Jean-Marc Azaïs
email: jean-marc.azais@math.univ-toulouse.fr

Céline Delmas

MEAN NUMBER AND CORRELATION FUNCTION OF CRITICAL POINTS OF ISOTROPIC GAUSSIAN FIELDS AND SOME RESULTS ON GOE RANDOM MATRICES

Keywords: Primary 60G60; secondary 60B20 Critical points of Gaussian fields, GOE matrices, Kac-Rice formula 1

Let X = {X

1. Introduction. Critical points of random fields play an important role in small or large dimension. In large dimension data they appear in the study of algorithms of maximisation of the likelihood [START_REF] Ben Arous | The Landscape of the Spiked Tensor Model[END_REF][START_REF] Ros | Complex energy landscapes in spiked-tensor and simple glassy models: Ruggedness, arrangements of local minima, and phase transitions[END_REF]. In smaller dimension they play a role in many applications from various areas: detection of peaks in a random field [START_REF] Cheng | Multiple testing of local maxima for detection of peaks in random fields[END_REF][START_REF] Taylor | Detecting sparse signals in random fields, with an application to brain mapping[END_REF][START_REF] Worsley | Searching scale space for activation in PET images[END_REF][START_REF] Worsley | A unified statistical approach for determining significant signals in images of cerebral activation[END_REF], optimisation of a response modelled by a random field [START_REF] Ginsbourger | Fast computation of the multi-points expected improvement with applications in batch selection[END_REF], modelling of random sea waves [START_REF] Longuet-Higgins | The statistical analysis of a random moving surface[END_REF][START_REF] Azaïs | Geometrical characteristic of Gaussian sea waves[END_REF][START_REF] Podgórski | Statistics for velocities of Gaussian waves[END_REF]. Critical points and their indexes are an important element of the geometry of random fields. They appear in the computation of the Euler characteristic [START_REF] Adler | Random Fields and Geometry[END_REF][START_REF] Cheng | The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments[END_REF]. They are a subject of study on their own as in [START_REF] Nicolaescu | A CLT concerning critical points of random functions on a Euclidean space[END_REF][START_REF] Auffinger | Complexity of random smooth functions on the highdimensional sphere[END_REF][START_REF] Beliaev | Two Point Function for Critical Points of a Random Plane Wave[END_REF][START_REF] Cheng | Expected number and height distribution of critical points of smooth isotropic Gaussian random fields[END_REF].

Let X = {X(t) : t ∈ R N } be an isotropic Gaussian random field with real values. If we look at the repartition of the critical points of X as a function of their index in dimension two, considerations of symmetry and of Euler characteristic (see, for example, [START_REF] Adler | The Geometry of Random Fields[END_REF], lemma 6.1.1) imply that if N c (S), N c 0 (S), N c 2 (S), N c 1 (S) are respectively the number of critical, minimum, maximum and saddle points on a given set S (1)

E(N c 0 (S)) = E(N c 2 (S)) = 1 2 E(N c 1 (S)) = 1 4 E(N c (S)).
In higher dimensions simulations suggest that such a simple result does not hold true. The purpose of Section 3 is to compute these expectations using random matrices tools. With this objective in mind, we obtain an exact expression for the probability density of the kth ordered eigenvalue of a N -GOE matrix (see Theorem 3.5 and ( 22)-( 28)). We deduce exact expressions for the mean number of critical points with a given index (Propositions 3.2, 3.7 and 3.8). In particular, for N = 3, if we denote by N c 0 (S), N c 1 (S), N c 2 (S) and N c 3 (S) respectively the number of minimum, the number of critical points with index 1 and 2 (see Section 3, equation [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] for the definition) and the number of maximum on a given set S, we obtain (see Proposition 3 Proposition 3.8 gives these expectations for N = 4.

On the other hand Section 4 studies how the critical points are spread in the space. In a pioneering work, Belyaev, Cammarota and Wigman [START_REF] Beliaev | Two Point Function for Critical Points of a Random Plane Wave[END_REF] study the attraction or repulsion (see Section 4.2 for a precise definition) of the point process of critical points of a particular random wave model in dimension 2. In the case of random processes (N = 1), it was generally admitted that repulsion between crossings or critical points occurs. In fact this result has never been written explicitly so it is the object of Section 4.4 with Proposition 4. [START_REF] Auffinger | Random matrices and complexity of spin glasses[END_REF]. With respect to this behaviour the result of [START_REF] Beliaev | Two Point Function for Critical Points of a Random Plane Wave[END_REF] is unexpected since no repulsion occurs between critical points. The object of Section 4.5 is to explore the phenomenon of attraction or repulsion for a large class of random fields, in all dimensions and for each type of indexes. A precise definition of attraction and repulsion is given in Section 4.2. Theorem 4.6 proves attraction between critical points when N > 2, neutrality for N = 2 and repulsion for N = 1. Theorem 4.9 shows that the attraction between critical points that occurs when the dimension is greater than 2 is due to the attraction between critical points with adjacent indexes. In Theorem 4.11 we prove a strong repulsion, growing with N , between maxima and minima. Finally Theorem 4.12 gives an upper bound to the correlation function between maxima (or equivalently minima) proving repulsion for N < 5.

In Appendix A we prove the validity of all the Kac-Rice formulas used in the paper. Appendix B is devoted to the proofs of Lemmas 2.1, 4.1 and 4.2.

2. Notation, main assumptions and background.

Notation.

• ϕ(•) and Φ(•) are respectively the probability density and the cumulative distribution function of a standard Gaussian variable, Φ(•) := 1 -Φ(•).

• If X and Y are random vectors, Var(X) denotes the variance-covariance matrix of X and Cov(X, Y ) := E X -E(X) Y -E(Y ) .

• For X and Y two random vectors p X (x) is the probability density of X at x and p X (x/Y = y) the probability density of X conditionally to Y = y when these densities exist.

• For a random field X = {X(t) : t ∈ R N }, we denote by ∇X(t) the gradient of X(t) and by ∇ 2 X(t) its Hessian, when they exist.

• i(M ) is the index of the matrix M : the number of its negative eigenvalues.

• z + and z -denote respectively sup(0, z) and -inf(0, z).

• X i1i2...in (t) denotes ∂ n X(t) ∂t i1 ∂t i2 . . . ∂t in .

• Id n is the identity matrix of size n; J n,p the (n × p)-matrix with all elements equal to 1.

• |S| is the Lebesgue measure of the Borel set S.

• For a random field X = {X(t) : t ∈ R N } the number of critical points is precisely defined for every Borel set S as N c (S) := #{t ∈ S : ∇X(t) = 0}.

2.2. Main assumptions. In the sequel we will use the following assumptions (Ak), with k = 2, 3 or 4.

ASSUMPTION (Ak).

• X = {X(t) : t ∈ R N } is a stationary and isotropic Gaussian field with real values. We assume, without loss of generality, that it is centred with variance 1. We set

E(X(s)X(t)) = c(s, t) = r(||s -t|| 2 ).
• r is of class C k . This is equivalent to the existence of a finite 2kth spectral moment λ 2k and it implies that X is k times differentiable in quadratic mean. Note that ∀ = 1, . . . , N and n = 1, . . . , k

1 = Var(X(t)) = r(0), λ 2n := Var ∂ n X(t) ∂t n = (-1) n (2n)! n! r (n) (0). (2)
• To avoid the trivial case of a constant random field, we assume that λ 2 = -2r (0) > 0.

• When N > 1 we know from Lemma 2.1 below that λ 2n λ 2n-4 > λ 2 2n-2 for n = 2, . . . , k. When N = 1, we have to assume these relations to avoid the particular case of the sinecosine process.

• When k = 2 we have to assume, in addition for the validity of the Kac-Rice formula, that X is C 2 which is slightly stronger than the finiteness of λ 4 . When k ≥ 3 the fact that X is C 2 is a consequence of the finiteness of λ 6 .

Note that, of course, (A4) is stronger than (A3) which is itself stronger than (A2).

2.3.

Background. We give two lemmas that will be needed in the next sections. The first one gives universal properties for X . The second one is more technical. LEMMA 2.1. Let X be a stationary and isotropic Gaussian field. When λ 2k is finite, the spectral moments satisfy for n = 2, . . . , k

(3) λ 2n λ 2n-4 ≥ (2n -1) (2n -3) (2n -4 + N ) (2n -2 + N ) λ 2 2n-2 .
Moreover when i 1 + j 1 , . . . , i N + j N are all even and when all the quantities hereunder are well defined

E ∂ |i| X(t) ∂t i1 1 , . . . , ∂t iN N ∂ |j| X(t) ∂t j1 1 , . . . , ∂t jN N = ∂ |i|+|j| c(s, t) ∂s i1 1 , . . . , ∂s iN N ∂t j1 1 , . . . , ∂t jN N s=t = (-1) |j| r (|β|) (0) N =1 (2β )! β ! = (-1) |β|+|j| λ 2|β| |β|! (2|β|)! N =1 (2β )! β ! , (4)
where |β| = |i| + |j| 2 and β = i + j 2 for = 1, . . . , N . In the other cases the covariances vanish.

REMARK 2.2. Note that the coefficients K(n, N ) := (2n -1) (2n -3) (2n -4 + N ) (2n -2 + N ) are greater than 1 when N > 1 and take the value 1 when N = 1.

The proof of Lemma 2.1 is given in Appendix B.

In the sequel we need a precise description of the distribution of X(t), ∇X(t), ∇ 2 X(t) and, for technical reasons in the last section, we need to add X 1ii , X 11 , X 1111 for 1 ≤ i ≤ N and 2 ≤ ≤ N . To get independence we have to partition this vector as follows. Set

ζ 1 := (X 2 (t), . . . , X -1 (t), X +1 (t), X N (t)) , ζ 2 := (X ij (t), i, j = 1, . . . , N and i = j) , ζ 3 := (X(t), X 1111 (t), X 11 (t), X 22 (t), X 33 (t), . . . , X N N (t)) , ζ 4 := (X 1 (t), X 111 (t), X 122 (t), X 133 (t), . . . , X 1N N (t)) , ζ 5 := (X (t), X 11 (t)) .
Var (ζ 1 ) = λ 2 Id N -2 , Var (ζ 2 ) = λ 4 3 Id N (N -1)/2 , Var (ζ 3 ) = M (N +2) , Var (ζ 4 ) = M(N+1) , Var (ζ 5 ) = λ 2 -λ 4 /3 -λ 4 /3 λ 6 /5 ,
where 5) and [START_REF] Azaïs | Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets[END_REF]. We have

M (N +2) =          1 λ 4 -λ 2 -λ 2 . . . -λ 2 λ 4 λ 8 -λ 6 -λ 6 /5 . . . -λ 6 /5 -λ 2 -λ 6 λ 4 λ 4 /3 . . . λ 4 /3 -λ 2 . . . -λ 2 -λ 6 /5 . . . -λ 6 /5 λ 4 /3 . . . λ 4 /3 2λ 4 3 Id N -1 + λ 4 3 J N -1          and M(N+1) =        λ 2 -λ 4 -λ 4 /3 . . . -λ 4 /3 -λ 4 λ 6 λ 6 /5 . . . λ 6 /5 -λ 4 /3 . . . -λ 4 /3 λ 6 /5 . . . λ 6 /5 2λ 6 15 Id N -1 + λ 6 15 J N -1        . Moreover we have (5) det (Var (X 11 (t), X 22 (t), . . . , X N N (t))) = (N + 2)2 N -1 λ 4 3 N , (6)
det (Var (X 1 (t), X 122 (t), . . . , X 1N N (t))) = 2λ 6 15 N -2 3(N + 1)λ 2 λ 6 -5(N -1)λ
Var(X 11 (t), X 22 (t), . . . , X N N (t)) = 2λ 4 3 Id N + λ 4 3 J N,N .
For x, y ∈ R it is well known that ( 7)

det(xId N + yJ N,N ) = x N -1 (x + N y).
So we obtain [START_REF] Azaïs | Bounds and asymptotic expansions for the distribution of the maximum of a smooth stationary Gaussian process[END_REF]. We have

det (Var (X 1 (t), X 122 (t), . . . , X 1N N (t))) = det λ 2 Ã12 Ã21 Ã22 , with Ã12 = -λ 4 /3 • • • -λ 4 /3 , Ã21 = Ã 12 and Ã22 = 2λ 6 15 Id N -1 + λ 6 15 J N -1,N -1 .
Then using the fact that for a partitioned matrix à = Ã11 Ã12 Ã21 Ã22 we have (see [START_REF] Searle | Linear Models[END_REF] p.46)

(8) det à = det Ã11 × det( Ã22 -Ã21 Ã-1 11 Ã12 ), we obtain det (Var (X 1 (t), X 122 (t), . . . , X 1N N (t))) = λ 2 det 2λ 6 15 Id N -1 + J N -1,N -1 3λ 2 λ 6 -5λ 2 4 45λ 2 .
Finally using [START_REF] Azaïs | Geometrical characteristic of Gaussian sea waves[END_REF] we obtain (6).

3. Mean number of critical points with a given index. In this section X is a random field satisfying Assumption (A2). Then it is proved in Appendix A that the sample paths are almost surely Morse and this implies that the number of critical points with index k in a Borel set S, N c k (S), is well defined. More precisely (9)

N c k (S) := #{t ∈ S : ∇X(t) = 0, i(∇ 2 X(t)) = k}. We define also the number of critical points of index k above the level u by [START_REF] Azaïs | Erratum to: A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail[END_REF] N c k (u, S) := #{t ∈ S : ∇X(t) = 0, i(∇ 2 X(t)) = k, X(t) > u}. The aim of this section is to derive explicit and exact expressions for the expectation of ( 9) and [START_REF] Azaïs | Erratum to: A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail[END_REF].

3.1. The general case. By Kac-Rice formulas (68) and (69) and Lemma 2.3 we get

E (N c k (S)) = |S| (2πλ 2 ) N/2 E | det(∇ 2 X(t))|1 i(∇ 2 X(t))=k . (11) E(N c k (u, S)) = |S| λ N/2 2 (2π) (N +1)/2 +∞ u exp - x 2 2 (12) × E | det(∇ 2 X(t))1 i(∇ 2 X(t))=k | X(t) = x dx.
Our main tool will be the use of random matrices theory and more precisely the GOE distribution. We recall that G n is said to have the GOE (Gaussian Orthogonal Ensemble) distribution if it is symmetric and all its elements are independent centred Gaussian variables satisfying

E(G 2 ii ) = 1 and E(G 2 ij ) = 1 2 .
The relation between GOE matrices and the study of critical points of stationary isotropic Gaussian fields is due to the following lemma initially due to [START_REF] Azaïs | A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail[END_REF] and that can be derived from Lemma 2.3. LEMMA 3.1. Let G N be a size N GOE matrix and Λ a N (0, 1/2) random variable independent of G N . Then ∇ 2 X(t) is equal in distribution to

2λ 4 3 G N -ΛId N ,
and under the assumption that λ 4 ≥ 3λ 2 2 , ∇ 2 X(t) conditionally to X(t) = x, is distributed as

2λ 4 3 G N -ΛId N ,
where

Λ is a N λ 2 x 3 2λ 4 , λ 4 -3λ 2 2 2λ 4 random variable independent of G N .
We recall that the joint density f N of the eigenvalues (µ 1 , . . . , µ N ) of a N -GOE matrix (see Theorem 3.3.1 of [START_REF] Mehta | Random Matrices[END_REF]) is given by:

(13) f N (µ 1 , . . . , µ N ) = k N exp - N i=1 µ 2 i 2 1≤i<k≤N |µ k -µ i | ,
where:

(14) k N := (2π) -N/2 (Γ(3/2)) N N i=1 Γ(1 + i/2) -1
.

Using Lemma 3.1, ( 11), ( 12) and ( 13) we get the following proposition. PROPOSITION 3.2. Let L p be the pth ordered eigenvalue of a (N + 1)-GOE matrix (L 1 < L 2 < . . . < L N +1 ). For X and S as above, under Assumption (A2)

(15) E (N c k (S)) = |S| π (N +1)/2 λ 4 3λ 2 N/2 k N k N +1 1 N + 1 E exp - L 2 k+1 2 .
When λ 4 -3λ 2 2 > 0,

(16) E(N c k (u, S)) = |S| π (N +1)/2 λ 4 3λ 2 N/2 k N k N +1 1 N + 1 × E exp - L 2 k+1 2 Φ λ 4 λ 4 -3λ 2 2 u -L k+1 √ 6λ 2 √ λ 4 ,
and when λ 4 -3λ 2 2 = 0

(17) E(N c k (u, S)) = |S| π (N +1)/2 k N k N +1 λ N/2 2 N + 1 E exp - L 2 k+1 2 1 Lk+1>u/ √ 2 .
REMARK 3.3. Such kind of result was first obtained by [START_REF] Auffinger | Random matrices and complexity of spin glasses[END_REF] for the p-spin spherical spin glass model defined on the Euclidean sphere of radius √ N of R N . This result can also be found in [START_REF] Cheng | Expected number and height distribution of critical points of smooth isotropic Gaussian random fields[END_REF] (Proposition 3.9) written in a slightly different way. In this paper we go further: in Theorem 3.5 we obtain an exact expression for the probability density of the kth ordered eigenvalue of a N -GOE matrix, denoted by q k N (l) (see [START_REF] Chiani | Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution[END_REF] and ( 22)-( 28)). We can deduce exact expressions for [START_REF] De Bruijn | On some multiple integrals involving determinants[END_REF], ( 16) and [START_REF] Cheng | Expected number and height distribution of critical points of smooth isotropic Gaussian random fields[END_REF] as in Propositions 3.7, and 3.8. REMARK 3.4. As remarked in [START_REF] Azaïs | Erratum to: A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail[END_REF] the condition λ 4 ≥ 3λ 2 2 is met if the covariance function r is a "Schoenberg covariance": it is a valid covariance function in every dimension. Note that more general cases have been studied by [START_REF] Cheng | Expected number and height distribution of critical points of smooth isotropic Gaussian random fields[END_REF].

PROOF OF PROPOSITION 3.2. Using [START_REF] Beliaev | Two Point Function for Critical Points of a Random Plane Wave[END_REF], Lemma 3.1 and ( 13)

E | det ∇ 2 X(t)|1 i(∇ 2 X(t))=k = 2λ 4 3 N/2 k N N ! µ1<µ2<•••<µk<λ<µk+1<•••<µN 1≤i<k≤N |µ k -µ i | 1≤i≤N |λ -µ i | exp - N i=1 µ 2 i 2 π -1/2 exp(-λ 2 /2) exp(-λ 2 /2)dµ 1 . . . dµ N dλ = 2λ 4 3 N/2 π -1/2 k N k N +1 N ! (N + 1)! E exp - L 2 k+1 2 ,
which proves [START_REF] De Bruijn | On some multiple integrals involving determinants[END_REF]. Using Lemma 3.1, by ( 12) and ( 13) we can write

E(N c k (u, S)) = |S| λ N/2 2 (2π) (N +1)/2 +∞ u exp - x 2 2 2λ 4 3 N/2 k N N ! × µ1<µ2<•••<µk<y<µk+1<•••<µN 1≤i<k≤N |µ k -µ i | 1≤i≤N |y -µ i | exp - N i=1 µ 2 i 2 × exp(-y 2 /2) exp(y 2 /2)p Y (y)dµ 1 . . . dµ N dy dx.
Integrating first with respect to x we obtain [START_REF] Cheng | Multiple testing of local maxima for detection of peaks in random fields[END_REF]. In the same way, when λ 4 -3λ 2 2 = 0, ∇ 2 X(t)/X(t) = x is distributed as 2λ 2 2 G N -λ 2 xId N and following the same lines as previously we obtain (17).

3.2. Exact expressions. Before giving exact expressions for E(N c k (S)) and E(N c k (u, S)) we obtain, in the following paragraph 3.2.1, some results concerning the probability density of the kth ordered eigenvalue of a N -GOE matrix. These results will be used in paragraph 3.2.2, with Propositions 3.2, to derive exact expressions for E(N c k (S)) and E(N c k (u, S)).

3.2.1.

Probability density of the kth ordered eigenvalue of a N -GOE matrix. We denote by L 1 ≤ L 2 ≤ . . . ≤ L N the ordered eigenvalues of a N -GOE matrix. We denote by q k N ( ) the probability density of L k . In this section we obtain an expression for q k N ( ). We give explicit expressions of q k N ( ) for N = 2, 3, 4, 5 and k = 1, . . . , N . In particular we obtain that q 2 3 ( ) is the density of a N (0, 1/2) random variable.

Let F p (n) be the set of the parts of {1, 2, . . . , n} of size p. Let I ∈ F p (n), let ∈ R, we define D I i ( ) = (-∞, ) when i ∈ I and D I i ( ) = ( , +∞) when i / ∈ I. We denote by s(.) the sign function. We define the matrix A α (I, ), for α = 1 or 2, as follows. When n is even, A α (I, ) is the n × n skew matrix whose elements are, ∀i, j = 1 . . . , n: (18)

a α i,j (I, ) = D I i ( ) dx D I j ( ) dy s(y -x)x i-1 y j-1 (x -) α (y -) α exp - x 2 + y 2 2 .
When n is odd , A α (I, ) is the (n + 1) × (n + 1) skew matrix whose elements are defined by ( 18) ∀i, j = 1 . . . , n and we add the extra terms, for i = 1, . . . , n:

a α i,n+1 (I, ) = -a α n+1,i (I, ) = D I i ( ) x i-1 (x -) α exp - x 2 2 (19)
dx and a I n+1,n+1 ( ) = 0.

We are now able to state the following theorem that gives an exact expression for q k N ( ), the probability density of L k , the kth ordered eigenvalue of a N -GOE matrix. THEOREM 3.5. Using notations above, for ∈ R and for k = 1, . . . , N we have

(20) q k N ( ) = k N N !(-1) k-1 exp - 2 2 I∈Fk-1(N -1) Pf A 1 (I, )
where k N is given by ( 14) and Pf A 1 (I, ) is the Pfaffian of the skew matrix A 1 (I, ) defined by ( 18) and ( 19) with n = N -1 and α = 1.

REMARK 3.6. The probability density of the kth ordered eigenvalue of a N -GOE matrix was only known for k = N , the largest eigenvalue (see [START_REF] Chiani | Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution[END_REF]). Theorem 3.5 gives an expression for all k = 1, . . . , N . 13), the probability density of L k is given by:

PROOF OF THEOREM 3.5. Let G N -1 be a (N -1)-GOE matrix with eigenvalues denoted by µ 1 , . . . , µ N -1 . For ∈ R, we set for k = 2, . . . , N -1, O k ( ) = {(µ i , i = 1, . . . , N -1) ∈ R N -1 : µ 1 ≤ µ 2 ≤ . . . ≤ µ k-1 ≤ ≤ µ k ≤ . . . ≤ µ N -1 }, with trivial adaptation when k = 1 and N . L 1 < L 2 < . . . < L N denote the ordered eigenvalues of a N -GOE matrix. Using (
q k N (l) =N ! Ok( ) f N (µ 1 , . . . , µ N -1 , )dµ 1 . . . dµ N -1 = k N N ! k N -1 Ok( ) (-1) k-1 det(G N -1 -Id N -1 )f N -1 (µ 1 , . . . , µ N -1 ) × exp - 2 2 dµ 1 . . . dµ N -1 . Thus q k N (l) = k N k N -1 N (-1) k-1 exp - 2 2 γ k-1 N -1,1 ( ) where γ k-1 N -1,1 ( ) = k N -1 (N -1)! × Ok( ) N -1 i=1 (µ i -) 1≤i<j≤N -1 (µ j -µ i ) exp - N -1 i=1 µ 2 i 2 dµ 1 . . . dµ N -1 .
We set h m (µ j , ) = µ m-1 j (µ j -) exp -µ 2 j /2 . We denote by H(µ µ µ, ) the matrix {h m (µ j , ), m, j = 1, . . . , N -1}. Then calculating det(H(µ µ µ, )) using a Vandermonde determinant we obtain:

γ k-1 N -1,1 ( ) = k N -1 (N -1)! Ok( ) det(H(µ µ µ, ))dµ 1 • • • µ N -1 .
We denote by I(µ µ µ, , k) the (N -1) × (N -1) diagonal matrix defined by

I(µ µ µ, , k) = diag(1 µ1≤ , . . . , 1 µk-1≤ , 1 µk≥ , . . . , 1 µN-1≤ )
with trivial adaptations when k = 1 or k = N . We set

H (µ µ µ, , k) := H(µ µ µ, )I(µ µ µ, , k). Then γ k-1 N -1,1 ( ) = k N -1 (N -1)! µ1≤...≤µN-1 det(H (µ µ µ, , k))dµ 1 • • • µ N -1 .
Now we need to introduce some more notation. Let I and J ∈ F p (n) and M a n × n matrix, we set ∆ I,J (M ) for det(M I,J ). We use the convention ∆ ∅,∅ (M ) = 1. K = {1, . . . , n}\K for any subset K.

Let I ∈ F p (n), σ I is the permutation of (1, 2, . . . , n) such that σ I | (1,2,...,p) (resp. σ I | (p+1,...,n)
) is an increasing one-to-one mapping from (1, 2, . . . , p) on I (resp. from (p + 1, . . . , n) on Ī). Finally we denote by (σ) the signature of the permutation σ of (1, 2, . . . , n).

For a (n + p) × (n + p) matrix D we have, for any J ∈ F p (n + p) fixed, [START_REF] Cramér | Stationary and Related Stochastic Processes[END_REF] det D =

I∈Fp(n+p) (σ I ) (σ J )∆ I,J (D)∆Ī , J (D).
We apply it for p = k -1, n = N -k and J = {1, . . . , k -1}. We get (σ J ) = 1 and

γ k-1 N -1,1 ( ) = k N -1 (N -1)! I∈Fk-1(N -1) (σ I ) × µ1≤...≤µN-1 ∆ I,J (H (µ µ µ, , k))∆Ī , J (H (µ µ µ, , k))dµ 1 • • • µ N -1 .
We denote by H I (µ µ µ, ) the matrix {h m (µ j , )1 µj∈D I m ( ) , m, j = 1, . . . , N -1}. We have ∆ I,J (H (µ µ µ, )) = ∆ I,J (H I (µ µ µ, )) and ∆Ī , J (H (µ µ µ, )) = ∆Ī , J (H I (µ µ µ, )).

To check this, note that, for example, for j ≤ k -1, the indicator function appearing in the entry (m, j) of H (µ µ µ, , k) is 1 µj≤ . For every I and for m ∈ I, this quantity is also equal to

1 µj∈D I m ( ) .
Then using [START_REF] Cramér | Stationary and Related Stochastic Processes[END_REF]

with p = k -1, n = N -k and J = {1, . . . , k -1}, we have ∆ Ĩ,J (H I (µ µ µ, )) = 0 when Ĩ = I and det(H I (µ µ µ, ))1 µ1≤...≤µN-1 = (σ I )∆ I,J (H I (µ µ µ, ))∆Ī , J (H I (µ µ µ, ))1 µ1≤...≤µN-1 .
We conclude that [START_REF] De Bruijn | On some multiple integrals involving determinants[END_REF], [START_REF] Mehta | Moments of the characteristic polynomial in the three ensembles of random matrices[END_REF] or equation [START_REF] Ben Arous | The Landscape of the Spiked Tensor Model[END_REF] in [START_REF] Chiani | Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution[END_REF]) we obtain [START_REF] Chiani | Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution[END_REF]. This concludes the proof.

γ k-1 N -1,1 ( ) = k N -1 (N -1)! I∈Fk-1(N -1) µ1≤...≤µN-1 det(H I (µ µ µ, ))dµ 1 • • • µ N -1 . Since Pf A 1 (I, ) = µ1≤...≤µN-1 det(H I (µ µ µ, ))dµ 1 • • • µ N -1 (see
The major drawback of the result above is its complicated form. However, for small values of N , we are able to get an explicit expression for q k N (l) and consequently (using ( 15), ( 16) and ( 17)) for E (N c k (S)) and E (N c k (u, S)). We give some examples below and we derive Propositions 3.7 and 3.8 which are new results.

Examples: After tedious calculations we obtain 1. For N = 2: q 1 2 (l) = q 2 2 (-l) and ( 22)

q 2 2 (l) = exp -l 2 2 2 √ π exp - l 2 2 + √ 2πlΦ(l) .
2. For N = 3:

q 1 3 (l) = q 3 3 (-l), q 2 3 (l) = exp -l 2 √ π and (23) q 3 3 (l) = exp -l 2 2 π √ 2 √ π(2l 2 -1)Φ l √ 2 + √ 2π exp - l 2 2 Φ(l) + l exp -l 2 .
3. For N = 4: q 1 4 (l) = q 4 4 (-l), q 2 4 (l) = q 3 4 (-l) and ( 24)

q 3 4 (l) = exp -l 2 2 2π 3l 2 exp - 3l 2 2 + √ 2π 1 - l 2 2 Φ(l) exp -l 2 - π(2l 3 -3l) √ 2 Φ l √ 2 Φ(l) + 3 √ π(1 + 2l 2 ) 2 Φ l √ 2 exp - l 2 2 . ( 25
) q 4 4 (l) = exp -l 2 2 2π 3l 2 exp - 3l 2 2 - √ 2π 1 - l 2 2 Φ(l) exp -l 2 + π(2l 3 -3l) √ 2 Φ l √ 2 Φ(l) + 3 √ π(1 + 2l 2 ) 2 Φ l √ 2 exp - l 2 2 .
4. For N = 5: q 1 5 (l) = q 5 5 (-l), q 2 5 (l) = q 4 5 (-l) and ( 26)

q 3 5 (l) = q 4 5 (l) -2q 5 5 (l) + √ 2 exp -l 2 2 3π 3/2 π 4l 4 -12l 2 + 3 Φ l √ 2 + √ π 3l 3 - 13l 2 exp -l 2 + √ 2π l 4 + 3l 2 + 3 4 exp - l 2 2 Φ(l) . ( 27
) q 4 5 (l) = √ 2 exp -l 2 2 3π 3/2 √ 2π exp - 3l 2 2 l 3 2 + 5l 4 + √ 2πΦ l √ 2 exp - l 2 2 l 4 + 3l 2 + 3 4 . ( 28
) q 5 5 (l) = √ 2 exp -l 2 2 3π 3/2 2l 4 -6l 2 + 3 2 πΦ 2 l √ 2 + l 4 + 3l 2 + 3 4 √ 2πΦ l √ 2 Φ(l) exp - l 2 2 + √ 2π l 3 2 + 5l 4 Φ(l) exp - 3l 2 2 + 3l 3 - 13l 2 √ πΦ l √ 2 exp -l 2 + l 2 -2 exp -2l 2 .
These probability densities are plotted in Figure 1. They all seem very close to a Gaussian density. But only one, q 2 3 (l), is exactly Gaussian.

Mean number of critical points.

For N = 2, using ( 15) and ( 23), after some calculations, we retreive (1) with

E (N c (S)) = 2|S| √ 3π λ 4 3λ 2 .
In Proposition 3.7 we give the exact expression of E (N c k (S)) when N = 3 for k = 0, 1, 2, 3. This proposition is a new result. PROPOSITION 3.7. Under the conditions of Proposition 3.2 when N = 3

E (N c 0 (S)) = E (N c 3 (S)) = |S| π 2 √ 2 λ 4 3λ 2 3/2 29 -6 √ 6 12 √ 6 E (N c 1 (S)) = E (N c 2 (S)) = |S| π 2 √ 2 λ 4 3λ 2 3/2 29 + 6 √ 6 12 √ 6 .
Consequently:

E (N c 0 (S)) E (N c (S)) = E (N c 3 (S)) E (N c (S)) = 29 -6 √ 6 116 0.1233 E (N c 1 (S)) E (N c (S)) = E (N c 2 (S)) E (N c (S)) = 29 + 6 √ 6 116
0.3767.

In the same way for N = 4, we obtain the expressions given in Proposition 3.8.

PROPOSITION 3.8.

Set I := E Φ(Y )Φ( √ 2Y
) , Y being a Gaussian centred variable with variance 1/3. Under the conditions of Proposition 3.2 when N = 4

E (N c 1 (S)) = E (N c 3 (S)) = |S| π 2 λ 4 3λ 2 2 25 24 √ 3 , E (N c 0 (S)) = E (N c 4 (S)) = |S| π 2 λ 4 3λ 2 2 I × 100π -57 48 √ 3π , E (N c 2 (S)) = |S| π 2 λ 4 3λ 2 2 50π(1 -2I) + 57 24 √ 3π .
Consequently,

E (N c 1 (S)) E (N c (S)) = E (N c 3 (S)) E (N c (S)) = 1 4 , E (N c 0 (S)) E (N c (S)) = E (N c 4 (S)) E (N c (S)) = I × 100π -57 200π 0.060, E (N c 2 (S)) E (N c (S)) = 50π(1 -2I) + 57 100π 0.380.
Note that, in the same way, it is possible to obtain exact expressions for the mean number of critical points above a level u for N = 2, 3 or 4 using ( 23), ( 24), ( 25) and ( 26), ( 27), [START_REF] Podgórski | Statistics for velocities of Gaussian waves[END_REF] with Proposition 3.2.

Correlation function between critical points.

In this section X is a random field satisfying Assumption (A2).

Correlation function, two points function. The correlation function of a point process P is defined by [14]

A(s, t) := lim where P(B(s, )) is the number of points in the ball with center s and radius and V ( ) its volume of this ball. When the considered process is stationary and isotropic, this function depends only on the norm ρ := s -t and by a small abuse of notation we will denote it by A(ρ). Suppose now that the point process P is the process of critical points of X . Under our conditions, the Kac-Rice formula of order two in sufficiently small sets is valid (see Appendix A, Proposition A.3, equation ( 70)) . In particular if S and T are disjoint, sufficiently close and sufficiently small (as B(s, ) and B(t, ) for s = t and s -t and sufficiently small), the Kac-Rice formula of order two (70) yields

(29) E[N c (S)N c (T )] = S×T E | det(∇ 2 X(s)) det(∇ 2 X(t))|/∇X(s) = ∇X(t) = 0 p ∇X(s),∇X(t) (0, 0)dsdt,
proving that for ρ sufficiently small, A(ρ) is well defined and given by ( 30)

A(ρ) = E | det(∇ 2 X(0)) det(∇ 2 X(t))|/∇X(0) = ∇X(t) = 0 p ∇X(0),∇X(t) (0, 0)
with, for example, t = ρe 1 and e 1 := (1, 0, . . . , 0). Some papers, as [START_REF] Beliaev | Two Point Function for Critical Points of a Random Plane Wave[END_REF] consider the behaviour as ρ → 0 of

T (ρ) := E(N c (B ρ )(N c (B ρ ) -1)
), B ρ is any ball with radius ρ.

It is elementary to see that if A(ρ) Cρ d then T (ρ) CV 2 N ρ d+2N
, where means, as in the rest of the paper, equivalence as ρ → 0 and V N is the volume of the unit ball.

Attraction, neutrality, repulsion.

• The reference of neutral point process is the Poisson process for which the correlation function A(ρ) is constant and T (ρ) behaves as ρ 2N . • The repulsion is defined by the fact that the correlation function A(ρ) tends to zero as ρ → 0. Equivalently T (ρ) = o(ρ 2N ). Note that determinental processes [START_REF] Hough | Determinantal processes and independence[END_REF] are a way of constructing repulsive point processes.

• The attraction is just the contrary:

as ρ → 0, A(ρ) → +∞, T (ρ) ρ 2N → +∞.
As already mentioned in the preceding section, under our assumptions the sample paths are a.s. Morse, and the index of each critical point can a.s. be defined. We can generalise the definition of the correlation function by

(31) A i1,i2 (s, t) := lim →0 1 V 2 ( ) E N c i1 (B(s, ))N c i2 (B(t,
)) , when it exists.

The Kac-Rice formula (71) yields that for ρ sufficiently small this function is well defined and (32) A i1,i2 (ρ) = p ∇X(0),∇X(t) (0, 0)

× E | det(∇ 2 X(0))1 i(∇ 2 X(0))=i1 det(∇ 2 X(t))1 i(∇ 2 X(t))=i2 |/∇X(0) = ∇X(t) = 0 ,
with again t = ρe 1 . In the same way, we can consider attraction, neutrality or repulsion between critical points with indexes i 1 and i 2 .

Before giving our main results concerning correlation functions between critical points, we give in the following paragraph some results concerning the conditional distribution of ∇ 2 X(0), ∇ 2 X(t) given ∇X(0) = ∇X(t) = 0.

4.3.

Conditional distribution of ∇ 2 X(0), ∇ 2 X(t). In this section, for short, r(ρ 2 ), r (ρ 2 ), r (ρ 2 ), r (ρ 2 ) and r (4) (ρ 2 ) are denoted by r, r , r , r and r (4) . We recall that t = ρe 1 . LEMMA 4.1. Let ξ(0) and ξ(t) be a representation of the distribution of ∇ 2 X(0) and ∇ 2 X(t) given ∇X(0) = ∇X(t) = 0. We note ξ d (t) the vector (ξ 11 (t), . . . , ξ N N (t)) and ξ u (t) the vector ξ 12 (t), ξ 13 (t), . . . , ξ (N -1)N (t) . The joint distribution of (ξ d (0), ξ u (0), ξ d (t), ξ u (t)) is centred Gaussian with variance-covariance matrix:

    Γ 1 0 Γ 3 0 0 Γ 2 0 Γ 4 Γ 3 0 Γ 1 0 0 Γ 4 0 Γ 2     .
Γ 1 is the N × N matrix:

Γ 1 =      12r (0) 4r (0) • • • 4r (0) 4r (0) . . . 4r (0) . . . . . . 4r (0) . . . 4r (0) 4r (0) • • • 4r (0) 12r (0)      + ρ 2 r (0) 2[r (0) 2 -(r + 2r ρ 2 ) 2 ] × M, with M =      (12r + 8ρ 2 r ) 2 4r (12r + 8ρ 2 r ) • • • 4r (12r + 8ρ 2 r ) 4r (12r + 8ρ 2 r ) 16r 2 • • • 16r 2 . . . . . . . . . . . . 4r (12r + 8ρ 2 r ) 16r 2 • • • 16r 2      . Γ 2 is the N (N -1) 2 × N (N -1)
2 diagonal matrix:

Γ 2 = D 1 0 0 D 2 with D 1 = 4r (0) + 8ρ 2 (r ) 2 r (0) r (0) 2 -(r ) 2 Id N -1 and D 2 = 4r (0) Id (N -1)(N -2) 2 .
We set a := 4r + 8ρ 2 r and d := 12r + 48ρ 2 r + 16ρ 4 r (4) . Γ 3 is the N × N matrix:

Γ 3 =        d a • • • • • • a a 12r 4r
       + ρ 2 (r + 2r ρ 2 ) 2 r (0) 2 -(r + 2r ρ 2 ) 2 × M. Γ 4 is the N (N -1) 2 × N (N -1) 2 diagonal matrix: Γ 4 = D1 0 0 D2 with D1 = a + 8ρ 2 (r ) 2 r r (0) 2 -r 2 Id N -1 and D2 = 4r Id (N -1)(N -2) 2 .
The proof of Lemma 4.1 is given in Appendix B.

In the following lemma, we give the equivalent of the variance-covariance matrix of ∇ 2 X(0), ∇ 2 X(t) given ∇X(0) = ∇X(t) = 0 as ρ → 0. (37) All the other covariances Cov(ξ il (t), ξ mn (t)) are zero, ∀i, l, m, n ∈ {1, . . . , N }.

∇X(t))) ρ 2N λ N 2 λ N 4 3 N -1 , Var (ξ 11 (t)) ρ 2 4 (λ 2 λ 6 -λ 2 4 ) λ 2 , Var (ξ 1j (t)) ρ 2 4 (9λ 2 λ 6 -5λ 2
We have of course the same relations for ξ(0).

Moreover we have ∀j, k ∈ {2, . . . , N } and ∀i ∈ {1, . . . , N }; as ρ → 0,

(38) Cov (ξ jk (0), ξ jk (t)) Var (ξ jk (t)) , Cov (ξ 1i (0), ξ 1i (t)) -Var (ξ 1i (t)) , (39)
Cov (ξ 11 (0), ξ jj (t)) = ρ 2 15λ 2 4 -7λ 2 λ 6 180λ 2 + o(ρ 2 ), Cov (ξ jj (0), ξ kk (t)) 2λ 4 9 for j = k, (40)
All the other covariances Cov(ξ il (0), ξ mn (t)) are zero, ∀i, l, m, n ∈ {1, . . . , N }.

Finally we also have

(41) det (Var (ξ 11 (t), ξ 11 (0))) ρ 6 144 (λ 4 λ 8 -λ 2 6 )(λ 2 λ 6 -λ 2 4 ) λ 2 λ 4 .
The proof of Lemma 4.2 is given in Appendix B. It is based on some lengthy computations but uses a trick that simplify drastically computations. We present it here in the example of the proof of (33). TRICK 4.3. Because a determinant is invariant by adding to some row (resp. column) a linear combination of other rows (resp. columns) as ρ → 0, det (Var (∇X(0), ∇X(t))) = det(Var(X 1 (0), . . . , X N (0), X 1 (t), . . . , X N (t))) = det(Var(X 1 (0), . . . , X N (0), X 1 (t) -X 1 (0), . . . , X N (t) -X N (0))) ρ 2N det(Var(X 1 (0), . . . , X N (0), X 11 (0), . . . , X 1N (0))).

Using Lemma 2.3 we obtain [START_REF] Worsley | Searching scale space for activation in PET images[END_REF]. 4.4. Correlation function between critical points for random processes. The result below appears in a hidden way in Proposition 4.5 or Section 5.2 of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] or in [START_REF] Azaïs | Bounds and asymptotic expansions for the distribution of the maximum of a smooth stationary Gaussian process[END_REF]. It is stated in term of crossings of a level and can be applied directly to critical points that are simply crossings of the level zero of the derivative. PROPOSITION 4.4. Let X = {X(t) : t ∈ R} be a stationary and isotropic Gaussian process with real values satisfying Assumption (A3). Let A(ρ) be the correlation function between critical points of X defined in [START_REF] Searle | Linear Models[END_REF]. Then, as ρ → 0

(42) A(ρ) (λ 2 λ 6 -λ 2 4 ) 8π λ 4 λ 3 2 ρ.
Under Assumption (A4), let A 1,1 (ρ) (resp. A 0,0 (ρ)) be the correlation function between maxima (resp. minima) defined in [START_REF] Taylor | Detecting sparse signals in random fields, with an application to brain mapping[END_REF]. Then as ρ → 0

(43) A 1,1 (ρ) = A 0,0 (ρ) (λ 4 λ 8 -λ 2 6 ) 3/2 1296π 2 (λ 2 4 )(λ 2 λ 6 -λ 2 4 ) 1/2 ρ 4 .
Note that all the coefficients above are positive. The interpretation of the proposition is that we have always repulsion between critical points and a very strong repulsion between maxima or between minima. As we will see, the surprising result is that does not remain true in higher dimension. Before beginning the proof we need to give the following lemma .

LEMMA 4.5. Let X, Y be two jointly Gaussian variables with common variance σ 2 and correlation c. Let r a positive real. Then as c → -1

E (X + Y + ) r K r σ -2(1+r) det(Var(X, Y )) (2r+1)/2 ,
where

K r = 1 2π +∞ 0 +∞ 0 x r y r exp - (x + y) 2 2 dxdy < +∞ , K 1 = 1 6π .
Moreover as c → -1

E X + Y - σ 2 2 .
PROOF OF LEMMA 4.5. We set Σ := Var(X, Y ). Then

E (X + Y + ) r = +∞ 0 +∞ 0 x r y r 1 2π √ det Σ exp - σ 2 (x 2 -2cxy + y 2 ) 2 det Σ dxdy 1 2π (det Σ) (2r+1)/2 σ -2(1+r) +∞ 0 +∞ 0 x r y r exp - (x + y) 2 2 dxdy,
where we have made the change of variables x = x det Σ σ 2 , y = y det Σ σ 2 and the passage to the limit is justified because the integrand is a monotone function of c. It is easy to check the convergence of the integral.

E X + Y -= - +∞ 0 0 -∞ xy 2π √ det Σ exp - σ 2 (x 2 -2cxy + y 2 ) 2 det Σ dydx.
Integrating first with respect to y we obtain

E X + Y -= (det Σ) 3/2 2πσ 4 -c +∞ 0 x 2 σ √ 2π exp - x 2 2σ 2 Φ -cx σ 2 det Σ dx. As c → -1, det Σ σ 2 = σ 2 (1 -c 2 ) → 0, therefore +∞ 0 x 2 σ √ 2π exp - x 2 2σ 2 Φ -cx σ 2 det Σ dx → σ 2 2 .
Moreover (det Σ)

A 0,0 (ρ) = p X (ρ),X (0) (0, 0) E X (0) + X (ρ) + /X (0) = X (ρ) = 0 . By (33) we have p X (ρ),X (0) (0, 0) ρ -1 2π λ 4 λ 2 -1 . Define w := Cov X (0), X (ρ)/X (0) = X (ρ) = 0 , v := Var X (0)/X (0) = X (ρ) = 0 = Var X (ρ)/X (0) = X (ρ) = 0 , D := det Var X (0), X (ρ)/X (0) = X (ρ) = 0 = v 2 -w 2 .
By (38), ( 34) and (41) we have w -v,

v ρ 2 λ 2 λ 6 -λ 2 4 4λ 2 and D ρ 6 (λ 4 λ 8 -λ 2 6 )(λ 2 λ 6 -λ 2 4 ) 144λ 2 λ 4 .
Now using Lemma 4.5 we get

E X (0) + X (ρ) + /X (0) = X (ρ) = 0 D 3/2 6πv 2 .
That concludes the proof.

Correlation function between critical points for random fields.

4.5.1. All the critical points. Consider X = {X(t) : t ∈ R N } a stationary and isotropic Gaussian field with real values. Let us consider two points s, t ∈ R N . Theorem 4.6 below, gives the asymptotic expression (as ρ → 0) of the correlation function between critical points (30) of any isotropic Gaussian field. It generalizes the result of [START_REF] Beliaev | Two Point Function for Critical Points of a Random Plane Wave[END_REF] and of [START_REF] Beliaev | No repulsion between critical points for planar Gaussian random fields[END_REF] to general fields in any dimension. THEOREM 4.6. Let X = {X(t) : t ∈ R N } be a stationary and isotropic Gaussian field with real values satisfying Assumption (A3). Let A(ρ) be the correlation function between critical points [START_REF] Searle | Linear Models[END_REF]. Then as ρ → 0,

(44) A(ρ) ρ 2-N γ N -1 2 3 3 (N -1)/2 π N λ 4 λ 2 N (λ 2 λ 6 -λ 2 4 ) λ 2 λ 4 ,
where γ N -1 is defined by

γ N -1 := E det 2 (G N -1 -ΛId N -1 ) ,
with G N -1 a (N -1)-GOE matrix and Λ an independent Gaussian random variable with variance 1/3.

REMARK 4.7. We set γ N -1,2 (x) := E det 2 (G N -1 -xId N -1
) . Note that formulas 2.2.16 and 2.2.17 in [START_REF] Mehta | Moments of the characteristic polynomial in the three ensembles of random matrices[END_REF] give explicit expressions for the calculation of γ N -1,2 (x) for x ∈ R. REMARK 4.8.

• Of course when N = 1, we retrieve (42). • In the particular case N = 2, our result agrees with the result of [START_REF] Beliaev | No repulsion between critical points for planar Gaussian random fields[END_REF], Theorem 1.2. • Theorem 4.6 means that, for N = 2, there is a neutrality between critical points and for N > 2 there is even attraction! This is quite different from the case N = 1. The next theorem will give an interpretation of this phenomenon.

• A first important consequence is finiteness of the second moment of the number of critical points. Indeed if S is a compact set of R N we can write the Kac-Rice formulas of order 1 and 2, ( 11), ( 29), [START_REF] Taylor | Detecting sparse signals in random fields, with an application to brain mapping[END_REF]. If N c (S) is the number of critical points that belong to S then

E(N c (S)) = |S| E | det ∇ 2 X(t)| p ∇X(t) (0)
and ( 45)

E (N c (S))(N c (S) -1) = S 2
A(||s -t||)ds dt.

Since λ 4 is assumed to be finite, the expectation is always finite. As for the second factorial moment, its finiteness is in general not systematic. In our case (44) implies the convergence of the integral in (45) on the diagonal implying in turn the finiteness of the second moment. Nevertheless our conditions are less general than [START_REF] Estrade | Number of critical points of a Gaussian random field: Condition for a finite variance[END_REF] or [START_REF] Azaïs | Necessary and sufficient conditions for the finiteness of the second moment of the measure of level sets[END_REF].

PROOF OF THEOREM 4.6. For the simplicity of the exposition we admit in a first step all passages to the limit. They will be justified at the end of the proof. By [START_REF] Searle | Linear Models[END_REF] (46)

A(ρ) = E (| det(ξ(0)) det(ξ(t))|) p ∇X(0),∇X(t) (0, 0),
with t = ρe 1 . We recall that ξ(0) and ξ(t) are a representation of the conditional distribution of the Hessian at 0 and t. Because of ( 33), (47)

p ∇X(0),∇X(t) (0, 0) ρ -N 3 (N -1)/2 λ -N/2 2 λ -N/2 4 (2π) -N .
It remains to study the expectation in (46). We denote by ξ -1 (t) the (N -1) × (N -1) matrix corresponding to the Hessian matrix ξ(t) without its first row and column. Let us develop det (ξ(t)) with respect to the first row, for example. The first term is ξ 11 (t) det (ξ -1 (t)).

Consider another term and develop it now with respect to the first column. Each of the N -1 terms that appear are products that include a factor ξ 1j ξ j 1 with j, j = 1 so because of (34) they are O p (ρ 2 ): divided by ρ 2 they are bounded in probability. As a consequence we have proved that

det (ξ(t)) = ξ 11 (t) det (ξ -1 (t)) + O p (ρ 2 ).
By [START_REF] Yaglom | Some classes of random fields in n-dimensional space, related to stationary random processes[END_REF] and [START_REF] Worsley | A unified statistical approach for determining significant signals in images of cerebral activation[END_REF] we have

(48) ξ -1 (t) 2λ 4 3 (G N -1 -ΛId N -1 ) ,
where Λ is a N (0, 1/3) random variable, G N -1 is a size (N -1) GOE matrix defined previously, Λ and G N -1 are independent. So ( 49)

det (ξ -1 (t)) 2λ 4 3 (N -1)/2 det (G N -1 -ΛId N -1 ) .
The order of magnitude of the first term in the development of the determinant is then ρ.

In conclusion we have proved that

det (ξ(t)) = O p (ρ) = ξ 11 (t) det (ξ -1 (t)) + O p (ρ 2 ) (50) det (ξ(0)) det (ξ(t)) = O p (ρ 2 ) = ξ 11 (0)ξ 11 (t) det (ξ -1 (0)) det (ξ -1 (t)) + O p (ρ 3 ).
By (38), ( 36) and (37):

Corr(ξ 11 (0), ξ 11 (t)) → -1, Corr(ξ jk (t), ξ jk (0)) → 1, ∀j, k ∈ 2, . . . , N , Cov(ξ 11 (t), ξ jj (t)) = O(ρ 2 ), Cov(ξ 11 (t), ξ jk (t)) = 0 ∀j = k ∈ 1, . . . , N . Then (51) det (ξ(0)) det (ξ(t)) -ξ 11 (t) 2 det 2 (ξ -1 (t)) E (| det (ξ(0)) det (ξ(t)) |) Var(ξ 11 (t)) E det 2 (ξ -1 (t)) ,
Finally equations ( 34) and ( 49) with ( 47) and ( 46) give (44).

Justification of the passages to the limit: Since we are in the Gaussian space generated by the random field X all the variables considered above are jointly Gaussian. So their absolute values are bounded by the maximum in absolute value of a bounded random field. General results about the maximum of random fields, for example the Borell-Sudakov-Tsirelson theorem ([9] Section 2.4) implies that the maximum of its absolute value has moments of every order, giving all the dominations needed.

Correlation function between critical points with adjacent indexes.

THEOREM 4.9. Let X = {X(t) : t ∈ R N } be a stationary and isotropic Gaussian field with real values satisfying Assumption (A3). Let A k,k+1 (ρ) be the correlation function [START_REF] Taylor | Detecting sparse signals in random fields, with an application to brain mapping[END_REF] between critical points with index k and critical points with index k + 1, k = 0, . . . , N -1. Then as ρ → 0,

(52) A k,k+1 (ρ) ρ 2-N γ k N -1 2 4 3 (N -1)/2 π N λ 4 λ 2 N (λ 2 λ 6 -λ 2 4 ) λ 2 λ 4 ,
where γ k N -1 is defined by

(53) γ k N -1 := E det 2 (G N -1 -ΛId N -1 ) 1 {i(GN-1-ΛIdN-1
)=k} , with G N -1 a (N -1)-GOE matrix and Λ an independent centred Gaussian random variable with variance 1/3. This theorem gives an interpretation to Theorem 4.6: the attraction for N ≥ 3 is in fact due to attraction between critical points with adjacent indexes.

Again, in the particular case of the random plane wave (N = 2), our result agrees with the result of [START_REF] Beliaev | Two Point Function for Critical Points of a Random Plane Wave[END_REF], formula [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF].

We set γ k N -1,2 (x) := E det 2 (G N -1 -xId N -1 ) 1 {i(GN-1-xIdN-1)=k} for x ∈ R.
Before proving Theorem 4.9 we give in the following lemma an exact expression for the calculation of γ k N -1,2 (x).

LEMMA 4.10. Using notation above and notation introduced in paragraph 3.2.1 we have

γ k N -1,2 (x) = k N -1 (N -1)! I∈Fk(N -1)
Pf A 2 (I, ) ,

where k N -1 is given by [START_REF] Berry | Phase singularities in isotropic random waves[END_REF] and Pf A 2 (I, ) is the Pfaffian of the skew matrix A 2 (I, ) defined by [START_REF] Cheng | The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments[END_REF] and [START_REF] Ginsbourger | Fast computation of the multi-points expected improvement with applications in batch selection[END_REF] with n = N -1 and α = 2.

PROOF OF LEMMA 4.10. The proof follows the same lines as in Theorem 3.5 with

γ k N -1,2 ( ) = k N -1 (N -1)! Ok+1( ) N -1 i=1 (µ i -) 2 1≤i<j≤N -1 (µ j -µ i ) exp - N -1 i=1 µ 2 i 2 dµ 1 . . . dµ N -1 .
Then we set H(µ µ µ, ) the matrix {h m (µ j , ), m, j = 1, . . . , N -1} with h m (µ j , ) =

µ m-1 j (µ j -) 2 exp -µ 2 j /2 .
PROOF OF THEOREM 4.9. By [START_REF] Taylor | Detecting sparse signals in random fields, with an application to brain mapping[END_REF] we have

A k,k+1 (ρ) = p ∇X(0),∇X(t) (0, 0) E | det ξ(0)|1 {i(ξ(0))=k} | det ξ(t)|1 {i(ξ(t))=k+1} = -p ∇X(0),∇X(t) (0, 0) E det (ξ(0)) 1 {i(ξ(0))=k} det ξ(t)1 {i(ξ(t))=k+1} .
We can modify the computation of the determinant of ξ(0) in (50) to compute in place det ξ(0) -λId N :

(54) det (ξ(0) -λId N ) = ξ 11 (0) -λ det (ξ -1 (0) -λId N -1 ) + O p (ρ 2 ).
Note that as ρ → 0, ξ(0) converges in L 2 (or a.s. if we use a Skorohod imbedding argument) to

     0 0 . . . 0 0 . . . ξ -1 (0) 0      .
Its eigenvectors converge to those of this last matrix. These eigenvectors are associated to different eigenvalues so we can define properly the eigenvalue µ 1 that tends to zero. Because of (54) with λ = µ 1 :

(ξ 11 (0) -µ 1 ) det (ξ -1 (0) -µ 1 Id N -1 ) = O p (ρ 2 ),
implying in turn that

µ 1 = ξ 11 (0) + O p (ρ 2 ).
On the other hand, the others eigenvalues, say µ 2 , . . . , µ N , converge in distribution to that given by the right-hand side of (48). This implies in turn that

1 {i(ξ(0))=k} 1 {ξ11(0)<0;i(ξ-1(0))=k-1} + 1 {ξ11(0)>0;i(ξ-1(0))=k} .
As a consequence, when computing E det (ξ(0)) 1 {i(ξ(0))=k} det ξ(t)1 {i(ξ(t))=k+1} , we have four cases to consider depending on the signs of ξ 11 (0), ξ 11 (t). First we consider the case ξ 11 (0) > 0, ξ 11 (t) < 0. We have the equivalent of ( 51)

det (ξ(0)) det (ξ(t)) 1 {ξ11(0)>0,ξ11(t)<0} 1 {i(ξ-1(t))=i(ξ-1(0))=k} = -ξ 11 (t) 2 1 {ξ11(t)<0} det 2 (ξ -1 (t)) 1 {i(ξ-1(t))=k} + O p (ρ 3 ), giving E | det (ξ(0)) det (ξ(t)) |1 {ξ11(0)>0,ξ11(t)<0} 1 {i(ξ-1(t))=i(ξ-1(0))=k} = 1/2 Var(ξ 11 (t)) E det 2 (ξ -1 (t)) 1 {i(ξ-1(t))=k} + O(ρ 3 ).
This gives (52) as soon as we have checked that the three other cases give smaller contributions which is direct. 4.5.3. Correlation function between critical points with extreme indexes. First, we give a bound for the correlation function between maxima and minima (Theorem 4.11) then a bound for the correlation function between maxima (or between minima) (Theorem 4.12). THEOREM 4.11. Let X = {X(t) : t ∈ R N , N > 1} be a stationary and isotropic Gaussian field with real values satisfying Assumption (A3). Let A 0,N (ρ) be the correlation function (32) between minima and maxima. Then for all > 0 there exists a positive constant K( ) > 0 such that as ρ → 0,

(55) A 0,N (ρ) ≤ K( ) × ρ 2N -1-.
Upper bound (55) proves strong repulsion between maxima and minima. THEOREM 4.12. Let X = {X(t) : t ∈ R N , N > 1} be a stationary and isotropic Gaussian field with real values satisfying Assumption (A4). Let A N,N (ρ) (respectively A 0,0 (ρ)) be the correlation function [START_REF] Taylor | Detecting sparse signals in random fields, with an application to brain mapping[END_REF] between maxima (respectively minima). Then ∀ > 0 there exists a constant K( ) > 0 such that as ρ → 0,

(56) A 0,0 (ρ) = A N,N (ρ) ≤ K( ) × ρ 5-N -.
Upper bound (56) proves repulsion between maxima (or minima) for N < 5.

In the particular case N = 1, note that, strictly speaking, Proposition 4.4 gives no equivalent of Theorem 4.11. In fact, since a maxima is followed by a minima and reciprocally it is easy to deduce from (42) that we have (55) with = 0. On the other hand (43) implies that the result of Theorem 4.12 holds true with = 0.

As this work was achieved we have discovered the work [START_REF] Beliaev | No repulsion between critical points for planar Gaussian random fields[END_REF] which is similar to the result of this section but limited to the dimension N = 2. In this last particular case Theorem 4.6 is the analogue of Theorem 1.2 of [START_REF] Beliaev | No repulsion between critical points for planar Gaussian random fields[END_REF]; Theorem 4.9 has no equivalent in [START_REF] Beliaev | No repulsion between critical points for planar Gaussian random fields[END_REF]; Theorems 4.11 and 4.12 are similar but a little weaker than Theorems 1.3 and 1.4 of [START_REF] Beliaev | No repulsion between critical points for planar Gaussian random fields[END_REF] that are obtained by a difficult diagonalization of the Hessian which is absolutely impossible in the case N > 2.

PROOF OF THEOREM 4.11. We use Hadamard's inequality. If M is a positive semidefinite matrix of size N

det(M ) ≤ M 11 × • • • × M N N .
As a consequence for any symmetric matrix M (57)

det(M )1 {i(M )=0} ≤ (M 11 ) + × • • • × (M N N ) + .
By Kac-Rice formula (32)

A 0,N (ρ) = p ∇X(0),∇X(t) (0, 0) E | det ξ(0)|1 {i(∇ 2 ξ(0))=0} | det ξ(t)|1 {i(∇ 2 ξ(t))=N } .
We have

A 0,N (ρ) ≤ (-1) N p ∇X(0),∇X(t) (0, 0) E ξ + 11 (0)ξ + 22 (0) . . . ξ + N N (0)ξ - 11 (t)ξ - 22 (t) . . . ξ - N N (t)
. By Cauchy-Schwarz inequality and symmetry of the role of 0 and t we obtain

A 0,N (ρ) ≤ (-1) N -1 p ∇X(0),∇X(t) (0, 0) E ξ + 11 (0) 2 ξ + 22 (0) . . . ξ + N N (0)ξ - 22 (t) . . . ξ - N N (t)
. By Hölder's inequality for every p > 1

A 0,N (ρ) ≤ p ∇X(0),∇X(t) (0, 0) × I ρ × E ξ + 11 (0) 2p/(p-1) (p-1)/p .
where

I ρ := E |ξ + 22 (0) . . . ξ + N N (0)ξ - 22 (t) . . . ξ - N N (t)| p 1/p . Since ξ 11 (0) is centred Gaussian with variance ρ 2 λ 2 λ 6 -λ 2 4 4λ 2
, there exists a constant K(p) > 0 depending on p such that

E ξ + 11 (0) 2p/(p-1) (p-1)/p ≤ ρ 2 K(p).
By (47) we have p ∇X(0),∇X(t) (0, 0) = O(ρ -N ). Now let us prove that (58)

I ρ = O ρ (N -1)(2+1/p) .
We set ξ := (ξ 22 (0), . . . , ξ N N (0), ξ 22 (t), . . . , ξ N N (t)), ξi the ith coordinate of ξ for i = 1, . . . , 2N -2, ξ-i (resp. ξ-(i,j) ) the vector ξ without its ith (resp. ith and jth) coordinate(s). We set Σ ρ := Var( ξ). We define

S = {x ∈ R 2N -2 : x 1 > 0, . . . , x N -1 > 0, x N < 0, . . . , x 2N -2 < 0}. Then (59) I ρ = S | ξ1 . . . ξ2N-2 | p (2π) (2N -2)/2 det Σ ρ exp - 1 2 ξΣ -1 ρ ξ d ξ 1/p .
• Equivalent of det(Σ ρ ) as ρ → 0

For short we set X i := X i (0) and X ij := X ij (0).

det Σ ρ = det Var(X 22 , . . . , X N N , X 22 (t), . . . , X N N (t), X 1 , . . . , X N , X 1 (t), . . . , X N (t)) det Var(X 1 , . . . , X N , X 1 (t), . . . , X N (t)) .

Using Trick 4.3, we obtain as ρ → 0 det Σ ρ ρ 2(N -1) det Var(X 22 , . . . , X N N , X 122 , . . . , X 1N N , X 1 , . . . , X N , X 11 , . . . , X 1N ) det Var(X 1 , . . . , X N , X 11 , . . . , X 1N ) .

Using Lemma 2.3, ( 5) and ( 6) we get, as ρ → 0 det Σ ρ ρ 2(N -1) det Var(X 11 , . . . , X N N ) det Var(X 1 , X 122 , . . . , X 1N N ) Var(X 1 ) Var(X 11 ) we deduce that g > 0 under our hypotheses.

ρ 2(N -1) (N + 2) 2λ 4 3 N -1 2λ 6
• Equivalent of Σ -1 ρ as ρ → 0.

Because of the exchangeability of the coordinates 2, . . . , N

Σ -1 ρ := α ρ Id N -1 + β ρ J N -1,N -1 γ ρ Id N -1 + δ ρ J N -1,N -1 γ ρ Id N -1 + δ ρ J N -1,N -1 α ρ Id N -1 + β ρ J N -1,N -1 .
Using the classical expression of conditional variance and covariance

Σ -1 ρ ii = 1 Var( ξi / ξ-i ) = det Var( ξ-i ) det Var( ξ) Σ -1 ρ ij = -Cov( ξi , ξj / ξ-(i,
Σ := αId N -1 + βJ N -1,N -1 -αId N -1 -βJ N -1,N -1 -αId N -1 -βJ N -1,N -1 αId N -1 + βJ N -1,N -1
where the matrix αId N -1 + βJ N -1,N -1 is positive definite under our conditions. Indeed by ( 7)

det(αId n + βJ n,n ) = α n-1 (α + nβ),
and we have α > 0 and

α + nβ = 15 2λ 6 3(N + 1 -n)λ 2 λ 6 -5(N -1 -n)λ 2 4 3(N + 1)λ 2 λ 6 -5(N -1)λ 2 4
> 0 ∀n = 1, . . . , N -1 under our conditions.

• Formal finite limit

We make the change of variables ξ = ρz in (59) to obtain (60)

I ρ = ρ (2N -2)+(N -1)/p ×   S |z 1 . . . z 2N -2 | p (2π) (2N -2)/2 ρ 2(1-N ) det Σ ρ exp - 1 2 z ρ 2 Σ -1 ρ z dz   1/p .
We set z1 := (z 1 , . . . , z N -1 ), z2 := (z N , . . . , z 2N -2 ) and

f ρ (z) := |z 1 . . . z 2N -2 | p (2π) (2N -2)/2 ρ 2(1-N ) det Σ ρ exp - 1 2 z ρ 2 Σ -1 ρ z . As ρ → 0, f ρ (z) converges to |z 1 . . . z 2N -2 | p (2π) (2N -2)/2 √ g exp - 1 2 z Σz equals to |z 1 . . . z 2N -2 | p (2π) (2N -2)/2 √ g exp - 1 2 (z 1 -z2 )(αId N -1 + βJ N -1,N -1 )(z 1 -z2 )
that is integrable on S since g > 0 and αId N -1 + βJ N -1,N -1 is positive definite.

• Domination

Let us choose g 0 , α 0 and β 0 such that 0 < g 0 < g, 0 < α 0 < α and β 0 < β such that 0 < α 0 + nβ 0 ∀n = 1, . . . , N -1 which is always possible. Then for ρ sufficiently small (61)

f ρ (z) ≤ |z 1 . . . z 2N -2 | p (2π) (2N -2)/2 √ g 0 exp - 1 2 (z 1 -z2 )(α 0 Id N -1 + β 0 J N -1,N -1 )(z 1 -z2 ) .
Since α 0 > 0 and α 0 + nβ 0 > 0 ∀n = 1, . . . , N -1, α 0 Id N -1 + β 0 J N -1,N -1 is positive definite and the right hand side of ( 61) is integrable on S. Finally we conclude the proof taking p = 1 -.

PROOF OF THEOREM 4.12. We follow the same lines as in the proof of Theorem 4.11. By Kac-Rice formula (32):

A 0,0 (ρ) = p ∇X(0),∇X(t) (0, 0) E | det ξ(0)|1 {i(∇ 2 ξ(0))=0} | det ξ(t)|1 {i(∇ 2 ξ(t))=0} .
Using (57), Cauchy-Schwarz inequality and symmetry of the role of 0 and t we get A 0,0 (ρ) ≤ p ∇X(0),∇X(t) (0, 0) E ξ + 11 (0)ξ + 11 (t)ξ + 22 (0) 2 . . . ξ + N N (0) 2 .

By (47) we have p ∇X(0),∇X(t) (0, 0) = O(ρ We now consider the term E ξ + 22 (0) 2p/(p-1) . . . ξ + N N (0) 2p/(p-1) (p-1)/p in (63).

We set ξ := (ξ 22 (0), ξ 33 (0), . . . , ξ N N (0)). ξ is a Gaussian vector centred with variancecovariance matrix

Var( ξ) = (a ρ Id N -1 + b ρ J N -1,N -1 )
explicitly given in Lemma 4.1. According to Lemma 4.2, as ρ → 0,

a ρ → 2λ 4 3 and b ρ → 2λ 4 9 .
Using Lebesgue's dominated convergence theorem we can deduce that there exists a constant C(p) > 0 such that, as ρ → 0 1) . . . ξ + N N (0) 2p/(p-1) C(p).

(66) E ξ + 22 (0) 2p/(p-
(66) and ( 65) give (62). Finally we conclude the proof taking p = 1 -.

APPENDIX A: VALIDITY OF KAC-RICE FORMULAS FOR STATIONARY GAUSSIAN PROCESSES AND FIELDS

Let X = {X(t) : t ∈ U ⊂ R N } be a zero-mean, stationary Gaussian random field defined on an open set U ⊂ R N . We follow chapter 6 of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]. So we need ∇X(•) to be C 1 . Recall that N c (S), N c k (S) and N c k (u, S) are respectively the number of critical points, the number of critical points with index k and the number of critical points with index k above the level u of the random field X(•) in a Borel set S ⊂ U . PROPOSITION A.1. Let X = {X(t) : t ∈ U ⊂ R N } be a zero-mean, stationary Gaussian random field defined on an open set U ⊂ R N . Suppose that X is C 2 and that the distribution of ∇ 2 X(t) does not degenerate, then the Kac-Rice formula of order one is always true in the sense that for every Borel set S ⊂ U

E (N c (S)) = |S|p ∇X(t) (0) E | det(∇ 2 X(t))| (67) E (N c k (S)) = |S|p ∇X(t) (0) E | det(∇ 2 X(t))|1 i(∇ 2 X(t))=k (68) (69) E (N c k (u, S)) = |S|p ∇X(t) (0) × +∞ u p X(t) (x) E | det(∇ 2 X(t))|1 i(∇ 2 X(t))=k /X(t) = x .
In addition the sample paths are a.s. Morse functions.

Note that Lemma 2.3 implies that the conditions of Proposition A.1 are met for a random field satisfying (A2). REMARK A.2. In the particular case N = 1, the Kac-Rice formulas (67), ( 68) and (69) are always true (both sides are equal, finite or infinite) when X is a zero-mean stationary Gaussian process with C 1 covariance function. This can be proved as in [START_REF] Cramér | Stationary and Related Stochastic Processes[END_REF]. These conditions are weaker than those in Proposition A.1. PROOF OF PROPOSITION A.1. We first consider the mean number of critical points. We need to apply Theorem 6.2 of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]. The conditions are clearly verified in our case except for the condition (iv) :

P{∃t ∈ S : ∇X(t) = 0, det(∇ 2 X(t)) = 0} = 0.
This last condition is equivalent to the fact that the sample paths are a.s. Morse functions. In the case of stationary random fields the simplest is to use Proposition 6.5 of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] with condition b). Note that because of stationarity, by an extension argument, we can get rid of the compactness condition. Since ∇X(t) and ∇ 2 X(t) are independent, this condition is equivalent to

P{| det(∇ 2 X(t))| < δ} → 0 as δ → 0. which is equivalent to P{| det(∇ 2 X(t))| = 0} = 0.
This is implied by assuming the non degeneracy of ∇ 2 X(t).

We consider now the case of Kac-Rice formula for the mean number of critical points with a given index and above the level u . We have to apply Theorem 6.4 of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]. More precisely we have to write their formula (6.6) with

g(t, Y t ) = g(∇ 2 X(t), X(t)),
here the process Y t is simply the Gaussian vector ∇ 2 X(t), X(t) which is Gaussian as required in Theorem 6.4 of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]. On the other hand, g is the indicator of the index multiplied by the indicator function that X(t) > u. It is not continuous and we must apply an approximation with a continuous function, followed by a monotone convergence argument. Then, since we have independence between X(t) and ∇X(t) and between ∇X(t) and ∇ 2 X(t) (see Lemma 2.3), we obtain (68) and (69).

For simplification we limit our second proposition to the isotropic case PROPOSITION A.3. Under Assumption (A2), the Kac-Rice formula of order 2 is valid for a sufficiently small set S in the sense that if S 1 and S 2 are sufficiently small and sufficiently close to each other,

(70) E (N c (S 1 )N c (S 2 )) = S1×S2 dt 1 dt 2 p ∇X(t1)∇X(t2) (0, 0) E | det(∇ 2 X(t 1 )) det(∇ 2 X(t 2 ))|/∇X(t 1 ) = ∇X(t 2 ) = 0 (71) E (N c k (S 1 )N c k (S 2 )) = S1×S2 dt 1 dt 2 p ∇X(t1)∇X(t2) (0, 0) E | det(∇ 2 X(t 1 ))1 i(∇ 2 X(t1))=k det(∇ 2 X(t 2 ))1 i(∇ 2 X(t2))=k |/∇X(t 1 ) = ∇X(t 2 ) = 0
PROOF. We need to apply Theorem 6.3 from [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]. With respect to Proposition A.1, we need in addition the distribution of ∇X(0), ∇X(ρe 1 ) to be non degenerated in R 2N (Condition (iii ) of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF]). From section B.2.1.1 we have Var (∇X(0)) = Var (∇X(ρe 1 )) = λ 2 Id N E ∇X(0)∇X(ρe 1 ) T = diag -2r (ρ 2 ) -4ρ 2 r (ρ 2 ), -2r (ρ 2 ), . . . , -2r (ρ 2 ) , which implies that the N vectors X i (0), X i (ρe 1 ) , i = 1, . . . , N are independent. It remains to check that the correlation between X i (0) and X i (ρe 1 ) cannot be 1. This is done by the Taylor expansions

-2r (ρ 2 ) = -2r (0) -2ρ 2 r (0) + o(ρ 2 ) -2r (ρ 2 ) -4ρ 2 r (ρ 2 ) = -2r (0) -6ρ 2 r (0) + o(ρ 2 ).
Since by hypothesis 0 < r (0) = λ 4 /12 < +∞ the result follows.

We consider now the case of Kac-Rice formula for the mean number of critical points with a given index. We proceed using Theorem 6.4 of [START_REF] Azaïs | Level Sets and Extrema of Random Processes and Fields[END_REF] as in the preceding case.

λ 2p = R N e 1 , λ 2p dF (λ) = +∞ 0 x 2p dG(x) S N -1 e 1 , u 2p dσ(u),
where F is the spectral measure; σ is the uniform probability on the sphere S N -1 and G is a measure deduced from F by change in polar coordinates. Using the fact that a standard normal variable is the independent product of a uniform variable on the sphere by a χ(N ) distribution and the fact that those two distributions have well known moments, we obtain giving (3). Now let us prove [START_REF] Auffinger | Random matrices and complexity of spin glasses[END_REF]. By (72), see also [START_REF] Schoenberg | Metric spaces and completely monotone functions[END_REF] and [START_REF] Yaglom | Some classes of random fields in n-dimensional space, related to stationary random processes[END_REF], r( s -t 2 ) can be written as That gives the first equality in (4). The second equality is obtained using (2). B.2. Proof of Lemma 4.1. We give the steps for the computation of the conditional variance of ∇ 2 X(0), ∇ 2 X(t) given ∇X(0) = ∇X(t) = 0 with t = ρe 1 . Some tedious but easy calculations are not detailed. r(ρ 2 ), r (ρ 2 ), r (ρ 2 ), r (ρ 2 ) and r (4) (ρ 2 ) are denoted by r, r , r , r and r (4) for short. The results below extend Lemma 2.1 to two times. Var(∇X(t)) = Var(∇X(0)) = -2r (0)Id N , Cov(X 1 (0), X 1 (t)) = -2r -4ρ 2 r and Cov(X i (0), X i (t)) = -2r for i = 1.

µ 2p := S N -1
Any other covariance is zero. B.2.1.2. Hessian. We define X d (t) as (X 11 (t), . . . , X N N (t)) and X u (t) as {X ij (t), 1 ≤ i < j ≤ N }. These two vectors are independent and Var(X d (0)) = 4r (0)(2Id N + J N,N ), Cov(X d (0), X d (t)) = 12r + 48ρ 2 r + 16ρ 4 r (4) (4r + 8ρ 2 r )J 1,(N -1) (4r + 8ρ 2 r )J (N -1),1 4r (2Id N -1 + J (N -1),(N -1) ) ,

Var(X u (0)) = 4r (0)Id N (N -1)/2 , Cov(X u (0), X u (t)) = diag (4r + 8ρ 2 r )Id N -1 , 4r Id (N -1)(N -2)/2 . B.2.1.3. Relation between gradient and Hessian.

E X 11 (0) X 1 (0), X 1 (t) = 1 4r (0) 2 -(2r + 4ρ 2 r ) 2 (0, 12ρr + 8ρ 3 r ) -2r (0) 2r + 4ρ 2 r 2r + 4ρ 2 r -2r (0) X 1 (0) X 1 (t) = (12ρr + 8ρ 3 r ) (2r + 4ρ 2 r )X 1 (0) -2r (0)X 1 (t) 4r (0) 2 -(2r + 4ρ 2 r ) 2

=: (12ρr + 8ρ 3 r )K 1 (t).

For i = 1 E X 1i (0) X i (0), X i (t) = 1 4r (0) 2 -4(r ) 2 (0, 4ρr ) -2r (0) 2r 2r -2r (0)

X i (0) X i (t)
= 4ρr 2r X i (0) -2r (0)X i (t) 4r (0) 2 -4(r ) 2 =: 4ρr K i (t).

E X ii (0) X 1 (0), X 1 (t) = 4ρr K 1 (t).

We have equivalent formulas, reversing time, for example: E(X 11 (t)/X 1 (t), X 1 (0)) = (12ρr + 8ρ 3 r ) (-(2r + 4ρ 2 r )X 1 (t) + 2r (0)X 1 (0)) 4(r (0)) 2 -(2r + 4ρ 2 r ) 2

=: (12ρr + 8ρ 3 r ) K1 (t).

Any other case corresponds to independence between gradient and Hessian.

B.2.2. Conditional distribution. Since the conditional variance-covariance matrix is equal to the unconditional variance-covariance matrix diminished by the variance of the conditional expectation, we compute this last term only. Let us consider for example the two terms X 11 (0) and X 11 (t) given ∇X(0) = ∇X(t) = 0. The 2 × 2 matrix to subtract is (12ρr + 8ρ 3 r ) 2 Var(K 1 (t), K1 (t)) with Var(K 1 (t), K1 (t)) = 1 4(r (0)) 2 -(2r + 4ρ 2 r ) 2 -2r (0) -(2r + 4ρ 2 r ) -(2r + 4ρ 2 r ) -2r (0) .

In the same way for i = 1 :

Var(K i (t); Ki (t)) = 1 4r (0) 2 -4(r ) 2 -2r (0) -2r -2r -2r (0) .

Giving the extra term to substract to get Lemma 4. det(Var(X 1 (0), . . . , X N (0), X 11 (0), X 1 (t), . . . , X N (t))) det (Var (∇X(0), ∇X(t)))

det(Var(X 1 (0), . . . , X N (0), X 11 (0), X 1 (t) -X 1 (0) -ρX 11 (0), X 2 (t) -X 2 (0), . . . , X N (t) -X N (0)))/ det (Var (∇X(0), ∇X(t)))

ρ 2 3 N -1 4λ N 2 λ N 4
det(Var(X 1 (0), . . . , X N (0), X 11 (0), . . . , X 1N (0), X 111 (0))).

LEMMA 2 . 3 .

 23 Under Assumption (A4) the random vectors ζ 1 , ζ 2 , ζ 3 , ζ 4 and ζ 5 defined above are Gaussian, independent, centred with variance matrices respectively given by

FIG 1 .

 1 FIG 1. Probability densities q k N (l), k = 1, . . . , N , of the ordered eigenvalues of a N GOE matrix for N = 2, 3, 4, 5.

→0 1 V 2

 12 ( ) E (P(B(s, ))P(B(t,))) when it exists,

LEMMA 4 . 2 .

 42 Using the same notation as in Lemma 4.1, for j = k and j, k = 1, as ρ → 0[START_REF] Worsley | Searching scale space for activation in PET images[END_REF] det (Var (∇X(0),

15 N -2 3 (N + 1 )λ 2 λ 6 - 5 (N - 1 )λ 2 4 135λ 2 . 3 N -1 2λ 6 15 N 4 135λ 2 .By ( 3 ) we have λ 2 λ 6 ≥ 5

 15316514231542365 Soρ 2(1-N ) det Σ ρ g := (N + 2) 2λ 4 -2 3(N + 1)λ 2 λ 6 -5(N -1)λ 2

1 ρ 11 , 4 - 4 ,

 11144 j) ) det Var( ξi , ξj / ξ-(i,j) ) = Var( ξi / ξ-(i,j) ) + Var( ξj / ξ-(i,j) ) -Var( ξi + ξj / ξ-(i,j) ) 2 det Var( ξi , ξj / ξ-(i,j) ) = det Var( ξ-j ) + det Var( ξ-i ) -det Var( ξi + ξj , ξ-(i,j) )2 det Var( ξ) , and using the same techniques as those previously used for the calculation of the equivalent of det(Σ ρ ) and for the proof of Lemma 4.2, we obtain the equivalents of Σ -3λ 2 λ 6 3(N + 1)λ 2 λ 6 -5(N -1)λ 2 ρ 2 γ ρ -α and ρ 2 δ ρ -β. So ρ 2 Σ -1 ρ converges to

  APPENDIX B: PROOFS OF LEMMAS B.1. Proof of Lemma 2.1. We start by proving (3). The spectral moment of order 2p has the following expressions (72)

e 1 2 2n- 2 .y

 122 , u 2p dσ(u) = (2p -1)!!Γ(N/2) 2 p Γ(N/2 + p) = Γ(N/2)Γ(1/2 + p) √ πΓ(N/2 + p) .We haveµ 2n µ 2n-4 = (2n -4 + N )(2n -1) (2n -2 + N )(2n -3) µUsing Cauchy-Schwarz inequality we conclude that λ 2n λ 2n-4 = µ 2n µ 2n-2n-2 dG(y) = K(n, N )λ 2 2n-2

r( s -t 2 0 (- 1 )

 201 k x 2k s -t 2 k 2 2k+(N -2)/2 k!Γ(k + 1 + (N -2)/2) dG(x). (73) Thus setting |β| = |i| + |j| 2 and β k = i k + j k 2 for k = 1, . . . ,N , we have ∂ |i|+|j| r( s -t 2 ) ∂s i1 1 , . . . , ∂s iN N ∂t j1 1 , . . . , ∂t jN N s=t = +∞ |β| x 2|β| 2 2|β|+(N -2)/2 |β|!Γ(|β| + 1 + (N -2)/2)

= (- 1 )xx

 1 |β| (-1) |j| |β|! 2 2|β|+(N -2)/2 |β|!Γ(|β| + 1 + (N -2)/2) 2|β| dG(x).Furthermore, using (73), we haver (|β|) (0) = (-1) |β| |β|! 2 2|β|+(N -2)/2 |β|!Γ(|β| + 1 + (N -2)/2) +∞ 0 2|β| dG(x).

B. 2 . 1 .

 21 Unconditional distribution. B.2.1.1. Gradient.

  PROOF OF LEMMA 2.3. The joint distribution of ζ 1 , ζ 2 , ζ 3 , ζ 4 is a direct consequence of (4). Now let us prove (

	45	2 4	.

  × ∂ |i|+|j| s -t 2 |β| ∂s i1 1 , . . . , ∂s iN N ∂t j1 1 , . . . , ∂t jN ! × . . . × k N ! (s 1 -t 1 ) 2 k1 × . . . × (s N -t N ) 2 kN .since the only term in the sum in the right side of (74) whose derivative is not null is the onefor which k 1 = i 1 + j 1 2 , . . . , k N = i N + j N2. We obtain∂ |i|+|j| r( s -t 2 )

						dG(x).
						N s=t
	By the multinomial formula
	(74) k 1 Thus s -t 2 |β| = k1+...+kN =|β|	|β|!
	∂ |i|+|j| s -t 2 |β| ∂s i1 1 , . . . , ∂s iN N ∂t j1 1 , . . . , ∂t jN N s=t	= (-1) |j| |β|!	N k=1	(2β k )! β k !	,
	∂s i1 1 , . . . , ∂s iN N ∂t j1 1 , . . . , ∂t jN N s=t

  1. B.3. Proof of Lemma 4.2. First note that (37) and (40) are deduced from Lemma 4.1. B.3.1. Proof of (34), (35) and (41). We now consider the case of ξ 11 (t). Using Trick 4.3 we have Var (ξ 11 (t)) =

. The other variances in [START_REF] Worsley | A unified statistical approach for determining significant signals in images of cerebral activation[END_REF] and [START_REF] Yaglom | Some classes of random fields in n-dimensional space, related to stationary random processes[END_REF] and the determinant (41) are obtained in the same way.

B.3.2. Proof of (36), ( 38) and (39) . • We consider Cov(ξ jj (t), ξ kk (t)) for j = k and j, k = 1. Since

and since, by [START_REF] Worsley | A unified statistical approach for determining significant signals in images of cerebral activation[END_REF], Var(ξ jj (t)) = Var(ξ kk (t)) = 8λ 4 9 , we just need to compute

det (Var (∇X(0), ∇X(t))) .

Using Trick 4.3 we obtain

Var (ξ jj (t) + ξ kk (t))

det(Var(X jj (0) + X kk (0), X 1 (0), . . . , X N (0), X 11 (0), X 12 (0), . . . , X 1N (0))) det (Var (X 1 (0), . . . , X N (0), X 11 (0), X 12 (0), . . . , X 1N (0))) .

By Lemma 2.3 we have

Finally

Cov(X 11 (0), X jj (t)) = 4r (ρ 2 ) + 8ρ 2 r (ρ 2 ).

With Section B.2.1.1 we deduce

Var(ξ jj (0)) = det(Var(X 1 (0), . . . , X N (0), X 1 (t), . . . , X N (t), X jj (0))) det (Var (∇X(0), ∇X(t)))

where ∇ -(1) X denotes the gradient without its first coordinate. Similarly

and

Using Lemma 2.3 we can verify that

where

) and α 2 = 4ρr (ρ 2 ). We deduce

det (Var (∇X(0), ∇X(t))) .

We can check that In the same way we prove Cov(ξ 11 (0), ξ jj (t)) = ρ 2 15λ 2 4 -7λ 2 λ 6 180λ 2 + o(ρ 2 ).