Catherine Donati-Martin

Alain Rouault
email: alain.rouault@math.uvsq.fr

I≤p

TRUNCATIONS OF HAAR DISTRIBUTED MATRICES, TRACES AND BIVARIATE BROWNIAN BRIDGE

Keywords: 2010 Mathematics Subject Classification. Primary 15B52, 60F17; Secondary 46L54 Random Matrices, unitary ensemble, bivariate Brownian bridge, invariance principle

published or not. The documents may come

Truncations of Haar distributed matrices, traces and bivariate Brownian bridges.

Introduction

Let σ be a random permutation uniformly distributed on the symmetric group S n . Define for p, q ≤ n X (n) p,q = #{1 ≤ i ≤ p , σ(i) ≤ q}.

In [START_REF] Chapuy | Random permutations and their discrepancy process[END_REF], G. Chapuy proved that a suitable normalization of X

p,q converges in distribution to the bivariate tied-down Brownian bridge. Note that X (n) p,q = Tr(Σ p,q Σ ⋆ p,q ) where Σ p,q is the truncated matrix of size p × q of Σ, the permutation matrix associated with σ, and ⋆ means adjoint. In this paper, we prove a similar result when the symmetric group is replaced by the unitary group or the orthogonal group, equipped with the Haar measure.

Let U be a Haar unitary, resp. orthogonal, matrix in U(n), resp. in O(n). We consider, for p ≤ n and q ≤ n, the upper-left p × q submatrix V p,q and the p × p Hermitian matrix H p,q = V p,q V ⋆ p,q . We are interested in the asymptotic behavior of

T p,q = Tr H p,q = i≤p,j≤q |U i,j | 2 .
(1.1) Setting Y (n) p,q = T p,q -ET p,q , we define a sequence of two-parameter processes W (n) by

W (n) := Y (n) ⌊ns⌋,⌊nt⌋ , s, t ∈ [0, 1]
Chapuy used the space C([0, 1] 2 ) completing its process in such a way that it is continuous and affine on each closed "lattice triangle". We prefer using the multidimensional generalization of Skorokhod space D([0, 1] 2 ) given by [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF]. It consists of functions from [0, 1] 2 to R which are at each point right continuous (with respect to the natural partial order of [0, 1] 2 ) and admit limits in all "orthants". The space D([0, 1] 2 ) is endowed with the topology of Skorokhod (see [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF] for the definition). Our main result is the following Theorem 1.1. The process W (n) converges in distribution in D([0, 1] 2 ) to a tied-down Brownian bridge 2 β W (∞) where W (∞) is a centered continuous Gaussian process on [0, 1] 2 of covariance

E[W (∞) (s, t)W (∞) (s ′ , t ′ )] = (s ∧ s ′ -ss ′ )(t ∧ t ′ -tt ′ ),
β = 2 in the unitary case and β = 1 in the orthogonal case.

Previous works are related to our problem. First, Borel in 1906 shows that for a uniformly distributed point on the (n -1)-dimensional (real) sphere, the scaled first coordinate converges in distribution to the standard normal. Since that time, many authors studied the entries and partial traces of matrices from the orthogonal and unitary group. In particular Diaconis and d'Aristotile ( [START_REF] D'aristotile | An invariance principle for triangular arrays[END_REF][START_REF] D'aristotile | Brownian motion and the classical groups, Probability, statistics and their applications: papers in honor of Rabi Bhattacharya[END_REF]) proved that the sequence of one-parameter processes

   ⌊ns⌋ i=1 U ii , s ∈ [0, 1]
   n converges in distribution to the complex Brownian motion. Besides, Silverstein [START_REF] Silverstein | Describing the behavior of eigenvectors of random matrices using sequences of measures on orthogonal groups[END_REF] proved that for q fixed, the sequence of one-parameter processes

   n 1/2 ( ⌊ns⌋ i=1 |U iq | 2 -s) , s ∈ [0, 1]    n (1.2)
converges in distribution to the Brownian bridge.

In the paper [START_REF] Silverstein | Weak convergence of random functions defined by the eigenvectors of sample covariance matrices[END_REF], Silverstein discussed the similarity between the matrix of eigenvectors of a (real) sample covariance matrix and a Haar distributed orthogonal matrix, with a one-parameter parameter process analogous to (1.2). We also refer to [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF]Chapter 10] for the behavior of eigenvectors of sample covariance matrices and universality conjectures. To extend this study and after reading a first draft of our result, Djalil Chafai conjectured that if M is a n × n matrix with i.i.d. entries having the same four first moments as the complex Gaussian standard and if U denotes the matrix of eigenvectors of N = M M * , then the sequence W (n) obtained by changing U ij into U ij converges to the tied-down Brownian bridge as in Theorem 1.1. At the moment of posting the present version, we are aware that Florent Benaych-Georges ( [START_REF] Benaych-Georges | Eigenvectors of Wigner matrices: Universality of global fluctuations hal-00583889[END_REF]) answered positively to the conjecture when N is a Wigner matrix, under a fourth moment hypothesis.

The rest of the paper is organized as follows. In section 2, we introduce the basic notions on permutations, partitions, classical cumulants. The paper is then split into two parts. Section 3 deals with the unitary case and Section 4 is devoted to the orthogonal case. In each part, we develop the combinatorics associated with the group1 (the Weingarten function and the associated cumulants).

In particular, we state a formula for the cumulants of variables of the form X = Tr(AU BU ⋆ ) for deterministic matrices A, B of size n. In the unitary case, the formula follows from the results of [START_REF] Mingo | Second order freeness and fluctuations of random matrices: II. Unitary random matrices[END_REF]. We then apply the above formula to the computation of the second and fourth cumulants of T p,q . The proof of Theorem 1.1 is then divided in two parts: tightness of the family of distributions of W (n) and convergence of the finite dimensional laws. To prove the tightness, we use a criterion of Bickel and Wichura for two-parameter processes, with the help of the estimates obtained in section 3.2, resp 4.3. The convergence of the finite dimensional distributions to Gaussian distributions relies on the computations of their cumulants and their asymptotics. The expression of the cumulants follows from the previous section and their limit follows from the asymptotics of cumulants of unitary, resp. orthogonal, Weingarten functions, obtained in [START_REF] Collins | Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability[END_REF], resp. [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF]. Let us mention that simultaneously and independently of the present paper, precise computations in the orthogonal case are presented in [START_REF] Redelmeier | Genus expansion for real Wishart matrices[END_REF] and [START_REF] Redelmeier | A definition for second-order freeness in the real case and the asymptotic real second-order freeness of several real matrix ensembles[END_REF]. In section 5, we give complementary remarks and connections with other problems.

Preliminaries

For n a positive integer we set [n] := {1, 2, • • • , n}.

Partitions, permutations. We call

A = {A 1 , • • • , A s } a partition of [k] if the A i (1 ≤ i ≤ s) are pairly disjoint, non-void subsets of [k] such that A 1 ∪ • • • ∪ A s = [k]. We call A 1 , • • • , A s the blocks of A.
The number of blocks of A is denoted by #(A). The set of all partitions of [k] is denoted by P(k). One partition A is said to refine another B, denoted A ≤ B provided every block of A is contained in some block of B. Given two partitions A and B, A ∧ B (resp. A ∨ B) is the largest (resp. smallest) partition which refines (resp. is refined by) both A and B. Under these operations, the partially ordered set P(k) is a lattice. We denote by 1 k the largest partition of [k] (one-block partition), and by 0 k the smallest one (k-blocks partition). For A, B ∈ P(k) with A ≤ B we denote by [A, B] the interval

[A, B] = {C ∈ P(k) | A ≤ C ≤ B} .
Since we will make some use of the Möbius function for partitions, let us just recall ([28] Sect 3.6) that for two real functions f, g defined on {(A, B) ∈ P(k) × P(k); A ≤ B}, we have:

f (A, B) = C∈[A,B] g(A, C) (2.1)
if and only if

g(A, B) = C∈[A,B] Möb(C, B)f (A, C) (2.2) with Möb(C, B) := i (-1) i-1 (i -1)! p i
where p i is the number of blocks of B that contain exactly i blocks of C. Let S k be the set of permutations on k elements. With σ ∈ S k , we associate the set C(σ) of its cycles, whose number is denoted by #(σ). We denote by 0 σ the partition whose blocks are the cycles of σ, or when the context is clear, just by σ. Given two pairings p 1 , p 2 , we define the graph Γ(p 1 , p 2 ) as follows. The vertex set is [2k] and the edge set consists of the pairs of p 1 and p 2 . Let loop(p 1 , p 2 ) the number of connected components of Γ(p 1 , p 2 ). 2.2. Cumulants. For r ≥ 1, κ r denotes the classical cumulant of order r (see [START_REF] Shiryayev | Probability. Graduate Texts in Mathematics[END_REF], [START_REF] Mingo | Second order freeness and fluctuations of random matrices: II. Unitary random matrices[END_REF] p.215). It is a multilinear function of r variables defined as follows: if a 1 , . . . , a r are random variables,

For π ∈ S k , a partition A = (A 1 , . . . A l ) of [k] is called π-invariant if π leaves invariant each block A i that is 0 π ≤ A (which we just write π ≤ A).
κ r (a 1 , . . . , a r ) = C∈P(r) Möb(C, 1 r ) E C (a 1 , . . . a r )
where for C = {C 1 , . . . , C k } ∈ P(r),

E C (a 1 , . . . , a r ) = r i=1 E( j∈C i a j ) .
(2.3) More generally, relative cumulants are defined, for A ≤ B ∈ P(r) as

κ A,B (a 1 , • • • , a r ) = C∈[A,B] E C (a 1 , • • • , a r ) Möb(C, B) . (2.4)
From the equivalence between (2.1) and (2.2) we have, for any A ≤ B ∈ P(r) Tr( j∈C M j ).

E B (a 1 , • • • , a r ) = C∈[A,B] κ A,C (a 1 , • • • , a r ) . ( 2
Let s be a fixed integer and let {M 1 , . . . , M s } n be a sequence of n × n deterministic matrices. We say that {M 1 , . . . M s } n has a limit distribution if there exists a non commutative probability space (A, ϕ) and a 1 , . . . a s ∈ A such that for any polynomial p in s non commuting variables, lim n→∞ tr(p(M 1 , . . . , M s )) = ϕ(p(a 1 , . . . , a s )).

The unitary group

3.1. Preliminary remarks: Some moments. Let U be a Haar distributed matrix on U(n) the unitary group of size n. We have the following important relations (see [START_REF] Hiai | The Semicircle Law, Free Random Variables and Entropy[END_REF], Proposition 4.2.3 and [START_REF] Jiang | A variance formula related to a quantum conductance problem[END_REF]). If U i,j is the generic element of U , then the random variable |U i,j | 2 follows the beta distribution on [0, 1] with parameter (1, n -1) of density (n -1)(1x) n-2 . Thus,

E|U i,j | 2 = 1 n , E|U i,j | 4 = 2 n(n + 1)
,

Var |U i,j | 2 = n -1 n 2 (n + 1) , (3.1)
and more generally

E|U ij | 2k = (n -1)!k! (n -1 + k)! . (3.2) If X = |U i,j | 2 and Y = |U i,k | 2 with k = j, then (X, Y ) follows the Dirichlet distribution on {0 ≤ x, y, x + y ≤ 1} with parameters (1, 1, n -2) of density (n -1)(n -2)(1 -x -y) n-3 . Thus E |U i,j | 2 |U i,k | 2 = 1 n(n + 1) . (3.3) Besides, if i = k, j = ℓ, E |U i,j | 2 |U k,ℓ | 2 = 1 n 2 -1 . (3.4) E|U ij | 2 = pqE|U 11 | 2 = pq n . (3.5)
and

Var T p,q = pq n 2 -n(p + q) + pq n 2 (n 2 -1) . (3.6)
Assume that p/n → s, q/n → t, then,

lim n 1 n ET p,q = st , lim n Var T p,q = st(1 -(s + t) + st) = st(1 -s)(1 -t).
Proof:

ET 2 p,q = i,k≤p,j,l≤q E|U ij | 2 |U kl | 2 = i≤p,j≤q E|U ij | 4 + i≤p,j =l≤q E|U ij | 2 |U il | 2 + i =k≤p,j≤q E|U ij | 2 |U kj | 2 + i =k≤p,j =l≤q E|U ij | 2 |U kl | 2 = pq 2 n(n + 1) + pq(q -1) 1 n(n + 1) +p(p -1)q 1 n(n + 1) + p(p -1)q(q -1) 1 n 2 -1 = pq p + q n(n + 1) + (p -1)(q -1) n 2 -1 .
This yields (3.6).

Remark 3.2. An easy consequence of the above Proposition is

lim n 1 n T ⌊ns⌋,⌊nt⌋ = st in probability.
Actually the convergence is uniform in s, t ∈ [0, 1] (see section 5).

3.2.

Combinatorics for the unitary group. Let U(n) denote the unitary group of size n endowed with the Haar probability measure. The generic element will be denoted by U and its (i, j) coefficient by U ij . In [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF], Collins and Sniady proved the following integration formula on U(n), see also [START_REF] Collins | Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability[END_REF].

Let M U 2k denote the set of pairings of [2k], pairing each element of [k] with an element of [ k]. Let G U(n) be the Gram matrix 2 G U(n) = (G U(n) (p 1 , p 2 )) p 1 ,p 2 ∈M U 2k := (n loop(p 1 ,p 2 ) ) p 1 ,p 2 ∈M U 2k .
Then the unitary Weingarten matrix Wg U(n) is the pseudo inverse of G U(n) . 2 The term of Gram matrix comes from the theory of representations of groups and algebras, see [START_REF] Collins | On some properties of orthogonal Weingarten functions[END_REF].

Proposition 3.3. For every choice of indices i = (i 1 , . . . , i k , i1, . . . , ik) and j = (j 1 , . . . , j k , j1, . . . , jk),

E U i 1 j 1 . . . U i k j k Ūi1j1 . . . Ūik,jk = p 1 ,p 2 ∈M U 2k δ p 1 i δ p 2 j Wg U(n) (p 1 , p 2 ) (3.7)
where δ p 1 i (resp. δ p 2 j ) is equal to 1 or 0 if i (resp. j) is constant on each pair of p 1 (resp. p 2 ) or not.

It is clear that with each pairing

p ∈ M U 2k we can associate a unique σ ∈ S k such that p = k i=1 (i, σ(i)). It is known that if p 1 is associated with α and p 2 with β, then Wg U(n) (p 1 , p 2 ) is a function of βα -1 denoted by Wg(n, βα -1 ), so that (3.7) becomes E U i 1 j 1 . . . U i k ,j k Ūi1j1 . . . Ūik,jk = α,β∈S k δα i δβ j Wg(n, βα -1 ) (3.8)
where δα

i = 1 if i(s) = i(α(s)) for every s ≤ k and 0 otherwise. In particular, if π ∈ S k , we have Wg(n, π) = E(U 11 . . . U kk Ū1π(1) . . . Ūkπ(k) ) . (3.9)
The Weingarten functions for k = 1, 2 are given by (see [START_REF] Collins | Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability[END_REF]): We set, for

Wg(n, (1)) = 1 n Wg(n, (1)(2)) = 1 n 2 -1 , Wg(n, ( 12
)) = - 1 n(n 2 -1) . ( 3
1 ≤ i ≤ r, X i = Tr(D i U DīU ⋆ ) . Then, κ r (X 1 , . . . , X r ) = α,β∈Sr A C βα -1 ,A Tr α ( D) Tr β -1 (D) (3.11)
where in the second sum A ∈ P(r) is such that

βα -1 ≤ A and A ∨ β ∨ α = 1 r , (3.12)
and C σ,A are the relative cumulants of the unitary Weingarten function (see [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF]) . Moreover, if the sequence {D, D} n has a limit distribution, then for r ≥ 3, lim n→∞ κ r (X 1 , . . . , X r ) = 0. ) is not given exactly on this form in [START_REF] Mingo | Second order freeness and fluctuations of random matrices: II. Unitary random matrices[END_REF] but in our particular case where X i = Tr(D i U DīU ⋆ ) with the D i deterministic, the formulas are equivalent. We shall prove a similar formula in the orthogonal case.

For the sake of completeness, let us give the meaning of C π,A . Let π ∈ S r and a i = U ii Ūiπ(i) . We denote for a π invariant partition Π ∈ P(r)

E Π (π) := E Π (a 1 , . . . , a r ) , (3.13)
i.e., owing to (3.9)

E Π (π) = s k=1 Wg(π| V k ) , (3.14)
where Π = {V 1 , . . . , V s }. Then, if we set

C π,A := κ π,A (a 1 , . . . , a r ) ,
the summation formula (2.4) yields

C π,A = π≤C≤A Wg(π| V 1 ) • • • Wg(π| V 1 ) Möb(C, A) (3.15)
and the reverse one (2.5)

E C (π) = A∈[π,C] C π,A . (3.16)
We will use these formulas in the orthogonal case.

3.3.

Computations of the second and fourth cumulants of T p,q .

3.3.1. The covariance of T p,q . The fundamental remark is that

H p,q = D 1 U D1U ⋆
with D 1 = I p , D1 = I q , where I k is the matrix of projection on the k first coordinates. Note that D 1 and D1 are commuting projectors and that if p/n → s, q/n → t, {D 1 , D1} n has a limit distribution with a 1 , a 2 commuting projectors on (A, ϕ) such that a 1 a 2 = a 1 if s < t and = a 2 if t < s, and ϕ(a 1 ) = s, ϕ(a 2 ) = t. Let p, p ′ , q, q ′ ≤ n. We now give an application of Proposition 3.4 to the computation of cov(T p,q , T p ′ ,q ′ ) = κ 2 (T p,q , T p ′ ,q ′ ). This can also be done, using the computations of Section 3.1. We set D 2 = I p ′ , D2 = I q ′ and apply formula (3.11) to X 1 = T p,q , X 2 = T p ′ ,q ′ , r = 2. The different possibilities for α, β and A satisfying (3.12) are gathered in the following table, where 0 and 1 stand for the convenient permutation or partitions on [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF]:

α β βα -1 A 0 0 0 1 1 0 1 1 0 1 1 1 1 1
0 0 or 1 The relative cumulants are given by (see (3.10), (2.4))

C 1,1 = - 1 n(n 2 -1) , C 0,1 = - 1 n 2 + 1 n 2 -1 = 1 n 2 (n 2 -1) , C 0,0 = 1 n 2
The different products of traces are quite obvious, so that plugging into (3.11), we get

κ 2 (T p,q , T p ′ ,q ′ ) = (3.17) (p ∧ p ′ )(q ′ ∧ q ′ ) n 2 -1 - (p ∧ p ′ )qq ′ n(n 2 -1) - pp ′ (q ∧ q ′ ) n(n 2 -1) + pp ′ q ′ q ′ n 2 (n 2 -1)
.

In the limit p/n → s, q/n → t, p ′ /n → s ′ , q ′ /n → t ′ , we get lim n κ 2 (T p,q , T p ′ ,q ′ ) = (s ∧ s ′ )(t ∧ t ′ ) -(s ∧ s ′ )tt ′ -ss ′ (t ∧ t ′ ) + ss ′ tt ′ = (s ∧ s ′ -ss ′ )(t ∧ t -tt ′ ). (3.18)
3.3.2. The fourth cumulant. We now give an estimate for κ 4 (T p,q ). From (3.11) with r = 4,

κ 4 = α,β∈S 4 A C βα -1 ,A Tr α ( D) Tr β -1 (D) (3.19)
where A runs over the partitions of [START_REF] Benaych-Georges | Eigenvectors of Wigner matrices: Universality of global fluctuations hal-00583889[END_REF] satisfying condition (3.12), and finally

D i = I p , Dī = I q (i ≤ 4) .
We have now Tr

β -1 (D) = p #(β) , Tr α ( D) = q #(α) .
In [8, Cor. 2.9], Collins proved that the order of C βα -1 ,A is at most n -8-#(βα -1 )+2#(A) . Finally,

C βα -1 ,A Tr β -1 (D) Tr α ( D) = O n -8-#(βα -1 )+2#(A) p #(β) q #(α) . (3.20)
From equation [START_REF] Mingo | Second order freeness and fluctuations of random matrices. I. Gaussian and Wishart matrices and cyclic Fock spaces[END_REF] in [START_REF] Mingo | Second order freeness and fluctuations of random matrices: II. Unitary random matrices[END_REF], we see that

2#(A) + #(α) + #(β) -#(βα -1 ) ≤ 6
and the expression in (3.20) is of order

n -8-#(π)+2#(A) p #(β) q #(α) ≤ p #(β) q #(α) n -2-#(α)-#(β) ≤ (p/n) #(β)-1 (q/n) #(α)-1 pqn -4 ≤ p 2 q 2 n -4 .
We conclude that (n) is in force as soon as the condition C(β, γ) with β > 1 is satisfied (see ( 2), (3) in [START_REF] Bickel | Convergence criteria for multiparameter stochastic processes and some applications[END_REF]):

κ 4 = O p 2 q 2 n -4 . ( 3
E(|W (n) (B)| γ 1 |W (n) (C)| γ 2 ) ≤ (µ(B)) β 1 (µ(C)) β 2 (3.22)
where γ = γ 1 +γ 2 > 0 and β = β 1 +β 2 > 1, B and C are two adjacent blocks in [0, 1] 2 and W (n) (B) denotes the increment of W (n) around B, given by

W (n) (B) = W (n) s ′ ,t ′ -W (n) s ′ ,t -W (n) s,t ′ + W (n) s,t for B =]s, s ′ ]×]t, t ′ ],
µ is a finite positive measure on [0; 1] 2 with continuous marginals.

From Cauchy-Schwarz inequality, (3.22) is implied by

E(|W (n) (B)| 2γ 1 ) ≤ (µ(B)) 2β 1 . (3.23)
Moreover, it is enough to consider blocks whose corner points are in

T n = { p n , 0 ≤ p ≤ n} × { q n , 0 ≤ q ≤ n} (see [5], p. 1665.) Let p ≤ p ′ ≤ n and q ≤ q ′ ≤ n and B =] p n , p ′ n ]×] q n , q ′ n ] W (n) (B) := ∆ (n) p,q (p ′ , q ′ ) = Y (n) p ′ ,q ′ -Y (n) p ′ ,q -Y (n) p,q ′ + Y (n) p,q = p+1≤i≤p ′ q+1≤i≤q ′ |U i,j | 2 -E(|U i,j | 2 ).
If we show that there exists a constant C, such that for all n,

E ∆ (n) p,q (p ′ , q ′ ) 4 ≤ C (p ′ -p) 2 (q ′ -q) 2 n 4 , (3.24)
then (3.23) is satisfied with γ 1 = 2, β 1 = 1 and µ is the Lebesgue measure. Since ∆

p,q (p ′ , q ′ ) has the same distribution as

Y (n) p ′ -p,q ′ -q , it is enough to show E Y (n) p,q 4 = O(p 2 q 2 n -4 ) . (3.25)
If X is a real random variable, an elementary computation gives

E(X -EX) 4 = κ 4 + 3κ 2 2 , (3.26)
where κ r is the r-th cumulant of X. Taking X = T p,q = Tr D 1 U D1U ⋆ , we saw above in (3.6) that

κ 2 = Var T p,q ≤ 2 pq n 2 . ( 3
) i≤k ∈ R, (s i , t i ) i≤k ∈ [0, 1] 2 . We must prove the convergence in distribution of X (n) := k i=1 a i W (n) s i ,t i to a Gaussian distribution. Let us denote p i = ⌊ns i ⌋, q i = ⌊nt i ⌋. Then X (n) = k i=1 a i Y (n) p i ,q i = k i=1 a i [Tr(D i U DīU ⋆ ) -E(Tr(D i U DīU ⋆ ))]
where D i = I p i , Dī = I q i . {D i , Dī, i = 1, . . . k} are commuting projectors with a limit distribution {q i , qī, i = 1, . . . k} on a probability space (A, φ 1 ) with φ 1 (q i ) = s i , φ 1 (qī) = t i and q i q j = q i if u i ≤ u j (and = q j otherwise) where u i = s i for i odd and

u i = t i for i even. Let r ≥ 3, then κ r (X (n) , . . . , X (n) ) = k i 1 ,...,ir=1 a i 1 . . . a ir κ r (Y (n) p i 1 ,q i 1 , . . . , Y (n) p ir ,q ir ) = k i 1 ,...,ir=1 a i 1 . . . a ir κ r (X i 1 , . . . , X ir )
where

X ip = Tr(D ip U Dī p U ⋆ ). From Proposition 3.4 lim n→∞ κ r (X i 1 , . . . , X ir ) = 0. (3.28)
Now, the second cumulant is given by

κ 2 (X (n) , X (n) ) = k i,j=1
a i a j κ 2 (Tr(D i U DīU ⋆ ), Tr(D j U DjU ⋆ )).

From (3.18)

lim n κ 2 (Tr(D i U DīU ⋆ ), Tr(D j U DjU ⋆ )) = (s i ∧ s j -s i s j )(t i ∧ t j -t i t j ).
Thus, we get the convergence of X (n) to a centered Gaussian distribution with variance k i,j=1

a i a j (s i ∧ s j -s i s j )(t i ∧ t j -t i t j ).
It follows that the finite-dimensional laws of the process W (n) converge to the finite-dimensional laws of the tied-down Brownian bridge. The generic element will be denoted by O and its (i, j) coefficient by O ij . In [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF], Collins and Sniady proved the following integration formula on O(n), see also [START_REF] Collins | On some properties of orthogonal Weingarten functions[END_REF]Theorem 2.1] for the following formulation. Let G O(n) be the Gram matrix 3

G O(n) = (G O(n) (p 1 , p 2 )) p 1 ,p 2 ∈M 2k := (n loop(p 1 ,p 2 ) ) p 1 ,p 2 ∈M 2k ,
where M 2k is the set of pairings of [2k] defined in Section 2.1. Then, the orthogonal Weingarten matrix Wg O(n) is the pseudo inverse of G O(n) .

Proposition 4.1. For every choice of indices i = (i 1 , . . . , i k , i1, . . . , ik) and j = (j 1 , . . . , j k , j1, . . . , jk),

E O i 1 j 1 . . . O i k j k O i1j1 . . . O ikjk = p 1 ,p 2 ∈M 2k δ p 1 i δ p 2 j Wg O(n) (p 1 , p 2 ) (4.1)
where δ p 1 i (resp. δ p 2 j ) is equal to 1 or 0 if i (resp. j) is constant on each pair of p 1 (resp. p 2 ) or not.

We now identify M 2k as the quotient set S 2k /H k where H k is a subgroup of S 2k known as the hyperoctahedral group and defined as follows (see [START_REF] Graczyk | The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution[END_REF] and [START_REF] Capitaine | Cumulants for random matrices as convolutions on the symmetric group[END_REF]). With each g ∈ S 2k , we associate the product of disjoint transpositions:

η(g) = k i=1 (g(i) g( ī)) ,
which can be identified as an element of M 2k .

We set γ = k i=1 (i ī) and define H k = {g ∈ S 2k , γg = gγ}, the centralizer of γ. We have the following equivalence

η(g) = η(g ′ ) ⇐⇒ ∃h ∈ H k , g = g ′ h , implying M 2k ∼ = S 2k /H k .
According to Proposition 3.3 in [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF], see also [START_REF] Collins | On some properties of orthogonal Weingarten functions[END_REF], Wg O(n) (η(g 1 ), η(g 2 )) depends only on the conjugacy class of η(g 1 )η(g 2 ) and we can define the orthogonal Weingarten function of S 2k , denoted by WΛ O(n) (see [6, p. 511]) by

W Λ O(n) (g) = Wg O(n) (η(Id), η(g))
and we have

Wg O(n) (η(g 1 ), η(g 2 )) = WΛ O(n) (g -1 1 g 2 ).
In the sequel, we shall drop the superscript O(n) keeping in mind for the asymptotics that W Λ depends on the size n. It is clear from the definitions that WΛ is invariant on the classes of the 3 See footnote 1.

double coset space H k \S 2k /H k . Then, formula (4.1) can be written as

E O i 1 j 1 . . . O i k j k O i1j1 . . . O ikjk = 1 |H k | 2 g 1 ,g 2 ∈S 2k δ η(g 1 ) i δ η(g 2 ) j WΛ(g -1 1 g 2 ).
(4.2) We now describe the generators of S 2k /H k following the presentation in [START_REF] Graczyk | The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution[END_REF], (see also [START_REF] Capitaine | Cumulants for random matrices as convolutions on the symmetric group[END_REF] with a slightly different definition for particular permutations). For ǫ = (ǫ 1 , . . . , ǫ k ) ∈ {-1, 1} k , we define τ ǫ by

τ ǫ = i,ǫ i =-1 (i ī) ∈ H k .
For π ∈ S k , we define t π ∈ S 2k by

t π (i) = i; t π ( ī) = π(i) .
We shall now parametrize S 2k /H k using special permutations. Definition 4.2. A pair (ǫ, π) ∈ {-1, 1} k × S k is particular if, for any cycle c of π, we have ǫ i = 1 where i is the smallest element of c. We say that the corresponding permutation g ǫ,π := τ ǫ t π in S 2k is particular. Proposition 4.3. (see Theorem 8 in [START_REF] Graczyk | The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution[END_REF]) The class S 2k /H k containing τ ǫ t π has exactely 2 #(π) elements of the form τ ǫ ′ t π ′ . Every class of S 2k /H k contains exactely one particular permutation of the form τ ǫ t π . There are

(2k)! 2 k k! particular permutations g l = τ ǫ l t π l , l ≤ (2k)! 2 k k! . Let Σ ∈ S 2k .
From the above Proposition, there exists a particular pair (ǫ, σ) such that Σ = τ ǫ t σ h with h ∈ H k . In particular,

Σ ∼ t σ in H k \S 2k /H k and WΛ(Σ) = WΛ(t σ ).
We now recall how to find σ from Σ (see [START_REF] Graczyk | The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution[END_REF]Proposition 18]). First consider the pairing η(Σ) and the 2-regular graph Γ(Σ) (i.e. with all the cycles having even length) defined by

Γ(Σ) = η(Σ) ∪ η(Id).
Then, any cycle Γ j of length 2q j of Γ(Σ) is of the form

Γ j = (i ± j1 , i ∓ j1 , . . . , i ± jq j , i ∓ jq j )
where the ordered couple i ± i ∓ is such that i ± i ∓ ∈ {i, ī} with i ± = i implies i ∓ = ī and i ± = ī implies i ∓ = i. By convention, the starting point of Γ j is i ± j1 = min{ ījl , l ≤ q j }. Set σ j = (i j1 , . . . , i jq j ). Then σ = σ j .

4.2.

The variance of T p,q . From Proposition 4.1 and the value of the Weingarten function for k = 2 ( [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF], [10, Examples 2.1-3.1], see also [START_REF] Jiang | A variance formula related to a quantum conductance problem[END_REF]), we have:

For i = k, j = ℓ, E O 2 ij O 2 kℓ = n + 1 n(n + 2)(n -1)
.

For j = k E O 2 ij O 2 ik = 1 n(n + 2) and EO 4 ij = 3 n(n + 2)
.

From these 3 relations, we easily get Proposition 4.4. The mean and the variance of T p,q are given by:

ET p,q = i≤p,j≤q EO 2 ij = pq EO 2 11 = pq n . (4.3)
and

Var T p,q = 2pq n 2 -n(p + q) + pq n 2 (n + 2)(n -1) . ( 4

.4)

Assume that p/n → s, q/n → t, then,

lim n 1 n ET p,q = st , lim n Var T p,q = 2st(1 -(s + t) + st) = 2st(1 -s)(1 -t).

4.3.

Mixed moments of the variables T p i ,q i . As in the unitary case, we need to compute cumulants κ r (X 1 , . . . X r ) where the variables X i are of the form X i = Tr(D i ODīO -1 ).

We shall first establish an analogue of Proposition 3.4. We assume in the following that the matrices D i are deterministic and symmetric. Even if we shall deal later only with diagonal matrices, the computations are the same for general matrices. Then,

κ r (X 1 , . . . , X r ) = (α,β,ǫ)∈Sr×Sr×{±1} r λ α,β,r A C σ,A Tr α (D) Tr β -1 ( D) (4.5)
where in the second sum

• σ ∈ S r is a function of α, β, ǫ satisfying t α -1 τ ǫ t β ∼ t σ in H k \S 2k /H k see (4.8), • A ∈ P(r) is such that σ ≤ A and A ∨ α ∨ β = 1 r , • C σ,A
are the relative cumulants of the orthogonal Weingarten function (see [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF])

• the combinatorial coefficient λ α,β,ǫ is 2 r-#(α)-#(β) .
Proof: We first give a formula for the mixed moments, following [START_REF] Capitaine | Cumulants for random matrices as convolutions on the symmetric group[END_REF]Eq (20)]:

E[ k i=1 Tr(D i ODīO -1 )] = 1 |H k | 2 g,g ′ ∈S 2k W Λ(g -1 g ′ )M + D (g)M - D ((g ′ ) -1 ) (4.6)
where the ± generalized moments M are functions on S 2k , resp. H k -right and H k -left invariant: for Y a k-tuple of random real matrices,

M + Y (g) = E[Tr π l (Y ǫ 1 (l) 1 , . . . , Y ǫ k (l) k )] if g ∈ g l H k M - Y (g) = E[Tr π l (Y ǫ 1 (l) 1 , . . . , Y ǫ k (l) k )] if g ∈ H k g l with Y -1 = Y t the transpose of Y .
Formula (4.6) follows from (4.2) and straightforward computations. Since the matrices D are symmetric, the moments do not depend of the sequence ǫ. Using the parametrisation g = τ ǫ 1 t α h and g ′ = τ ǫ 2 t β h ′ with h, h ′ ∈ H k , we can rewrite (4.6) as

E[ k i=1 Tr(D i ODīO -1 )] = α,β∈S k ,ǫ∈{±1} k λ α,β,k W Λ(t α -1 τ ǫ t β ) Tr α (D) Tr β -1 ( D) (4.7)
The coefficient λ α,β,k comes from the fact that we do not impose the sequences ǫ 1 and ǫ 2 associated with α, resp. β to be particular and ǫ = ǫ 1 ǫ 2 . From Proposition 4.3, λ α,β,k = 2 k-#(α)-#(β) As recalled before, with the triplet (α, β, ǫ), we can associate a particular pair (σ, ǫ ′ ) such that:

t α -1 τ ǫ t β = τ ǫ ′ t σ h (4.8)
with h ∈ H. Then, WΛ(t α -1 τ ǫ t β ) = WΛ(t σ ), denoted below by WΛ(σ). In the following, we will denote by σ := σ(α, β, ǫ) the permutation constructed above.

We now compute the cumulant, following the same scheme as in [START_REF] Mingo | Second order freeness and fluctuations of random matrices: II. Unitary random matrices[END_REF]. Let C = {V 1 , . . . V k } ∈ P(r). We denote by S V i the permutations on V i .

E C (X 1 , . . . , X r ) = k i=1 E   j∈V i X j   = (α i ,β i ,ǫ i )∈S V i ×S V i ×{±1} |V i | ;i≤k ( k i=1 λ α i ,β i ,|V i | )W Λ(σ 1 ) . . . W Λ(σ k ) . . . . . . Tr α 1 (D) Tr β -1 1 ( D) . . . Tr α k (D) Tr β -1 k ( D) = (α, β, ǫ) ∈ S r × S r × {±1} r α, β ≤ C λ α,β,r WΛ C (σ) Tr α (D) Tr β -1 ( D)
where σ ∈ S r is such that σ ≤ C and (see [11, section 3.4])

WΛ C (σ) = k i=1 WΛ(σ |V i ).
Collins and Sniady defined the cumulants of orthogonal Weingarten functions that we will denote by C σ,A for A a partition σ-invariant. They satisfy like (3.15) and (3.16):

WΛ C (σ) = A∈[σ,C] C σ,A .
Thus,

E C (X 1 , . . . X r ) = α, β ≤ C ǫ ∈ {±1} r λ α,β,r A∈[σ,C] C σ,A Tr α (D) Tr β -1 ( D). Now, κ r (X 1 , . . . , X r ) = C Möb(C, 1 r ) E C (X 1 , . . . X r ) = C Möb(C, 1 r ) (α, β, ǫ) ∈ S r × S r × {±1} r α, β ≤ C λ α,β,r A∈[σ,C] C σ,A Tr α (D) Tr β -1 ( D) = (α,β,ǫ)∈Sr×Sr×{±1} r λ α,β,r σ≤A {C;A,α,β≤C} Möb(C, 1 r )C σ,A Tr α (D) Tr β -1 ( D) = (α,β,ǫ)∈Sr×Sr×{±1} r λ α,β,r σ≤A;A∨α∨β=1r C σ,A Tr α (D) Tr β -1 ( D)
where the last equality follows from

C;A,α,β≤C Möb(C, 1 r ) = 1 if A ∨ α ∨ β = 1 r , 0 otherwise.
We now study the asymptotic of (4.5) when n → ∞. We assume that the family of n × n matrices (D i ) has a limit distribution. It is known (see [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF]Theorem 3.16]) that the order of C σ,A is n -2r-#(σ)+2#(A) . In [START_REF] Collins | Integration with respect to the Haar measure on unitary, orthogonal and symplectic group[END_REF], the asymptotic of the cumulant is given in terms of a metric l on pairings. We use that l(p σ , p Id ) = |σ| := r -#(σ) where p σ = (i, σ(i)) is the pairing associated with σ and p Id = γ. Now,

Tr α (D) Tr β -1 ( D) = n #(α)+#(β) tr α (D) tr β -1 ( D) = O(n #(α)+#(β) ).
Therefore, for given α, β, ǫ, the corresponding term in the cumulant is of order n -2r-#(σ)+2#(A)+#(α)+#(β) , where σ and A satify the conditions quoted in Proposition 4.5, i.e. σ is the permutation defined by (4.8), σ ≤ A and A ∨ α ∨ β = 1 r .

(4.9)

Proposition 4.6. Under the conditions (4.9), for r ≥ 3,

-2r -#(σ) + 2#(A) + #(α) + #(β) + 1 ≤ 0 . (4.10) Corollary 4.7. Let {D i , i ∈ [2r]
} n be a sequence of deterministic and symmetric matrices of size n which has a limit distribution and X i = Tr(D i ODīO -1 ).

For r ≥ 3, lim n→∞ κ r (X 1 , . . . , X r ) = 0 (4.11)

We first recall the following lemma (see [START_REF] Mingo | Second order freeness and fluctuations of random matrices: II. Unitary random matrices[END_REF] and the proof therein for the second assertion below). Lemma 4.8. For A, B ∈ P(k) we have

#(A) + #(B) ≤ k + #(A ∨ B) .
Moreover, if there exists a block A i of A and B j of B such that #(A i ∩B j ) = l, then,

#(A) + #(B) ≤ k -l + 1 + #(A ∨ B) .
Proof of Proposition 4.6 From the above lemma and Condition (4.9), we have: #(α) + #(β) ≤ r + #(α ∨ β) , (4.12) #(A) + #(α ∨ β) ≤ r + 1 , (4.13) #(A) ≤ #(σ) .

(4.14) The proof relies on the following property: Lemma 4.9. If α or β has a fixed point, then there is a strict inequality in (4.13) or in (4.14) Proof of Lemma 4.9 We denote by Σ the permutation of S 2r given by t α -1 τ ǫ t β .

1) Assume that i is a fixed point of α and β. Then (Σ(i), Σ( ī)) = (i, ī) or ( ī, i) depending on the sign of ǫ(i) and therefore, i is a fixed point of σ. In this case, σ ∨ α ∨ β = 1 r and therefore #(A) < #(σ).

2) Assume that i is a fixed point of β and α(i) = i. Then {Σ(i), Σ( ī)} = {i, α -1 (i)} and therefore the elements {i, α -1 (i)} are in the same cycle of σ (thus in the same block of A), thery are obviously in the same block of α. From Lemma 4.8, the inequality is strict in (4.13).

3) Assume that i is a fixed point of α and β(i) = i. Σ(β -1 (i)) = i or ī according to the sign of ǫ i and

Σ(β -1 (i)) = β -1 (i) or α -1 β -1 (i).
Thus, we find two distinct elements (i, β -1 (i)) (or (i, α -1 β -1 (i))) belonging to a cycle of σ (thus to the same block of A) and belonging to a block of α∨β (which is β -1 -invariant and α -1 β -1 -invariant). Therefore, the inequality (4.13) is strict.

Let α and β two permutations in S r . If #(α) + #(β) ≤ r -2 + #(α ∨ β), then, (4.10) is satisfied, using (4.13) and (4.14). From (4.12), it remains to study the two cases:

1) #(α) + #(β) = r + #(α ∨ β), 2) #(α) + #(β) = r -1 + #(α ∨ β).
Case 1) It is not difficult to see that in this case, there exist two different fixed points for α or β. For example, a unique fixed point for both α and β would imply:

#(α) + #(β) ≤ 1 + r -1 2 + 1 + r -1 2 = r + 1.
This is not possible since in this case, α ∨ β has at least two blocks. So, let i = j the two fixed points. Several situations can occur: a) i, j are fixed points of both α and β. Then, as seen above, i, j are fixed points of σ. In this case, #(A) ≤ #(σ) -2, leading to (4.10).

b) i is a fixed point of α and β and j is a fixed point of one of them. From the previous Lemma, #(A) ≤ #(σ) -1 and #(A) + #(α ∨ β) ≤ r -1 + 1 = r leading to (4.10). c) i, j are fixed points of one of the two permutations. For example, i, j are fixed points of β. If #{i, j, α -1 (i), α -1 (j)} ≥ 3, then, from Lemmas 4.8 and 4.9, it is not difficult to see that #(A) + #(α ∨ β) ≤ r -2 + 1 (either a block of σ and a block of α have 3 common points or two blocks of σ and two blocks of α have two common points, etc.), leading to (4.10). If #{i, j, α -1 (i), α -1 (j)} = 2, in this case, α and σ have a 2-cycle (i, j). This yields #(A) + #(α ∨ β) ≤ r -1 + 1 but we also have #(A) < #(σ) since σ ∨ (α ∨ β) = 1 r . The other cases are similar, leading to

-#(σ) + 2#(A) + #(α ∨ β) ≤ r -2 + 1 = r -1 proving (4.10). Case 2): #(α) + #(β) = r -1 + #(α ∨ β).
If there exists a fixed point, then one of the inequality in (4.13) or (4.14) is strict and we are done. If there is no fixed point, this implies that r is even, #(α) = #(β) = r 2 and α ∨ β = 1 r . An equality in (4.13) is true only for A = 0 r , the partition in singletons. This would imply that σ = Id. Let i ≤ r,

Σ(i) = i or α -1 (i) and Σ( ī) = β(i) or α -1 β(i).
Therefore, σ = Id implies that α -1 β(i) = i or α -1 (i) = β(i) and α and β have a common 2-cycle. This is not possible (α ∨ β = 1 r ) except if r = 2. .

4.4.

The fourth cumulant. We give an estimate for κ 4 (T p,q ). From the previous section,

κ 4 (T p,q ) = (α,β,ǫ)∈S 4 ×S 4 ×{±1} 4 λ α,β,4 A;A∨α∨β=1 4 C σ,A Tr α (D) Tr β -1 ( D)
where D = (I p , I p , I p , I p ) and D = (I q , I q , I q , I q ). From the asymptotic behavior of the cumulant, each term is of order

n -8-#(σ)+2#(A) p #(α) q #(β) (4.15)
where σ, A, α, β satisfy condition (4.9). First assume that #(α) = #(β) = 1. Then the order of (4.15) is at most: pq n 4 ≤ p 2 q 2 n 4 . Now assume that α or β has at least two blocks. From Proposition 4.6, the order of (4.15) is at most

n -1-#(α)-#(β) p #(α) q #(β) .
It is easy to see that this is at most of order p 2 q 2 n 4 . For example, α and β has at least two blocks,

n -1-#(α)-#(β) p #(α) q #(β) ≤ p 2 q 2 n 5 p n #(α)-2 q n #(β)-2 .
In the remaining cases, we find a majoration by pq 2 n 4 or p 2 q n 4 . Therefore, we obtain

κ 4 (T p,q ) = O p 2 q 2 n 4 . ( 4
|W (n) (s, t)| → sup s,t∈[0,1] |W (∞) (s, t)| in distribution, which implies that sup s,t∈[0,1] | 1 n T ⌊ns⌋,⌊nt⌋ -st| → 0 (5.1)
in probability.

2) The definition of our process focusses on the trace of a random matrix H p,q . This trace is a linear statistic of its empirical spectral distribution, i.e.

T p,q = Tr H p,q = p xdµ(x) , and the λ k 's are the eigenvalues of H p,q . If we are interested only in marginals (p = ⌊ns⌋, q = ⌊nt⌋, with s, t ∈ (0, 1) fixed), we can look directly at the asymptotic behavior of µ when n → ∞ and then deduce results from the continuity of the mapping µ → xdµ(x) on M 1 ([0, 1]). It is known that, if p ≤ q and p + q ≤ n, the random matrix H p,q belongs to the Jacobi unitary/orthogonal ensemble ([9] Theorem 2.2, [1] Prop. 4.1.4), which entails that the joint distribution of eigenvalues has a density proportional to

p k=1 λ a-1 i (1 -λ i ) b-1 1≤i<j≤p |λ i -λ j | β (5.2)
where a = (qp + 1) β 2 and b = (npq + 1) β 2 . The sequence of empirical spectral distributions converges to the generalized Kesten-McKay distribution. When s ≤ min(t, 1t), this distribution has a density which can be parametrized by s, t or by the endpoints of its support (u -, u + ) with 0 ≤ u -< u + ≤ 1:

π u -,u + (x) = C u -,u + (x -u -)(u + -x) 2πx(1 -x) (5.3)
where

C -1 u -,u + := 1 2 1 - √ u -u + -(1 -u -)(1 -u + ) .
The relation between (s, t) and u ± is

u ± = s(1 -t) ± (1 -s)t 2 .
By continuity, in all cases, we recover a weak form of (5.1), i.e. from the known results on the fluctuations of linear statistics of µ. Actually, the result of Johansson [START_REF] Johansson | On fluctuations of eigenvalues of random Hermitian matrices[END_REF] is not specific of the Jacobi ensemble, but uses a model of random matrices invariant by conjugation, with polynomial external field. Here, the ensemble is invariant but the potential is logarithmic (see (5.2)). The result is a Gaussian limit with the good variance. At another level, in the same asymptotics as above, Hiai and Petz [START_REF] Hiai | Large deviations for functions of two random projection matrices[END_REF] proved that the family of empirical spectral distributions satisfies the Large Deviation Principle in M 1 ([0, 1]) with speed βn 2 /2 and good rate function, which in the case s < t < 1/2 is

I(ν) = -s 2
log |x -y|dν(x)dν(y)

-s ((1st) log(1x) + (ts) log x) dν(x) + I 0 (s, t) .

where I 0 (s, t) is some constant (the limiting free energy). Appealing again to the continuity of the mean, we deduce from the contraction principle that n -1 T ⌊ns⌋,⌊nt⌋ satisfies the LDP at scale n -2 with good rate function

I(c) = inf{I(ν); ν ∈ M 1 ([0, 1]), 1 0 xdν(x) = c} .
3) In multivariate (real) analysis of variance, the random variable T p,q is known as the Bartlett-Nanda-Pillai statistics. The exact distribution of T p,q is known by its Laplace transform which is an hypergeometric function of matrix argument ( [START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF] p.479). Various asymptotic studies have been performed, essentially p, q fixed, n → ∞ (large sample framework), or highdimensional framework with q fixed, n, p → ∞ and p/n → s < 1 (see for instance [START_REF] Fujikoshi | Asymptotic results in canonical discriminant analysis when the dimension is large compared to the sample size[END_REF]). The asymptotic regime of the present paper (p/n → s, q/n → t) is considered in Section 4.4 of the book [START_REF] Bai | Spectral Analysis of Large Dimensional Random Matrices[END_REF] and a CLT for the statistic T p,q may be deduced from Theorem 2.2 of [START_REF] Bai | Corrections to LRT on Large Dimensional Covariance Matrix by RMT[END_REF].

Finally, we

  define M 2k as the set of pairings of [2k], i.e. of partitions where each block consists of exactly two elements. It is then convenient to encode the set [2k] by [2k] ∼ = {1, . . . , k, 1, . . . , k} .

. 5 ) 2 . 3 .

 523 Matrices. For a matrix M = (M ij ) i,j≤n , we denote by Tr the trace and by tr the normalized trace Tr(For π ∈ S k and M = (M 1 , . . . M k ) a k-tuple of n × n matrices, we set Tr π (M ) = Tr π (M 1 , . . . , M k ) = C∈C(π)

. 10 )

 10 From these equations, we can recover (3.3),(3.4). We can now state a proposition which is a particular case of[21, Theorem 3.10]. Proposition 3.4. Let U be Haar distributed on U(n). Let D = (D 1 , . . . D k ) and D = (D1, . . . Dk) be two families of deterministic matrices of size n.

Remark 3 . 5 .

 35 When writing Tr α ( D) in (3.11), we consider α as a permutation acting on [ k]. The formula (3.11

.21) 3 . 4 . 3 . 4 . 1 .

 34341 Proof of Theorem 1.1. Tightness. According to Bickel and Wichura [5, Theorem 3], since our processes are null on the axes, the tightness of the family of distributions of W

4. The orthogonal case 4 . 1 .

 41 Combinatorics for the orthogonal group. Let O(n) denote the orthogonal group of size n endowed with the Haar probability measure.

Proposition 4 . 5 .

 45 Let O be Haar distributed on O(n). Let D = (D 1 , . . . D k ) and D = (D1, . . . Dk) be two families of deterministic and symmetric matrices of size n. We set, for 1 ≤ i ≤ r, X i = Tr(D i ODīO -1 ).

.16) 4 . 5 .5. Complementary remarks 1 )

 451 Proof of Theorem 1.1. The proof is similar to the proof in the unitary case, using the asymptotic vanishing cumulants of order r ≥ 3 (Corollary 4.7) and the estimate (4.16) for the fourth cumulant, ensuring the tightness of the family of distributions. Since the sup norm is continuous for the Skorokhod topology, Theorem 1.1 implies that sup s,t∈[0,[START_REF] Anderson | An introduction to random matrices[END_REF]

lim n 1 n

 1 T ⌊ns⌋,⌊nt⌋ = s xπ u -,u + (x)dx = st , in probability.It could also be possible to recover the limiting fluctuations of the marginal distribution of T ⌊ns⌋,⌊nt⌋ with s, t fixed, i.e.T ⌊ns⌋,⌊nt⌋ -ET ⌊ns⌋,⌊nt⌋ law -→ N (0, 2 β s(1s)t(1t))

We will deal with the symplectic case in a forthcoming paper.

Acknowledgement This work is supported by the ANR project Grandes Matrices Aléatoires ANR-08-BLAN-0311-01. C. D-M. thanks J.A. Mingo for the references [23] and [24].


Raphaël Soulard
email: raphael.soulard@univ-poitiers.frcarre@sic.univ-poitiers.fr

Philippe Carré

COLOR MONOGENIC WAVELETS FOR IMAGE ANALYSIS

Keywords: Color Wavelets, Analytic, Monogenic, Wavelet transforms, Image analysis

We define a color monogenic wavelet transform. This is based on the recent grayscale monogenic wavelet transform and a nonmarginal extension to color signals. To our knowledge, wavelet based color image processing schemes have always been made by using a grayscale tool separately on color channels. This may have some unexpected effect on colors because those marginal schemes are not necessarily justified. Here we propose a definition that considers a color (vector) image right at the beginning of the mathematical definition and so brings an actual color wavelet transformwhich has not been done so far to our knowledge. This so provides a promising multiresolution color geometric analysis of images.

INTRODUCTION

Wavelets have been widely used for handling images for more than 20 years. It seems that the human visual system sees images through different channels related to particular frequency bands and directions; and wavelets provide such decompositions. Since 2001, the analytic signal and its 2D generalizations have brought a great improvement to wavelets [START_REF] Selesnick | The dual-tree complex wavelet transform[END_REF][START_REF] Chan | Coherent multiscale image processing using dual-tree quaternion wavelets[END_REF][START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF]] by a natural embedding of an AM/FM analysis in the subband coding framework. This yields an efficient representation of geometric structures in grayscale images thanks to a local phase carrying geometric information complementary to an amplitude envelope having good invariance properties. So it codes the signal in a more coherent way than standard wavelets. The last and seemingly most appropriate proposition [START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF] of analytic wavelets for image analysis is based on the monogenic signal [START_REF] Felsberg | Low-level image processing with the structure multivector[END_REF].

In parallel a color monogenic signal was proposed [START_REF] Demarcq | The color monogenic signal: a new framework for color image processing[END_REF] as a mathematical extension of the monogenic signal; paving the way to nonmarginal color tools especially by using geometric algebra and above all by considering a color signal right at the foundation of the mathematical construction.

We define here a color monogenic wavelet transform that extends the monogenic wavelets of [START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF] to color. These analytic wavelets are defined for color 2D signals (images) and avoid the classical pitfall of marginal processing (grayscale tool used separately on color channels) by relying on a sound mathematical definition. We may so expect to handle coherent information of multiresolution color geometric structure; which would make easier any wavelet based color image processing. To our knowledge color wavelets have not been proposed so far.

We first give a technical study of analytic signal/wavelets with the intent to popularize them since they rely on non-trivial concepts This work is part of the French ANR project CAIMAN. of geometric algebra, complex/harmonic analysis, as well as nonseparable wavelet frames. Then we describe our color monogenic wavelet transform.

Notations : 2-vector coordinates : x = (x, y) , ω = (ω1, ω2) ∈ R 2 ; k ∈ Z 2 Euclidean norm : x = x 2 + y 2
Complex imaginary number : j ∈ C Argument of a complex number : arg Convolution symbol : * Fourier transform : F

ANALYTIC SIGNAL AND 2D GENERALIZATION

An analytic signal sA is a multi-component signal associated to a real signal s to analyze. The definition is well known in the 1D case where sA(t) = s(t) + j (h * s)(t) is the complex signal made of s and its Hilbert transform (with h(t) = 1 πt ). The polar form of the 1D analytic signal provides an AM/FM representation of s with |sA| being the amplitude envelope and ϕ = arg (sA) the instantaneous phase. This classical tool can be found in many signal processing books and is used in communications for example.

Interestingly we can also interpret the phase in terms of signal shape i.e. there is a direct link between the angle ϕ and the local structure of s. Such a link between a 2D phase and local geometric structures of images would be very attractive in image processing. That is why there were several attempts to generalize it for 2D signals; and among them the monogenic signal [START_REF] Felsberg | Low-level image processing with the structure multivector[END_REF] seems the most advanced since it is rotation invariant.

The Monogenic Signal

Without going beyond strictly needed details we here review the key points of the fundamental construction of the monogenic signal; which will be necessary to understand the color extension.

The definition of the 1D case given above can be interpreted in terms of signal processing : the Hilbert transform makes a "pure π 2 -dephasing". But such a dephasing is not straightforward to define in 2D (same issue with many 1D signal tools) so let us look at the equivalent complex analysis definition of the 1D analytic signal. It says that sA is the holomorphic extension of s restricted to the real line. But complex algebra is impeding for generalizations to higher dimensions. To bypass this limitation we can see a holomorphic function like a 2D harmonic field that is an equivalent harmonic analysis concept involving the 2D Laplace equation ∆f = 0. It so can be generalized within the framework of 3D harmonic fields by using the 3D Laplace operator ∆3 = δ δx + δ δy + δ δz . The whole generalization relies on this natural choice and remaining points are analogous to the 1D case (see [START_REF] Felsberg | Low-level image processing with the structure multivector[END_REF] for more details). Note that in Felsberg's thesis this construction is expressed in terms of geometric algebra but here we avoided it for simplicity's sake. Finally the 2D monogenic signal sA associated to s is the 3-vector valued signal :

sA(x) =   s(x) sr1(x) = x 2π x 3 * s(x) sr2(x) = y 2π x 3 * s(x)   (1)
Where sr1 and sr2 are analogous to the imaginary part of the complex 1D analytic signal. Interestingly, this construction reveals the two components of a Riesz transform (that is convolution of a function by the two kernels x 2π x 3 and y 2π x 3 ) in the same way that the 1D case exhibits a Hilbert transform. Note that we get back to a signal processing interpretation since the Riesz transform can also be viewed like a pure 2D dephasing. In the end, by focusing on the complex analysis definition of the analytic signal we end up with a convincing generalization of the Hilbert transform.

Now recall that the motivation to build 2D analytic signals arises from the strong link existing between the phase and the geometric structure. To define the 2D phase related to the Riesz transform the actual monogenic signal must be expressed in spherical coordinates that yield the following amplitude envelope and 2-angle phase : Amplitude :

A = s 2 +s 2 r1 +s 2 r2 s = A cos ϕ Orientation : θ = arg (sr1 + j sr2) sr1 = A sin ϕ cos θ 1D Phase : ϕ = arccos ( s A ) sr2 = A sin ϕ sin θ (2)
Felsberg shows a direct link between the angles θ and ϕ and the geometric local structure of s. The signal is so expressed like an "Astrong" 1D structure with orientation θ. ϕ is analogous to the 1D local phase and indicates if the structure is rather a line or an edge. A direct drawback is that intrinsically 2D structures are not handled. Yet this tool found many applications in image analysis from contour detection to motion estimation (see [START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF] and references therein p. 1). From a signal processing viewpoint the AM/FM representation provided by an analytic signal is accordingly well suited for narrowband signals. That is why it seems natural to embed it in a wavelet transform that performs subband decomposition. We now present the monogenic wavelet analysis proposed in [START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF].

MONOGENIC WAVELETS

So far there is one proposition of computable monogenic wavelets in the literature [START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF]. It provides 3-vector valued monogenic subbands consisting of a rotation-covariant magnitude and this new 2D phase. This representation -specially defined for 2D signals -is a great theoretic improvement of the complex and quaternion wavelets [START_REF] Selesnick | The dual-tree complex wavelet transform[END_REF][START_REF] Chan | Coherent multiscale image processing using dual-tree quaternion wavelets[END_REF]; as well as the monogenic signal itself is an improvement of its complex and quaternion counterparts.

The proposition of [START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF] consists of one real-valued "primary" wavelet transform in parallel with an associated complex-valued wavelet transform. Both transforms are linked each other by the Riesz transform so they carry out a multiresolution monogenic analysis. We end up with 3-vector coefficients forming subbands that are monogenic.

Primary transform

The primary transform is real-valued and relies on a dyadic pyramid decomposition tied to a wavelet frame. Only one 2D wavelet is needed and the dyadic downsampling is done only at the low frequency branch; leading to a redundancy of 4:3. The scaling function ϕγ and mother wavelet ψ are defined in the Fourier domain :

ϕγ F ←→ 4(sin 2 ω 1 2 + sin 2 ω 2 2 ) -8 3 sin 2 ω 1 2 sin 2 ω 2 2 γ 2 ω γ (3)
ψ(x) = (-∆) γ 2 ϕ2γ(2x) (4)
Note that ϕγ is a cardinal polyharmonic spline of order γ and spans the space of those splines with its integer shifts. It also generatesas a scaling function -a valid multiresolution analysis. In this paper we set γ = 1.00001 that gave satisfying experimental results. This particular construction is made by an extension of a wavelet basis (non-redundant) related to a critically-sampled filterbank. This extension to a wavelet frame (redundant) adds some degrees of freedom used by the authors to tune the involved functions. In addition a specific subband regression algorithm is used at the synthesis side. The construction is fully described in [START_REF] Unser | The pairing of a wavelet basis with a mildly redundant analysis via subband regression[END_REF].

The monogenic transform

The second "Riesz part" transform is a complex-valued extension of the primary one. We define the associated complex-valued wavelet by including the Riesz components :

ψ = - x 2π x 3 * ψ(x) + j y 2π x 3 * ψ(x) (5)
It can be shown that it generates a valid wavelet basis and that it can be extended to the pyramid described above. The joint consideration of both transforms form monogenic subbands from which can be extracted the amplitude and phase for an overall redundancy of 4:1. So far no applications of the monogenic wavelets have been proposed. In [START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF] a demonstration of AM/FM analysis is done with fine orientation estimation and gives very good results in terms of coherency and accuracy. Accordingly this tool may be rather used for analysis tasks than processing.

Motivated by the powerful analysis provided by the monogenic wavelet transform we propose now to extend it for color images.

COLOR MONOGENIC WAVELETS

We define here our proposition that combines a fundamental generalization of the monogenic signal to color with the monogenic wavelets described above. The challenge is to avoid the classical marginal definition that would be applying a grayscale monogenic transform on each of the three color channels of a color image. We believe that the monogenic signal has a favorable theoretical framework for a color extension and this is why we propose to start from this particular wavelet transform rather than from a more classical one.

The color generalization of the monogenic signal is expressed within the geometric algebra framework. This algebra is very general and embeds the complex and quaternion as subalgebras. Its elements are "multivectors" naturally linked with various geometric entities. The use of this fundamental tool is gaining popularity in the literature because it allows rewriting sophisticated concepts with simpler algebraic expressions and so paves the way to innovative ideas and generalizations in many fields.

For simplicity's sake and since anyway we would not have enough space to present the fundamentals of geometric algebra we here express the construction in classical terms; as we already did section 2. Yet we may sometimes point out some necessary specific mechanisms but we refer the reader to [START_REF] Felsberg | Low-level image processing with the structure multivector[END_REF][START_REF] Demarcq | The color monogenic signal: a new framework for color image processing[END_REF] for further details.

The Color Monogenic Signal

Starting from Felsberg's approach that is originally expressed in the geometric algebra of R 3 ; the extension proposed in [START_REF] Demarcq | The color monogenic signal: a new framework for color image processing[END_REF] is written in the geometric algebra of R 5 for 3-vector valued 2D signals of the form (sR, sG, sB). By simply increasing the dimensions we can embed each color channel along a different axis and the original equation from Felsberg involving a 3D Laplace operator can be generalized in 5D with ∆5 = δ δx 1 + δ δx 2 + δ δx 3 + δ δx 4 + δ δx 5 . Then the system can be simplified by splitting it into three systems with a 3D Laplace equation, reducing to applying Felsberg's condition to each color channel. At this stage appears the importance of geometric algebra since an algebraic simplification between vectors leads to a 5-vector color monogenic signal that is non-marginal. Instead of naively applying the Riesz transform to each color channel, this fundamental generalization carries out the following color monogenic signal : sA = (sR, sG, sB, sr1, sr2) where sr1 and sr2 are the Riesz transform applied to sR+sG+sB.Note that this simple sum of color channels is obtained by calculation. This is coherent with the idea that sr1 and sr2 carry structural information -independent of color. Now the color extension of Felsberg's monogenic signal is defined let us construct the color extension of the monogenic wavelets.

The Color Monogenic Wavelet Transform

We can now define a wavelet transform whose subbands are color monogenic signals. The goal is to obtain vector coefficients of the form (cR, cG, cB, cr1, cr2) such that cr1 = x 2π x 3 * (cR + cG + cB) and cr2 = y 2π x 3 * (cR + cG + cB) . It turns out that we can very simply use the transforms presented above by applying the primary one on each color channel and the Riesz part on the sum of the three. The five related color wavelets illustrated Fig. 1 and forming one color monogenic wavelet ψA are :

ψR =   ψ 0 0   ψG =   0 ψ 0   ψB =   0 0 ψ   (6) ψr1 =    x 2π x 3 * ψ x 2π x 3 * ψ x 2π x 3 * ψ    ψr2 =    y 2π x 3 * ψ y 2π x 3 * ψ y 2π x 3 * ψ    (7)
ψA = (ψR , ψG , ψB , ψr1 , ψr2) (

We then get 5-vector coefficients verifying our conditions and so forming a color monogenic wavelet transform. The associated decomposition is described by the diagram Fig. 2. This provides a multiresolution color monogenic analysis made of a 5-vector valued pyramid transform.The construction can be viewed like a vector extension of the scalar wavelet ψ. The 5 decompositions of two images are shown Fig. 3 from left to right. Each one consists of 4 juxtaposed image-like subbands resulting from 3-level decomposition.

Let us look at the first 3 graymaps. These are the 3 primary transforms cR, cG and cB where white (resp. black) pixels are high positive (resp. negative) values. Note that our transform is non-separable and so provides at each scale only one subband related to all orientations. We are not subjected to the arbitrarily separated horizontal, vertical and diagonal analyses of usual wavelets. This advantage is even greater in color. Whereas marginal separable transforms show 3 arbitrary orientations within each color channel -which is not easily interpretable -the color monogenic wavelet transform provides a more compact energy representation of the color image content regardless of the local orientation. The color information is well separated through cR, cG and cB : see that blue contours of first image are present only in cB. And in each of the 3 decompositions it is clear that every orientation is equally represented all along the round contours. That is different from separable transform that privileges particular directions. The multiresolution framework makes the horizontal blue low frequency structure of second image be coded mainly in the third scale of cB.

But the directional analysis is not lost thanks to the Riesz part that completes this representation. Now look at the "2-in-1" last decomposition forming the Riesz part. It is displayed in one color map where the geometric energy c 2 r1 + c 2 r2 is encoded into the intensity (with respect to the well known HSV color space) and the orientation arg (cr1 +jcr2) (π) is encoded in the hue (e.g. red is for {0, π} and cyan is for ± π 2 ). This way of displaying the Riesz part well reveals the provided geometric analysis of the image.

The Riesz part makes a precise analysis that is local both in space and scale. If there is a local color geometric structure in the image at a certain scale the Riesz part exhibits a high intensity in the corresponding position and subband. This is completed with an orientation analysis (hue) of the underlying structure. For instance a horizontal (resp. vertical) structure in the image will be coded by a cyan (resp. red) intense point in the corresponding subband. The orientation analysis is strikingly coherent and accurate. See for example that color structures with constant orientation (second image) Note that low intensity corresponds to "no structure" i.e. where the image has no geometric information. It is coherent not to display the orientation (low intensity makes the hue invisible) for these coefficients since this data has no sense in those cases.

In short the color and geometric information of the image are well separated from each other and the orientation analysis is very accurate. In addition the invariance properties of the primary and Riesz wavelet transforms are kept in the color extension for a slight overall redundancy of 20 : 9 ≈ 2.2.

CONCLUSION

We define a color extension of the recent monogenic wavelet transform proposed in [START_REF] Unser | Multiresolution monogenic signal analysis using the riesz-laplace wavelet transform[END_REF]. This extension is non-marginal since it takes care of considering a vector signal at the very beginning of the fundamental construction and leads to a definition basically different from the marginal approach. This has not been proposed before to our knowledge. The use of non-separable wavelets joint with the monogenic framework allows for a good orientation analysis well separated from the color information. This color transform can be a great color image analysis tool thanks to this good separation of information through various data. We are currently working on color denoising by using this transform.

Although it is not marginal the color generalization has a marginal style since it reduces to apply the Riesz transform on the intensity of the image. So the geometric analysis is done without considering the color information and it would be much more attractive to have a complete representation of the color monogenic signal into magnitude and phase(s) with color/geometric interpretation.

Another limit is the restriction to analysis applications. The redundancy of the grayscale transform makes it not well suited a priori to image compression.

Fig. 1 .

 1 Fig. 1. Space representation of the 5 color wavelets.

Fig. 2 .

 2 Fig.2. Color MWT scheme. Each color channel is analyzed with primary wavelet transform ψ; and the sum "R + G + B" is analyzed with the "Riesz part" wavelet transform (ψr1 and ψr2 blocs).

Fig. 3 .

 3 Fig. 3. Color MWT of images. The two components of the Riesz part are displayed in the same graphic with the magnitude of cr1 +jcr2 encoded in the intensity and argument (local orientation) encoded in the hue.


Julie Vallée
email: valleej@yahoo.fr

Laurence Marrama
email: marrama@pasteur.sn

Didier Fontenille
email: didier.fontenille@mpl.ird.fr

Alioune Badara Ly
email: ly@pasteur.sn

Jean-Francois Trape
email: francois.trape@ird.sn

Fatoumata Diene Sarr
email: fdsarr@pasteur.sn

Christophe Rogier
email: christophe.rogier@wanadoo.fr

Adama Tall
email: tall@pasteur.sn

Pierre Nabeth
email: nabeth@pasteur.sn

THE GEOGRAPHICAL STUDY OF ANOPHELINE DENSITIES ON A SMALL SPACE, USING SATELLITE IMAGERY AND GEOGRAPHICAL INFORMATION SYSTEMS

Keywords: Malaria, transmission, anopheles, GIS, Remote sensing, scale, Dielmo, Senegal

To predict the spatial distribution of anopheles in the Dielmo village (located in the southeastern part of Senegal), we used residual fauna collected from 104 different rooms during four separate trips conducted in 1994 and 1995. Thanks Generalized Estimating Equations, we were able to identify factors influencing the distribution of Anopheles in the village. Several variables, such as the number of persons sleeping in the room, population density around the hut, construction materials, presence of mosquito nets, were found to be significant, while many spatial variables relevant to the scale of a region (vegetation index, distance to larval sites…) were not found to be significant on the village level. As a result, it became clear that it is difficult to correctly predict the anopheline density for each house even with precise spatial data created with Satellite imagery and Geographical Information Systems (GIS). This work highlights the complexity of the geographical study of anopheline density and its limits on a small space.

INTRODUCTION

Satellite imagery and Geographical Information Systems (GIS) analysis techniques have been used to identify areas at risk of vector-borne diseases, such as malaria, but, most of the time, they were carried out at a regional scale [START_REF] Beck | Remote sensing as a landscape epidemiologic tool to identify villages at high risk for malaria transmission[END_REF][START_REF] Salem | Espace urbain et risque anophélien à Pikine (Sénégal)[END_REF][START_REF] Roberts | Predictions of malaria vector distribution in Belize based on multispectral satellite data[END_REF][START_REF] Thomson | The ecology of malaria -as seen from Earth-observation satellites[END_REF][START_REF] Beck | Assessment of a remote sensing-based model for predictiong malaria transmission risk in villages of Chiapas, Mexico[END_REF][START_REF] Connor | The contribution of satellite derived informations to malaria stratification, monitoring and early warning[END_REF][START_REF] Manguin | Apport de la télédétection en santé publique: l'exemple du paludisme et autres perspectives[END_REF][START_REF] Beck | Remote sensing and human health : New sensors and new opportunities[END_REF][START_REF] Thomas | Local-scale variation in malaria infection amongst rural Gambian children estimated by satellite remote sensing[END_REF] A longitudinal epidemiological and entomological follow-up began in 1990 to evaluate malaria infections and the mechanisms of protective immunity in a population living in Dielmo, a village in Senegal. Every month, an average entomological inoculation rate is calculated for the entire village and analyses are based on the assumption that the person's level of exposure to infected bites is identical [START_REF] Rogier | Malaria attacks in children exposed to high transmission: who is protected[END_REF]Trape, 1994.;[START_REF] Rogier | Evidence for an age-dependant pyrogenic threshold of Plasmodium falciparum parasitemia in highly endemic populations[END_REF][START_REF] Rogier | Plasmodium Falciparum clinical malaria: lessons from longitudinal studies in Senegal[END_REF][START_REF] Sokhna | Host factors affecting the delay of reappearance of Plasmodium falciparum after radical treatment among a semi-immune population exposed to intense perennial transmission[END_REF] However, variations in anopheles densities may be important within a same village [START_REF] Smith | Mapping the densities of malaria vectors within a single village[END_REF][START_REF] Minakawa | Influence of host and larval habitat distribution on the abundance of afriacn malaria vectors in Western Kenya[END_REF]. In Dielmo, we demonstrated with analysis of variance that anopheline densities collected by hut are significantly different (Vallee, personnal communication). To predict anopheline densities in the differents huts in Dielmo village, we used spatial data. Our aim was to test the relevance of spatial data (issued from Satellite imagery and Geographical Information Systems techniques) applied to a small area.

MATERIALS AND METHODS

Study Area

Dielmo (13°45'N, 16°25'W), a village of about 300 inhabitants, is situated 280 km on the southeast of Dakar in the Sahelo-Soudanian region of Senegal (Map 1).

Map 1: Location of Dielmo in Senegal

The Nema, a small and permanent river, borders the village and its marshy banks provide breeding sites to anopheles throughout the year. Malaria is holo-endemic (parasite prevalence > 75% among infants) and transmission is high and perennial, reaching its peak during the rainy season (approximately 200 infected bites/person/year). In Dielmo, the major vectors transmitting malaria are An. funestus, and An. gambiae that account respectively for 63% and 36% of anopheles captured on humans (Konate, 1994;[START_REF] Fontenille | High annual and seasonal variations in malaria transmissions by anophelines and vacetor sepcies composition in Dielmo, a holoendemic area in Senegal[END_REF]. We used a GPS (Global Positioning System) to localize every house and produce an accurate map of the village. To remedy to the GPS inaccuracy which is within the scope of 10 meters, we choose to combine the use of GPS with a Quickbird satellite image that has a very fine spatial resolution (61 cm in panchromatic and 2.44 meters in multispectral). For every room, the following data have been collected: number of people sleeping in the room, roofing materials, wall materials, presence of space between the wall and roof, presence of at least a hole of 25 cm2 in the wall, the use of fumigations and mosquito nets. Thanks to the Quickbird satellite image (captured on March 9, 2004) that has a very fine spatial resolution, we calculated the vegetation index (NDVI for Normalized Difference Vegetation Index). To take into consideration the environment surrounding each habitation, we calculated the average vegetation density within the radiuses of 50 meters and 100 meters around each habitation (Map 2) and within a 25 meters thick polygon linking each house to the nearest point of the Nema River. We calculated the average number of persons sleeping in each room and we produced a map of the population's distribution in Dielmo in 1995. To take into consideration the population density around each habitation, the number of persons sleeping within the radiuses of 50 meters and 100 meters around each hut, as well as the number of persons between the studied hut and the main mosquito breeding place (Nema river) were calculated. We localized cattle existing in 1995 and we separated two categories of cattle to distinguish them with many animals. Following the same way than person density, we estimated the presence of livestock within the radiuses of 50 meters and 100 meters around each hut, as well as between the studied hut and the Nema River.

Statistical analysis to predict anopheline densities

We used the Generalized Estimating Equations (GEE) to test the predictor variables' pertinence for the whole of the 273 available entomological observations which are repeated data, meaning that we had several observations (between 1 and 4) of a same variable (number of anopheles) for each room. (Liang, 1986;Zeger, 1986) The variable to be explained was on one hand, the number of An. funestus, and on the other hand, the number of An. gambiae caught in a room. We replaced these variables with their square roots to correspond with a distribution model of Poisson. To evaluate the prediction's quality, we established groups of anopheline densities based on square roots. We compared, for each observation (among the 273), the predicted number of anopheles with the corresponding observed number of anopheles. The prediction is said correct if the predicted number is affected in the same group than the observed number. Then we calculated the number of predictions that are correctly affected.

RESULTS

Variables influencing the anopheline densities

We obtained two GEE models, one for each species of anopheles, containing only the variables significantly associated to the number of anopheles (tables 1 and 2). The Incidence Rate Ratio (IRR) measures the relative risk for the transformed variable (square root of mosquitoes). The IRR2 gives the value of risk relating to the variable of interest (number of mosquitoes).

There is an important "mission" effect. With regards to mission 1 (December 1994), missions 2 (March 1995), 3 (June 1995) and 4 (September 1995) multiplied the number of An. gambiae by 8.3, 4.7 and 7and divide the number of An. funestus by 1.4, 3.3 and 2.5. The "person/room" effect is also significant. An additional person sleeping in a room multiplied the total number of vectors, whether An. gambiae or An. funestus, by 1.2. An additional person within a radius of 50 meters multiplied by 1.012 the number of An. gambiae and by 1.01 the number of An. funestus. This multiplying coefficient is very low, but should be compared with the number of persons within 50 meters from the hut (varying between 4 and 96), which gives therefore weight to this variable, or up to 2.6 times more vectors because of this variable alone. The presence of a space between the wall and roof (the architectural variable retained for An. gambiae) multiplied the number of Anopheles by 2.1. For An. funestus, the roofing material is more pertinent as a roof made of hay multiplied the number of Anopheles by 2.1. The use of mosquito nets (in good condition) divided the number of An. gambiae by 1.8 and the number of An. funestus by 2. However, use of fumigation is not a variable significantly associated to the number of mosquitoes.

We tested the relevance of other spatial variables but without significant results. Distance to the Nema, vegetation density within 50 and 100 meters from houses, as well as vegetation density between the Nema and habitations, the number of persons between habitations and the Nema and the presence if livestock within 50 and 100 meters from habitations and between the Nema and habitations are not significantly associated with the number of collected Anopheles.

Quality of prediction of anopheline densities

3.2

Based on coefficients affected to each significant variable, we have attempted to predict the number of Anopheles collected in a specific hut during a given mission. The percentage of correctly affected observations at 41% is low ( The coefficient of common correlation helps to appreciate the hut's proper effect. The value of correlation is 0.1886 for the model performed on the square roots of An. gambiae and 0.2814 for the one performed on the square roots of An. funestus, which means that the hut's residual effect explains the 3.5% (0.18862) and 7.9% (0.28142) of the variability of the number of collected Anopheles. If the variables explain all the data's variability, the proper effect of the hut should be null. This means that some variables describing the hut are missing in our model, particularly for An. funestus. To predict the anopheline distribution at the regional scale, it is usual to take into consideration the distance between breeding sites and habitations in urban areas [START_REF] Trape | Vector density gradients and the epidemiology of urban malaria in Dakar, Senegal[END_REF][START_REF] Salem | Espace urbain et risque anophélien à Pikine (Sénégal)[END_REF] as well as in rural areas [START_REF] Beck | Remote sensing as a landscape epidemiologic tool to identify villages at high risk for malaria transmission[END_REF][START_REF] Fontenille | High annual and seasonal variations in malaria transmissions by anophelines and vacetor sepcies composition in Dielmo, a holoendemic area in Senegal[END_REF][START_REF] Thomas | Local-scale variation in malaria infection amongst rural Gambian children estimated by satellite remote sensing[END_REF][START_REF] Hii | Spatial and temporal variation in abundance of Anopheles in malaria endemic area in Papua New-Guinea[END_REF] At village scale, in Kenya [START_REF] Minakawa | Influence of host and larval habitat distribution on the abundance of afriacn malaria vectors in Western Kenya[END_REF] and in Uganda [START_REF] Staedke | Short Report: Proximity to mosquitoes breeding sites as a risk factors for clinical malaria episodes in an urban cohort og Ugandan children[END_REF] distance from a house to its nearest larval habitats showed a significant correlation with anopheline densities. Our study did not demonstrate the relevance of this variable, maybe because Dielmo is a very limited space; every hut is < 300 m from the stream. A study in Sri Lanka using small distance categories (<250m/>500m) from stream can't reach statistical significance (Van der Hoeck, 1998). Landscape description with remotely sensed data may allow locating habitats (un)favorable to Anopheles and predicting anopheline densities [START_REF] Beck | Assessment of a remote sensing-based model for predictiong malaria transmission risk in villages of Chiapas, Mexico[END_REF][START_REF] Wekesa | Spatial distribution of adult mosquitoes in habitats associated with the rice agroecosystem of Northern California[END_REF][START_REF] Grillet | Factors associated with distribution of An. aquaqalis and An. oswaldoi in a malarious area, Northeastern Venezuela[END_REF][START_REF] Eisele | Linking field-based ecological data with remotely sensed data using a geographic information system in two malaria endemic urban areas of Kenya[END_REF]. In our study, we calculated Normalized Difference Vegetation Index (NDVI) from satellite imagery with high spatial resolution, but could not prove its relevance to the Anopheles density over a limited space such as Dielmo village. As shown by an article synthesizing the potential of remote sensing for human health [START_REF] Herbreteau | Sizing up human health through remote sensing : uses and misuses[END_REF], many studies used the NDVI index (a quantitative variable that is quite commonly used to describe the type of landscape and vegetation density) at a scale that is inappropriate to qualify the environment of a vector.

Even if some variables proved to be significant, it is not possible to estimate the distribution of Anopheles based on these variables only. This study raises the more theoretical question on the limitations of a health geography analysis: Is the geography of anopheline densities relevant over such a limited space as Dielmo?

  We collected morning residual fauna of An. funestus and An. gambiae in 104 different rooms of Dielmo village by pyrethrum indoor spray collection. Anopheles were gathered once during each mission in December 1994, March 1995, June 1995 and September 1995. Data consisted in 273 observations: only 33 rooms were surveyed 4 times.

Map 2 :

 2 Calculated average vegetation densities around a habitation* based on the NDVI index. * For this habitation, the average vegetation density is 0,194 within a radius of 50 meters around the hut and 0,284 within 100 meters, indicating a low vegetation density close to the hut.

Table 1 :

 1 Model obtained for square root of An. gambiae

	Variables Coef. Std. Err.	z	P>z	IRR IRR²
	mission 1 Ref.
	mission 2 1.056 0.11357 9.3 < 0.001 2.87 8.26
	mission 3 0.771 0.12059 6.3 < 0.001 2.16 4.67
	mission 4 0.975 0.11500 8.4 < 0.001 2.65 7.03
	Persons /room	0.087 0.0193	4.5 < 0.001 1.09 1.19
	Pop. 50m radius	0.005 0.0019	3	0.003	1.00 1.01
	Space	0.378 0.09314	4	< 0.001 1.46 2.13
	Mosquito net	-0.289 0.10036 -2.8	0.004	0.74 0.56
	Variables Coef. Std. Err.	z	P>z	IRR IRR²
	mission 1 Ref.
	mission 2 -0.21 0.0908	-2	0.019 0.80 0.65
	mission 3 -0.560 0.1048	-5 < 0.001 0.57 0.32
	mission 4 -0.637 0.1022	-6 < 0.001 0.52 0.27
	Persons /room	0.076 0.0247	3.1	0.002 1.07 1.16
	Pop. 50m radius	0.005 0.0024	2.1	0.034 1.00 1.01
	Hay	0.369 0.1072	3.4	0.001 1.44 2.09
	Mosquito net	-0.334 0.1300	-3	0.01	0.71 0.51

Table 2 :

 2 Model obtained the square root of An. funestus

  table3). It was therefore not possible, based on the selected variables, to project the Anopheles densities in a relevant manner.

	Difference in affectation	An. gambiae	An. funestus
	0	112 (41%)	110 (41%)
	|1|	136 (50%)	138 (50%)
	|2|	24 (9%)	24 (9%)
	|3|	1 (0%)	1(0%)
	Total	273 (100%)	273 (100%)
	Table 3: Percentage of observations correctly affected
	to the groups of densities
	4. DISCUSSION AND CONCLUSION

Acknowledgments: We thank Maps Geosystem and Digital Globe for having graciously made available the Quickbird image. We also thank Florence Fournet for her advices.


Julien Moreau

Vincent Loriette

Confocal thermal-lens microscope

The use of a confocal detection scheme in a dual-beam thermal-lens microscope is presented. The scheme allows the measurement of absorption factors down to 1.2 3 10 27 in a 0.35-mm 3 volume by use of a heating laser power of 100 mW incident upon the sample. Results are presented that prove that a 450-nm axial resolution is possible when a 1.2 water immersion objective lens with a N.A. of 1.2 is used.

Thermal lensing provides high-sensitivity measurements for thermal 1 and spectroscopic 2 characterization or detection 3 -5 and imaging 6 of low-absorption samples. Thermal-lens microscopy 7 combines the high sensitivity of photothermal techniques [START_REF] Bialkowski | Photothermal Spectroscopy Methods for Chemical Analysis[END_REF] with the lateral and axial resolution of conventional microscopy. The detection method used in thermal-lens microscopy does not differ from conventional thermal-lens techniques [START_REF] Franko | [END_REF] : The thermal lens generated by a heating beam induces a change in the geometric parameters (divergence and beam-waist position) of a probe beam that are detected with the help of a diaphragm placed in the path of the beam in front of a photodetector. With this detection method the signal is maximum when the thermal lens is located at a distance of one heating beam's Rayleigh range from the probe beam's waist; it is null when the thermal lens lies in the plane of the probe beam's waist. 7 The axial resolution (FWHM) of the thermal lens microscope is given by the following formula:

Dz standard ഠ 2.8 3 z R , (1)
where z R is the heating beam's Rayleigh range at the focal point. Thermal lens microscope signals exhibit two main drawbacks when one needs to resolve structures in the axial direction. First, the signal's shape (see Fig. 1), which shows two extrema that correspond to focusing or defocusing by the thermal lens, and a zero point crossing make it diff icult to use efficient deconvolution algorithms. Second, it can be shown that the relative height of the two extrema strongly depends on the distance between the two beam waists, and it is quite diff icult to control that parameter, which depends on the beams' different geometric parameters and on the chromatic characteristics 7 of the microscope objective lens that is used. Our detection scheme aims at eliminating these two drawbacks. The thermal lens microscope is not an imaging, but a scanning, instrument, so a detection scheme similar to confocal microscopy can be used. The axial resolution can then be improved to

Dz confocal ഠ 1.0 3 z R . ( 2
)
In this case the signal is now maximum when the thermal lens lies in the plane of the probe beam's waist and so exhibits only one extremum (see Fig. 1). This increase of resolution and the single maximum signal shape can be used to study transparent stratif ied samples such as dielectric multilayer coatings. It permits discrimination of the photothermal signal generated by the different layers and possibly by the different interfaces. It is worth noting that this significant increase of axial resolution is not obtained to the detriment of sensitivity. It can be shown, on the contrary, that a signal increase of a factor 2 can be obtained at the same time. Figure 2 shows our confocal dual-beam thermal-lens microscope. The heating beam is an intensitymodulated l 514 nm argon laser, and the probe beam is a 1-mW unmodulated red laser diode emitting at 638 nm. The heating power is limited to 100 mW to prevent damaging the sample. The modulation frequency of 1 MHz was chosen such that the thermal diffusion length in the sample is smaller than the heating beam's spot size. Both beams pass through a beam expander before entering the focusing objective lens. After crossing the sample, the beams are collected by a second objective lens. The sample is f ixed on a motorized stage that permits axial displacements in 74-nm steps. The sample is the only moving part of the experiment, so, except for geometric modifications of the beam parameters induced by aberrations caused by the sample, both beams' characteristics, in particular their beam-waist positions, do not change during scanning. After crossing the sample, the heating beam is then directed into a light trap while the probe beam passes through a pinhole and then goes onto a silicon photodetector. Pinhole diameter r D is chosen to give the best shot-noise-limited signal-to-noise ratio; its theoretical value is

r D ഠ 0.57 3 w D , (3)
where w D is the probe beam's radius on the pinhole plane. The position of the pinhole is fixed by the condition

z 1 1 z 2 1 2 ͑z 2 ͞f ͒ z R0 , (4)
where z 1 is the distance from the probe beam's waist in the sample to the imaging objective lens, z 2 is the objective lens -pinhole distance, and z R0 is the probe beam's Rayleigh range in the sample. This condition states that the pinhole plane is conjugate not with the plane of the probe beam's waist but with a plane that is one Rayleigh range away. One can understand why this arrangement is the correct one by looking at the effect of a lens of focal length f with f . . z R0 placed at the probe beam's waist (for typical values take z R0 1 mm, and, for low-absorption samples, f 1 m). In the paraxial approximation, the presence of this lens changes the beam's spot size w at a distance z after the lens by an amount equal to

dw w ͑z͒ ∑µ 1 2 z f ∂ 2 1 z 2 z R0 2 ∏ 1͞2 2 µ 1 1 z 2 z R0 2 ∂ 1͞2 µ 1 1 z 2 z R0 2 ∂ 1͞2 . ( 5
)
In the confocal case one can look at a plane where dw͞w is an extremum; this happens when z z R0 , whatever the sign of f . In this case Eq. ( 5) gives

dw w ഠ z R0 f . ( 6
)
The signal is null when z 0. This condition differs from that of conventional confocal microscopy in which the pinhole plane is conjugate with the object plane. Fulfillment of this condition is critical because it fixes the confocal behavior of the instrument. One can in practice maintain the confocality condition only by limiting z scans to a few tens of micrometers and by using identical focusing and imaging objectives. With these restrictions, few-millimeter-thick optical samples, for example, structured thin-film coatings upon transparent substrates, can be investigated.

The axial resolution was measured by use of a test sample made from a single 180-nm-thick tantalum pentoxide layer deposited by ion-beam sputtering onto a synthetic fused-silica sample. The extinction and Figure 3 shows the result of this measurement. The focusing objective lens was a water immersion lens with a N.A. of 1.2. (Olympus Model UPLanApo 360). The experimental signal's FWHM was Dz͑exp͒ 450 nm. Using a mathematical model of the instrument, we calculated that the confocal signal's FWHM with this objective lens should be Dz͑theoretical͒ 465 nm. We checked the expected gain in axial resolution by moving the pinhole a few millimeters and performing the same measurement with a nonconfocal conf iguration. The result is presented in Fig. 4. The measured FWHM is Dz͑nonconfocal͒ 1.5 mm. We then estimated the instrument's sensitivity by measuring volume absorption in a neutral-density f ilter with an a 255 m 21 absorption coefficient. The fraction of absorbed power that gives rise to a thermal lens is roughly

P abs ͞P p ഠ aDz , (7)
where P p is the incident heating power. With the 1.2-N.A. objective lens we have aDz 2.1 3 10 24 .

We measured a signal-to-noise ratio of 1830 when we integrated the signal for 1 s, so the smallest detectable absorption factor is estimated to be 1.15 3 10 27 .

V. Loriette's e-mail address is loriette@opitque. espci.fr.

  Figure 1 illustrates this point by comparing calculations of classical and confocal thermal-lens signals performed in the paraxial approximation (i.e., by considering Gaussian beam propagation in the microscope). Another interesting feature of the confocal scheme is that, unlike in the nonconfocal case, the signal's shape is nearly insensitive to the amount of chromatic aberration caused by the focusing objective lens.

Fig. 1 .

 1 Fig. 1. Theoretical calculations of the confocal (solid curve) and the nonconfocal (dashed curve) thermal-lens signals for surface absorption. The parameters used in the calculations were l probe 638 nm for the probe beam and l heat 514 nm for the pump beam, a N.A. of 1.2 for the objective lens, index of refraction n 1.33 for the immersion medium. 0146-9592/04/131488-03$15.00/0 © 2004 Optical Society of America

Fig. 2 .

 2 Fig. 2. Schematic of the confocal thermal-lens microscope.The heating beam is an argon laser (Ar), 100 mW of which is focused into the sample (S). Its intensity is modulated with an electro-optic modulator (Mod). The probe laser (LD) is a 1-mW 638-nm laser diode. Before entering the microscope, both beams pass through a 53 beam expander (X). M1, M2, focusing and collecting objective lenses; D1, D2, dichroic mirrors used to direct the Ar beam into a beam stop (LT) and the probe beam through a calibrated pinhole (H) and onto a photodetector (PD). We performed axial scans by moving the sample.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Experimental confocal surface absorption signal amplitude and model. The sample is a single layer of 180-nm-thick Ta 2 O 5 sputtered onto a fused-silica substrate. The layer's absorption factor was estimated to be 1.8 3 10 26 . The 900-nm resolution was achieved with a water immersion objective of 1.2 N.A.