In [START_REF] Mossel | Noise stability of functions with low influences: Invariance and optimality[END_REF], the authors were motivated by solving two conjectures, namely the Majority Is Stablest conjecture from theoretical computer science and the It Ain't Over Till It's Over conjecture from social choice theory. It is worthwhile noting that there is another striking consequence of Theorem 1.1, more in the spirit of the classical central limit theorem. Indeed, in article [START_REF] Nourdin | Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos[END_REF] Nourdin, Peccati and Reinert combined Theorem 1.1 with the celebrated Fourth Moment theorem of Nualart and Peccati [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF], and deduced that multilinear homogenous sums of general centered independent random variables with unit variance enjoy the following universality phenomenon. Theorem 1.2 (Nourdin-Peccati-Reinert). Let (Ω, F , P ) be a probability space (in the classical sense). Let G 1 , G 2 , . . . be a sequence of i.i.d. N (0, 1) random variables. Fix d ≥ 2 and consider a sequence of functions f N : {1, . . . , N } d → R satisfying the following three assumptions for each N and each i 1 , . . . , i d = 1, . . . , N :

(i) (full symmetry) f N (i 1 , . . . , i d ) = f N (i σ(1) , . . . , i σ(d) ) for all σ ∈ S d ;

(ii) (vanishing on diagonals) f N (i 1 , . . . , i d ) = 0 if i k = i l for some k = l; (iii) (normalization) d! N j1,...,j d =1 f N (j 1 , . . . , j d ) 2 = 1. Also, let Q N (x 1 , . . . , x N ) be given by [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF]. Then, the following two conclusions are equivalent as N → ∞:

(A) Q N (G 1 , . . . , G N ) law → N (0, 1); (B) Q N (X 1 , . . . , X N ) law
→ N (0, 1) for any sequence X 1 , X 2 , . . . of i.i.d. centered random variables with unit variance and all moments.

In the present paper, our goal is twofold. We shall first extend Theorem 1.1 in the context of free probability and we shall then investigate whether a result such as Theorem 1.2 continues to hold true in this framework. We are motivated by the fact that there is often a close correspondence between classical probability and free probability, in which the Gaussian law (resp., the classical notion of independence) has the semicircular law (resp., the notion of free independence) as an analogue.

Free probability in a nutshell. Before going into the details and for the sake of clarity, let us first introduce some of the central concepts in the theory of free probability. (See [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF] for a systematic presentation.)

A non-commutative probability space is a von Neumann algebra A (i.e., an algebra of operators on a real separable Hilbert space, closed under adjoint and convergence in the weak operator topology) equipped with a trace ϕ, that is, a unital linear functional (meaning preserving the identity) which is weakly continuous, positive (meaning ϕ(X) ≥ 0 whenever X is a non-negative element of A; i.e., whenever X = Y Y * for some Y ∈ A), faithful (meaning that if ϕ(Y Y * ) = 0 then Y = 0), and tracial (meaning that ϕ(XY ) = ϕ(Y X) for all X, Y ∈ A, even though in general XY = Y X).

In a non-commutative probability space, we refer to the self-adjoint elements of the algebra as random variables. Any random variable X has a law : this is the unique probability measure µ on R with the same moments as X; in other words, µ is such that

R Q(x) dµ(x) = ϕ(Q(X)) (3)
for any real polynomial Q.

In a non-commutative probability setting, the central notion of free independence (introduced by Voiculescu in [START_REF] Voiculescu | Symmetries of some reduced free product C * -algebras[END_REF]) goes as follows. Let A 1 , . . . , A p be unital subalgebras of A. Let X 1 , . . . , X m be elements chosen among the A i 's such that, for 1 ≤ j < m, two consecutive elements X j and X j+1 do not come from the same A i , and such that ϕ(X j ) = 0 for each j. The subalgebras A 1 , . . . , A p are said to be free or freely independent if, in this circumstance,

ϕ(X 1 X 2 • • • X m ) = 0. ( 4
)
Random variables are called freely independent if the unital algebras they generate are freely independent. If X, Y are freely independent, then their joint moments are determined by the moments of X and Y separately as in the classical case.

The semicircular distribution S(m, σ 2 ) with mean m ∈ R and variance σ 2 > 0 is the probability distribution

S(m, σ 2 )(dx) = 1 2πσ 2 4σ 2 -(x -m) 2 1 {|x-m|≤2σ} dx.
If m = 0, this distribution is symmetric around 0, and therefore its odd moments are all 0. A simple calculation shows that the even centered moments are given by (scaled) Catalan numbers: for non-negative integers k,

m+2σ m-2σ (x -m) 2k S(m, σ 2 )(dx) = C k σ 2k ,
where C k = 1 k+1 ( 2k k ) (see, e.g., [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF], Lecture 2). Our main results. We are now in a position to state our first main result, which is nothing but a suitable generalization of Theorem 1.1 in the free probability setting.

Theorem 1.3. Let (A, ϕ) be a non-commutative probability space. Let X 1 , X 2 , . . . (resp., Y 1 , Y 2 , . . .) be a sequence of centered free random variables with unit variance (i.e., such that ϕ(X 2 i ) = ϕ(Y 2 i ) = 1 for all i), satisfying moreover

sup i≥1 ϕ(|X i | r ) < ∞ resp., sup i≥1 ϕ(|Y i | r ) < ∞ for all r ≥ 1,
where |X| = √ X * X. Fix d ≥ 1, and consider a sequence of functions f N : {1, . . . , N } d → R satisfying the following three assumptions for each N and each i 1 , . . . , i d = 1, . . . , N :

(i) (mirror-symmetry) f N (i 1 , . . . , i d ) = f N (i d , . . . , i 1 ); (ii) (vanishing on diagonals) f N (i 1 , . . . , i d ) = 0 if i k = i l for some k = l; (iii) (normalization) N j1,...,j d =1 f N (j 1 , . . . , j d ) 2 = 1. Also, set Q N (x 1 , . . . , x N ) = N i1,...,i d =1 f N (i 1 , . . . , i d )x i1 • • • x i d ( 5
)
and

Inf i (f N ) = d l=1 N j1,...,j d-1 =1
f N (j 1 , . . . , j l-1 , i, j l , . . . , j d-1 ) 2 , i = 1, . . . , N.

Then, for any integer m ≥ 1,

ϕ(Q N (X 1 , . . . , X N ) m ) -ϕ(Q N (Y 1 , . . . , Y N ) m ) = O(τ 1/2 N ), ( 6
)
where τ N = max 1≤i≤N Inf i (f N ).

Due to the lack of commutativity of the variables involved, the proof of Theorem 1.3 raises new difficulties with respect to its commutative counterpart. Moreover, it is worthwhile noting that it contains the free central limit theorem as an immediate corollary. Indeed, let us choose d = 1 (in this case, assumptions (i) and (ii) are of course immaterial), Y 1 , Y 2 , . . . ∼ S(0, 1) and f N (i) = 1 √ N , i = 1, . . . , N . We then have Q N (Y 1 , . . . , Y N ) ∼ S(0, 1) law = Y 1 (thanks to (iii) as well as the fact that a sum of freely independent semicircular random variables remains semicircular) and τ N → 0 as N → ∞, so that, thanks to (6),

ϕ X 1 + • • • + X N √ N m → ϕ(Y m 1 )
for each m ≥ 1 as N → ∞, which is exactly what the free central limit theorem asserts. When d ≥ 2, by combining Theorem 1.3 with the main finding of [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF], we will prove the following free counterpart of Theorem 1.2.

Theorem 1.4. Let (A, ϕ) be a non-commutative probability space. Let S 1 , S 2 , . . . be a sequence of free S(0, 1) random variables. Fix d ≥ 2 and consider a sequence of functions f N : {1, . . . , N } d → R satisfying the following three assumptions for each N and each i 1 , . . . , i d = 1, . . . , N :

(i) (full symmetry) f N (i 1 , . . . , i d ) = f N (i σ(1) , . . . , i σ(d) ) for all σ ∈ S d ; (ii) (vanishing on diagonals) f N (i 1 , . . . , i d ) = 0 if i k = i l for some k = l; (iii) (normalization) N j1,...,j d =1 f N (j 1 , . . . , j d ) 2 = 1.
Also, let Q N (x 1 , . . . , x N ) be the polynomial in non-commuting variables given by [START_REF] Kemp | Strong Haagerup inequalities for free R-diagonal elements[END_REF]. Then, the following two conclusions are equivalent as N → ∞:

(A) Q N (S 1 , . . . , S N ) law → S(0, 1); (B) Q N (X 1 , . . . , X N ) law → S(0, 1) for any sequence X 1 , X 2 , . .

. of free identically distributed and centered random variables with unit variance.

Although a weak 'mirror-symmetry' assumption would have been undoubtedly more natural, we impose in Theorem 1.4 the same 'full symmetry' assumption (i) than in Theorem 1.2. This is unfortunately not insignificant in our non-commutative framework. But we cannot expect better by using our strategy of proof, as is illustrated by a concrete counterexample in Section 2.

Theorem 1.4 may be seen as a free universality phenomenon, in the sense that the semicircular behavior of Q N (X 1 , . . . , X N ) is asymptotically insensitive to the distribution of its summands. In reality, this is more subtle, as the following explicit situation well illustrates in the case d = 2 (quadratic case). Indeed, let us consider

Q N (x 1 , . . . , x N ) = 1 √ 2N -2 N i=2 (x 1 x i + x i x 1 ), N ≥ 2,
let S 1 , S 2 , . . . be a sequence of free S(0, 1) random variables and let X 1 , X 2 , . . . be a sequence of free Rademacher random variables (i.e., the law of X 1 is given by 1

2 δ 1 + 1 2 δ -1 ). Then Q N (X 1 , . . . , X N ) law → S(0, 1) as N → ∞, but Q N (S 1 , . . . , S N ) law → 1 √ 2 (S 1 S 2 + S 2 S 1 ) ∼ S(0, 1).
(See Section 2 for the details.) This means that it is possible to have Q N (X 1 , . . . , X N ) converging in law to S(0, 1) for a particular centered distribution of X 1 , without having the same phenomenon for every centered distribution with variance one. The question of which are the distributions that enjoy such a universality phenomenon is still an open problem. (In the commutative case, it is known that the Gaussian and the Poisson distributions both lead to universality, see [START_REF] Nourdin | Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos[END_REF][START_REF] Peccati | Universal Gaussian fluctuations on the discrete Poisson chaos[END_REF]. Yet there are no other examples.)

Organization of the paper. The rest of our paper is organized as follows. In Section 2, we deduce from Theorem 1.3 several results connected with the universality phenomenon and we study the limitations of Theorem 1.4. Section 3 is devoted to the proof of Theorem 1.3.

Free universality

In this section, we show how Theorem 1.3 leads to several results connected with the universality phenomenon. We also study the limitations of Theorem 1.4: Can we replace the role played by the semicircular distribution by any other law? Can we replace the full symmetry assumption (i) by a more natural one?

To do so, we first need to recall some facts proven in references [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF][START_REF] Kemp | Wigner chaos and the fourth moment[END_REF].

Convergence of Wigner integrals. For 1 ≤ p ≤ ∞, we write L p (A, ϕ) to indicate the L p space obtained as the completion of A with respect to the norm A p = ϕ(|A| p ) 1/p , where |A| = √ A * A, and • ∞ stands for the operator norm. For every integer q ≥ 2, the space L 2 (R q + ) is the collection of all real-valued functions on R q + that are square-integrable with respect to the Lebesgue measure. Given f ∈ L 2 (R q + ), we write f * (t 1 , t 2 , . . . , t q ) = f (t q , . . . , t 2 , t 1 ), and we call f * the adjoint of f . We say that an element of L 2 (R q + ) is mirror symmetric whenever f = f * as a function. Given f ∈ L 2 (R q + ) and g ∈ L 2 (R p + ), for every r = 1, . . . , p ∧ q we define the rth contraction of f and g as the element of

L 2 (R p+q-2r + ) given by f r ⌢g(t 1 , . . . , t p+q-2r ) (7) = R p+q-2r + f (t 1 , . . . , t p-r , x 1 , . . . , x r )g(x r , . . . , x 1 , t p-r+1 , . . . , t p+q-2r ) dx 1 • • • dx r .
One also writes f ⌢ 0 g(t 1 , . . . , t p+q ) = f ⊗ g(t 1 , . . . , t p+q ) = f (t 1 , . . . , t q )g(t q+1 , . . . , t p+q ).

In the following, we shall use the notation f ⌢ 0 g and f ⊗ g interchangeably. Observe that, if p = q, then f ⌢ p g = f, g * L 2 (R q + ) . A free Brownian motion S on (A, ϕ) consists of: (i) a filtration {A t : t ≥ 0} of von Neumann sub-algebras of A (in particular, A u ⊂ A t for 0 ≤ u < t), (ii) a collection S = (S t ) t≥0 of self-adjoint operators such that:

-S t ∈ A t for every t; -for every t, S t has a semicircular distribution S(0, t); -for every 0 ≤ u < t, the increment S t -S u is freely independent of A u , and has a semicircular distribution S(0, tu).

For every integer q ≥ 1, the collection of all random variables of the type I q (f ), f ∈ L 2 (R q + ), is called the qth Wigner chaos associated with S, and is defined according to [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF], Section 5.3, namely:

-first define I q (f ) = (S b1 -S a1 ) • • • (S bq -S aq ) for every function f having the form f (t 1 , . . . , t q ) = 1 (a1,b1) (t 1 ) × • • • × 1 (aq,bq) (t q ), (8)
where the intervals (a i , b i ), i = 1, . . . , q, are pairwise disjoint; -extend linearly the definition of I q (f ) to simple functions vanishing on diagonals, that is, to functions f that are finite linear combinations of indicators of the type (8); -exploit the isometric relation

I q (f 1 ), I q (f 2 ) L 2 (A,ϕ) = ϕ(I q (f 1 ) * I q (f 2 )) = ϕ(I q (f * 1 )I q (f 2 )) = f 1 , f 2 L 2 (R q + ) , (9)
where f 1 , f 2 are simple functions vanishing on diagonals, and use a density argument to define I q (f ) for a general f ∈ L 2 (R q + ).

Observe that relation [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF] continues to hold for every pair f 1 , f 2 ∈ L 2 (R q + ). Moreover, the above sketched construction implies that I q (f ) is self-adjoint if and only if f is mirror symmetric. We recall the following fundamental multiplication formula, proven in [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF]. For every f ∈ L 2 (R p + ) and g ∈ L 2 (R q + ), where p, q ≥ 1, we have

I p (f )I q (g) = p∧q r=0 I p+q-2r (f r ⌢g). (10)
Let S 1 , S 2 , . . . ∼ S(0, 1) be freely independent, fix d ≥ 2, and consider a sequence of functions f N : {1, . . . , N } d → R satisfying assumptions (ii) and (iii) of Theorem 1.4 as well as

f N (i 1 , . . . , i d ) = f N (i d , . . . , i 1 )
for all N ≥ 1 and i 1 , . . . , i d ∈ {1, . . . , N }.

Let also Q N (x 1 , . . . , x N ) be the polynomial in non-commuting variables given by [START_REF] Kemp | Strong Haagerup inequalities for free R-diagonal elements[END_REF]. Set

e i = 1 [i-1,i] ∈ L 2 (R + ), i ≥ 1.
For each N , one has

Q N (S 1 , . . . , S N ) law = Q N (I 1 (e 1 ), . . . , I 1 (e N )). (12)
By applying the multiplication formula [START_REF] Nourdin | Yet another proof of the Nualart-Peccati criterion[END_REF] and by taking into account assumption (ii), it is straightforward to check that

Q N (I 1 (e 1 ), . . . , I 1 (e N )) = I d (g N ), (13)
where

g N = N i1,...,i d =1 f N (i 1 , . . . , i d )e i1 ⊗ • • • ⊗ e i d . (14)
The function g N is mirror-symmetric (due to [START_REF] Nourdin | Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos[END_REF]) and has an L 2 (R d + )-norm equal to 1 (due to (iii)). Using both Theorems 1.3 and 1.6 of [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF] (see also [START_REF] Nourdin | Yet another proof of the Nualart-Peccati criterion[END_REF]), we deduce that the following equivalence holds true as N → ∞:

Q N (S 1 , . . . , S N ) law → S(0, 1) ⇐⇒ g N r ⌢g N L 2 (R 2d-2r + ) → 0 (15) for all r ∈ {1, . . . , d -1}. For r = d -1, observe that g N d-1 ⌢ g N L 2 (R 2 + ) = N i,j=1 N k2,...,k d =1 f N (i, k 2 , . . . , k d )f N (k d , . . . , k 2 , j) e i ⊗ e j L 2 (R 2 + ) = N i,j=1 N k2,...,k d =1 f N (i, k 2 , . . . , k d )f N (k d , . . . , k 2 , j) 2 (16) ≥ N i=1 N k2,...,k d =1 f N (i, k 2 , . . . , k d ) 2 2
(by setting j = i and using ( 11))

≥ max i=1,...,N N k2,...,k d =1 f N (i, k 2 , . . . , k d ) 2 .
Proof of Theorem 1.4. Of course, only the implication (A) → (B) has to be shown. Assume that (A) holds. Then, using (15) (condition (i) implies in particular ( 11)), we get that 16) and since f N is fully-symmetric, we deduce that the quantity τ N of Theorem 1.3 tends to zero as N goes to infinity. This, combined with assumption (A) and ( 6), leads to (B).

g N ⌢ d-1 g N L 2 (R 2 + ) → 0 as N → ∞. Using (
A counterexample. In Theorem 1.4, can we replace the role played by the semicircular distribution by any other law? The answer is no in general. Indeed, let us take a look at the following situation. Fix d = 2 and consider

Q N (x 1 , . . . , x N ) = 1 √ 2N -2 N i=2 (x 1 x i + x i x 1 ), N ≥ 2.
Let S 1 , S 2 , . . . be a sequence of free S(0, 1) random variables and let X 1 , X 2 , . . . be a sequence of free Rademacher random variables (i.e., the law of X 1 is given by 1 2 δ 1 + 1 2 δ -1 ). Then, using the free central limit theorem, it is clear on one hand that

Q N (X 1 , . . . , X N ) = 1 √ 2 X 1 1 √ N -1 N i=2 X i + 1 √ 2 1 √ N -1 N i=2 X i X 1 law → 1 √ 2 (X 1 S 1 + S 1 X 1 ) as N → ∞,
with X 1 and S 1 freely independent. By Proposition 1.10 and identity (1.10) of Nica and Speicher [START_REF] Nica | Commutators of free random variables[END_REF], it turns out that 1 √ 2 (X 1 S 1 + S 1 X 1 ) ∼ S(0, 1). But, on the other hand,

Q N (S 1 , . . . , S N ) = 1 √ 2 S 1 1 √ N -1 N i=2 S i + 1 √ 2 1 √ N -1 N i=2 S i S 1 law = 1 √ 2 (S 1 S 2 + S 2 S 1 ).
The random variable 1 √ 2 (S 1 S 2 + S 2 S 1 ) being not S(0, 1) distributed (its law is indeed the so-called tetilla law, see [START_REF] Deya | Convergence of Wigner integrals to the tetilla law[END_REF]), we deduce that one cannot replace the role played by the semicircular distribution in Theorem 1.4 by the Rademacher distribution.

Another counterexample. In Theorem 1.4, can we replace the full symmetry assumption (i) by the mirror-symmetry assumption? Unfortunately, we have not been able to answer this question. But if the answer is yes, what is sure is that we cannot use the same arguments as in the fully-symmetric case to show such a result. Indeed, when f N is fully-symmetric we have

τ N = d × max i=1,...,N N k2,...,k d =1 f N (i, k 2 , . . . , k d ) 2 ,
allowing us to prove Theorem 1.4 by using the following set of implications: as N → ∞,

Q N (S 1 , . . . , S N ) law → S(0, 1) (15) =⇒ g N d-1 ⌢ g N L 2 (R 2 + ) → 0 (16) =⇒ τ N → 0 (17) Theorem 1.3 =⇒ Q N (X 1 , . . . , X N ) law → S(0, 1).
Unfortunately, when f N is only mirror-symmetric the implication

g N d-1 ⌢ g N L 2 (R 2 + ) → 0 =⇒ τ N → 0, (18)
that plays a crucial role in (17), is no longer true in general. To see why, let us consider the following counterexample (for which we fix d = 3). Define first a sequence of functions f ′ N : {1, . . . , N } 2 → R according to the formula

f ′ N (i, i + 1) = f ′ N (i + 1, i) = 1 √ 2N -2 ,
and f ′ N (i, j) = 0 whenever i = j or |j -i| ≥ 2. Next, for i, j, k ∈ {1, . . . , N }, set

f N (i, j, k) (19)
= 0, if j ≥ 2 or (j = 1 and i = 1) or (j = 1 and k = 1),

f ′ N -1 (i -1, k -1), otherwise.
Easy-to-check properties of f N include mirror-symmetry, vanishing on diagonals property,

N i,j,k=1 f N (i, j, k) 2 = N -1 i,k=1 f ′ N -1 (i, k) 2 = 1 and N i,j=1 N k,l=1 f N (i, k, l)f N (l, k, j) 2 (20) = N i,j=1 N -1 l=1 f ′ N -1 (i, l)f ′ N -1 (l, j) 2 → 0.
Let g N be given by ( 14), that is,

g N = 1 √ 2N -4 N -2 i=1 (e i+1 ⊗ e 1 ⊗ e i+2 + e i+2 ⊗ e 1 ⊗ e i+1 ).
The limit (20) can be readily translated into

g N ⌢ 2 g N 2 L 2 (R 2 + ) → 0 as N → ∞.
On the other hand, we have

τ N = max 1≤j≤N Inf j (f N ) = max 1≤j≤N N i,k=1 {f N (i, j, k) 2 + f N (j, i, k) 2 + f N (i, k, j) 2 } ≥ max 1≤j≤N N i,k=1 f N (i, j, k) 2 = N i,k=1 f N (i, 1, k) 2 = 1,
which contradicts (18), as announced.

is also worth noting that the sequence of functions f N defined by ( 19) provides an explicit counterexample to the so-called Wiener-Wigner transfer principle (see [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF], Theorem 1.8) in a non-fully-symmetric situation. Indeed, on one hand, we have

g N 1 ⌢g N 2 L 2 (R 2 + ) = g N 2 ⌢g N 2 L 2 (R 2 + ) → 0 as N → ∞,
which, due to (15), entails that Q N (S 1 , . . . , S N ) law → S(0, 1). On the other hand, let G 1 , . . . , G N ∼ N (0, 1) be independent random variables defined on a (classical) probability space (Ω, F , P ). One has

Q N (G 1 , . . . , G N ) = G 1 × 2 √ 2N -4 N -1 i=2 G i G i+1 ,
and it is easily checked that

2 √ 2N -4 N -1 i=2 G i G i+1 law
→ N (0, 2) (apply, e.g., the Fourth Moment theorem of [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]). As a result, the sequence Q N (G 1 , . . . , G N ) converges in law to √ 2G 1 G 2 , which is not Gaussian. This leads to our desired contradiction.

Free CLT for homogeneous sums. As an application of Theorem 1.3, let us also highlight the following practical convergence criterion for multilinear polynomials, which can be readily derived from (15). Theorem 2.1. Let (A, ϕ) be a non-commutative probability space. Let X 1 , X 2 , . . . be a sequence of centered free random variables with unit variance satisfying sup i≥1 ϕ(|X i | r ) < ∞ for all r ≥ 1. Fix d ≥ 1, and consider a sequence of functions f N : {1, . . . , N } d → R satisfying the three basic assumptions (i)-(ii)-(iii) of Theorem 1.3. Assume moreover that, as N tends to infinity, max 1≤j≤N Inf j (f N ) → 0 and g N ⌢ r g N 2 (R 2d-2r + ) → 0 for all r ∈ {1, . . . , d -1}, where g N is defined through [START_REF] Voiculescu | Symmetries of some reduced free product C * -algebras[END_REF]. Then one has

N i1,...,i d =1 f N (i 1 , . . . , i d )X i1 • • • X i d law → S(0, 1). ( 21
)
For instance, thanks to this result one can easily check that, given a positive integer k, one has

1 √ N N -k i=1 {X i X i+1 • • • X i+k + X i+k X i+k-1 • • • X i } law → S(0, 1) as N → ∞
for any sequence (X i ) of centered free random variables with unit variance satisfying sup i≥1 ϕ(|X i | r ) < ∞ for all r ≥ 1.

Proof of Theorem 1.3

As in [START_REF] Mossel | Noise stability of functions with low influences: Invariance and optimality[END_REF], our strategy is essentially based on a generalization of the classical Lindeberg method, which was originally designed for linear sums of (classical) random variables (see [START_REF] Lindeberg | Eine neue Herleitung des exponential-Gesetzes in der Warscheinlichkeitsrechnung[END_REF]). Before we turn to the details of the proof, let us briefly report the two main differences with the arguments displayed in [START_REF] Mossel | Noise stability of functions with low influences: Invariance and optimality[END_REF] for commuting random variables. First, in this non-commutative context, we can no longer rely on some classical Taylor expansion as a starting point of our study. This issue can be easily overcome though, by resorting to abstract expansion formulae (see ( 24)) together with appropriate Höldertype estimates (see ( 28)). As far as this particular point is concerned, the situation is quite similar to what can be found in [START_REF] Kargin | A proof of a non-commutative central limit theorem by the Lindeberg method[END_REF], even if the latter reference is only concerned with the linear case, that is, d = 1.

Another additional difficulty raised by this free background lies in the transposition of the hypercontractivity property, which is at the core of the procedure. In [START_REF] Mossel | Noise stability of functions with low influences: Invariance and optimality[END_REF], the proof of hypercontractivity for multilinear polynomials heavily depends on the fact that the variables do commute (see, e.g., the proof of [START_REF] Mossel | Noise stability of functions with low influences: Invariance and optimality[END_REF], Proposition 3.11). Hence, new arguments are needed here and we postpone this point to Section 3.2.

General strategy

For the rest of the section, we fix two sequences (X i ), (Y i ) of random variables in a non-commutative probability space (A, ϕ), two integers N, m ≥ 1, as well as a function f N : {1, . . . , N } d → R giving rise to a polynomial Q N through (1), and we assume that all of these objects meet the requirements of Theorem 1.3. In accordance with the Lindeberg method, we are first prompted to introduce some additional notation.

Notation. For every i ∈ {1, . . . , N + 1}, let us consider the vector

Z N,(i) := (Y 1 , . . . , Y i-1 , X i , . . . , X N ).
In particular, Z N,(1) = (X 1 , . . . , X N ) and

Z N +1,(N ) = (Y 1 , . . . , Y N ), that Q N (X 1 , . . . , X N ) m -Q N (Y 1 , . . . , Y N ) m = N i=1 [Q N (Z N,(i) ) m -Q N (Z N,(i+1) ) m ]. ( 22
)
Since the only difference between the vectors Z N,(i) and Z N,(i+1) is their ith-component, it is readily checked that

Q N (Z N,(i) ) = U (i) N + V (i) N (X i ) and Q N (Z N,(i+1) ) = U (i) N + V (i) N (Y i ),
where

U (i)
N stands for the multilinear polynomial

U (i) N := j1,...,j d ∈{1,...,N }\{i} f N (j 1 , . . . , j d )Z N,(i) j1 • • • Z N, (i) j d , and V (i)
N : A → A is the linear operator defined, for every x ∈ A, by f N (j 1 , . . . , j l-1 , i, j l . . . , j d-1 )Z N,(i) j1

V (i) N (x)
• • • Z N,(i) j l-1 xZ N,(i) j l • • • Z N,(i) j d-1 .
Expansion. Once endowed with the above notation, the problem reduces to examining the differences ϕ((U

(i) N + V (i) N (X i )) m ) -ϕ((U (i) N + V (i) N (Y i )) m ) (23)
for i ∈ {1, . . . , N -1}. In a commutative context, this could be handled with the classical binomial formula. Although such a mere formula is not available here, one can still assert that for every A, B ∈ A,

(A + B) m = A m + m n=1 (r,ir+1,jr)∈Dm,n c m,n,r,ir+1,jr A i1 B j1 A i2 B j2 • • • A ir B jr A ir+1 , (24)
where

D m,n := (r, i r+1 , j r ) ∈ {1, . . . , m} × N r+1 × N r : r+1 l=1 i l = n, r l=1 j l = m -n
and the c m,n,r,ir+1,jr 's stand for appropriate combinatorial coefficients (independent of A and B). The sets D m,n must of course be understood as follows: given (r, i r+1 , j r ) ∈ D m,n , the product

A i1 B j1 A i2 B j2 • • • A ir B jr A ir+1
contains A exactly n times and B exactly (mn) times, both counted with multiplicity.

Let us go back to (23) and let us apply formula (24) in order to expand (U

(i) N + V (i) N (X i )) m (resp., (U (i) N + V (i) N (Y i )) m
). The first and second order terms (i.e., for n = 1, 2 in (24)) of the resulting sum happen to vanish, as a straightforward use of the following lemma shows. Lemma 3.1. Let Y and Z be two centered random variables with unit variance. Then, for every integer k ≥ 1 and every sequence (X i ) of centered freely independent random variables independent of Y and Z, one has

ϕ(X i1 • • • X ir Y X ir+1 • • • X i k ) = ϕ(X i1 • • • X ir ZX ir+1 • • • X i k ) = 0 (25) and ϕ(X i1 • • • X ir Y X ir+1 • • • X is Y X is+1 • • • X i k ) = ϕ(X i1 • • • X ir ZX ir+1 • • • X is ZX is+1 • • • X i k ) (26) for all 0 ≤ r ≤ s ≤ k and (i 1 , . . . , i k ) ∈ N k .
Proof. Let us first focus on (25). For k = 1, this is obvious. Assume that the result holds true up to k -1 and write

ϕ(X i1 • • • X ir Y X ir+1 • • • X i k ) = ϕ(X m1 i ′ 1 • • • X m r ′ i ′ r ′ Y X m r ′ +1 i ′ r ′ +1 • • • X m s ′ i ′ s ′ ) with i ′ p+1 = i ′ p for p ∈ {1, . . . , s ′ -1} \ {r ′ }, i ′ s ′ = i ′
1 and m p ≥ 1 for every p ∈ {1, . . . , s ′ }. Center successively every random variable X

mp 1 i ′ p 1 , . . . , X mp t i ′ p t
for which m pi ≥ 2: together with an induction argument, this yields

ϕ(X m1 i ′ 1 • • • X m r ′ i ′ r ′ Y X m r ′ +1 i ′ r ′ +1 • • • X m s ′ i ′ s ′ ) = ϕ(X i ′ 1 • • • X i ′ p 1 -1 (X mp 1 i ′ p 1 -ϕ(X mp 1 i ′ p 1
))X

mp 1 +1 i ′ p 1 +1 • • • X m r ′ i ′ r ′ Y X m r ′ +1 i ′ r ′ +1 • • • X m s ′ i ′ s ′ ) = ϕ(X i ′ 1 • • • X i ′ p 1 -1 (X mp 1 i ′ p 1 -ϕ(X mp 1 i ′ p 1 ))X i ′ p 1 +1 • • • X i ′ p 2 -1 (X mp 2 i ′ p 2 -ϕ(X mp 2 i ′ p 2
))

X mp 2 +1 i ′ p 2 +1 • • • X m r ′ i ′ r ′ Y X m r ′ +1 i ′ r ′ +1 • • • X m s ′ i ′ s ′ ) = • • • = 0
owing to free independence. Identity (26) can be easily derived from a similar induction procedure.

Let us go back to the proof of Theorem 1.3. As a consequence of the previous lemma, it now suffices to establish that, either for W = X i or for W = Y i , one has, as soon as

l j l ≥ 3, |ϕ((U (i) N ) i1 (V (i) N (W )) j1 (U (i) N ) i2 (V (i) N (W )) j2 • • • (U (i) N ) ir (V (i) N (W )) jr )| ≤ c m,d Inf i (f N ) 3/2 (27)
for some constant c m,d . Indeed, in this case, by combining ( 22), ( 24) and ( 27) with the identities in the statement of Lemma 3.1, we get

|ϕ(Q N (X 1 , . . . , X N ) m ) -ϕ(Q N (Y 1 , . . . , Y N ) m )| ≤ C m,d N i=1 Inf i (f N ) 3/2 ≤ C m,d τ 1/2 N N i=1 Inf i (f N ) = C m,d τ 1/2 N ,
which is precisely the expected bound of Theorem 1.3.

In order to prove (27), let us first resort to the following Hölder-type inequality, borrowed from [START_REF] Kargin | A proof of a non-commutative central limit theorem by the Lindeberg method[END_REF], Lemma 12:

|ϕ((U (i) N ) i1 (V (i) N (W )) j1 • • • (U (i) N ) ir (V (i) N (W )) jr )| (28) ≤ ϕ((U (i) N ) 2 r i1 ) 2 -r ϕ((V (i) N (W )) 2 r j1 ) 2 -r • • • ϕ((U (i) N ) 2 r ir ) 2 -r ϕ((V (i) N (W )) 2 r jr ) 2 -r
. Now, let the key (forthcoming) Proposition 3.5 come into the picture. Thanks to it, we can simultaneously assert that, for every p ≥ 1,

ϕ((U (i) N ) 2p ) ≤ C p,d and ϕ(V (i) N (X i ) 2p ) ≤ C p,d • Inf i (f N ) p
for some constant C p,d . Going back to (28), we deduce that for every (j l ) such that

l j l ≥ 3, |ϕ((U (i) N ) i1 (V (i) N (X i )) j1 • • • (U (i) N ) ir (V (i) N (X i )) jr )| ≤ C ′ r,d • Inf i (f N ) 2 -1 (j1+•••+jr ) ≤ C ′ r,d • Inf i (f N ) 3/2
since Inf i (f N ) ≤ 1, and so the proof of Theorem 1.3 is done.

Hypercontractivity

In order to prove the forthcoming Proposition 3.5 (which played an important role in the proof of Theorem 1.3), we first need a technical lemma. To state it, a few additional notation must be introduced.

c * (n 1 ⊗ • • • ⊗ n r ) the subset of complete graphs in G * (n 1 ⊗ • • • ⊗ n r ).
Now, given a graph γ of {1, . . . , n} with p vertices (p ≤ n) and a function f : {1, . . . , N } n → R, we call contraction of f with respect to γ the function C γ (f ): {1, . . . , N } n-p → R defined for every (j 1 , . . . , j n-p ) by the formula

C γ (f )(j 1 , . . . , j n-p ) := N i1,...,ip=1
f (j 1 , . . . , i 1 , . . . , i p , . . . , j n-p ) • δ(γ, j 1 , . . . , i 1 , . . . , i p , . . . , j n-p ), where:

• the (fixed) positions of the i k 's in (j 1 , . . . , i 1 , . . . , i p , . . . , j n-p ) correspond to the positions of the vertices of γ; • δ(γ, j 1 , . . . , i 1 , . . . , i p , . . . , j n-p ) = 1 if all i k , i l in a same block of γ are equal, and 0 otherwise.

With these notation in hand, we can prove the following lemma.

Lemma 3.3. For every γ ∈ G * (n 1 ⊗ • • • ⊗ n r ) and all f i ∈ ℓ 2 ({1, . . . , N } ni ) (i = 1, . . . , r), one has C γ (f 1 ⊗ • • • ⊗ f r ) ℓ 2 ≤ r i=1 f i ℓ 2 .
Proof. We use an induction procedure on r.

When r = 1, C γ (f 1 ) = f 1 . Fix now r ≥ 2 and γ ∈ G * (n 1 ⊗ • • • ⊗ n r ). Denote by γ ∈ G * (n 2 ⊗ • • • ⊗ n r ) the restriction of γ to n 2 ⊗ • • • ⊗ n r (i.e.
, the graph that one obtains from γ by getting rid of the blocks with vertices in {1, . . . , n 1 }). If γ has no vertex in {1, . . . , n 1 }, then

C γ (f 1 ⊗ • • • ⊗ f r ) = f 1 ⊗ C γ (f 2 ⊗ • • • ⊗ f r )
and we can conclude by induction. Otherwise, it is easily seen that C γ (f

1 ⊗ • • • ⊗ f r ) 2 ℓ 2
can be decomposed as where:

C γ (f 1 ⊗ • • • ⊗ f r )
• l (resp., m) is the number of points in {1, . . . , n 1 } (resp., {n 1 + 1, . . . , n 1 + • • • + n r }) which are not assigned by γ;

• in f 1 (i 1 , . . . , k 1 , . . . , k q , . . . , i l ), the (fixed) positions of the k i 's correspond to the positions of the q vertices of γ in {1, . . . , n 1 }; • σ: {1, . . . , p} → {1, . . . , q} (p ≥ q) is a surjective mapping, meaning that each k i appears at least once in (k σ(1) , . . . , k σ(p) ). Here, we use the fact that γ respects n 1 ⊗ • • • ⊗ n r and contains no singleton.

Then, by applying Cauchy-Schwarz inequality over the set of indices (k 1 , . . . , k q ), we get

C γ (f 1 ⊗ • • • ⊗ f r ) 2 ℓ 2 ≤ f 1 2 ℓ 2 C γ (f 2 ⊗ • • • ⊗ f r ) 2 ℓ 2 ,
where we have used (possibly several times) the trivial property: for any g: {1, . . . , N } 2 → R, N k=1 g(k, k) 2 ≤ N k1,k2=1 g(k 1 , k 2 ) 2 . We can now conclude by induction.

Let us finally turn to the proof of Proposition 3.5, which is the hypercontractivity property for homogeneous sums of free random variables. We shall use Lemma 3.3 as a main ingredient. The following elementary lemma will also be needed at some point. Lemma 3.4. For every integer r ≥ 1 and every sequence X = (X i ) of random variables, one has |ϕ(X i1 • • • X i2r )| ≤ µ X 2 r-1 , where µ X k := sup 1≤l≤k,i≥1 ϕ(X 2l i ).

Proof. For r = 1, this corresponds to Cauchy-Schwarz inequality (see [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]). Assume that the result holds true up to r -1 (r ≥ 2) for any sequence of random variables. By using Cauchy-Schwarz inequality, we first get

|ϕ(X i1 • • • X i2r )| = |ϕ((X i1 • • • X ir )(X ir+1 • • • X i2r ))| (29) ≤ ϕ(X 2 i1 • • • X ir-1 X 2 ir X ir-1 • • • X i2 ) 1/2 ϕ(X 2 ir+1 • • • X i2r-1 X 2 i2r X i2r-1 • • • X ir+2 ) 1/2 .
Denote by X 2 the sequence X 1 , X 2 1 , X 2 , X 2 2 , . . . . Then by induction, we deduce from (29) that |ϕ(X i1 • • • X i2r )| ≤ µ X 2 2 r-2 ≤ µ X 2 r-1 , which concludes the proof.

Proposition 3.5. Let X 1 , . . . , X N be centered freely independent random variables and denote by (µ N k ) the sequence of larger even moments, that is, µ N k := sup 1≤i≤N,1≤l≤k ϕ(X 2l i ). (30)

  ...,j d-1 ∈{1,...,N }\{i}

2 ℓ 2 =

 22 i1,...,i l ,j1,...,jm k1,...,kqf 1 (i 1 , . . . , k 1 , . . . , k q , . . . , i l ) × C γ (f 2 ⊗ • • • ⊗ f r )(j 1 , . . . , k σ(1) , . . . , k σ(p) , . . . , j m ) 2 ,

Fix d ≥ 1 ,

 1 and consider a sequence of functions f N : {1, . . . , N } d → R satisfying the three basic assumptions (i)-(ii)-(iii) of Theorem 1.3. Define Q N through[START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF]. Then for every r ≥ 1, there exists a constant C r,d such thatϕ(Q N (X 1 , . . . , X N ) 2r ) ≤ C r,d µ N 2 rd-1 N j1,...,j d =1f N (j 1 , . . . , j d ) 2 r .

  Definition 3.2. Fix integers n 1 , . . . , n r ≥ 1. Any set of disjoint blocks of points in {1, . . . , n1 + • • • + n r } is called a graph of {1, . . . , n 1 + • • • + n r }. A graph is complete if the union of its blocks covers the whole set {1, . . . , n 1 + • • • + n r }. Besides, a graph is said to respect n 1 ⊗ • • • ⊗ n r if each of its blocks contains at most one point in each set {1, . . . , n 1 }, {n 1 + 1, . . . , n 2 }, . . . , {n 1 + • • • + n r-1 + 1, . . . , n 1 + • • • + n r }.Finally, we denote by G * (n 1 ⊗ • • • ⊗ n r ) the set of graphs respecting n 1 ⊗ • • • ⊗ n r and containing no singleton (i.e., no block with exactly one element), and by G

Acknowledgements

We are grateful to Todd Kemp for helpful comments and references about hypercontractivity. Special thanks go to Roland Speicher, who suggested a shorter proof for the hypercontractivity property (Proposition 3.5). Finally, we thank an anonymous referee for a careful reading and for his/her positive comments and constructive remarks. This work has been supported in part by the two following (french) ANR grants: 'Exploration des Chemins Rugueux' [ANR-09-BLAN-0114] and 'Malliavin, Stein and Stochastic Equations with Irregular Coefficients' [ANR-10-BLAN-0121].

Proof. The argument is in spirit quite close to ideas of [START_REF] Kemp | Strong Haagerup inequalities for free R-diagonal elements[END_REF]. Owing to Lemma 3.1, it holds that

where we have set, for every R ≥ 1,

Bounding each term of the form ϕ((X

Recall the notation G c * (d ⊗2r ) and C γ from the beginning of Section 3.2. By taking into account that f N is assumed to vanish on diagonals, it is easily seen that the above sum is equal to

Therefore, we may apply Lemma 3.3 so as to deduce that


HAL : halshs

GALINON-MELENEC Béatrice, THE "SIGNE-TRACE PARADIGM". FRAGMENTS. translation by Laura Kraftowitz, from « Fragments théoriques du signe-trace, propos sur le corps communicant », in L'Homme trace, Perspectives anthropologiques des traces contemporaines, Paris, CNRS éditions, série L'Homme-trace, tome 1, 2011, pp. 191-213, by Laura Kraftowitz, Sterling Height, USA, 2012. 1 Mots-Clés : Anthropology and communication -The limits of rationality -The signtraces of the body -The sign-trace paradigm -The echo effect of sign-traces.

INTRODUCTION

This chapter delves into one aspect of the signe-trace paradigm -the signes-traces of the communicating body. This paradigmatic fragment falls within the framework drawn by Paul Watzlawick, J. Helmick Beavin, and Don D. Jackson 2 , who, following the theoretical context of Gregory Bateson, disseminated the idea that in order to understand communication between two people, one cannot merely consider its spoken content, but must also account for the relationship that determines the meaning of that content.

Under the same filiation, popularised as the "Palo Alto School," other authors like E.T. Hall and E. Goffman demonstrated the importance of cultural, social, and interactional context in utterance, body language, and social representation. The psychological, sensorial, cognitive or cultural differences between individuals in a communication situation have been addressed, thanks to advances in psychology in general and cognitive psychology in particular 3 . For my part, with the concept of signes-traces I am attempting to interweave these different levels, which have been widely used in research within the field of information and communication sciences (ICS) in France since 1975. To do this, I build upon the rapprochement between ICS and other fields, including sociology -in particular, Pierre Bourdieu's concept of habitus 4 -and semiology, with Roland Barthes's concept of the sign 5 .

At the end of a series of interdisciplinary seminars that preceded what would eventually become the "Homme-trace" research team and the aforementioned RIGHT network, I introduced the term "signe-trace" in order to better highlight the ways in which the past acted upon the relationship between individuals in a communicative situation. While my work indirectly questioned Greimas 6 and Hjelmslev's 7 definitions of semiotics, the "signetrace paradigm" claimed no affiliation to semiotics, nor to any of the above-mentioned disciplines (sociology, psychology, semiotics, etc.). Rather, I maintained that my work belonged within the realm of information and communication sciences, which were founded on, and are constantly renewed by, interdisciplinary synergy. 6 "Semiotic theory should first be presented for what it is: a theory of meaning. Its primary concern will thus be to make explicit, in the form of a conceptual construction, the conditions of the entry and production of meaning." Translation by L. Kraftowitz. 7 Hjelmslev considers semiotics to be a network of relationships that has a dual mode of paradigmatic and syntagmatic existence, which can be understood as a system and thus as a semiotic process.

The term "signe-trace" is constructed upon the association of two terms that, taken separately, have themselves been analysed -either frequently (as with "sign"), or less frequently (as with "trace"). The signe-trace paradigm results from a double process : the appropriation of the aforementioned research, and the progressive objectivation of a new conceptual difference that offers to a non-specialised public the possibility of understanding the complexity of what occurs when individuals "recognise" one another through an echoing of signes-traces, or (so as to mention only two borderline cases of communication) when they "reject" one another to the point of introducing an objectivation of the social processes of discrimination.

In a first definitional proposal 8 , I will state that for me, associating the two terms -"signe" and "trace" -aims to signify that : -the process that produced the sign is present within the sign ; -to examine the interpretation given to a sign implies acknowledging that an assumption exists that brings attention to one sign rather than another ; -the interpretation itself is a signe-trace, in that it contains within itself the implicit acceptance of an interpretive system ; and that, -there exists a circular and continuous process 9 from the trace to the sign, and from the sign to the trace.

By becoming a "signe-trace," the parameters that come into play refer, infra, to the history that constructed them, and integrate, supra, into a matrix whose resolution 10 produces the resultant signe-trace of communication.

Associating the terms "sign" and "trace" differs from other associations used by contributors to the book L'Homme trace (Volume 1, 2011). Take, for example, the association of "sign" and "artifact" in "sign-artifact", or the association of "sign" and "symbol" in "signsymbol". In my view, "signe-trace" refers to that which is transversal to all signs, whereas "sign-artifact" and "sign-symbol" refer only to certain among them. I provided several examples of this in "L'universalité de la trace, le XXIe siècle, siècle de la trace" 11 , and in "Prolégomènes illustrés de la trace ; l'exemple du 20 juillet 1969" 12 . So as to avoid repeating the details of points previously articulated, below is a consolidated text 13 .

GENERAL PRINCIPLES

Signes, Traces, System

In 1991, Derrida and Bennigton articulated the relationships between sign, trace, and system as follows:

8 Based on field observations. 9 In the systemic sense. This circularity induces changes in what is sequentially considered as "input" and "output". 10 Here, I mean the matrix of mathematics, where matrices are used to solve systems of complex equations. 11 Galinon-Mélénec B. (Ed.), CNRS éditions, 2011. 12 Ibid. 13 Ibid.

-"To begin with the sign is to begin with the secondary itself." (Derrida and Bennington, 1991: 26); -"Every trace is a trace of a trace. No element is ever anywhere present (nor simply absent); there are only traces." (Derrida and Bennington, 1991: 74), -"No element of the system has any identity, except in its difference to the other elements; each element is thus marked by that which it is not, whose trace it carries" (Derrida and Bennington, 1991: 73) 14 .

My proposed term "signe-trace" takes these initial proposals into account, and reads as follows :

Everything is a sign15 (the body and all material outside the body, that is, humans and their human and nonhuman environment), and all signs are the result of interactions. In this interpretation, by connecting a sign to that which produced it, the sign becomes a "signetrace". This association connects traces of the past to their present interpretations. In this sense, every sign is signe-trace of the process that constructed it.

For example, stating that, "A sign does not exist prior to its interpretation"16 , or that, "A sign is indeterminate with respect to its interpretation"17 , does not remove the sign's intrinsic status as a signe-trace 18 . By way of this term, I suggest that every sign is a result and product of the elements that allowed for its emergence and construction 19 . In sum, for me, although not every sign is a sign-signal -in the sense that it does not necessarily attract attention, and that its presence is not perceived, identified20 , or interpreted -every sign is always a signe-trace.

The Interpretation of Signes-traces Results from a System of Signe-trace Interaction

In examining the meaning given to signes-traces, it becomes necessary to accumulate collections of traces through which evidence 21 can argue for one interpretation or another. The accumulation of such evidence, which proves more essential than the origin of that which produced the trace, is located at the intersection of multiple systems because we must somehow reconstruct the trace's origin 22 . We can thus say that human interpretation of traces is at the heart of a system of signes-traces.

The Signe-trace: A Dynamic and Systemic Concept of Trace/Sign Interaction

According to this approach, if I perceive a sign and give it meaning, this implies a circular approach: from the trace to the sign -that is, to the signal 23 , since the sign is signaled to the receiver -but also, from the sign to the trace of the sign's internalisation, since the sign only takes on meaning if, beforehand, signs and their meanings have been internalised in a complexity and continuum that escape the conscious mind. Consequently, by not separating the two terms "sign" and "trace", and by bringing them them together into the single term "signe-trace", a triple meaning of the trace emerges: that of the "presence of the past in its absence from the present" (Jeanneret, 2011) 24 ; the "projection of the signe-trace in the interpretative process -unconscious or conscious -of the received signe-trace"; and of "communication as the interaction of signes-traces".

THE COMMUNICATING BODY

Behaviours Interpreted as the Body's Signs-Traces 25

Individual behaviour can be understood as a magma of signs constituting a sort of readable text. This text is constructed from the signes-traces of an individual's life history26 . 22 Certain signs are consciously put forward (highlighted) to catch the attention of the recipient, referred to as the "target". This approach fits the (by now basic) logic of Shannon and Weaver's canonical model (Shannon and Weaver, 1975). This model which, to self-identify as communication, is hardly anymore used as such, except in select professional fields that are researching a strategy for manipulating signs. In this context, choosing a sign-signal borrowed from a repertoire of signs that is "written nowhere and shared by all" (Sapir), allows the "emitter" to cull significant benefits from its target. This seducer emits signs with a pretended passion, knowing that the signs are those that the target secretly hopes for. This practice is familiar to politicians who, whether through reading or by other means, have absorbed Mazarin's Breviary For Politicians, and have learned to select the signs that will affect the receiver: placing a hand on the heart, for example, or reaching out their arms. They have internalised the idea that politics is an affair of seduction, not deduction. In this, they follow the rhetorical lesson that gesture must constantly underscore verbal communication. Mimesis plays an important role in forming repertoires of seduction-oriented behaviours, since the seduction of an individual (or a group), requires gestures that echo the target(s)' individual (and social) histories. 23 This signe-trace can become a sign-signal for an individual (human or animal) who observes the signs. This does not mean that all signes-traces become sign-signals, because the sign receiver does not detect all the signs present, but rather sorts through them ; this sorting and subsequent interpretation are themselves signes-traces. This leads to consequences that we can apply the body. I should specify that "sign perceived-sign-received" is not a one-way function, except when the sign is part of a "sign repertoire" that is shared by a social or cultural group, or by all humanity. Desmond Morris illustrates this "silent language" in his book Peoplewatching, whose French translation is subtitled, "All the Keys to Decrypting Silent Expressions, Gestures and Attitudes" (Morris, 2002). As he specified in the introduction for his 1994 edition: "[E]very seasoned traveler will have discovered [that w]hat is friendly here, is hostile there." On the other hand, "Some gestures, intriguingly, do not suffer from these local variations. They appear to be universal and make us feel at least partially at home even when we are on the other side of the globe. A smile is a smile is a smile, the world over … but … even these global signals can vary in style and intensity. We all laugh, but in some places a loud laugh is considered rude." I should specify, then, that the sign that becomes a signal in the eyes of the receiver has the indirect consequence of placing a shadow over other signs, which lessens their visibility without fully obliterating them.

This inscription of signes-traces in bodily material is an automatic and inherent 27 part of life. The bearer of these signes-traces is only conscious of a small number of them, and forgets the details of the thousands of events that built them.

Nevertheless, the body and its behaviour allow signs to be seen that are interpreted or produced, for example by the "impulses towards interaction" 28 (Montagner, 2002) 29 or conversely, by more and less conscious distancing 30 . The complexity of individualenvironment interactions, internalised as each person develops, seeps out through behaviours that are the signes-traces of internalised complexity. The continuum, and the complexity of this internalisation, produce behaviours that are constructed from a magma of signes-traces. An individual's communicative experience of the individual can lead to an attempt to retrace the path that led to this construction, via psychoanalysis. The ability to achieve this goal is limited by the low traceability of that path.

Behaviour is multifactorial. However, it is always possible to identify factors, such as: -the signe-trace of the interactions an individual has had, is having, and expects to have with his or her environment ; -the signe-trace of representation, in the broadest sense ; -the signe-trace that an individual has of self and environs ; and, more generally, the "signetrace of the permanent capital of representations" 31 (of self), both conscious and unconscious ; and both of the immediate situation and of the entirety of parameters 32 that come into play in that situation.

Corporeal signes-traces: Interfaces Within and Without

In the perspective defined above, behaviour is defined as a signe-trace of the "inbetween" that is both external (that which can be observed) and internal (the generator of behaviour, on which one must impose interpretative hypotheses).

Nevertheless, daily observation shows us that behaviour can lead to commentary and judgment. This assumes that the commentator/judge is capable of sorting -of paying attention 27 Descartes's mind-body split deeply marked modern thought (for example, in the field of medicine). Yet even in its era era, this split was being challenged by Spinoza, who saw emotions as "the basis for human survival and culture" Translation by L. Kraftowitz. In Descartes' Error, Antonio R. Damasio explains "the way in which the body and brain produce emotion and feeling. For him, joy and sorrow, in particular, are the keys to life, not only for preserving and ensuring basic needs, but for stimulating artistic production and the highest achievements" (Damasio, 1995 ; emphasis added). The body is not simply a mechanical, chemical and sensorial intermediary. With the development of the field of neuroscience the mind-body split has become an artifact. Translation by L. Kraftowitz. 28 As articulated by H. Montagner in "Communication Homme-animal [Human-Animal Communication]", a conference organised by Béatrice Galinon-Mélénec, CDHET/GREC/O, Bordeaux 3, 2002. 29 Ibid. 30 When a person is unable to trace the emergence of his or her own judgment-which would take the place of justificationthat person encloses the experience of unconscious recognition within an undifferentiated magma of signs residing under the term "intuition". For more information on the link between interaction and signes-traces ; echoing and signes-traces ; intuition, and communication, see the illustration in the below chapter, "From Medical Diagnosis to Societal Diagnosis". 31 Here, "the permanent capital of representations" means that past experience, the present situation, and projections for the future are integrated. 32 Here, I am not returning to previously enumerated parameters, nor to the chapters illustrating the situation of doctor-patient communication and the communication of the politician.

to few signes-traces among all that constitute the an Other's behavior -and on the other hand, that he/she possesses a set of assumptions underlying the process that produced them. The interpreter's ability to transcend appearances, along with the relevance of the interpretative hypotheses (which are themselves behaviours and thus signes-traces of the same process) affect interpretative accuracy.

In fact, attributing meaning to a gesture expresses that it functions much as a word does. On that basis, without placing doubt upon an interpretation's validity, one could reference a book on the language of gestures, which could be consulted much as one consults a dictionary. Such an equivalence hides the fact that the body sends out a magma of signs ; to limit oneself to a single sign would produce an artifact. A sign that summons 33 the gaze does not intrinsically produce this effect. Rather, attention stems as much from the sign's beholder as from the sign itself. Moreover, corporeal signes-traces are not only numerous, but in perpetual motion. Unlike words on a page, they are not static, unless they are so rendered artificially 34 . In effect, when a person plays a role on the world stage, that person may voluntarily show sign-symbols that are immediately interpretable (Goffman, 1973) 35 . This practice not only reduces the number of signs to which the target pays attention, but also the target's interpretive open-mindedness 36 . The increased visibility of certain signs narrows the gaze to those signs alone. Ceteris paribus 37 , the signs then lead to a forgetfulness of the set of norms influencing an exchange, as well as the origin of the criteria for assessment and judgment 38 . The sign's interpretive immediacy appears as the signe-trace of a semiotic relationship to an overly rudimentary world 39 .

To my mind, there exists no fixity of the body's signs, or their interpretation. Because of this, we should encourage people, at minimum, to remember to differentiate between immediate visibility, and the presence of bodily signs: interpretation reflects as much on the interpreter as the interpreted. Here, I use the term "sign" because it is most often used in this interpretive context; but, as stated above, signs are also traces. I feel that it makes sense to substitute the term "signe-trace", which constantly recalls this bifacial characteristic; and to create a rupture with the aforementioned interpretative models, which focus only on the sign.

Consequences of the Continuum between the Individual and its Human and Non-Human Environment

33 "What we call signs must be seen as the result of complex operations, over the course of which various modalities of production and recognition come into play" ; "even in perception, semiotic processes are at play", Eco, U., p. 7. Translation by L. Kraftowitz. 34 For a better understanding of the general techniques of manipulation, see Petit traité de manipulation à l'usage des honnêtes gens [A Brief Treatise on the Manipulation of Honest People], Beauvois J. L. and Vincent J. R., Grenoble, PUG, 1987. 35 Goffman, E. The Presentation of Self in Everyday Life, Edinburg, Anchor Books, 1959. 36 Morris D., Peoplewatching, London, Vintage, 2002. 37 For Desmond Morris, every gesture is accompanied by four rubrics: "Meaning, Action, Circumstance, and Region". 38 In sum, the individual trace cannot be understood without being placed within the collective, in its center . Translation by L. Kraftowitz . 39 It seems difficult to keep to this approach of decoding gestures, facial expressions and general behaviour, and to "go against what the evolutionary sciences have taught us about the phylogenetic continuity of organisms, thereby devaluing all sorts of biological mechanisms that we share with other oganised beings. Our uniqueness in relation to other existing beings is relative, just as the human awareness of it is relative." (Descola, 2001). Translation by L. Kraftowitz.

But to what does this concept of the individual refer ? Isn't the individual/environment disconnect an artifact, inasmuch as no individual exists outside of an environment, and that every society is composed of individuals? My response starts from the above stated principle that, on one hand, everything is continuum and interaction; and that on the other hand, everything is a sign, and that every sign a signe-trace. After applying the designation "signestraces" to "corporeal signes-traces", I gradually extended it to all signs as a reminder that all signs bear the traces of the complexity of the interweaving 40 of the living (human or otherwise), and the non-living, which combine at the intersection of representations which bear upon the past, present, and projections on the future.

The rub is that by becoming more complex, the paradigm of signes-traces moves away from fast thinking. At issue, then, is knowing to what extent this approach is acceptable by actors who are caught up in short-term constraints. This is why, to facilitate its gradual assimilation, I rely below on applied examples.

"Lifestyle", the Signes-traces of a Common History

Beyond what is transversal to human needs, every society and social group directs its attention to elements that differ from reality. Consequently, behaviours may be shared within a group 41 . This is how people externalize -often unconsciously -signes-traces of common social and/or individual histories: even dressing styles, ways of behaving publicly, of speaking, of sharing "living quarters", and "lifestyles" (Bourdieu, 1979: 191) 42 on the whole. These behavioural signs can become sign-symbols and promote a sense of belonging within to a community, to a world 43 that shares the same tastes, indeed the same values. They can also serve as identifying signs for individuals who are exogenous to the group. These behavioural signs can be internalised from social learning, like that provided by school; or more generally, the values and culture delivered by a country's institutions, as well as a person's relationship to time and space. Thus, without necessarily understanding the circular process of internalising externality and externalising interiority through behavioural signes-traces, people associate specific behaviours with "the English", "the French", "the Italians", "the Spanish", and more 40 See below, "General Theory of Signes-traces." 41 The example of the behaviour of national presidents is interesting to analyse. It makes sense to place them within the behaviours that are sign-signals that relate to the presidential office. In the 1970s, these titularies of the Stae's highest office had to adopt behaviours that served as sign-signals of their awareness of their choices' consequences. They were not asked to be close to the citizen. They were expected to exhibit a sleek façade-an example of erasing a layer of secondary indicators that referred to signes-traces of mood, of doubt, of personal problems related to emotion or health. We have to place ourselves in this context to understand the interpretation of the period's contemporaries. Today, our relationship to politicians (see Annick Monseigne in Galinon-Mélénec, CNRS, 2011) differs from that of the 1970s: the distance has lessened, and those who govern seek a mimesis that promotes sign echoing that can in turn foster empathy with political leaders. In this case, as before, the "staging" of these behaviours responds to the expectations of the voters and seeks to strengthen that image. The natural magma of corporeal signes-traces is in this case meticulously worked on to put forward sign-signals that voters will interpret positively. This treatment functions as a signe-trace of the professionalism of those in power. 42 Citation ? 43 One inevitably thinks of Boltanski and Thevenot's On Justification: Economies of Worth, Princeton, Princeton University Press, 2006. generally, with all peoples [START_REF]On this topic, I refer to E.T. Hall's body of work, particularly Beyond Culture[END_REF] , which produces identifier signes-traces (Hall, 1959) [START_REF] See | Hall's The Silent Language (1973) and The Hidden Dimension[END_REF] . Sometimes, voluntary behaviours establish a set of socio-symbolic devices to treat signals, symbols, and formal signs.

Individuals Cannot Not Have Signes-traces That Communicate With One Another In a Silent Language

In our present context, Paul Watzlawick's famous dictum that "One cannot not behave; one cannot not communicate" 46 becomes: Individuals cannot not have corporeal signes-traces that communicate with one another in a silent language; or again: Since corporeal signes-traces are inherent to humans 47 , whose very existence is associated with corporeal life, they are present in every situation of communication in copresence.

The form -fluidity, tension, indifference, etc. -that such a communication in copresence takes on is the signe-trace of the inductive impact of the interaction of corporeal signes-traces between the individuals themselves, and between those individuals and all human and non-human parameters that come into play in the interaction -consciously or not. Every person incorporates the continuum 48 of a life story, and carries its signes-tracesconsciously or unconsciously -in all behaviours 49 . These signes-traces, while present, are not necessarily visible on the body surface, but all "bodily material" bears its footprint.

Wanting to Hide Signes-traces is Also a Signe-trace

Wanting to hide traces is the representational signe-trace of a situation by an individual and author of this erasure 50 . If this practice spreads in the collective², this is also a signe-trace; but this time, of the societal evolution that itself is nothing but the fruition of interactions between individuals. The way in which the individual self-positions with regards to these changes is also a signe-trace. This position drives the person to self-situate in an environment that he or she internalises, and this process of internalisation again produces signes-traces. In this systemic approach, everything -human and nonhuman -is interacting, and the interaction produces signes-traces that in turn produce interactions, and so on.

The Echoing of Signes-traces

When signes-traces echo positively, fluid communication is established. The other appears open to receiving it. In principle, the signes-traces in question are those of "neighbourly" life experiences [START_REF]Le poids des apparences[END_REF] . When there is a mimetic 52 synchronisation of signes-traces, a phenomenon of empathy is created. This synchronisation may be spontaneous. The closeness felt by individuals then reveals a closeness of prior life experiences. This mimetic synchronisation may be artificial, as when one of the individuals modifies behaviour to mirror the other's in order to induce that person's receptiveness. This manipulative approach can be analysed as a self-interest signe-trace. Its frequency increases in all situations where financial benefit or the obtention of power are expected 53 .

JUDGMENT AS SIGNE-TRACE The Parasitic Filters

The parameters that determine what meaning a receiver assigns to signes-traces are multidimensional (contextual weight of every kind, projections, etc.), and are in every way contingent upon the receiver's "filters" 54 . These filters are themselves signes-traces (as is the meaning assigned to the signe-trace). Each individual's interpretation of signs perceived, consciously or not (in the case of intuition, the process is not conscious) bears the traces of the complex of individual-environment interactions that surrounded that person since birth 55 .

Individual interpretations that come together are a signe-trace of cultural internalisation in the broadest sense; of the prior apprehension and internalisation of "common sense" 56 values; of a signe-trace of shared practices among group of individuals. Even as social scientists conduct work on the genesis of collective representations; and even as the construction of "common sense", of group stereotypes, judgments, and representations; are identified and rendered traceable, most individuals ignore them most of the time. Therefore, they give the weight of hard evidence to the convergence of meanings that they attribute to signs. In this sense, judgment, whether individual or collective, is a signe-trace ; and the induction process that opens, closes, or restricts communication results from interactions between signes-traces. 52 For example: If a person reminds us (hence the term "echo") exactly (hence the term "mimetic") of the message (verbal or non-, conscious or un-) that we direct their way, we have the impression that this message is understood and, at times, that we in general are understood. This impression creates trust. It is precisely because this "echoing" engenders trust that it is produced artificially by "sign manipulators". They mirror the interlocutor's gestures or words. At issue, then, is how to distinguish artifice from sincerity. This is what drives my wish to draw attention to the other, less visible, that is nearly invisible, signs-signs that help to determine whether we should doubt the authenticity of an echo. 53 The expansion of these practices into all domains of everyday life is what leads me to draw attention on the way in which they are implemented. 54 Thus, the monster is it not inherently monstrous, but rather is monstrous in the eyes of its beholder. By reversing the gaze back to the beholder, that person's judgment means he or she is not seeing from the heart but rather is comparing that Other to norms. However, as J. J. Courtine shows, these norms evolved hand-in-hand with the field of medicine, and these very signs used to be located in classifications of disability, then handicap, and finally in difference from norms. See Courtine J. J., "Histoire et anthropologies culturelles de la difformité [Cultural History and Anthropology of Deformity]" in Alain Corbin A., Courtine J. J., and Vigarello G. (Eds.), Histoire du corps, volume 3. Les mutations du regard. Le XXe siècle [History of the Body, volume 3: Mutations of the Gaze, The Twentieth Century], Paris, Seuil, 2006. 55 The consciously expressed sign should be fundamentally distinguished from unconsciously expressed signs. Indeed, an unconsciously expressed sign that is received positively is not an artificial "call to interaction". Rather, it is a spontaneous "echoing of signes-traces". Two individuals "recognize" one another. The intertwining of signs is so complex that it is difficult to identify them one at a time. In the famous words of Montaigne, "[B]ecause he was he and I was I". That says it all. The history of individuals as told by signes-traces cannot be expressed in words. Each minute that creates it exists in the interstitial spaces of visible signs. 56 "Common Sense" is the title of the collection edited by Pierre Bourdieu for Editions de Minuit.

Deconstructing the Processes that Underpin Judgment

Deconstructing the processes that underpin judgment 57 can reduce intolerance to difference. This is because signe-trace recognition essentially functions through the mimesis of the echoing of signs of the Other, in conscious or unconscious mnemonic traces 58 left by our life history within our corporeal matter 59 .

Past experiences leave signes-traces in our memory, conscious or not, of individuals and constitute embedded mnemonic signes-traces 60 . They interact with an event (or person) that is the object of judgment, producing a judgment that appears as the signe-trace of the interaction at hand.

Media's Role in the Construction of Collective Representations

Some interpretations can impose themselves upon a larger or smaller collectivity of people over a longer or shorter period of time, depending on the communal processes of constructing collective representations. The media plays a fundamental role, guiding the way we view and interpret signes-traces. With the media, the signe-trace becomes an indicator of meaning; a meaning that is seen within a "phenomenon seen-phenomenon interpreted" relationship 61 . Thus, media workers may bring an ideological or political discourse that validates the interpretations desired by existing powers; or they may set themselves up as "secret agents" (Jeanneret, 2011) revealing hidden information. They collect signes-traces and bestow upon them indicative value, allowing them to be interpreted either as a trace of an event's realness -or of its fabrication -according to the connotation they give to the evidence. 57 A sign can be borrowed. Etiquette manuals allow their readers to learn the signes-traces of the living conditions of social groups to which they do not belong, and that correspond to a strong "economic capital" and "cultural capital", to borrow Bourdieu's terminology in Distinction: A Social Critique of the Judgment of Taste (1984). This conscious borrowing of signs does not erase interstitial (unconscious) signs that signal the traces of an individual's history for those who can see them. Two types of people are more apt than others to identify these borrowed interstitial signs: -Those in the social group from which the signs are borrowed. This is an unconscious process ; the receiver intuitively feels a lack of echoing in the signs' subtility. -Decoders of signs ; specialists on interstitial signs. 58 Memory (from where the term "mnemonic" comes) records lived situations over the course of life. Work done on memory function is seeing advances thanks to research into Alzheimer's disease. I don't presume to present its results, but wish to remind the reader that most adults believe they have forgotten parts of their childhood or various moments of their livesmoments that later resurface unexpectedly. 59 We should extend this debate by integrating the systemic dimenstion of corporeal function that transcends the artificial distinction between body and mind. 60 Sometimes they get stuck in a sort of closed vault, and lead to behaviours that we wish to eliminate. As an example, consider traumatic experiences. They "remain beyond symbolisation (mental assimilation). They are incorporated and remain active, yet they act upon the unwitting subject from a mental vacuole like a separate vessel, a vault. The trauma, enclosed within symbolic impossibility of recognition, produces a kind of internal cleavage between assimilable and often worked through experiences that produce a plastic identity and the interior of the vault where the trauma lies". The vault's existence, while completely hidden from others and sometimes from the individual, leaves its traces in behaviour. These traces, unconsciously recognized by those with identical traumas, produce a mimetic outburst, an impetus to interaction, an orchestration without a conductor, to paraphrase Bateson (Bateson, 1980). Daniel Bougnoux later spoke of "heat" (Bougnoux, 2002), Fabienne Martin-Juchat of the "body as medium" (MARTIN-JUCHAT, 2008). The signe-trace produces an interior interactional ecology whose finesse and complexity keep rapid interpretation at bay by breaking down the sign's social interpretation. The signe-trace does not limit meaning to the "utterable". If, in the structuralist paradigm's continuity, nonverbal behaviours can sometimes be described in an expressive logic of content and bodily surface, they cannot be reduced to this. The body is also a matter of invisible flesh and internal states whose complex interweaving cannot yet be grasped by any scientific field. On these matters, see also Jean-Jacques Boutaud's bibliography. 61 By the perceiver or, more broadly, by the receiver's subconscious.

The conditions of media expression (type of program, form of language62 , presence of experts, etc.) also give interpretive statements a feeling of legitimacy-or illegitimacy. This result is itself the signe-trace of processes giving legitimacy, the topics of programs in question: the images and associated commentary concerning a past inquiry are presented as representative of reality: the linkages between facts, their recording and broadcasting and reviews, are not highlighted. Thus the expertly-orchestrated but misleading coverage is internalised within the cognitive schemas of those who read, listen to, and view them. The image acts as an instrument of proof. It is redoubtable because it is not put into doubt, except by a well-informed public63 . For this latter group, reality's image-trace relationship is constantly questioned. The critical mass of evidence varies with a person's experience and level of familiarity with information manufacturing. In my opinion, the Ministry of Education should integrate literacy programs not only for reading written material, but also for reading images and for reading digital material.

RISKS, DECISIONS, INTUITION

The use of the term "trace" is commonplace, and is spreading yet further with the dissemination of traces obtained through video surveillance, biometric identification (fingerprints, DNA, etc.), and websites, allowing a human traceability that quite rightly provokes concern. In this text, I have not considered these aspects which other authors explored in the series of work that I coordinated for L'Homme-trace. In this chapter, on occasion, in my analysis of interpersonal communication, I hoped to decentre the obsession with the risks associated with traceability, by demonstrating that it is less the dissemination of traces that is at stake-since for me, they are inherent to humans-than their interpretation.

To respond to the concerns arising from our inability to determine the complexity of the real, several approaches are possible. Without excluding other paths, I propose for the case presently under consideration-that of interpersonal face-to-face communication which integrates the interactions of the body's signes-traces-to valorise intuition. Indeed, beyond its power to simplify, intuition can also illustrate the meaning of the term "signe-trace". Intuition, then, is understood as an experiential signe-trace taken in its temporal continuity, a continuity that escapes our conscious memory, associated with moments that artificially partition duration64 .

Intuition keeps the trace within itself as a process, flux, continuity and movement. In this respect, more than any other signe-trace, intuition provides its listener with a judgment of the human and nonhuman environment that is less subject to the different filters brought on by the introduction of discontinuity into the continuum of time and space. For we mustn't forget that the necessities of logical thought and action lead to classifications, that is, they introduce discontinuity to the world's continuity. These divisions appear as filters to our understanding of "the substantific marrow 65 " of the real, which, with all its levels combined, appear to us as the continuum.

The analysis of the internal complexity of intuition's construction disappears, in a situation of communication, in favor of the pragmatic efficacy of its apparent simplicity. Insofar as it resolves all rational questions that, in any case, can always be questioned because of humanity's limited rationality, intuition produces a feeling of calm and a form of certitude. It also provides justifications beyond its very existence 66 . Intuition, focused on the past, and the instant brilliance of understanding what is happening here and now, appears to be a signetrace surpassing all others both in its content and its potential for understanding and communication.

CONCLUSION

The term "signe-trace" aims to produce a rupture with the ordinary understanding of the sign. Simply to understand the reasons for this choice of terminology, I can draw a simplified analogy. In its most ordinary meaning, the sign is like a coin, with a head, a tails, and a side. The side indicates the appellation (sign). The "heads" (of the sign) corresponds to a signal, the "tails" to an indicator. In the signe-trace paradigm, the signal side of the sign, and its interpretation, become "signes-traces" of differences or similarities in the (past) life history of individuals. This is a systemic process. Applied to "corporeal signes-traces", the paradigm is visualised in a simplified manner in a graph (see annex).

The concept of "corporeal signes-traces" leads us progressively towards the paradigm of a Homme-trace that, through its material body, is a construction of signes-traces, and through its practices is a producer of signes-traces in the environment, all operating in a system of internalisation of externality and externalisation of interiority.

In the following chapters, we will return to this conclusion and to the parameters that have been considered to arrive at them. These illustrations will lead us to an exploration of contemporary situations, both frequent (medical diagnosis, fatigue in the workplace) and rare (the social integration of digital natives with schizophrenia). We will take a look at research on the brain that uses magnetic resonance imaging (MRI). We will examine the relationship between my present discussion on intuition, and MRI, which shows that even when the brain is supposedly at rest, in Resting State Activity (RSA), it shows significant activity that translates into an energetic expenditure by the organ that never decreases by more than 5%. I 65 An expression bh François Rabelais. Here = core content. 66 "Whereof one cannot speak, thereof one must be silent", see the bibliography in L. Wittgneistein. will explore whether, as J. D. Vincent proposes (Vincent, 2012) 67 , this means that, when making complex choices, the decision seems to respond automatically (without specific attention given) to clues from our environment. This question of the brain's signes-traces will appear equally fundamental in remedying the observable changes in pathologies affecting the mentally impaired. 67 Vincent J. D., "Le cerveau, un organe qui ne chôme jamais", in L' Express, N° 32,000 °, 31 October 2012, p. 108.

  Béatrice Galinon-Mélénec 1 Équipe de recherche : Equipe Homme-trace, direction Pr. B.Galinon-Mélénec Pr. B. Galinon-Mélénec est également directrice du CDHET/IUT (Communication & Développement des Hommes, Entreprises, Territoires), et co-directrice de RIGHT (Research : International Group of the Human Trace) : http://rightunivlehavre.wordpress et responsable de Human-trace Complex Systems DC UNESCO Discipline : Sciences de l'Homme et Société/Sciences de l'information et de la communication Domaine : Anthropologie de la communication 1 See so : GALINON-MELENEC Béatrice, « Des signes-traces à l'Homme-trace. La production et l'interprétation des traces placées dans une perspective anthropologique », dans Mille A. (dir.), De la trace à la connaissance à l'ère du web, Intellectica, 2013/1, n°59, pp. 89-113. Publié en anglais sous le titre From "Sign-Traces" to "Human-Trace", L. Brown, Hong Kong, 2013. halshs.archives-ouvertes.fr/halshs-00935346

2

  Watzlawick P., Beavin J..H. Jackson D. J., Pragmatics of Human Communication: A Study of Interactional Patterns, Pathologies and Paradoxes, New York: Norton, 1967. 3 For example, Ghiglione R. (Ed.), Traité de psychologie cognitive [Treatise on Cognitive Psychology], Paris, Dunod, 2 volumes, 1989-1990. 4 Bourdieu P., Distinction: A Social Critique of the Judgement of Taste [orig. La distinction. Critique sociale du jugement], Boston, Harvard University Press, 1984. 5 Barthes R., The Fashion System [orig. Systèmes de mode], and Eléments of Semiology [orig. Éléments de sémiologie].

Translation by L. Kraftowitz.

For heuristic reasons, consider the (quite reductive) example of the sign's intentionality. Then, add a further reduction: the verbal sign ; and another: the situation of communication in copresence. Now, to further reduce this example to the point of absurdity, let's suppose the body is no longer material, a magma of signes-traces. When a person does not wish to speak, that is, to emit intententional sound signs, the other person perceives the sign of silence. To this end, Barthes stated that, "That which is produced against and outside of signs is very quickly identified as a sign"[START_REF] Barthes R | Le Neutre, cours dispensés en 1977-78 au collège de France[END_REF]. Translation by L. Kraftowitz.

Meaning that it is understood as a sign-signal, the sign-signal being a stimulus that provokes attention.

To highlight the strictly individual part of interpretation. But of course, agreement on the meaning of signs lifts the initial uncertainty.

In the sense that nothing is ahistorical.

In a general manner, the sign is necessarily a "signe-trace", since for me the process that produced the sign is present within the sign.

This formulation, "sign-signal", is obviously quite far from the technicist definition of the intentional signal developed by Shannon and Weaver[START_REF] Shannon | La théorie mathématique de la communication[END_REF]) and Pavlov's classical conditioning. These approaches-whose scope of operation makes them successful-are set aside from this context because I am attempting to demonstrate a signe-trace that is placed at the intersection of multiple systems.

In the text below, as in that above, I have italicized key words.

See Yves Jeanneret's chapter on this topic in Galinon-Mélénec, B. (Ed.), L'Homme-trace, CNRS Editions, 2011.

This paragraph partly returns to the initial personal project of L'Homme trace, which aimed to demonstrate that in interpersonal communication, everything is a signe-trace.

See Galinon-Mélénec, B., "Penser autrement la communication. Du sens commun vers le sens scientifique. Du sens scientifique vers la pratique [Rethinking Communication: From Common Sense to Scientific Sense ; from Scientific Sense to Scientific Practice]", pages 37-61.

See these developments in the chapter, "Traces langagières en psychologie sociale de la communication [Language Traces in the Social Psychology of Communication]", Lepastourel, N. and Testé, B., in Galinon-Mélénec B. (Ed.), Paris, CNRS Editions, 2011.

This is the paradox of television programs that analyse images transmitted by television (for example, the show Arrêt sur images). At the same time that these programs respond to journalistic ethics by revealing images that have been manipulated, in what ways and to what aims, they introduce doubt in the viewer, who now questions the authenticity of the images and photos shown on TV. These TV shows train the viewer to search for small signs, the same ones that I wish to highlight here, in view of delivering the highest possible accuracy in trace interpretation.

On these matters, see H.Bergson, Philisophical Intuition (1911), taken up again in La Pensée et le mouvant [Thought and Motion], Paris, 1934. Articles et conférences datantde 1903de à 1923de , PUF, 1969. .