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

Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles

Keywords: 2000 Mathematics Subject Classification. Primary 15 A 52, 15 A 15; Secondary 60F 10, 60F 17, 62 H 10 Random matrices, Wishart ensemble, Laguerre ensemble, Jacobi ensemble, Gram ensemble, Hadamard ratio, determinant, invariance principle, large deviations. 1

We consider properties of determinants of some random symmetric matrices issued from multivariate statistics: Wishart/Laguerre ensemble (sample covariance matrices), Uniform Gram ensemble (sample correlation matrices) and Jacobi ensemble (MANOVA). If n is the size of the sample, r ≤ n the number of variates and X n,r such a matrix, a generalization of the Bartlett-type theorems gives a decomposition of det X n,r into a product of r independent gamma or beta random variables. For n fixed, we study the evolution as r grows, and then take the limit of large r and n with r/n = t ≤ 1. We derive limit theorems for the sequence of processes with independent increments {n -1 log det X n,⌊nt⌋ , t ∈ [0, T ]} n for T ≤ 1 : convergence in probability, invariance principle, large deviations. Since the logarithm of the determinant is a linear statistic of the empirical spectral distribution, we connect the results for marginals (fixed t) with those obtained by the spectral method. Actually, all the results hold true for log gases or β models, if we define the determinant as the product of charges. The classical matrix models (real, complex, and quaternionic) correspond to the particular values β = 1, 2, 4 of the Dyson parameter.

Introduction

Random determinants of symmetric matrices are of constant use in random geometry to compute volumes of parallelotopes (see [START_REF] Nielsen | The distribution of volume reductions[END_REF], [START_REF] Mathai | Random p-content of a p-parallelotope in Euclidean n-space[END_REF]) and in multivariate statistics to build tests. Twenty years after the book of [START_REF] Girko | Theory of random determinants[END_REF] Motivated by basis reduction problems, [START_REF] Schnorr | A hierarchy of polynomial time basis reduction algorithms. Theory of algorithms[END_REF] defined the orthogonality defect as the quantity 1/ h(B) (see also [START_REF] Akhavi | Random lattices, threshold phenomena and efficient reduction algorithms[END_REF] and references therein). [START_REF] Abbott | How tight is Hadamard bound? Experiment[END_REF] and [START_REF] Dixon | How good is Hadamard's inequality for determinants?[END_REF] are concerned with the tightness of the bound h(B) ≤ 1 when B is random and n = r. For these authors, the random vectors b i are sampled independently and uniformly on the unit sphere

S n R = {(x 1 , . . . , x n ) ∈ R n : x 2 1 + • • • + x 2 n = 1}
. It is known that then the variables b i 2 are independent and Beta distributed with varying parameters. When the entries of the matrix B are independent and N (0, 1), the variables b i 2 are independent and Gamma distributed with varying parameters [START_REF] Bartlett | On the theory of statistical regression[END_REF].

Writing B n,r instead of B to stress on dimensions and X n,r = B ′ n,r B n,r , we are interested in this paper in the asymptotic behavior of det X n,r when n and r both tends to infinity, in the regime r/n → c ∈ [0, 1]. Since the construction of the b i is recursive, it is possible (for fixed n) to consider the whole sequence of variables {det X n,r , r = 1, . . . , n} at the same time.

It corresponds to the decomposition of the determinant of a symmetric positive matrix A as det A = r j=1 det A [j] det A [j-1] ,

where A [j] is the j × j upper-left corner of A with the convention det A [0] = 1. When using this approach, we will refer to it as the decomposition method.

The decomposition method is also valid when entries of the matrix are complex, considering the Hermitian conjugate B ⋆ and then B ⋆ B, and also when the entries are real quaternions, considering the dual B † and then B † B.

In these three cases, a Bartlett-type theorem gives the determinant as a product of independent variables, with Gamma or Beta distributions. Passing to logarithms, it is then possible to consider a triangular array of variables and a process with independent increments {n -1 log det X n,⌊nt⌋ , t ∈ [0, T ]} n for T ≤ 1 indexed by the "time" t = r/n. Thanks to the additive structure of the log det, we obtained limit theorems : convergence in probability, invariance principle and large deviations.

The same is true for random matrices following the Jacobi (or MANOVA) distribution. Actually, the whole construction is possible in the so-called β-models, which are an extension of the above ones, which correspond to the three-fold way β = 1, 2, 4 of Dyson. For other values of β they are not defined as matrix models but log gases models, in which the eigenvalues are replaced by charges and determinants by products of charges. It has be shown recently that they correspond also to models of tri-diagonal random matrices (see [START_REF] Dumitriu | Matrix models for beta-ensembles[END_REF], [START_REF] Killip | Matrix models for circular ensembles[END_REF], [START_REF] Edelman | The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems[END_REF]).

Of course, for r fixed, there is also another underlying structure of product: the determinant as the product of eigenvalues. We may use the asymptotical behavior of empirical spectral distributions, i.e. convergence to the Marčenko-Pastur distribution in the Wishart/Laguerre case and to the generalized McKay distribution in the Jacobi case. However, this structure is not "dynamic": if you change r, the whole set of eigenvalues is changing. When using this approach, we will refer to it as the spectral method.

In Section 2 we set the framework. We begin with the matrix models (Wishart-Laguerre, Uniform Gram and Jacobi), and proceed with the β-models and processes of determinants.

The main results of this paper are in Section 3: laws of large numbers and fluctuations, large deviations and variational problems. The comparison of results obtained by the two methods (decomposition and spectral) deserves interest and is the topic of Section 4. Some extensions to other models are given in Subsection 4.4.

Sections 5, 6 and 7 are devoted to the proofs. In Appendix 1 we gather some details on the Binet formula on the Gamma function which are of constant use in this paper, and Appendix 2 gives identification of the McKay distribution.

Notation and known facts

In this long section, we present our different models whose common feature is to introduce processes of random determinants with independent multiplicative factors. The distribution of these factors are recorded in Proposition 2.1 for real matrix models, and settled in formulae (2.2), (2.3) and (2.4) for the (other) βmodels.

Throughout, |A| stands for det A, and I n for the n × n identity matrix. If X, Y are real random variables and µ a distribution on R, we write

X (d) = Y (resp. X (d) = µ)
if X and Y have the same distribution (resp. if the distribution of X is µ).

2.1. Real matrix models and Bartlett-type theorems. In the basic model, we consider independent random vectors b i , i ≥ 1 with the same distribution ν n in R n . The most important example is the Gaussian one with ν n = N (0,

I n ). If B = [b 1 , . . . , b r ],
all the entries of B are independent N (0, 1) and the distribution of W = B ′ B is denoted by W r (n, R) and called the Wishart ensemble. For r ≤ n, its density on the space S r of symmetric positive matrix is 1 2 rn/2 Γ r (n/2) |W | (n-r-1)/2 exp -1 2 tr W where Γ r is the multivariate Gamma function Γ r (α) = π r(r-1)/4 Γ(α)Γ α -

1 2 • • • Γ α - r -1 2 ,
It is the matrix variate extension of the Gamma distribution. Recall that for a, c > 0, the Gamma(a, c) distribution has density c a Γ(a)

x a-1 e -cx (x > 0) .

For r > n, the matrix is singular. Motivated by Hadamard inequality (1.1), we may choose ν n to be the uniform distribution on the unit sphere S n R . The corresponding ensemble for B is called Uniform Spherical Ensemble by [START_REF] Donoho | Breakdown of equivalence between the minimal ℓ 1 -norm solution and the sparsest solution[END_REF]. The matrix ensemble for B ′ B is called the Gram ensemble by [START_REF] Cock | On quantum dynamics and statistics of vectors[END_REF], since B ′ B is the Gram matrix built from the b i 's. To stress on the distribution, we call it Uniform Gram ensemble. The diagonal entries are one and for r ≤ n, the joint density of the non-diagonal entries (r ij , 1 ≤ i < j ≤ r) of the matrix

G = B ′ B is [Γ (n/2)] r Γ r (n/2)
|G| (n-r-1)/2 (-1 < r ij < 1) (2.1) (see [START_REF] Gupta | 'une question relative aux déterminants[END_REF] Theorem 3.3.24 p.107, [START_REF] Mathai | A handbook of generalized special functions for statistical and physical sciences[END_REF] Example 1.25 and [START_REF] Mathai | Jacobians of matrix transformations and functions of matrix argument[END_REF] p.58).

Let us now introduce Jacobi ensembles. For n 1 , n 2 ≥ 1 and r ≤ n := n 1 + n 2 , we can decompose every (n 1 + n 2 ) × r matrix M in two blocks

M = M 1 M 2 with M 1 of type n 1 × r and M 2 of type n 2 × r. If the entries of M are independent N (0, 1), then W 1 := M ′ 1 M 1 and W 2 := M ′ 2 M 2 are independent Wishart matrices of distribution W r (n 1 , R) and W r (n 2 , R), respectively. It is well known that W 1 + W 2 is W r (n 1 + n 2 , R
) distributed and a.s. invertible. Let us denote by (W 1 + W 2 ) 1/2 the symmetric positive square root of (W 1 + W 2 ). The r × r matrix

X := (W 1 + W 2 ) -1/2 W 1 (W 1 + W 2 ) -1/2
has a distribution denoted by J r (n 1 , n 2 , R) and called the Jacobi ensemble.

If T is upper triangular with positive diagonal entries and [START_REF] Olkin | Multivariate Beta distribution and independence properties of the Wishart distribution[END_REF], [START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF] p.108).

W 1 + W 2 = T ′ T (Cholesky decomposition) then Z = (T ′ ) -1 W 1 T -1 is also J r (n 1 , n 2 , R) distributed, (see
Another occurrence of the Jacobi ensemble is interesting (see [START_REF] Doumerc | Matrices aléatoires, processus stochastiques et groupes de réflexion[END_REF], [START_REF] Collins | Products of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF]). If M is as above, its singular value decomposition is

M = U DV , D = ∆ 0
with D of type n × r, with ∆ diagonal with nonnegative entries, with U ∈ O(n) and V ∈ O(r) (the orthogonal groups). Although U and V are not uniquely determined, one can choose them according to the Haar distribution on their respective group and such that U, V, ∆ are independent. Then

M ′ M = V ′ ∆ 2 V and (W 1 + W 2 ) 1/2 = (M ′ M ) 1/2 = V ′ ∆V . Let Y r = U [n1,r] be the n 1 × r upper-left corner of U . Since M 1 = Y r ∆V we have M ′ 1 M 1 = V ′ ∆Y ′ r Y r ∆V = (M M * ) 1/2 (V ′ Y ′ r Y r V )(M M * ) 1/2 and then X = (Y r V ) ′ (Y r V ) (d) = Y ′ r Y r . In other words, Y := U [n1,r] ′ U [n1,r] is also J r (n 1 , n 2 , R) distributed. If r ≤ min(n 1 , n 2 ), the distribution J r (n 1 , n 2 , R) has a density on S r which is 1 β r n1 2 , n2 2 |Z| n 1 -r-1 2 |I r -Z| n 2 -r-1 2 1 0<Z<Ir , (2.2)
where

β r (a, b) = Γ r (a) Γ r (b) Γ r (a + b) ,
(see for example [START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF] Theorem 3.3.1). It is the matrix variate extension of the beta distribution. Recall that for a > 0, b > 0, the Beta(a, b) distribution has density

Γ(a + b) Γ(a)Γ(b) x a-1 (1 -x) b-1 (x > 0) . (2.3)
Until now, we had r fixed. Our purpose is now to consider all values of r simultaneously to give a "sample path" study of determinants.

For

an n × n matrix B = [b 1 , . . . , b n ], we have for r ≤ n (B ′ B) [r] = B [n,r] ′ B [n,r] ,
and for every j ≤ n, the quantity

ρ j,n := |(B ′ B) [j] | |(B ′ B) [j-1] | (2.4)
is a measurable function of (b 1 , . . . , b j ) and

|(B ′ B) [r] | = r j=1 ρ j,n .
(2.5)

The same occurs with b

i := b i / b i instead of b i (i = 1, . . . , n) and B := [ b 1 , . . . , b n ] instead of B.
Let us note that ρ 1,n = 1 and

ρ j,n = | W [j] | | W [j-1] | = |W [j] | |W [j-1] |W jj = ρ j,n b j 2 , j = 2, . . . , n , (2.6) so that |( B ′ B) [r] | = r j=1 ρ j,n .
(2.7)

The Wishart case and the Uniform Gram case corresponds to (2.5) and (2.7) respectively, for r = 1, . . . , n.

In the Jacobi case, r ∈ {1, . . . , n 1 }.

If M = [b 1 , . . . , b n1 ]
, and if T , W 1 , Z are defined as above with n 1 instead of r, then

Z [r] = T [r] ′ -1 W [r] 1 T [r] -1 .
For every j, the quantity

ρ Z j,n1,n2 := |Z [j] | |Z [j-1] |
is a measurable function of (b 1 , . . . , b j ) and

|Z [r] | = r j=1 ρ Z j,n1,n2 .
It can be noticed that

ρ Z j,n1,n2 = |W [j] 1 | |W [j] 1 + W [j] 2 | × |W [j-1] 1 + W [j-1] 2 | |W [j-1] 1 | .
Besides, the construction with the symmetric square root is different. If

ρ X j,n1,n2 := |X [j] | |X [j-1] | we have X [r] = W [r] 1 + W [r] 2 -1/2 W [r] 1 W [r] 1 + W [r] 2 -1/2 . (Take n 1 = n 2 = 2, W 1 = I 2 , W 2 = 1 s s 1 and r = 1 then W [1] 1 + W [1] 2 -1/2 W [r] 1 W [r] 1 + W [r] 2 -1/2 = 2/(4 -s 2 ) ,
and X [1] = 1/2). Moreover we cannot say that ρ X j,n1,n2 is measurable with respect to b 1 , . . . , b j .

Let us consider the construction from contraction of Haar matrices. Since

U [n1] ′ U [n1] [n1,r] = U [n1,r] ′ U [n1,r] ,
we see that the quantity

ρ Y n1,n2,j := Y [j] Y [j-1]
depends only on the j first columns of the matrix U , and

|Y [r] | = r j=1 ρ Y n1,n2,j .
It is possible to introduce a probability space on which all Uniform Gram and Wishart matrices are defined for all values of n simultaneously. It is enough to consider the infinite product space generated by a double infinite sequence of independent N (0, 1) variables {b i,j } ∞ i,j=1 , and for every n to perform the above constructions with b i = (b 1i , . . . , b n,i ) ′ . To embed the Jacobi matrices in this framework, we have to restrict ourselves to the X -type and Z-type ones; however, only the Z one gives a natural meaning to the dynamic study.

The starting point of our study of random determinants is the following proposition which gathers known results about the factors entering in the above decompositions.

Proposition 2.1.

1) (Bartlett) The random variables ρ j,n , j = 1, . . . , n are independent and

ρ j,n (d) = Gamma n -(j -1) 2 , 1 2 , where (d)
= stands for equality in distribution.

2) The random variables ρ j,n , j = 2, . . . , n are independent and

ρ j,n (d) = Beta n -j + 1 2 , j -1 2 .
3) For J = X (resp. Y, Z), the random variables ρ J j,n1,n2 , j = 1, . . . , n 1 are independent and

ρ J j,n1,n2 (d) = Beta n 1 -j + 1 2 , n 2 2 .
The first claim is known as the celebrated Bartlett decomposition (stated with χ 2 distributions) [START_REF] Bartlett | On the theory of statistical regression[END_REF]. It is quoted in many books and articles in particular [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF]) pp.170-172, Muirhead (1982) Theorem 3.2.14 p.99, Kshirsagar (1972), Gupta and Nagar (2000) Theorem 3.3.4 p.91 and ex. 3.8 p.127. The second claim may be found in [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF] Theorem 9.3.3. In the third claim, we first note that it is enough to get the proof for Z since the three random matrices have the same distribution. It is a consequence of a result quoted in [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF], due to Kshirsagar, is proved in [START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF] Theorem 3.3.1 p.110 under the assumption r ≤ n 1 , n 2 and in [START_REF] Rao | Linear statistical inference and its application[END_REF] p.541 under the only assumption r ≤ n 1 . Actually (see Muirhead (1982) ex. 3.24 and[START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF] Theorem 8.4.1), some proofs use probabilistic arguments (as [START_REF] Rao | Linear statistical inference and its application[END_REF] and [START_REF] Anderson | An introduction to multivariate statistical analysis[END_REF]), Jacobian arguments (as in [START_REF] Gupta | 'une question relative aux déterminants[END_REF] Theorem 5.3.24 p.181), or Mellin transform arguments (as in [START_REF] Mathai | Random p-content of a p-parallelotope in Euclidean n-space[END_REF] Theorem 2).

2.2. Distribution of eigenvalues and β-models. In the study of sttionary processes, random matrices of the Wishart type with complex entries play an important role [START_REF] Goodman | Statistical analysis based on a certain multivariate complex gaussian distribution[END_REF]). Less often, quaternionic entries are considered (see [START_REF] Hanlon | Some combinatorical aspects of the spectra of normally distributed random matrices[END_REF]). We do not give details on the complex and quaternionic cases but jump to a general framework. Popularized by physicists, the modern point of view consists in introducing a parameter β taking value 1 when real, 2 when complex, and 4 when quaternionic, this parameter playing the role of an inverse temperature. The above constructions can be extended when replacing the transpose (R case) by the adjoint (C case) or the dual (H case). Many of the above results are then true when replacing in displays the factor 1/2 by the factor β/2.

Actually there are two ways to reach the law of determinants : a) directly from the distribution of matrices, and using the decomposition method quoted above, b) from the joint distribution of eigenvalues.

The second way has been used to define the so-called β-models, see for instance [START_REF] Forrester | Log-gases and random matrices[END_REF] Chap.2. The idea of extending the range of β to (0, ∞) is quite natural. As mentioned in Section 1, they correspond also to models of tri-diagonal random matrices (see [START_REF] Dumitriu | Matrix models for beta-ensembles[END_REF], [START_REF] Killip | Matrix models for circular ensembles[END_REF], [START_REF] Edelman | The beta-Jacobi matrix model, the CS decomposition, and generalized singular value problems[END_REF]).

Because of the connections with orthogonal polynomials in the complex case, the extended families are called β-Laguerre ensemble (or just Laguerre ensemble) instead of Wishart ensemble and β-Jacobi ensemble (or just Jacobi ensemble) instead of MANOVA or Beta ensemble.

Throughout, we use the symbol β ′ for β/2 to simplify displays.

2.2.1. Laguerre. When β = 1, 2, 4 the joint probability density of the eigenvalues λ j , j = 1, . . . , r of W on the orthant λ j > 0, j = 1, . . . , r is 1

Z L,β r (n) r j=1 λ β ′ (n-r+1)-1 j e -β ′ λj 1≤j<k≤r |λ k -λ j | 2β ′ , (2.8)
and the normalizing constant is

Z L,β r (n) = 1 β ′ β ′ rn r j=1 Γ (1 + β ′ j) Γ (β ′ (n -j + 1)) Γ (1 + β ′ ) .
This is the Selberg integral (see for instance [START_REF] Edelman | Random matrix theory[END_REF], formula 4.6 and references therein).

When β > 0 is not 1, 2, 4, we consider the density (2.8) on (0, ∞) r . We also denote the product r j=1 λ j by |W |. This gives the Mellin transform

E|W | β ′ s = Z L,β r (n + s) Z L,β r (n) = 1 β ′ β ′ rs r k=1 Γ (β ′ (n -k + 1 + s)) Γ (β ′ (n -k + 1)) .
Remembering that if X

= Gamma(a, 1/2) then

EX µ = 2 µ Γ(µ + a) Γ(a) (µ > -a) ,
we deduce the following proposition from the uniqueness of Mellin transform.

Proposition 2.2. We have

|W | (d) = r j=1 ρ L,β j,n ,
where the variables ρ L,β j,n , j = 1, . . . , r are independent and

ρ L,β j,n (d)
= Gamma (β ′ (nj + 1), 1/2) .

(2.9)

We stress that our point of view is not compatible with the construction by [START_REF] Dumitriu | Matrix models for beta-ensembles[END_REF]) of matrix models for the (general) β-Laguerre ensemble. Actually, they define a random r × r matrix B (r) where only diagonal and subdiagonal terms are nonzero, independent and satisfy (for n fixed):

B (r) ii (d) = Gamma (β ′ (n -i + 1), 1/2) (1 ≤ i ≤ r) , B (r) i,i-1 (d) = Gamma (β ′ (r -i + 1), 1/2) (2 ≤ i ≤ r) .
They prove that the distribution of eigenvalues of B (r) B (r) ′ is precisely (2.8). Of course we recover the determinant as a product of elements with the good distribution, but the problem is that we cannot consider all r simultaneously in their framework, since

B (r) B (r) ′ [r-1]
= B (r-1) B (r-1) ′ .

2.2.2. Uniform Gram. It is useful in the study of correlations. A correlation matrix is a positive definite matrix with diagonal entries equal to one. Here, there is no explicit expression for the law of eigenvalues. However, the expression 1

Z G,β r (n) |G| β ′ (n-r+1)-1 with Z G,β r (n) = π β ′ r(r-1) r j=1 Γ (β ′ (n -j + 1)) Γ (β ′ n)
is a density on the space of symmetric (resp. Hermitian, resp. self-dual) positive matrices with diagonal entries equal to one, and it fits with the distribution of correlation matrix in the real (see (2.1)), complex and quaternion case, for the appropriate values of β. This yields [START_REF] Gupta | 'une question relative aux déterminants[END_REF] ex. 3.26 p.130) the Mellin transform

E|G| β ′ s = Z G,β r (n + s) Z G,β r (n) = r j=1 Γ (β ′ (n -j + 1 + s)) Γ (β ′ n) Γ (β ′ (n -j + 1)) Γ (β ′ (n + s)) From (2.3), it is clear that if X (d) = Beta(a, b) then EX µ = Γ(a + µ)Γ(a + b) Γ(a)Γ(a + b + µ) (µ > -a) .
(2.10)

Again the uniqueness of the Mellin transform leads to the proposition.

Proposition 2.3. We have

|G| (d) = r j=2 ρ G,β j,n
where the variables ρ G,β j,n , j = 2, . . . , r are independent and

ρ G,β j,n (d)
= Beta (β ′ (nj + 1), β ′ (j -1)) .

(2.11) 2.2.3. Jacobi. If Z is distributed as in (2.2), the joint density of eigenvalues on the set (0 < λ j < 1 j = 1, . . . , r) is given by

1 Z r (n 1 , n 2 ) r i=1 λ n 1 -r-1 2 i (1 -λ i ) n 2 -r-1 2 1≤i<j≤r |λ j -λ i | ,
where Z r (n 1 , n 2 ) is a normalizing constant (see for example [START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF] Theorem 3.3.4).

For n 2 < r < n 1 , the matrix W 2 is singular, and the Jacobi matrix I -Z has 1 as an eigenvalue with multiplicity rn 2 . The distribution of Z has no density. Nevertheless we may study its determinant. Indeed, the matrix I -Z has 0 as an eigenvalue of multiplicity rn 2 . Actually the density of the law of the nonzero eigenvalues of this matrix is known (see [START_REF] Srivastava | Singular Wishart and multivariate Beta distributions[END_REF] and Diaz-Garcia and Gutierrez Jaimez (1997)), so that the non-one eigenvalues of Z have the joint density

1 Z r (n 1 , n 2 ) n2 i=1 λ n 1 -r-1 2 i (1 -λ i ) r-n 2 -1 2 1≤i<j≤n2 |λ j -λ i | ,
where the normalizing constant is Z r (n 1 , n 2 ) = Z n2 (n 1 + n 2r, r) .

We now consider matrices with elements in X = C or H. When r ≤ min(n 1 , n 2 ), the distribution of Z has a density proportional to

|Z| β ′ (n1-r+1)-1 |I -Z| β ′ (n2-r+1)-1 1 0<Z<I .
where β ′ = 1 or 2. The distribution of the eigenvalues of Z has the density (on [0, 1] r ) :

f β r,n1,n2 (λ 1 , . . . , λ r ) = (2.12) 1 Z (J,β) r (n 1 , n 2 ) r i=1 λ β ′ (n1-r+1)-1 i (1 -λ i ) β ′ (n2-r+1)-1 1≤i<j≤r |λ j -λ i | 2β ′ ,
where

Z (J,β) r (n 1 , n 2 ) = r j=1 Γ (1 + β ′ j) Γ (β ′ (n 1 + j -r)) Γ (β ′ (n 2 + j -r)) Γ (1 + β ′ ) Γ (β ′ (n 1 + n 2 + j -r)) , (2.13)
is the value of the Selberg integral (see [START_REF] Hiai | The Semicircle Law, Free Random Variables and Entropy[END_REF] p.118 and also [START_REF] Edelman | Random matrix theory[END_REF] p.19 and references therein).

In the "singular" case (n 2 ≤ r ≤ n 1 ), the density of the non-one eigenvalues is f β n2,n1+n2-r,r (λ 1 , . . . , λ n2 ).

We consider an extension of the above models. For every β > 0, we define a family of distribution densities f β r,n1,n2 on [0, 1] min(n2,r) :

f β r,n1,n2 = f β r,n1,n2 if r ≤ min(n 1 , n 2 ) f β n2,n1+n2-r,r if n 2 ≤ r ≤ n 1 .
(2.14)

We set by convention

|Z n1,n2,r | = min(n2,r) i=1 λ i
in all cases, and we call it the determinant, even if we do not define any matrix. For r ≤ n 1 , n 2 , using (2.12) and (2.13) we obtain

E |Z n1,n2,r | β ′ s = Z J,β r (n 1 + s, n 2 ) Z J,β r (n 1 , n 2 ) = r j=1 Γ (β ′ (n 1 + n 2 + j -r)) Γ (β ′ (n 1 + j -r + s)) Γ (β ′ (n 1 + j -r)) Γ (β ′ (n 1 + n 2 + j -r + s)) .
(2.15)

If n 2 < r ≤ n 1 we start directly from (2.14) and (2.13) we have

E |Z n1,n2,r | β ′ s = Z J,β n2 (n 1 + n 2 -r + s, r) Z J,β n2 (n 1 + n 2 -r, r) = n2 j=1 Γ (β ′ (n 1 + j)) Γ (β ′ (n 1 + j -r + s)) Γ (β ′ (n 1 + j -r)) Γ (β ′ (n 1 + j + s)) .
(2.16)

Multiplying up and down by

r k=n2+1 Γ (β ′ (n 1 + k -r)) Γ (β ′ (n 1 + k -r + s))
we get again the right hand side of (2.15). Going back to (2.10), we have the proposition Proposition 2.4. For r ≤ n 1 :

|Z n1,n2,r | (d) = r j=1 ρ β,J j,n1,n2 ,
where ρ β,J j,n1,n2 , j = 1, . . . , r are independent and

ρ β,J j,n1,n2 (d)
= Beta (β ′ (n 1j + 1), β ′ n 2 ) .

(2.17) 2.3. Processes. In the three ensembles defined above, we have met arrays of independent variables with remarkable distributions. In Section 2.1, we have discussed the interest of studying all values of r simultaneously in the matrix cases (β = 1, 2, 4). Since the structure remains the same in the β-models, it is meaningful to consider the processes (indexed by r) of partial sums. A now classical asymptotic regime is n, r → ∞ with fixed ratio in the Laguerre and Uniform Gram case, and n 1 , n 2 , r → ∞ with fixed ratios in the Jacobi case. It means that we consider the asymptotic behavior determinants in a dynamic (or path wise) way.

For the Laguerre model, we define

log ∆ L,β n,p := p k=1 log ρ L,β k,n βn (p ≤ n) (2.18)
and the process

∆ L,β n (t) := ∆ L,β n,⌊nt⌋ , t ∈ [0, 1] . (2.19)
For the Uniform Gram model, we define log ∆ G,β n,p :=

p k=1 log ρ G,β k,n (p ≤ n) (2.20)
and the process

∆ G,β n (t) := ∆ G,β n,⌊nt⌋ , t ∈ [0, 1] . (2.21)
For the Jacobi model, we fix τ 1 and τ 2 > 0, set

n 1 = ⌊nτ 1 ⌋, n 2 = ⌊nτ 2 ⌋, and define log ∆ J,β n,p = p k=1 log ρ J,β k,n1,n2 (p ≤ n 1 ) (2.22)
and the process

∆ J,β n (t) = ∆ J,β n,⌊nt⌋ , t ∈ [0, τ 1 ] . (2.23)
There are some connections between the above processes. For instance, in the real matrix model (β = 1) we saw in (2.6) that

ρ L,1 j,n = ρ G,1 j,n b j 2 ,
that the two random variables in the right hand side are independent and b j

2 (d) = Gamma(n/2, 1/2).
To see these connections in the general case, we use the so-called "beta-gamma" algebra that will be really helpful in the sequel. Details can be found in [START_REF] Chaumont | Exercises in probability[END_REF] pp.93-94. In the following relation, γ(a) denotes a random variable with distribution Gamma(a, 1), and β(a, b) denotes a random variable with distribution Beta(a, b). The relation is

γ(a), γ(b) (d) = β(a, b)γ(a + b), (1 -β(a, b))γ(a + b) , (2.24)
where, on the left hand side the random variables γ(a) and γ(b) are independent and on the right hand side the random variables β(a, b) and γ(a + b) are independent. It entails in particular

γ(a) γ(a) + γ(b) (d) = β(a, b) .
(2.25)

Let us note that this relation can be extended at the matrix variate level.

From the definitions (2.18) and (2.20) and owing to the equalities in distribution (2.9) and (2.11), we have then

log ∆ L,β n (d) = log ∆ G,β n + S n , (2.26)
where S n is independent of log ∆ G,β n , and specified by

S n (t) = ⌊nt⌋ k=1 log ε (n) k , t ∈ [0, 1] (2.27)
where ε

(n) k , k = 1, . . . , n are independent and satisfy ε (n) k (d) = Gamma (β ′ n, β ′ n) .
In the sequel, we begin by setting the claims for the Uniform Gram process and then deduce the corresponding results for the Laguerre process.

Using the definitions (2.18) and (2.22) and the equalities in distribution (2.9) and (2.17), we get, by another application of (2.24) log ∆ L,β n1,r

(d) = log ∆ J,β n,r + log ∆ L,β n1+n2,r -r log n 1 n 1 + n 2 , (2.28)
where this equality holds for all indices r = 1, . . . , n 1 simultaneously, and the two processes log ∆ J,β n and log ∆ L,β n1+n2 are independent. It allows to deduce asymptotic results for the Jacobi model from those of the Laguerre model.

Main results

In this section, we state first a law of large numbers and fluctuations for our three models, and then the corresponding LDP for processes and marginals.

Let

D T = {v ∈ D([0, T ]) : v(0) = 0} the set of càdlàg functions on [0, T ] and D = {v ∈ D([0, 1)) : v(0) = 0}
the set of càdlàg functions on [0, T ] and [0, 1), respectively, starting from 0.

We use often the following entropy function

J (u) =      u log u -u + 1 if u > 0 1 if u = 0 +∞ if u < 0 (3.1)
and its primitive:

F (t) = t 0 J (u) du = t 2 2 log t - 3t 2 4 + t, (t ≥ 0) . (3.2)
We use also the function defined in [START_REF] Hiai | Large deviations for functions of two random projection matrices[END_REF], for s, t ≥ 0:

B(s, t) := (1 + s) 2 2 log(1 + s) - s 2 2 log s + (1 + t) 2 2 log(1 + t) - t 2 2 log t - (2 + s + t) 2 2 log(2 + s + t) + (1 + s + t) 2 2 log(1 + s + t) . (3.3)
which may also be written as

B(s, t) = F (1 + s) -F (s) + F (1 + t) -F (t) -F (2 + s + t) + F (1 + s + t) - 7 4 .
3.1. Law of large numbers and fluctuations.

3.1.1. Uniform Gram ensemble. Let us define a drift and a diffusion coefficient by

d G,β (t) := 1 β + 1 2 - 1 β 1 1 -t , σ G,β (t) := 2t β(1 -t) . (3.4) Theorem 3.1. (1) As n → ∞, lim n sup p≤n 1 n E log ∆ G,β n,p + J 1 - p n = 0 . (3.5) (2) For every t ∈ [0, 1), as n → ∞, E log ∆ G,β n (t) + nJ 1 - ⌊nt⌋ n → t 0 d G,β (s) ds (3.6)
and

E log ∆ G,β n (1) + n + 1 β - 1 2 log n → K 1 β , (3.7)
where

K 1 β := 1 2 log(2π) + 1 -γ β - ∞ 0 sf (s) e βs/2 -1 ds , (3.8
)

and γ = -Γ ′ (1) is the Euler constant. (3) For every t ∈ [0, 1), as n → ∞, Var log ∆ G,β n (t) → t 0 σ G,β (s) 2 ds (3.9) Var log ∆ G,β n (1) - 2 β log n → K 2 β , (3.10)
where

K 2 β := 2(γ -1) β + ∞ 0 s(sf (s) + 1 2 ) e βs/2 -1 ds . (3.11) (4) As n → ∞, lim n sup t∈[0,1] log ∆ G,β n (t) n + J (1 -t) = 0 . (3.12)
in probability.

For β = 1, the formulae (3.7) and (3.10) are due to Abbott and Mulders (2001) (see their lemmas 4.2 and 4.4), using a variant of the decomposition method.

Theorem 3.2.

(1

) Let for n ≥ 1 η G,β n (t) := log ∆ G,β n (t) + nJ 1 - ⌊nt⌋ n , t ∈ [0, 1) . Then as n → ∞ η G,β n (t); t ∈ [0, 1) ⇒ X G,β t ; t ∈ [0, 1) , (3.13)
where X G,β is the (Gaussian) diffusion solution of the stochastic differential equation :

dX G,β t = d G,β (t) dt + σ G,β (t) dB t , (3.14)
with X G,β 0 = 0, B is a standard Brownian motion and ⇒ stands for the weak convergence of distributions in D endowed with the Skorokhod topology.

(2) Let

η G,β n = log ∆ G,β n (1) + n + 1 β -1 2 log n 2 β log n . Then as n → ∞, η G,β n
⇒ N where N is N (0, 1) and independent of B, (and ⇒ stands for the weak convergence of distribution in R).

3.1.2. Laguerre ensemble. Let us define a drift and a diffusion coefficient by

d L,β (t) := 1 2 - 1 β 1 1 -t , σ L,β (t) := 2 β(1 -t) . (3.15) Theorem 3.3. (1) As n → ∞, lim n sup p≤n 1 n E log ∆ L,β n,p + J 1 - p n = 0 (3.16) (2) For every t ∈ [0, 1), as n → ∞, E log ∆ L,β n (t) + nJ 1 - ⌊nt⌋ n → t 0 d L,β L (s) ds , (3.17)
and

E log ∆ L,β n (1) + n + 1 β - 1 2 log n → K 1 β , (3.18)
(3) For every t ∈ [0, 1), as n → ∞,

Var log ∆ L,β n (t) → t 0 σ L,β (s) 2 ds (3.19) Var log ∆ L,β n (1) - 2 β log n → K 2 β . (3.20) (4) As n → ∞, sup t∈[0,1] 1 n log ∆ L,β n (t) + J (1 -t) → 0 (3.21)
in probability.

Remark 3.4. In the Uniform Gram and Laguerre ensembles, when all the variables are defined on the same space (i.e. β = 1, 2, 4), an application of the Borel-Cantelli lemma leads to almost sure convergence.

Theorem 3.5. Let

η L,β n (t) := log ∆ L,β n (t) + nJ 1 - ⌊nt⌋ n , t ∈ [0, 1) , η L,β n = log ∆ L,β n (1) + n + 1 β -1 2 log n 2 β log n . Then as n → ∞ η L,β n (t); t ∈ [0, 1) ⇒ X L,β t , t ∈ [0, 1) (3.22) η L,β n ⇒ N
where X L,β is the Gaussian diffusion solution of the stochastic differential equation:

dX L,β t = d L,β (t) dt + σ L,β (t) dB t , (3.23)
with X L,β 0 = 0, where B is a standard Brownian motion and N is N (0, 1) and independent of B.

The convergence of η L,1 n (t), for fixed t and of η L,1 n were proved by Jonsson (1982) Theorem 5.1a. Recently and independently the convergence of η L,1 n was proved in Theorem 4 of [START_REF] Rempa | Asymptotics for products of independent sums with an application to Wishart determinants[END_REF].

3.1.3. Jacobi ensemble. In this part we use new auxiliary functions. Let

E(x, y, z) = x log x -(x + y) log(x + y) + (x + y -z) log(x + y -z) -(x -z) log(x -z) or using J defined in (3.1) E(x, y, z) = J (x) -J (x -z) -J (x + y) + J (x + y -z) .
(3.24)

The partial derivative of E with respect to x is :

E 1 (x, y, z) := ∂ ∂x E(x, y, z) = log x(x + y -z) (x -z)(x + y) . (3.25) Let for 0 ≤ t < τ 1 σ 2 (t) := ∂ ∂t E 1 (τ 1 , τ 2 , t) = τ 2 (τ 1 -t)(τ 1 + τ 2 -t) . (3.26)
Again we define drift and diffusion coefficients:

d J,β (t) = 1 2 - 1 β σ 2 (t) , σ J,β (t) = 2 β σ(t) .
Theorem 3.6.

(1) As n → ∞,

sup t∈[0,τ1] 1 n E log ∆ J,β n (t) -E (τ 1 , τ 2 , t) → 0 . (3.27) (2) For every t ∈ [0, τ 1 ), as n → ∞, E log ∆ J,β n (t) -E(⌊τ 1 n⌋, ⌊τ 2 n⌋, ⌊tn⌋) -→ t 0 d J (s) ds , (3.28)
and1

E log ∆ J,β n (τ 1 ) -E(⌊τ 1 n⌋, ⌊τ 2 n⌋, ⌊τ 1 n⌋) + 1 β - 1 2 log n -→ 1 2 - 1 β log τ 1 τ 2 τ 1 + τ 2 + K 1 β . (3.29) . (3) For every t ∈ [0, τ 1 ), as n → ∞, Var log ∆ J,β n (t) → t 0 σ J,β (s) 2 ds , (3.30)
and 1

Var log ∆ J,β n (τ 1 ) -

2 β log n -→ 2 β log τ 1 τ 2 τ 1 + τ 2 + K 2 β . (3.31) (4) As n → ∞, sup t∈[0,τ1] 1 n log ∆ J,β n (t) -E(τ 1 , τ 2 , t) → 0 (3.32)
in probability.

Remark 3.7. For β = 1, 2, 4, when all variables are on the same probability space, the convergence in (4) may be strengthened to almost sure convergence.

Theorem 3.8. Let for n ≥ 1

η J,β n (t) := log ∆ J,β n (t) -E(⌊τ 1 n⌋, ⌊τ 2 n⌋, ⌊tn⌋) , t ∈ [0, τ 1 ) , η J,β n := log ∆ J,β n (τ 1 ) -nE(τ 1 , τ 2 , τ 1 ) + 1 2 -1 β log n 2 β log n . Then as n → ∞ η J,β n (t); t ∈ [0, τ 1 ) ⇒ X J t ; t ∈ [0, τ 1 ) , (3.33) η J,β n ⇒ N
where X J,β is the (Gaussian) diffusion solution of the stochastic differential equation :

dX J t = d J,β (t)dt + σ J,β (t) dB t , (3.34)
with X J,β 0 = 0, B is a standard Brownian motion and N is a standard normal variable independent of B.

3.2. Large deviations. All along this section, we use the notation of [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. In particular we write LDP for Large Deviation Principle. The reader may have some interest in consulting [START_REF] Dette | Asymptotic properties of the algebraic moment range process[END_REF] where a similar method is used for a different model, but here we use a slightly different topology to be able to catch the marginals in T .

For T < 1, let M T be the set of signed measures on [0, T ] and let M < be the set of measures whose support is a compact subset of [0, 1).

We provide D with the weakened topology σ(D, M < ). So, D is the projective limit of the family, indexed by T < 1 of topological spaces (D T , σ(D T , M T )).

Let V ℓ (resp. V r ) be the space of left (resp. right) continuous R-valued functions with bounded variations. We put a superscript T to specify the functions on [0, T ]. There is a bijective correspondence between V T r and M T : -for any v ∈ V T r , there exists a unique µ

∈ M T such that v = µ([0, •]); we denote it by v , -for any µ ∈ M T , v = µ([0, •]) stands in V r .
For v ∈ D, let v = va + vs be the Lebesgue decomposition of the measure v ∈ M([0, 1)) in absolutely continuous and singular parts with respect to the Lebesgue measure and let µ be any bounded positive measure dominating vs .

For A ⊂ [0, 1) and v ∈ D let

I A (v) = A L a t, d va dt (t) dt + A L s t, d vs dµ (t) dµ(t) if v ∈ V r , (3.35)
and

I A (v) = ∞ if v ∈ D \ V r
, where functions L a (t, x) and L s (t, x) will be defined later for each of the ensembles of interest.

3.2.1. Uniform Gram ensemble. For the following statement, we need some notation. Let H be the entropy function :

H(x|p) = x log x p + (1 -x) log 1 -x 1 -p , and put 1 L G a (t, y) = H(1 -t|e y ) δ(y|(-∞, 0)) , L G s (t, y) = -(1 -t)y δ(y|(-∞, 0)) . (3.36) Theorem 3.9. The sequence {n -1 log ∆ G,β n (t), t ∈ [0, 1)} n satisfies a LDP in (D, σ(D, M < )) in the scale 2β -1 n -2 with good rate function I G [0,1)
. That means, roughly speaking, that

P(log ∆ G,β n ≃ nv) ≈ e -βn 2 2 I G [0,1) (v) .
The proof, in Section 6.1, needs several steps. Let Θ

G n = n -1 log ∆ G,β n , so that ΘG n = 1 n n j=1 log ρ G,β n,j δ j/n . (3.37)
First we show that { ΘG n } satisfies a LDP in M T equipped with the topology σ(M T , V ℓ ). Then we carry the LDP to (D T , σ(D T , M T )) with good rate function:

I G [0,T ] (v) = [0,T ] L G a t, d va dt (t) dt + [0,T ] L G s t, d vs dµ (t) dµ(t) . (3.38)
To end the proof we apply the Dawson-Gärtner theorem on projective limits [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] Theorem 4.6.1, see also [START_REF] Léonard | Large deviations for Poisson random measures and processes with independent increments[END_REF] Proposition A2).

Let us note that I G [0,T ] (v) vanishes only when v satisfies (essentially)

d va dt (t) = log(1 -t) , d vs dµ (t) = 0 , (3.39)
i.e. for v(t) = -J (1t), which is consistent with the result (3.12).

The LDP for marginals is given in the following theorem, where a rate function with affine part appears.

Theorem 3.10. For every T < 1, the sequence n -1 log ∆ G,β n (T ) n satisfies a LDP in R in the scale 2β -1 n -2 with good rate function denoted by

I G T (ξ) = inf{I G [0,T ] (v) ; v(T ) = ξ} . (3.40) (1) If ξ ∈ [-T, 0) the equation J (1 + θ) -J (1 -T + θ) -T log(1 + θ) = ξ , (3.41)
has a unique solution, and we have

I G T (ξ) = θξ + T J (1 + θ) (3.42) + (F (1) -F (1 -T ) -F (1 + θ) + F (1 -T + θ)) .
(2) If ξ < -T , we have

I G T (ξ) = I G T (-T ) -(1 -T )(ξ + T ) . (3.43) (3) If ξ ≥ 0, I G T (ξ) = ∞. 3.2.2. Laguerre ensemble. Let L L a (t, y) = (e y -1) -(1 -t)y + J (1 -t) L L s (t, y) = -(1 -t)y δ(y|(-∞, 0)) . (3.44) Theorem 3.11. The sequence {n -1 log ∆ L,β n (t), t ∈ [0, 1)} n satisfies a LDP in (D, σ(D, M < )), in the scale 2β -1 n -2 with good rate function I L [0,1)
. That means, roughly speaking, that

P(log ∆ L n ≃ nv) ≈ e -βn 2 2 I L [0,1) (v) .
The proof uses the above result for the Uniform Gram process and the beta-gamma algebra.

Let us note that I L [0,T ] (v) vanishes only when v satisfies (3.39) (again) i.e. for v(t) = -J (1t), which is consistent with the result (3.21).

The LDP for marginals is given in the following theorem.

Theorem 3.12. For every T < 1, the sequence {n -1 log ∆ L,β n (T )} n satisfies a LDP in R in the scale 2β -1 n -2 with good rate function denoted by I L T .

I L T (ξ) = inf{I L [0,T ] (v) ; v(T ) = ξ} . (3.45) (1) If ξ ≥ ξ T := J (T ) -1 the equation J (1 + θ) -J (1 -T + θ) = ξ . (3.46)
has a unique solution, and we have

I L T (ξ) = θξ + F (1) -F (1 -T ) -F (1 + θ) + F (1 -T + θ) .
(3.47)

(2) If ξ < ξ T , we have

I L T (ξ) = I L T (T ) + (1 -T )(ξ T -ξ) .
(3.48) 3.2.3. Jacobi ensemble. In this subsection, the endpoint 1 of subsections 3.2.1 and 3.2.2 is replaced by τ 1 . Let, for t < τ 1 ,

L J a (t, y) = (τ 1 + τ 2 -t) H τ 1 -t τ 1 + τ 2 -t e y L J s (t, y) = -(τ 1 -t)y if y < 0 . (3.49) Theorem 3.13. The sequence {n -1 log ∆ J,β n (t), t ∈ [0, τ 1 )} n satisfies a LDP in (D, σ(D, M < )) in the scale 2β -1 n -2 with good rate function I [0,τ1) .
That means, roughly speaking, that

P(log ∆ J,β n ≃ nv) ≈ e -βn 2 2 I J [0,1) (v) . Let us note that I J [0,T ] (v) vanishes only when v satisfies (essentially) d va dt (t) = log τ 1 -t τ 1 + τ 2 -t , d vs dµ (t) = 0 ,
i.e. for v(t) = E(τ 1 , τ 2 , t), which is consistent with the result (3.32).

The LDP for marginals is given in the following theorem.

Theorem 3.14. Let T ∈ [0, τ 1 ), and ξ J T = J (τ 2 ) + J (T ) -J (T + τ 2 ) -1. (1) The sequence {n -1 log ∆ J,β n (T )} n satisfies a LDP in R in the scale 2β -1 n -2 with good rate function I J T where

I J T (ξ) := inf{I J [0,T ] (v) ; v(T ) = ξ} . (3.50) (2) If ξ ∈ [ξ J T , 0), the equation E(θ + τ 1 , τ 2 , T ) = ξ (3.51)
has a unique solution θ ≥ Tτ 1 , and we have

I J T (ξ) = θξ -[F (θ + τ 1 ) -F (θ + τ 1 -T )] -[F (τ 1 + τ 2 ) -F (τ 1 + τ 2 -t)] + [F (τ 1 ) -F (τ 1 -T )] + [F (θ + τ 1 + τ 2 ) -F (θ + τ 1 + τ 2 -T )] . (3.52) (3) If ξ < ξ J T , we have I J T (ξ) = I J T (ξ J T ) + (ξ J T -ξ)(τ 1 -T ) . (3.53) (4) If ξ ≥ 0, then I J T (ξ) = ∞.

Connections with the spectral method

The logarithm of the determinant of a non singular matrix is a linear statistic of the empirical distribution of its eigenvalues, so that we may compare the above result with those obtained by this spectral approach.

4.1. Laguerre/Wishart. We start with

1 n log ∆ L,β n,r = r n (log x) dµ n,r (x)
where µ n,r the so called empirical spectral distribution (ESD) is

µ n,r = 1 r r k=1 δ λ k (4.1)
For c > 0 and σ > 0, let π c σ 2 be the distribution on R defined by

π c σ 2 (dx) = (1 -c -1 ) + δ 0 (dx) + (x -σ 2 a(c))(σ 2 b(c) -x) 1/2 + 2πσ 2 cx dx , (4.2)
where δ 0 is the Dirac mass in 0, x + = max(x, 0) and

a(c) = (1 - √ c) 2 , b(c) = (1 + √ c) 2 . (4.3)
It is called the Marčenko-Pastur distribution with ratio index c and scale index σ 2 (Bai (1999) p.621).

It is well known [START_REF] Marčenko | Distribution of eigenvalues of some sets of random matrices[END_REF], Bai (1999) section 2.1.2 for the cases β = 1 and β = 2) that as n, r → ∞ with r/n → T ∈ (0, ∞), the family of ESD (µ n,r ) converges a.s. weakly to π T 1 . If we replace the common law N (0, 1) by N (0, σ 2 ) then the limiting distribution is π T σ 2 . To conclude that lim n (log x) dµ n,r (x) = (log x) dπ T 1 (x) , (4.4)

an additional control is necessary, since x → log x is not bounded. Actually, the largest and the smallest eigenvalue converge a.s. to b(T ) < ∞ and a(T ) > 0, respectively. For comments on these results and references, one may consult Bai (1999) sections 2.1.2 and 2.2.2., (see also [START_REF] Johnstone | On the distribution of the largest eigenvalue in principal component analysis[END_REF]). In our context, this implies easily that a.s.

1 n log ∆ L,β n,r = r n (log x) dµ n,r (x) → T log x dπ T 1 (x) (4.5)
Moreover, it is known [START_REF] Jonsson | Some limit theorems for the eigenvalues of a sample covariance matrix[END_REF] p.31 and [START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF] p.596-597) that :

T (log x) dπ T 1 (x) = b(T ) a(T ) log x 2πx (x -a(T ))(b(T ) -x) dx = (T -1) log(1 -T ) -T = -J (1 -T ) (4.6)
which implies that claim (4.5) is consistent with (3.21).

Recently, [START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF] proved a CLT for linear statistics of sample covariance matrices (non necessarily Gaussian), with the meaningful example of determinants. They consider the real and complex case, and their results (Theorem 1.1 ii) and iii) are consistent with the marginal version of (3.22). It is likely that β = 4 can also be handled under their assumptions.

Let us end with the large deviations. [START_REF] Hiai | Eigenvalue density of the Wishart matrix and large deviations[END_REF], (see also [START_REF] Hiai | The Semicircle Law, Free Random Variables and Entropy[END_REF] section 5.5) proved2 that if n → ∞ and r/n → T < 1, then {µ n,r } satisfies a LDP in M 1 ([0, ∞)) in the scale 2β -1 n -2 with some explicit good rate function I spL T given below in (4.8, 4.9, 4.10). If the contraction µ → log x dµ(x) were continuous, we would claim that {n -1 log ∆ L n,⌊nT ⌋ } n satisfies a LDP in R in the same scale, with good rate function

I L T (ξ) = inf I spL T (µ) ; T log x dµ(x) = ξ . (4.7)
Actually,

I spL T (µ) = -T 2 Σ(µ) + T (x -(1 -T ) log x) dµ(x) + 2B(T ) (4.8)
where

Σ(µ) := log |x -y| dµ(x)dµ(y) (4.9)
is the so-called logarithmic entropy and for T ∈ (0, 1)

2B(T ) = - 1 2 3T -T 2 log T + (1 -T ) 2 log(1 -T ) . (4.10)
We do not know if the contraction µ → log x dµ(x) does work, although not continuous. However we will prove the following result, where for u ∈ R we put Remark 4.2.

A(u) = {µ : (log x) dµ(x) = u} .
(1) The endpoint is ξ T = J (T ) -1, with σ 2 = T . (2) For ξ < ξ T we do not know what happens. We can imagine that the infimum in (4.12) has a solution in some extended space.

4.2. Uniform Gram. Let λ k , k = 1, . . . , r be the (real) eigenvalues of G in the Uniform Gram ensemble, and set

µ n,r = 1 r r k=1 δ λ k . (4.13)
For β = 1, De [START_REF] Cock | On quantum dynamics and statistics of vectors[END_REF] proved that, as n → ∞ and r/n → T ∈ (0, ∞), the family ( µ n,r ) converges a.s. to π T 1 . More recently [START_REF] Jiang | The limiting distributions of eigenvalues of sample correlation matrices[END_REF] proved that the same result holds true in a complex Gram ensemble not necessarily uniform. Again, like in Section 4.1, we may write

1 n log ∆ G,β n,r = r n (log x) d µ n,r (x)
and use the weak convergence of µ n,r towards π T 1 . Recently, [START_REF] Jiang | The limiting distributions of eigenvalues of sample correlation matrices[END_REF] proved that the largest and the smallest eigenvalue converge a.s. as r/n → T < 1 to b(T ) < ∞ and a(T ) > 0 respectively. So, we have lim

n (log x) d µ n,r (x) = (log x) dπ T 1 (x) . (4.14)
In view of (4.6), this matches with the result (3.12).

No result on fluctuations or large deviations seems to be known on µ n,r .

4.3. Jacobi. In the matrix models (β = 1, 2 or 4), take r ≤ n 1 and let λ k , k = 1, . . . , r be the eigenvalues of Z n1,n2,r (they are real nonnegative). The ESD is

ν n1,n2,r = 1 r r k=1 δ λ k .
When n 2 ≤ r ≤ n 1 we have

ν n1,n2,r = n 2 r µ n1,n2,r + 1 - n 2 r δ 1 ,
where µ n1,n2,r is the ESD built with eigenvalues different from 1. We can write in all cases log ∆ J,β n,r = min(r, n 2 ) (log x) µ n1,n2,r (dx) .

(4.15)

It is then possible to carry asymptotical results of this empirical distribution to log ∆ J,β n,r . [START_REF] Capitaine | Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices[END_REF] studied the complex case in the asymptotical regime n 1 /r → u ′ , n 2 /r → v ′ with u ′ + v ′ ≥ 1. They prove 3 that Eν n1,n2,r converges (in moments hence) in distribution. To give the expression of the limiting distribution, which we denote CC u ′ ,v ′ and to compare with known results in some other contexts with coherent notation, we will use in the following, four functions : for (b, c) ∈ (0, 1) × (0, 1) we put

σ ± (b, c) = 1 2 1 + √ bc ± (1 -b)(1 -c) , (4.16)
and for (x, y) ∈ (0, 1) × (0, 1)

a ± (x, y) = (1 -x -y + 2xy) ± 2 x(1 -x)y(1 -y) = (1 -x)(1 -y) ± √ xy 2 .
(4.17)

The mappings σ ± and a ± are inverse in the following sense :

{(b, c) : 0 < b < c < 1} (σ-,σ+) -----⇀ ↽ ----- (a-,a+)
{(x, y) : 0 < x < y < 1 and x + y > 1} (4.18)

For 0 < a -< a + < 1, let π a-,a+ be the distribution on R defined by

π a-,a+ (dx) = C a-,a+ (x -a -)(a + -x) 2πx(1 -x) 1 [a-,a+] (x) dx , (4.19)
3 They use the notation α and β but we change not to confuse with β already defined.

where C a-,a+ is the normalization constant. Since we found some mistakes in the literature, let us compute explicitly the constant C a-,a+ . From the obvious decomposition 1

x(1 -x) = 1 x + 1 1 -x we get (C a-,a+ ) -1 = I(a -, a + ) + I(1 -a + , 1 -a -)
where, for 0 < u < v

I(u, v) = v u (x -u)(v -x) 2πx dx
This last integral could be calculated by elementary method, but it is shorter to connect it with the Marčenko-Pastur distribution. Taking

σ 2 = √ v + √ u 4 , √ c = √ v - √ u √ v + √ u
the simple fact that π c σ 2 , given in (4.2), is a probability distribution yields

I(u, v) = ( √ v - √ u) 2 4 (0 < u < v) .
Finally, we get:

(C a-,a+ ) -1 = 1 2 1 - √ a -a + -(1 -a -)(1 -a + ) . (4.20) The distribution CC u ′ ,v ′ is then (recall u ′ + v ′ ≥ 1) : CC u ′ ,v ′ := (1 -u ′ ) + δ 0 + (1 -v ′ ) + δ 1 (4.21) + 1 -(1 -u ′ ) + -(1 -v ′ ) + π a-,a+ , where (a -, a + ) = a ± u ′ u ′ + v ′ , 1 - 1 u ′ + v ′ . (4.22)
Remark 4.3. The case (v ′ < 1) corresponds to r > n 2 , the second matrix W 2 is singular and the case (v ′ ≥ 1) corresponds to r ≤ n 2 , the second matrix is nonsingular.

For particular values of the parameters and up to an affine change to make the distribution symmetric, the distribution π a-,a+ was introduced by Kesten (1959) as limit distribution for random walks on some classical groups. It was (independently) introduced by McKay (1981) as a limit distribution in a graph problem. It is sometimes called the generalized McKay distribution. Some important connections are in Section 9.

For the LLN, the same remarks as above are relevant. Let us recall the notation

r ≤ n 1 , n → ∞ , r n → T , n 1 n → τ 1 , n 2 n → τ 2 , u ′ = τ 1 T , v ′ = τ 2 T .
The weak convergence of the ESD [START_REF] Capitaine | Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices[END_REF]) and the control on the extremal eigenvalues [START_REF] Ledoux | Differential operators and spectral distributions of invariant ensembles from the classical orthogonal polynomials. The continuous case[END_REF], [START_REF] Collins | Products of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF] and references therein),

yield, if u ′ ≥ 1 lim n 1 r log ∆ J,β n (T ) = (log x) CC u ′ ,v ′ (dx) = min(v ′ , 1) (log x) π a-,a+ (dx) (4.23)
where a ± are in (4.22). Nevertheless a computation of this integral by elementary methods is not so easy. After some attempts, we choose to consider the above result as an indirect way to compute this integral and we obtain the following result.

Proposition 4.4. For 0 < a -< a + < 1,

(log x) π a-,a+ (dx) = (4.24) = σ + log σ + + σ -log σ --(σ + + σ --1) log(σ + + σ --1) 1 -σ +
where σ ± are specified by (4.16).

Proof:

From (3.32), lim n 1 r log ∆ J,β n (T ) = 1 T lim n 1 n log ∆ J,β n (T ) = 1 T E(τ 1 , τ 2 , T ) = E(u ′ , v ′ , 1) ,
where for the last equality we noticed that E is homogenous. With the help of (4.23) we get

min(v ′ , 1) (log x)π a-,a+ (dx) = E(u ′ , v ′ , 1) . (4.25) From (4.18) we see that if u ′ ≥ 1 then {σ -, σ + } = u ′ u ′ + v ′ , u ′ + v ′ -1 u ′ + v ′
We have two cases. When v ′ > 1,

σ -= u ′ u ′ + v ′ , σ + = u ′ + v ′ -1 u ′ + v ′ , so that (4.25) yields (log x)π a-,a+ (dx) = E(u ′ , v ′ , 1) = E σ - 1 -σ + , 1 -σ - 1 -σ + , 1 (4.26) When v ′ < 1, σ + = u ′ u ′ + v ′ , σ -= u ′ + v ′ -1 u ′ + v ′ , so that (4.25) yields (log x)π a-,a+ (dx) = 1 v ′ E(u ′ , v ′ , 1) = 1 -σ - 1 -σ + E σ + 1 -σ - , 1 -σ + 1 -σ - , 1 (4.27)
and together (4.26-4.27) provide (4.24). This ends the proof.

Let us remark that the first case above (v ′ < 1) corresponds to T > τ 2 (i.e. r > n 2 , the second matrix W 2 is singular) and the second one (v ′ ≥ 1) corresponds to T ≤ τ 2 (i.e. r ≤ n 2 , the second matrix is non-singular).

Let us end with the large deviations. In the complex case (β = 2), [START_REF] Hiai | Large deviations for functions of two random projection matrices[END_REF]

proved that if n → ∞, n 1 /n → τ 1 , n 2 /n → τ 2 > τ 1 , r/n → T < τ 1 , then {µ n1,n2,r } n satisfies a LDP in M 1 ([0, 1]
) the set of probability measures on [0, 1] endowed with the weak convergence topology, in the scale n -2 , with the good rate function

I spJ T (µ) := -T 2 Σ(µ) -T 1 0 ((τ 1 -T ) log x + (τ 2 -T ) log(1 -x)) dµ(x) + T 2 B τ 1 -T T , τ 2 -T T , (4.28)
where B is defined in (3.3) (it is the limiting free energy).

A computation similar to p.10 of [START_REF] Hiai | Large deviations for functions of two random projection matrices[END_REF] gives the same result for general β.

Proposition 4.5. If T < τ 1 ≤ τ 2 , the family {µ ⌊nτ1⌋,⌊nτ2⌋,⌊nT ⌋ } satisfies a LDP in M 1 ([0, 1]) in the scale 2β -1 n -2 and good rate function I spJ T . If the contraction µ → log x dµ(x) from the set M 1 ([0, 1]) to R were continuous, we would claim that {n -1 log ∆ J,β n (T )} n satisfies a LDP in R with good rate function I J T where

I J T (ξ) = inf I spJ T (µ) ; µ ∈ A(ξT -1 ) (4.29)
with A(u) as defined in (4.11).

Like in the Laguerre case we will prove the following result.

Proposition 4.6. Let T < min(τ 1 , τ 2 ), ξ ∈ [ξ J T , 0) and θ solution of (3.51). Then the infimum of I spJ T (µ) over A(ξT -1 ) is uniquely achieved at µ = π ã-,ã+ where (ã -, ã+ ) = a ± (s -, s+ )

with

s-= τ 1 + θ τ 1 + τ 2 + θ , s+ = τ 1 + θ + τ 2 -t τ 1 + τ 2 + θ , (4.30)
and

I J T (ξ) = I spJ T (π ã-,ã+ ) = inf{I spJ T (µ); µ ∈ A(ξT -1 )} . (4.31) Remark 4.7. The endpoint is ξ J T , which corresponds to θ = T -τ 1 , i.e. ã-= 0 , ã+ = 4τ 2 T (τ 2 + T ) 2 .
For ξ < ξ J T we do not know what happens. We can imagine that the infimum in (4.29) has a solution in some extended space.

Remark 4.8. In the range τ 2 ≤ T < τ 1 we have a similar result, exchanging s-and s+ in (4.30). We omit the details. 4.4. Extensions. We already mentioned that in the Wishart and Gram models, limiting results exist for marginals when we leave the Gaussian/Uniform world, in particular for fluctuations in [START_REF] Bai | CLT for linear spectral statistics of large-dimensional sample covariance matrices[END_REF]. The Bartlett decomposition is not possible in the general case. Nevertheless, a product formula for the determinant is well known (see for example Lemma 3.1 p.9 and formula 4.3 p.15 in [START_REF] Friedland | Concentration of permanent estimators for certain large matrices[END_REF]), but nothing can be said about the distribution of the components of the product in general.

Nevertheless, if the columns (or the rows) of the matrix B are i.i.d. and isotropic, the previous results extend easily.

Let us begin with the "column" case. The beta-gamma algebra allowed us to pass from the Uniform Gram ensemble to the Wishart ensemble. The polar decomposition allows to obtain similar results as for the Wishart ensemble under convenient assumptions on the radial distribution. Let ε n = log b 1 2log E b 1 2 (remember that we omit the dimension index n). To get convergence and fluctuations it is enough to assume

nEε n → a 1 , nVar ε n → a 2 , nE(ε n -Eε n ) 4 → 0 . (4.32)
To get large deviations, it would be sufficient to assume that, for some convenient functions ϕ, the quantity n -2 n k=1 log E exp (nϕ(k/n)ε n ) has a limit . [START_REF] Akhavi | Random lattices, threshold phenomena and efficient reduction algorithms[END_REF] uses the uniform distribution in the unit ball, so that the distribution of b 1 2 is Beta(n/2, 1) and (4.32) is satisfied with a 1 = -2, a 2 = 0. The contribution of the radial part is then roughly "deterministic" since E b 1 2 is bounded.

In the "row" case, we can use the results of the "column" case since the eigenvalues of BB ′ are (except 0 with multiplicity nr) the same as those of B ′ B.

5. Proofs of Theorems of Section 3.1 5.1. Proof of Theorem 3.1. We will use Mellin transforms and their first two derivatives at θ = 0. From the decomposition (2.20) we have

log E|∆ G,β n,r | β ′ θ = r k=1 Λ G,β n,k (θ) (5.1) with Λ G,β n,k (θ) := log E ρ G,β n,k β ′ θ (5.2)
and from (2.11)

Λ G,β n,k (θ) = ℓ β ′ (n -k + 1 + θ) -ℓ β ′ (n -k + 1) +ℓ(β ′ n)-ℓ (β ′ (n + θ)) (5.
3) where we set ℓ(x) = log Γ(x) . Proof of 1) and 2) Differentiating once, we get

E log ∆ G,β n,r = r j=1 [Ψ (β ′ (n -j + 1)) -Ψ (β ′ n)] ,
and from Binet formula (8.5),

E log ∆ G,β n,r = log (n) r n r + 1 β (H n-r -H n ) + r βn -δ 1 n,r . (5.4) in which 1) (p) r = p(p -1) • • • (p -r + 1) is the falling factorial 2) H 0 = 0 and H p = 1 + 1 2 + • • • + 1 p are the harmonic numbers 3) the delta term is δ 1 n,r = ∞ 0 sf (s) r k=1
[e -β ′ (n-k+1)se -ns ] ds .

(5.5)

Using Binet formula (8.1) twice, we have for r < n (n-r) e -sn ]ds .

log (n) r n r = -n -r + 1 2 log 1 - r n -r - ∞ 0 f (s)[e -s(n-r) -e -sn ]ds = -nJ 1 - r n - 1 2 log 1 - r n - ∞ 0 f (s)[e -s
For r = n the Stirling formula gives

log (n) n n n = -n + 1 2 log(2πn) + o(1) ,
The harmonic contribution in (5.4) is

H n-r -H n = log 1 - r n + o(1)
as soon as nr → ∞. For r = n, we have H 0 -H n =log nγ + o(1) . Applying the dominated convergence theorem and (8.4), we see that the delta contribution satisfies:

sup r≤n δ 1 n,r = δ 1 n,n → ∞ 0 sf (s) e β ′ s -1 ds ,
and lim n δ 1 n,⌊nt⌋ = 0 for t < 1. Gathering all these estimates, and applying again the dominated convergence theorem, we get (for nr → ∞)

E log ∆ G,β n,r = -nJ 1 - r n + r βn + 1 β - 1 2 log 1 - r n + o(1) ,
and for r = n

E log ∆ G,β n,n = -n - 1 β - 1 2 log n + K 1 β + o(1) .
Moreover, for the supremum, we have

sup r≤n E log ∆ G,β n,r -log (n) r n r = O(log n) sup r≤n log (n) r n r + nJ 1 - r n = O(log n)
so that (3.5), (3.6) and (3.7) are proved.

3) Taking logarithms in (5.1) and differentiating twice, we get

Var log ∆ G,β n,r = r j=1 Ψ ′ (β ′ (n -j + 1)) -Ψ ′ (β ′ n)
and owing to (8.9)

Var log ∆ G,β n,r = 1 β ′ (H n -H n-r ) - r β ′ n + ε where |ε| ≤ n n-r+1 2 β ′2 j 2 Moreover Var log ∆ G,β n,n = 1 β ′ H n - 1 β ′ + δ 2 n where δ 2 n = ∞ 0 s sf (s) + 1 2 n k=1 [e -β ′ (n-k+1)s -e -β ′ ns ] ds .
Applying again the dominated convergence theorem and (8.4), we get

lim n δ 2 n = ∞ 0 s sf (s) + 1 2 e β ′ s -1 ds
Using (8.4) and dominated convergence we deduce easily (3.9) and (3.10).

To prove 4), let us note that since J is uniformly continuous on [0, 1] we have

lim n sup t∈[0,1] J 1 - ⌊nt⌋ n -J (1 -t) = 0 ,
so that, owing to (3.5), it is enough to prove that in probability

sup 1≤p≤n log ∆ G,β p,n -E log ∆ G,β p,n = o(n) .
Actually this convergence is a consequence of Doob inequality and of the variance estimate Var n -1 ∆ G,β n,n = O(n -2 log n) coming from (3.10).

5.2. Proof of Theorem 3.2. Let us first note that, thanks to the estimations of expectations in (3.6) and (3.7), we can reduce the problem to the centered process and centered variable :

δ n (t) := log ∆ G,β n (t) -E log ∆ G,β n (t) , δ n = δ n (1)/ (2/β) log n . 1) We have δ n (t) = ⌊nt⌋ k=1 η n,k where η n,k := (log ρ G,β k,n ) -E(log ρ G,β k,n ), k ≤ n (5.6)
is a row-wise independent arrow. To prove (3.13) it is enough to prove the convergence in distribution in D([0, T ]), for every T < 1, of δ n to a centered Gaussian process with independent increments, and variance

t 0 σ G,β (s) 2 ds.
To this purpose we apply a version of the Lindeberg-Lévy-Lyapunov criteria (see [START_REF] Dacunha-Castelle | Probability and Statistics[END_REF]Duflo (1986) Volume II Theorem 7.4.28 , or Jacod andShiryaev (1987, Chap. 3 c)). For t < 1, from (3.9) it is enough to prove that lim

n ⌊nt⌋ k=1 E (η 4 n,k ) = 0 .
(5.7)

We have from definitions (5.6) and ( 5.2)

β ′4 E(η 4 n,k ) = (Λ G,β n,k ) (4) (0) + 3[(Λ G,β n,k ) (2) (0)] 2 , (5.8)
On the one hand, from expression (5.3)

(Λ G,β n,k ) (4) (0) = β ′4 [Ψ (3) (β ′ (n -k + 1)) -Ψ (3) (β ′ n)]
and Binet estimates (8.8), (8.9) for q = 4 yield

p k=1 (Λ G,β n,k ) (4) (0) -6β ′ p k=1 1 (n -k + 1) 3 - 1 n 3 ≤ 6 p k=1 1 (n -k + 1) 4 ,
(5.9) the limiting distribution does not depend on ε 1 , . . . , ε r , we have proved that δ n converges in distribution to a random variable which is N (0, 1) and independent of W.

5.3. Proof of Theorems 3.3 and 3.5. It is of course possible to follow the same schemes of proof. Actually we prefer, at least for the beginning, exploit the betagamma algebra and the fundamental relation (2.26). So, for instance (5.13) which provides estimates for the expectation and the variance. Differentiating once and taking θ = 0, we see that

E ε (n) k β ′ θ = 1 β ′ n β ′ θ Γ(β ′ (n + θ)) Γ(β ′ n) hence log E ε (n) k β ′ θ = ℓ (β ′ (θ + n)) -ℓ (β ′ n) -β ′ θ log (β ′ n) ,
E log ε (n) k = Ψ (β ′ n) -log (β ′ n) = - 1 βn - ∞ 0 e -sβ ′ n sf (s) ds = - 1 βn + O 1 n 2
(see (8.5), (8.4)), which gives

sup p≤n ES n,p + p nβ = O 1 n .
(5.14)

Besides, differentiating (5.13) twice and taking θ = 0 again, we have Var log ε

(n) k = Ψ ′ (β ′ n) = 1 β ′ n + O 1 n 2
(see (8.9)), which yields

sup p≤n Var S n,p - 2p βn = O 1 n .
(5.15)

From (5.14) and (5.15) it is easy to check (via a fourth moment estimate) that S n converges in distribution in D([0, 1]) to

-(t/β) + 2/β B t , t ∈ [0, 1]
where B is a Brownian motion independent of (∆ G,β n , n ∈ N). Finally the family of processes ∆

L,β n = ∆ G,β n + S n converges in distribution towards X G,β t -(t/β) + 2/β B t , t ∈ [0, 1) .
It is a Gaussian process, whose drift and variance coefficients are

d G,β (t) - 1 β = 1 2 - 1 β 1 1 -t = d L,β (t) , σ G,β (t) 2 + 2 β = σ L,β (t) 2 .
which identify the process X L,β . Besides, we have

η L,β n (1) = η G,β n (1) + S n (1) √ 2 log n ,
so that the convergence of η L,β n (1) is clear. Moreover the independence properties seen in Theorem 3.2 remain true. 5.4. Proof of Theorem 3.6. Again, we could follow the same schemes as in the Gram section. Actually we take again the benefit of beta-gamma algebra. Let us delete the superscript β for the sake of simplicity.

Let us recall the equality in law (2.28)

log ∆ L n1,r (d)
= log ∆ J n,r + log ∆ L n1+n2,rr log

n 1 n 1 + n 2
with independence in the left hand side.

We deduce easily

E log ∆ J n,r = E log ∆ L n1,r -E log ∆ L n1+n2,r + r log n 1 n 1 + n 2 and Var log ∆ J n,r = Var log ∆ L n1,r -Var log ∆ L n1+n2,r
The results are now straightforward. We let the proof to the reader. We just note that since r/n 1 → t/τ 1 and r/(

n 1 + n 2 ) → t/(τ 1 + τ 2 ) then E log ∆ L n1,r + n 1 J 1 - r n 1 → t/τ1 0 d L,β (s)ds E log ∆ L n1+n2,r + (n 1 + n 2 )J 1 - r n 1 + n 2 → t/(τ1+τ2) 0 d L,β (s)ds hence d J,β (t) = 1 τ 1 d L t τ 1 - 1 τ 1 + τ 2 d L t τ 1 + τ 2 .
In the same vein

σ J (t) 2 = 1 τ 1 σ L t τ 1 2 - 1 τ 1 + τ 2 σ L t τ 1 + τ 2 2 .
5.5. Proof of Theorem 3.8. Again, it is possible to follow the classical scheme. Instead let us look at the situation we are faced to. Put

U n = log ∆ L n1,r -E∆ L n1,r , V n = log ∆ L n1+n2,r -E log ∆ L n1+n2,r , W n = log ∆ J n,r -E log ∆ J n,r .
(5.16) so that U n = V n + W n with U n ⇒ U and V n ⇒ V , where U and V are Gaussian processes with independent increments, and V n and W n are independent. Looking for instance at characteristic functions, it is clear that W n converges in the sense of finite distributions to a Gaussian process with independent increments. Its drift and variance are the difference of the corresponding ones. Moreover, since {U n } n and {V n } n are tight, {U n -V n } n is tight.

6. Proofs of Theorems of Section 3.2 6.1. Proof of Theorem 3.9. Recall the notation Θ G n = n -1 log ∆ G,β n . As mentioned after the statement of the theorem, we are going to prove at first the LDP for the restriction of ΘG n to [0, T ], viewed as an element of M T , in the scale β ′-1 n -2 with rate function

I G [0,T ] (m) := T 0 L G a t, dm a dt (t) dt + T 0 L G s t, dm s dµ (t) dµ(t) . (6.1)
Let V ℓ be the set of functions from [0, T ] to R which are left continuous and have bounded variation, and let V * ℓ be its topological dual when V ℓ is equipped with the uniform convergence topology.

Actually ΘG n ∈ M T may be identified with an element of V * ℓ (see [START_REF] Léonard | Large deviations for Poisson random measures and processes with independent increments[END_REF] Appendix B): owing to (3.37) its action on ϕ ∈ V ℓ is given by

< ΘG n , ϕ >:= 1 n ⌊nT ⌋ k=1 ϕ(k/n) log ρ G,β n,k .
The proof of Theorem 3.9 is based on the ideas of Baldi theorem [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] p.157). The main tool is the normalized cumulant generated function (n.c.g.f.) which here takes the form

L G,β n,⌊nT ⌋ (ϕ) := 1 β ′ n 2 log E exp β ′ n 2 < ΘG n , ϕ > . (6.2)
Owing to (6.1) we have

L G,β n,⌊nT ⌋ (ϕ) = 1 β ′ n 2 ⌊nT ⌋ k=1 Λ G,β n,k (nϕ(k/n)) (6.3)
and from (5.2) it is finite iff ϕ(k/n) > -(nk + 1)/n for every 1 ≤ k ≤ ⌊nT ⌋.

In Subsection 6.1.1, we prove the convergence of this sequence of n.c.g.f. for a large class of functions ϕ. It will be sufficient, jointly to the variational formula given in Subsection 6.1.2 to get the upper bound for compact sets. Then Subsection 6.1.3 is devoted to exponential tightness, which allows to get the upper bound for closed sets. However, since the limiting n.c.g.f. is not defined everywhere, the lower bound (for open sets) is more delicate than in Baldi theorem. Actually a careful study of exposed points as in [START_REF] Gamboa | A functional large deviation principle for quadratic forms of Gaussian stationary processes[END_REF] is managed in Subsection 6.1.4. We end the proof in Subsection 6.1.5.

6.1.1. Convergence of the n.c.g.f. Let, for t ∈ [0, 1] and θ > -(1 -t) g G (t, θ) := J (1 -t + θ) -J (1 -t) -J (1 + θ) . (6.4) Lemma 6.1. If ϕ ∈ V ℓ satisfies ϕ(t) + 1 -t > 0 for every t ∈ (0, T ], then lim n L G,β n,⌊nT ⌋ (ϕ) = Λ G [0,T ] (ϕ) := T 0 g G (t, ϕ(t)) dt . (6.5) Proof:
The key point is a convergence of Riemann sums. From (5.2) and (8.1) we have, for every θ >

-n-k+1 n , Λ G,β n,k (nθ) = β ′ (n -k + nθ) log 1 - k n + θ + 1 n -β ′ (n -k) log 1 - k n + 1 n -β ′ (n -1 + nθ) log(1 + θ) + R n,k (θ)
where the quantity

R n,k (θ) = ∞ 0 f (s)e -β ′ s e -β ′ (n-k+nθ)s -e -β ′ (n-k)s ds - ∞ 0 f (s) e -β ′ (n-1+nθ)s -e -β ′ s e -β ′ (n-1)s ds is bounded by 2 ∞ 0 e -β ′ s f (s) ds. If we set Φ n (t) := (1 -t + ϕ(t)) log 1 -t + ϕ(t) + 1 n -(1 -t) log(1 -t + 1 n ) -1 - 1 n + ϕ(t) log(1 + ϕ(t))
then, making θ = ϕ(k/n), and adding in k, we get from (6.3)

1 β ′ n 2 L G n,⌊nt⌋ (ϕ) - ⌊nt⌋ 2 R n,k (ϕ(k/n)) = 1 n ⌊nt⌋ 1 Φ n k n = = ⌊nt⌋/n 1/n Φ n ⌊ns⌋ n ds + 1 n Φ n ⌊nt⌋ n .
On the one hand, since ϕ is left continuous, lim n Φ n ⌊nt⌋ n = g(t, ϕ(t)) for every t ∈ [0, T ]. On the other hand the following double inequality holds true:

Φ n (t) ≥ (1 -t + ϕ(t)) log (1 -t + ϕ(t)) -(1 -t) log(2 -t) -(1 + ϕ(t)) log (1 + ϕ(t)) -| log (1 -t + ϕ(t)) | Φ n (t) ≤ (1 -t + ϕ(t)) log (2 -t + ϕ(t)) -(1 -t) log(1 -t) -(1 + ϕ(t)) log (1 + ϕ(t)) + | log (1 -t + ϕ(t)) | ,
and with our assumptions on ϕ, these bounds are both integrable. This allows to apply the dominated convergence theorem which ends the proof of Lemma 6.7.

If there exists s < T such that ϕ(s) < -(1s) then for n large enough, L n,⌊nT ⌋ (ϕ) = +∞ and we set Λ G [0,T ] (ϕ) = ∞. In the other cases we do not know what happens, but as in [START_REF] Gamboa | A functional large deviation principle for quadratic forms of Gaussian stationary processes[END_REF], we will study the exposed points. Before, we need another expression of the dual of Λ G [0,T ] .

6.1.2. Variational formula. Let us define Λ G [0,T ] (ϕ) = +∞ if ϕ does not satisfy the assumption of Lemma 6.1. The dual of Λ G [0,T ] is then Λ G [0,T ] ⋆ (ν) = sup ϕ∈V ℓ < ν, ϕ > -Λ G [0,T ] (ϕ) (6.6) for ν ∈ V * ℓ .
Mimicking the method of [START_REF] Léonard | Large deviations for Poisson random measures and processes with independent increments[END_REF] p. 112-113, we get

Λ G [0,T ] ⋆ (ν) = sup ϕ∈C < ν, ϕ > -Λ G [0,T ] (ϕ) (6.7)
where C is the set of continuous functions from [0, T ] into R vanishing at 0. Then we apply Theorem 5 of [START_REF] Rockafellar | Integrals which are convex functionals[END_REF] and get

Λ G [0,T ] ⋆ (ν) = T 0 g ⋆ t, dν a dt dt + T 0 r t, dν s dµ (t) dµ(t)
where

g ⋆ (t, y) = sup λ λy -g G (t, λ)δ(λ|(-1, ∞)) , (6.8)
and r is the recession function :

r(t, y) = lim κ→∞ g ⋆ (t, κy) κ .
Actually, if y < 0, the supremum is achieved for

λ G (t, y) := -1 - t 1 -e y (6.9)
and we have

g ⋆ (t, y) = λ G (t, y)y -g G t, λ G (t, y) = -y(1 -t) + (1 -t) log(1 -t) + t log t -t log(1 -e y ) = H (1 -t|e y ) . (6.10) If y ≥ 0, g ⋆ (t, y) = ∞. The recession is now r(t, y) = -(1 -t)y if y ≤ 0, and = ∞ si y > 0. As a result g ⋆ (t, y) = L G a (t, y) , r(t, y) = L G s (t, y) . (6.11) So we proved the identification Λ G [0,T ] ⋆ = I G [0,T ]
(recall (6.1)).

6.1.3. Exponential tightness. In this paragraph and in Section 6.2 we use the function defined for θ > -(1 -T ) by

L G T (θ) := T 0 g G (t, θ) dt . (6.12)
If V * ℓ is equipped with the topology σ(V * ℓ , V ℓ ), the set

B a := {µ ∈ V * ℓ : |µ| [0,T ] ≤ a}
is compact according to the Banach-Alaoglu theorem. Now -ΘG n is a positive measure and its total mass is -ΘG n ([0, T ]) = -Ξ n (T ). We have then

P ΘG n / ∈ B a = P Θ G n (T ) < -a . Now for θ < 0 P Θ G n (T ) < -a ≤ e β ′ θn 2 a E exp{n 2 β ′ θΘ G n (T )}
so that, taking logarithm and applying Lemma 6.1 we get, for θ

∈ (-(1 -T ), 0) lim sup n 1 β ′ n 2 log P Θ G n (T ) < -a ≤ θa + L G T (θ) .
It remains to let a → ∞ and we have proved the exponential tightness.

Let us note that the restriction T < 1 is crucial in the above proof.

6.1.4. Exposed points. Let R be the set of functions from [0, T ] into R which are positive, continuous and with bounded variation. Let F be the set of those m ∈ V * ℓ (identified with M T as in [START_REF] Léonard | Large deviations for Poisson random measures and processes with independent increments[END_REF]) which are absolutely continuous and whose density ρ is such that -ρ ∈ R. Let us prove that such a m is exposed, with exposing hyperplane f m (t) = λ(t, ρ(t)) (recall (6.9)). Actually we follow the method of [START_REF] Gamboa | A functional large deviation principle for quadratic forms of Gaussian stationary processes[END_REF]. For fixed t, g ⋆ (t, .) is strictly convex on (-∞, 0) so that, if z = ρ(t), we have

g ⋆ (t, ρ(t)) -g ⋆ (t, z) < λ(t, ρ(t))(ρ(t) -z) . Let dξ = l(t)dt + ξ ⊥ the Lebesgue decomposition of some element ξ ∈ M T such that I G [0,T ] (ξ) < ∞.
Taking z = l(t) and integrating, we get

T 0 g ⋆ (t, ρ(t)dt - T 0 g ⋆ (t, l(t))dt < T 0 λ(t, ρ(t))ρ(t)dt - T 0 λ(t, ρ(t)) l(t) dt and since T 0 g ⋆ (t, l(t))dt = T 0 L G a (t, l(t))dt ≤ I G [0,T ] (ξ) this yields I G [0,T ] (m) -I G [0,T ] (ξ) < T 0 f m dm - T 0 f m dξ .
The following lemma says that this set of exposed points is rich enough.

Lemma 6.2. Let m ∈ V r such that I G [0,T ] (m) < ∞.
There exists a sequence of functions l n ∈ R such that

(1) lim n l n (t)dt = -m in V * ℓ with the σ(V * ℓ , V ℓ ) topology, (2) lim n I G [0,T ] (-l n (t)dt) = I G [0,T ] (m) .
Proof: The method may be found in [START_REF] Gamboa | A functional large deviation principle for quadratic forms of Gaussian stationary processes[END_REF] and in [START_REF] Dette | Asymptotic properties of the algebraic moment range process[END_REF]. The only difference is in the topology because we want to recover marginals. We will use the basic inequality which holds for every ǫ ≤ 0 : From (3.38) and (3.36) it is clear that -m a and -m s must be positive measures.

L G a (t, v + ǫ) ≤ L G a (t, v) -ǫ(1 -t) (6.13) Let m = m a + m s such that I G [0,T ] (m) < ∞.
First step We assume that m = -l(t)dtη with l ∈ L 1 ([0, T ]; dt) and η a singular positive measure. One can find a sequence of non negative continuous functions h n such that h n (t)dt → η for the topology σ(V * ℓ , V ℓ ). Indeed every function ψ ∈ V ℓ may be written as a difference ψ 1ψ 2 of two increasing functions. There exists a unique (positive) measure α 1 such that ψ 1 (t) = α 1 ([t, T ]) for every t ∈ [0, T ]. Moreover, the function g = η([0, •]) ∈ V r is non decreasing and may be approached by a sequence of continuously derivable and non decreasing functions (g n ) such that g n ≤ g. Setting h n := g ′ n and ν n = h n (t)dt, the dominated convergence theorem gives

ψ 1 , ν n = T 0 ν n ([0, t])α 1 (dt) → T 0 η([0, t])α 1 (dt) .
With the same result for ψ 2 we get

ψ, ν n = T 0 ν n ([0, t])α 1 (dt) - T 0 ν n ([0, t])α 2 (dt) → T 0 η([0, t])α 1 (dt) - T 0 η([0, t])α 2 (dt) .
or lim n ψ, ν n = ψ, η . On the one hand, the lower semi-continuity of

I G [0,T ] yields lim inf n I G [0,T ] (-(l(t) + h n (t))dt) ≥ I G [0,T ] (m) .
On the other hand, integrating (6.13) yields

I G [0,T ] (-(l(t) + h n (t))dt) ≤ T 0 L G a (t, -l(t))dt + T 0 (1 -t)h n (t)dt → T 0 L G a (t, -l(t))dt + T 0 (1 -t)η(dt) = I G [0,T ] (m) .
Second step Let us assume that m = -l(t)dt with l ∈ L 1 ([0, T ]; dt) and for every n, let us set l n = max(l, 1/n). It is clear that as n → ∞, then l n ↓ l. On the one hand the lower semi-continuity gives

lim inf A I G [0,T ] (-l n (t)dt) ≥ I G [0,T ] (-l(t)dt) .
On the other hand, by integration of inequality (6.13), since l nl ≤ 1/n

I G [0,T ] (-l n (t)dt) ≤ I G [0,T ] (-l(t)dt) + 1 n .
It is then possible to reduce the problem to the case of functions bounded below.

Third step Let us assume that m = -l(t)dt with l ∈ L 1 ([0, T ]; dt) and bounded below by A > 0. One can find a sequence of continuous functions (h n ) with bounded variation such that h n ≥ A/2 for every n and such that h n → l a.e. and in L 1 ([0, T ], dt). We have h n (t)dt → l(t)dt in σ(V * ℓ , V ℓ ) and since L G a (t, •) is uniformly Lipschitz on (-∞, -A/2], say with constant κ, we get

| I G [0,T ] (-h n (t)dt) -I G [0,T ] (-l(t)dt)| ≤ κ T 0 |h n (t) -l(t)|dt → 0 .
Actually, h n ∈ R and ϕ n (t) := λ(t, -h n (t)) satisfies the assumption of Lemma 6.1 since

1 + ϕ n (t) -t ≥ t 1 -e -A/2 .
6.1.5. End of the proof of Theorem 3.9. The first step is the upper bound for compact sets. We use Theorem 4.5.3 b) in [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] and the following lemma.

Lemma 6.3. For every δ > 0 and m ∈ V * ℓ , there exists ϕ δ fulfilling the conditions of Lemma 6.1 and such that

T 0 ϕ δ dm -Λ G T (ϕ δ ) ≥ min I G [0,T ] (m) -δ, δ -1 . (6.14)
The second step is the upper bound for closed sets : we use the exponential tightness. The third step is the lower bound for open sets. The method is classical (see [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] Theorem 4.5.20 c)), owing to Lemma 6.2.

To prove Lemma 6.3, we start from the definition (6.6) or (6.7). One can find φδ ∈ V ℓ satisfying (6.14). If φδ does not check assumptions of the lemma we add ε > 0 to φδ which allows to check them and satisfy (6.14) up to a change of δ.

I G T (ξ) = inf{I G [0,T ] (v) ; v(T ) = ξ} . Since the process Θ G
n takes its values in (-∞, 0] (remember Hadamard inequality), it is clear that I G T (ξ) = ∞ for ξ > 0. Fixing ξ < 0, we can look for optimal v. Let θ > -(1 -T ) (playing the role of a Lagrange multiplier).

By the duality property (6.8)

g ⋆ t, d va dt (t) ≥ θ d va dt (t) -g G (t, θ) .
Integrating and using (6.1), (6.11) and (6.5) we get

I G [0,T ] (v) ≥ θ va ([0, T ]) -L G T (θ) - T 0 (1 -t) d vs (t) , (6.15)
For every v such that v(T ) = ξ it turns out that

I G [0,T ] (v) ≥ θξ -L G T (θ) - T 0 (1 -t + θ) d vs (t) ≥ θξ -L G T (θ) . (6.16)
Besides, from (6.9) the ordinary differential equation

λ G (t, φ ′ (t)) = θ φ(0) = 0 , admits for unique solution in C 1 ([0, T ]) t → φ(θ; t) := J (1 + θ) -J (1 -t + θ) -t log(1 + θ) . Now, since ∂ ∂θ φ(θ; T ) = -log 1 - T 1 + θ + T 1 + θ > 0 we see that the mapping θ → φ(θ; T ) is bijective from [-(1 -T ), ∞) onto [-T, 0).
Moreover, by duality

g ⋆ t, ∂ ∂t φ(θ, t) = θ ∂ ∂t φ(θ, t) -g G (t, θ) .
There are two cases.

• If ξ ∈ [-T, 0), there exists a unique θ ξ such that φ(θ ξ , T ) = ξ (i.e. the relation (3.41) is satisfied). For v ξ := φ(θ ξ , •), we get from (6.1), (6.11) and (6.12) again Let us note that at the end point ξ = -T , we have

I G [0,T ] (v ξ ) = θ ξ ξ -L G T (θ ξ ) so that v ξ
θ ξ = -(1 -T ) , v ξ (t) = J (T ) -J (T -t) -t log T , (v ξ ) ′ (t) = log(1 -t/T ) . Finally I G T (-T ) = 2T (1 -T ) + F (1) -F (1 -T ) -F (T ) + T 2 log T = T (1 -T ) 2 + T 2 log T 2 - (1 -T ) 2 log(1 -T ) 2 + 3 4 . • Let us assume ξ = -T -ε with ε > 0. Plugging θ = -(1 -T ) in (6.16) yields, for every v such that v(T ) = ξ I G [0,T ] (v) ≥ -(1 -T )ξ -L G T (-(1 -T )) = (1 -T )ε + I G T (-T ) ,
and this lower bound is achieved by the measure v = (v -T ) ′ (t)dtεδ T (t), since

T 0 L G a t, (v -T ) ′ (t) dt = I G T (T ) , T 0 (1 -t) ε dδ T (t) = (1 -T ) ε .
It remains to look at ξ = 0. Taking ξ = 0 in (6.16), we get

I G T (0) ≥ -L G T (θ)
for every θ ≥ -(1 -T ). Now, from (6.4) and (6.12) we may write

-L G T (θ) = T 0 (1 -t) log(1 -t) dt + T 0 (1 + θ) log 1 - t 1 + θ dt + T 0 t log(1 -t + θ) dt .
When θ tends to infinity, the second term tends to zero and the third, which is bounded above by (T 2 /2) log(1 -T + θ) tends to infinity. Finally I G T (0) = ∞. That ends the proof of the second statement of Theorem 3.10. Remark 6.4. It is possible to try a direct method to get (3.42), (3.43) using Gärtner-Ellis' theorem (Dembo and Zeitouni (1998), Theorem 2.3.6). From Lemma 6.1 the limiting n.c.g.f. of Θ

G n (T ) is L G T which is analytic on (-(1 -T ), ∞). When θ ↓ -(1 -T ), we have (L G T ) ′ (θ) ↓ -T .
We meet a case of so called non steepness. To proceed in that direction we could use the method of time dependent change of probability (see [START_REF] Dembo | Large deviations via parameter dependent change of measure, and an application to the lower tail of Gaussian processes[END_REF]). We will not give details here. Nevertheless, this approach allows to get one-sided large deviations in the critical case T = 1. Actually we get

lim n 1 β ′ n 2 log P(Θ G n (1) ≥ nx) = -I G 1 (x)
for x ≥ -1. The value x = -1 corresponds to the limit of Θ G n (1). note that the second (right) derivative of I G 1 at this point is zero (or equivalently lim(L G 1 ) ′′ (θ) = ∞ as θ ↓ 0) , which is consistent with previous results on variance. I do not know the rate of convergence to 0 of P(Θ G n (1) ≤ nx) for x < -1.

6.3. Proof of Theorem 3.11 and Theorem 3.12. Again, the three routes are possible to tackle the problem of large deviations for determinant of Wishart matrices. A direct method would use the cumulant generating function from (5.13) and would meet computations similar to those seen in the Uniform Gram case.

To avoid repetitions, we use the decomposition (2.26), drawing benefit from an auxiliary study of S n,r .

Lemma 6.5. The sequence {n -1 S n (t), t ∈ [0, 1)} n satisfies a LDP in the space (D, σ(D, M < )) in the scale 2β -1 n -2 with good rate function

I S [0,1) (v) = [0,1) L S a d va dt (t) dt + [0,1) L S s d vs dµ (t) dµ(t) (6.17)
where L S a (y) = (e yy -1) , L S s (y) = -yδ(y|(-∞, 0)) , (6.18)

and µ is any measure dominating vs .

Let us stress that the instantaneous rate functions are time homogeneous and then we may write [0, 1] instead of [0, 1). 6.3.1. Proof of Lemma 6.5. It is a route similar to the proof of Theorem 3.9 in Section 6.1 (see also [START_REF] Najim | A Cramér type theorem for weighted random variables[END_REF]). We start from (2.27) so that

1 n Ṡn = n j=1 log ε (n) k δ j/n .
Withe help of (5.13) this yields :

log E exp < β ′ n Ṡn , γ >= n k=1 ℓ β ′ n 1 + γ k n -β ′ γ k n log(β ′ n) -nℓ(β ′ n) if γ(s) + 1 > 0 for every s ∈ [0, 1].
A little computation shows that the limiting n.c.g.f. is

L S (γ) = 1 0 J (1 + γ(t))dt , (6.19)
which yields (6.18) by duality (see [START_REF] Rockafellar | Integrals which are convex functionals[END_REF] again).

6.3.2. Proof of Theorem 3.11. Let Θ L n = n -1 log ∆ L,β n . We deduce from Lemma 6.5 and Theorem 3.9 that the sum ΘL n = ΘG n + 1 n Ṡn satisfies a LDP in the same scale with good rate function I G [0,T ] I S [0,T ] where denotes the infimum convolution :

(f g)(x) = inf{f (x 1 ) + g(x 2 ) | x 1 + x 2 = x} .
The two characteristics of the rate function are then

L L a = inf v {L G a (v) + L S a (u -v)} L L s = inf v {L G s (v) + L S s (u -v)} .
which yield (3.44) by an explicit computation.

Alternatively, it is possible to sum the two n.c.g.f. ((6.5) and (6.19)) and get the rate function by duality. We claim : if γ(s) + 1 > 0 for every s ∈ [0, 1] (6.20) where

1 β ′ n 2 log E exp β ′ n 2 ΘL n , γ → T 0 g L (t, γ(t)) dt ,
g L (t, γ) = g G (t, γ) + J (1 + γ) = J (1 -t + γ) -J (1 -t) .
(6.21) 6.3.3. Proof of Theorem 3.12. We may either use the contraction Θ L n → Θ L n (T ) or establish a LDP for the marginal S n (T ) and then perform an inf-convolution. We leave the details of the proof to the reader. We just give the expression of the optimal path when it exists.

For θ > -(1 -T ), the function ∞), where ξ T = J (T ) -1. Fixing ξ ≥ ξ T , we can look for optimal v. There exists a unique θ ξ such that φ(θ ξ , T ) = ξ. Then v ξ := φ(θ ξ , •) is the optimal path (v ξ realizes the infimum in (3.45). Let us note that at the end point ξ = ξ T , we have

t → φ(θ; t) := J (1 + θ) -J (1 -t + θ) . is in C 1 ([0, T ]) and the mapping θ → φ(θ; T ) is bijective from [-(1 -T ), ∞) onto [ξ T ,
θ ξ = -(1 -T ) , v ξ (t) = J (T ) -J (T -t) , (v ξ ) ′ (t) = log(T -t) .
Remark 6.6. It is possible to get (3.47), (3.48) using Gärtner-Ellis' theorem [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], Theorem 2.3.6). We are in the same situation as in Remark 6.4. This approach allows to get one-sided large deviations in the critical case T = 1. Actually we get lim n 2 βn 2 log P(log ∆ L,β n (1) ≥ nx) = -I L 1 (x) for x ≥ -1. The value x = -1 corresponds to the limit. Note that the second (right) derivative of I L 1 at this point is zero (or equivalently lim(L L 1 ) ′′ (θ) = ∞ as θ ↓ 0), which is consistent with previous results on variance. We do not know the rate of convergence to 0 of P(log ∆ L,β n (1) ≤ nx) for x < -1.

6.4. Proof of Theorem 3.13 and Theorem 3.14. We may try again to use the betagamma algebra, but we do not succeed to go until the end. Let as in Subsection 5.5, U n and V n be the two Laguerre variables. From the exponential tightness of U n and V n , we deduce easily the exponential tightness of W n . From Puhalskii (2001), the sequence W n contains subsequences satisfying LDP. If for such a subsequence we call I p the rate function, the independence gives

I U = I V I p
This equation has many solutions and only one convex solution, which is [START_REF] Mazure | Equations inf-convolutives et conjugaison de Moreau-Fenchel[END_REF]). But we do not know a priori that I p is convex.

I p = I U ⊟ I V defined by (f ⊟ g)(x) = sup{f (x 1 ) -g(x 2 ) | x 1 -x 2 = x} ( Mazure
We choose to use the beta-gamma trick to study the n.c.g.f. For the remaining we do not give details since it is similar to the above cases and again based on the ideas of Baldi theorem [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]) and a variational formula. 6.4.1. Convergence of the n.c.g.f. Put Θ J n = n -1 log ∆ J,β n so that

ΘJ n = 1 n n1 k=1 log ρ J,β j,n δ j/n ,
and put for T ≤ τ 1 and ϕ ∈ V T ℓ : (6.22) where, for θ + τ 1s > 0

L J n,⌊nT ⌋ (ϕ) = 2 βn 2 log E exp n ΘJ n , ϕ . Lemma 6.7. If ϕ ∈ V τ1 ℓ satisfies ϕ(s) + τ 1 -s > 0 for every s ∈ (0, T ], then lim n L J n,⌊nT ⌋ (ϕ) = Λ J [0,T ] (ϕ) := T 0 g J (s, ϕ(s)) ds ,
g J (s, θ) = E (τ 1 -s + θ, τ 2 , θ) . (6.23) Proof: From (2.28) we have n ΘJ n , γ + (n 1 + n 2 ) ΘL n1+n2 , γ((n 1 + n 2 ) • /n) = n 1 ΘL n1 , γ(n 1 • /n) + log n 1 n 1 + n 2 ⌊nT ⌋ k=1 γ(k/n) and then, by independence, log E exp β ′ n 2 ΘJ n , γ = log E exp β ′ nn 1 ΘL n1 , γ(n 1 • /n) -log E exp β ′ n(n 1 + n 2 ) ΘL n1+n2 , γ((n 1 + n 2 ) • /n) +n log n 1 n 1 + n 2 ⌊nT ⌋ k=1 γ(k/n) .
By a slight modification of (6.20) we have, for p/n → τ (6.24) so that taking τ = τ 1 and τ = τ 1 + τ 2 and subtracting, we get

1 β ′ p 2 log E exp β ′ np ΘL r , γ(p • /n) → 1 τ T 0 g L s τ , γ (s) τ ds ,
1 β ′ n 2 log E exp β ′ n 2 ΘJ n , ϕ → T 0 g J (s, γ(s)) ,
where

g J (s, θ) = τ 1 g L s τ 1 , θ τ 1 -(τ 1 + τ 2 )g L s τ 1 + τ 2 , θ (τ 1 + τ 2 ) +θ log τ 1 τ 1 + τ 2 ,
and this is equivalent to (6.23).

6.4.2. Duality. Let us define Λ J [0,T ] (ϕ) = +∞ if ϕ does not satisfy the assumption of Lemma 6.7. The dual of Λ J [0,T ] is then

Λ J [0,T ] ⋆ (ν) = sup ϕ∈V ℓ < ν, ϕ > -Λ J [0,T ] (ϕ) (6.25) for ν ∈ V * ℓ .
Mimicking the method of [START_REF] Léonard | Large deviations for Poisson random measures and processes with independent increments[END_REF] p. 112-113, we get

Λ J [0,T ] ⋆ (ν) = sup ϕ∈C < ν, ϕ > -Λ J [0,T ] (ϕ) (6.26)
where C is the set of continuous functions from [0, T ] into R vanishing at 0. Then we apply Theorem 5 of [START_REF] Rockafellar | Integrals which are convex functionals[END_REF]. We get

Λ J [0,T ] ⋆ (ν) = T 0 g J ⋆ t, dν a dt dt + T 0 r J t, dν s dµ (t) dµ(t) (6.27)
where

g J ⋆ (s, y) = sup λ λy -g J (s, λ)δ(λ|(-τ 1 , ∞)) . (6.28)
This supremum is achieved by

λ J (s, y) = -(τ 1 -s) + τ 2 e -y -1 (6.29)
and we have

g J ⋆ (s, y) = λ J (s, y)y -g J (s, λ J (s, y)) (6.30) = (τ 1 + τ 2 -s) H τ 1 -s τ 1 + τ 2 -s e y . (6.31)
The recession is r J (s; y) = -(τ 1s)y if y < 0.

6.4.3. Proof of Theorem 3.10. We use the contraction from the LDP for paths.

Since the mapping m → m([0, T ]) is continuous from D to R, the family {Θ J n (T )} n satisfies the LDP with good rate function given by (3.50). Since the process Θ n takes its values in (-∞, 0] , it is clear that I J T (ξ) = ∞ for ξ > 0. Fixing ξ < 0, we can look for optimal v, i.e. a path (v(t), t ∈ [0, T ]) such that v(T ) = ξ and v achieves the infimum in (3.50). Fix θ ≥ Tτ 1 (playing the role of a Lagrange multiplier). In view of (6.27), (6.28) and (6.29), it is clear that (in the generic case) the Euler-Lagrange equation is

λ J (s, φ ′ (s)) = θ φ(0) = 0 .
This ordinary differential equation admits for unique solution in C 1 ([0, T ])

s → φ J (θ; s) := E(θ + τ 1 , τ 2 , s) .
To know if the path φ J may have ξ as its terminal value, look at

E ′ (θ + τ 1 , τ 2 , T ) = ∂ ∂τ E(τ, τ 2 , T )| τ =θ+τ1 = log 1 - T θ + τ 1 + τ 2 -log 1 - T θ + τ 1 ;
since it is positive, we see that the mapping θ -→ E(θ + τ 1 , τ 2 , T ) is continuous and increasing from [Tτ 1 , ∞) onto D T = [ξ J T , 0). If ξ ∈ [ξ J T , 0), we call θ ξ the unique solution of φ J (θ, T ) = ξ or in other words, E(θ ξ + τ 1 , τ 2 , T ) = ξ , and we set v ξ := φ J (θ ξ , •).

To end the proof, let us now consider some inequalities. The duality property (6.28) gives, for every v and t g For every v such that v(T ) = ξ it turns out that

I J [0,T ] (v) ≥ θξ -L J T (θ) - T 0
(τ 1 -T + θ) d vs (t) ≥ θξ -L J T (θ) . (6.34)

There are three cases.

• If ξ ∈ [ξ J T , 0) , we get I J [0,T ] (v ξ ) = θ ξ ξ -L J T (θ ξ ) so that v ξ realizes the infimum in (3.50). A simple computation leads to (3.42) which ends the proof of the first statement of Theorem 3.10.

Let us note that at the end point ξ = ξ J T , we have (τ 1t)εdδ T (t) = (τ 1 -T )ε .

• It remains to look at ξ = 0. Taking ξ = 0 in (6.34), we get I J [0,T ] (0) ≥ -L J T (θ) for every θ ≥ Tτ 1 . Now, from (6.23) and (6.33), we may write (after some calculation) -L J T (θ) = We see from (7.1) that if θ > -1 we can take: and transform π c σ 2 to π c 1 using the dilatation. In particular, (4.9) yields Σ(π c σ 2 ) = log σ 2 + Σ(π c 1 ) and Σ(π c 1 ) may be picked from formula (13) p.10 in [START_REF] Hiai | Eigenvalue density of the Wishart matrix and large deviations[END_REF] :

σ 2 = 1 + θ, c = T σ 2 = T 1 + θ .
Σ(π c 1 ) = -1 + 1 2 c -1 + log c + (c -1 -1) 2 log(1 -c) .
Besides we have easily x dπ c 1 (x) = 1. After some tedious but elementary computations we get exactly the RHS of (3.47), which yields I spL T (π c σ 2 ) = I L T (ξ) , and ends the proof of (4.12). 7.2. Proof of proposition 4.6. Let θ < tτ 1 (Lagrangian multiplier). We begin by minimizing We use the following lemma.

I
Lemma 7.1. For ζ 1 , ζ 2 > 0, the infimum of -Σ(µ) -2ζ 1 log x dµ(x) -2ζ 2 log(1x) dµ(x)

among the probability measures µ on [0, 1] is achieved by π a-,a+ where (a -, a + ) = λ ± (s -, s + )

with s -= 1 + 2ζ 1 2(1 + ζ 1 + ζ 2 ) , s + = 1 + 2ζ 1 + 2ζ 2 2(1 + ζ 1 + ζ 2 ) .
The infimum in (7.2) is achieved by π ξ,η , where ( ξ, η) = λ ± (s -, s+ ) , s-=

τ 1 + θ τ 1 + τ 2 + θ , s+ = τ 1 + θ + τ 2 -t τ 1 + τ 2 + θ .
It should be clear that Σ(π ξ,η ) + τ 1 + θt t log x dπ ξ,η (x) + τ 2t t log(1x) dπ ξ,η (x) = B τ 1 + θt t , τ 2t t and then, on A(ξT -1 ) the infimum is uniquely realized in π ξ,η and its value is

θξ + t 2 B τ 1 -t t , τ 2 -t t -B τ 1 + θ -t t , τ 2 -t t .
Finally a small computation leads to (3.52) and (4.31).

Proof of Lemma 7.1 In [START_REF] Saff | Logarithmic potentials with external fields[END_REF]

θ i = ζ i 1 + ζ 1 + ζ 2 , i = 1, 2 , ∆ = 1 -(θ 1 + θ 2 ) 2 1 -(θ 1 -θ 2 ) 2 .
and K(b -, b + ) is a normalizing constant. With the push-forward by the function x → (x + 1)/2, we get the result.

8. Appendix 1 : Some properties of ℓ = log Γ and Ψ From the Binet formula [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] where the function f is defined by As easy consequences, we have, for every x > 0 0 < x x -Ψ(x)) ≤ 1 , (8.6)

f (s) = 1 2 - 1 s + 1 e s -1 1 s = 2 ∞ k=1 1 s 2 + 4π 2 k 2 , ( 8
0 < log x -Ψ(x) - 1 2x ≤ 1 12x 2 .
(8.7)

Differentiating again we see that for q ≥ 1 Ψ (q) (z) = (-1) q-1 q!z -q + (-1) q-1 ∞ 0 e -sz s q sf (s) + 1 2 ds (8.8) and then |Ψ (q) (z) -(-1) q-1 q!z -q | ≤ z -q-1 q! . (8.9)

Appendix 2 : Identification of the McKay distribution

The reader is recalled that, for u ′ and v ′ positive numbers 4 such that u ′ + v ′ > 1, [START_REF] Capitaine | Asymptotic freeness by generalized moments for Gaussian and Wishart matrices. Application to beta random matrices[END_REF] defined the probability measure

CC u ′ ,v ′ := (1 -u ′ ) + δ 0 + (1 -v ′ ) + δ 1 + 1 -(1 -u ′ ) + -(1 -v ′ ) + π a-,a+ , where (a -, a + ) = a ± u ′ u ′ + v ′ , 1 - 1 u ′ + v ′ .
We present now three identifications of this distribution connected with free probability.

For k = 0, let D k the dilatation operator by factor k. For p ≤ 1, let b p denote the Bernoulli distribution of parameter p. At last, let ⊞ (resp. ⊠) denote the additive (resp. multiplicative) free convolution.

1) Rewriting the distribution with the notation of [START_REF] Demni | Free Jacobi processes[END_REF], we find four cases 3) Finally, we quote the correspondence with the results of [START_REF] Collins | Products of random projections, Jacobi ensembles and universality problems arising from free probability[END_REF] who claimed that for 0 < p -< p + < 1 b p-⊠ b p+ = (1p -)δ 0 + (p -+ p + -1) + δ 1 + C -1 a-,a+ π a-,a+ where a ± = a ± (1p -, p + ). In [START_REF] Hiai | Large deviations for functions of two random projection matrices[END_REF], formula (2.8) the authors consider the same distribution.

• Situation II : p -+ p + -1 < 0, σ -= p + , σ

+ = 1 -p - b p-⊠ b p+ = σ + δ 0 + (1 -σ + )π a-,a+ = σ -CC u ′ ,v ′ + (1 -σ -)δ 0 • Situation IV : p -+ p + -1 > 0, σ -= 1 -p -, σ + = p + b p-⊠ b p+ = σ -δ 0 + (σ + -σ -)δ 1 + (1 -σ + )π a-,a+ = σ + CC u ′ ,v ′ + (1 -σ + )δ 0 .

  , recent developments in Random Matrix Theory add a new interest to the study of their asymptotical behavior and invite to a new insight. Let B = [b 1 , . . . , b r ] be the n × r matrix with r column vectors b 1 , . . . , b r of R n . If B ′ denotes its transpose, the determinant of the r × r Gram matrix B ′ B satisfies the well known Hadamard inequality : det B ′ B ≤ b 1 and only if b 1 , . . . , b r are orthogonal (Hadamard, 1893). It means that the volume (or r-content) of the parallelotope built from b 1 , . . . , b r is maximal when the vectors are orthogonal. The quantity h(B) = det B ′ B b 1 2 • • • b r 2 is usually called the Hadamard ratio. If we replace sequentially b i by its projection b i on the orthogonal of the subspace spanned by b 1 , . . . , b i-1 (Gram-Schmidt orthogonalization), we have det B ′ B =

T

  For ξ ≥ ξ T and θ solution of (3.46), let σ 2 = 1 + θ. Then the infimum of I spL T (µ) over A(ξ/T ) is uniquely achieved for π

6. 2 .

 2 Proof of Theorem 3.10. We use the contraction from the LDP for paths. Since the mapping m → m([0, T ]) is continuous from D to R, the family Θ G n (T ) satisfies the LDP with good rate function specified by (3.40):

  realizes the infimum in (3.40). A simple computation ends the proof of the first statement of Theorem 3.10.

  32) and using (3.35), (3.49) and (6.33) we getI J [0,T ] (v) ≥ θ va ([0, T ]) -L J T (θ) -T 0 (τ 1t) d vs (t) .

θ•

  ξ = (Tτ 1 ) , v ξ (t) = E(T, τ 2 , t) , (v ξ ) ′ (t) = log Tt τ 1 + τ 2t . Let us assume ξ = ξ J Tε with ε > 0. Plugging θ = Tτ 1 in (6.34) yields, for every v such that v(T ) = ξ I J [0,T ] (v) ≥ (Tτ 1 )ξ J T -L J T (Tτ 1 )ε(Tτ 1 ) = I J T (ξ J T )ε(Tτ 1 ) ,and this lower bound is achieved by the measure ṽ = v ξ J T ′ (t)dtεdδ T (t), since

  infinity as θ → ∞. We conclude I J [0,T ] (0) = ∞.7. Proofs of Theorems of Section 47.1. Proof of Proposition 4.1. Let θ ∈ R be a Lagrangian factor. We begin by minimizingI spL T (µ) -θT log x dµ(x) = T 2 -Σ(µ) + 2 q λ,s (x) dµ(x) + 2B(T )where q λ,s (x) = λxs log x, λ of[START_REF] Saff | Logarithmic potentials with external fields[END_REF] p.43 example 5.4, it is stated that for λ > 0 and 2s + 1 > 0 fixed, the infimuminf µ -Σ(µ) + 2 q λ,s (x) dµ(x)is achieved by the unique extremal measure π c σ

  Now it remains to look for θ such that the constraint µ ∈ A(ξ/T ) is saturated. Since log x dπ c σ 2 (x) = log σ 2 + log x dπ c 1 (x)dx , and thanks to (4.6) we see that θ must satisfyξ = T log σ 2 -T J (1c) c = J (1 + θ) -J (1 -T + θ) ,which is exactly exactly (3.46). To compute I spL T (π c σ 2 ), we start from the definition (4.8):I spL T (π c σ 2 ) = -T 2 Σ(π c σ 2 ) + T (x -(1 -T ) log x) dπ c σ 2 (x) + 2B(T ) ,

  p.241, it is proved that the infimum oflog |x -y|dµ(x)dµ(y) -2ζ 1 log(1x) dµ(x) -2ζ 2 log(1 + x) dµ(x)among the probability measures µ on [-1, +1] is achieved bydµ(y) = K(b -, b + ) (yb -)(b +y) 2π(1y 2 ) 1 [b-,b+] (y) dy ,

  pp. 258-259 or Erdélyi et al. (1981) p.21), we have ℓ(x) = (x -

  .3) and satisfies for every s ≥ 00 < f (s) ≤ f (0) = 1/12 , 0 < sf (s

where K 1 β (resp. K

β ) was defined in (3.8) (resp. (3.11))

we set δ(y|A) = 0 if y ∈ A and = ∞ if y / ∈ A

Their β is our β ′ .

Acknowledgement

I thank Catherine Donati-Martin for introducing me to some references. I am also grateful to the anonymous referee for a careful reading of the manuscript.

(Λ G,β n,k ) ′′ (0) .

(5.10)

We already know, from (3.9) that

and from (8.9) (again), this term tends to 0. We just checked (5.7), which proves that the sequence of processes {δ n (t), t ∈ [0, 1)} n converges to a Gaussian centered process W with independent increments and the convenient variance. It is now straightforward to get equation (3.14).

2) When t = 1, most of the sums studied above explode when n tends to infinity and we need a renormalisation. In fact, for every n, the process (δ (5.12)

We start again with the decomposition (5.8). From the above estimate (5.9), the sum n k=1 (Λ G,β n,k ) (4) (0) is bounded. In (5.10), we have

which is equivalent to 2 log n (see (3.10)) and the supremum in (5.10) with p = n is bounded. This yields

which proves (5.12). Then n

[ntr]+1 η k,n converges in distribution to N (0, 1), and the same is true for the conditional distribution of δ n knowing δ n (t 1 ) = ε 1 , . . . , δ n (t r ) = ε r . Since

Dirac masses (at 0 and at 1)

2) There is a connection with the family of free Meixner law [START_REF] Bozejko | On a class of free Lévy laws related to a regression problem[END_REF], [START_REF] Bryc | Cauchy-Stieltjès kernel families[END_REF], [START_REF] Bryc | Approximation operators, exponential, q-exponential and free exponential families[END_REF]). Indeed, computing the mean m and variance V of the distribution

We see that fixing u ′ + v ′ = s -1 , we get V = s 2 m(1m), and then up to an affine transformation we find the "free binomial type law" as in [START_REF] Bryc | Approximation operators, exponential, q-exponential and free exponential families[END_REF] example vi p.18 or [START_REF] Bozejko | On a class of free Lévy laws related to a regression problem[END_REF] example 6 p.8. It could also be seen starting from the above formulae using dilatations and free convolutions and comparing with formula (7) page 6 in [START_REF] Bozejko | On a class of free Lévy laws related to a regression problem[END_REF].