MOMENTS OF THE HERMITIAN JACOBI PROCESS

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1 MOMENTS OF THE HERMITIAN JACOBI PROCESS LUC DELEAVAL AND NIZAR DEMNI Abstract. In this paper, we compute the expectation of (Jt n), n, where (Jt) t 0 is the hermitian Jacobi process of size m n. To proceed, we first derive the semi-group density of the eigenvalues process of (J t) t 0 as a bilinear series of symmetric Jacobi polynomials. Next, we use the expansion of the power sum in the Schur polynomial basis and the integral Cauchy-Binet formula in order to determine the partitions having non zero contributions after integration. These are hooks of weight less than n and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized Beta distribution. For special parameters on which the process (J t) t 0 depends, the last integral reduces to the Cauchy determinant leading to a more tractable formula. Finally, we determine the asymptotic behavior as m of all the terms involved in this formula.. Reminder and motivation Given three integers d, p, m such that p, m < d, the hermitian Jacobi process (J t ) t 0 of parameters (p, q := d p) was defined in [3, page 4] as the radial part of the m p upper-left corner of a d d unitary Brownian motion. Equivalently, if P m and Q p are two d d diagonal projections of ranks m and p respectively, then J t 0 d m := P m Y t Q p Yt P m, where (Y t ) t 0 is a (left or right) Brownian motion on the unitary group U(d) ([20]). With this matrix representation in hands and from the independence of the increments of the Lévy process (Y t ) t 0, it follows that if d = d(m) and p = p(m) depend on m such that p(m) () := θ ]0, [, d(m) m := η > 0, with ηθ ]0, [, p(m) exist, then the non commutative moments of (J t/d(m) ) t 0 with respect to the normalized trace converge as m ([9]). In particular, if E denotes the expectation of the probability space where (Y t ) t 0 is defined, then the following it (2) M n (t, η, θ) := m E ( tr[(j t/d(m) ) n ] ) exists for any n 0, t 0, and the sequence (M n (t, η, θ)) n 0 determines the spectral distribution of the so-called free Jacobi process ([9]). Furthermore, the it M n (, η, θ) := M n (t, η, θ), n 0, t is the moment sequence of the spectral distribution of the large m-it of P m UQ p(m) U P m, where U is a d(m) d(m) Haar unitary matrix. In other words, this iting distribution describes the spectrum of the large m-it of matrices drawn from the Jacobi unitary ensemble and its Lebesgue decomposition follows readily from freeness considerations ([4], [6], [9]). An explicit expression of M n (, η, θ) obtained from large m-asymptotics of the moments of the multivariate Beta distribution figures in [5, Theorem 4.4]. However, no general formula is known for M n (t, η, θ) up to our best knowledge, and the recent papers [] and [2] are attempts toward establishing such a formula for particular values of (η, θ). For instance, it was proved in [] that (3) M n (t,, /2) = 2 2n ( ) 2n + n 2 2n n k= ( ) 2n n k k L k (2kt)e kt, 200 Mathematics Subject Classification. 5B52, 33C45, 60H5. Key words and phrases. Hermitian Jacobi process, Schur polynomial, symmetric Jacobi polynomial, hook.

2 where L k is the k-th Laguerre polynomial of index ([, chapter 6]). In this formula k L k (2kt)e kt, k, is the k-th moment of the so-called free unitary Brownian motion at time 2t ([3], [9], [24]), which arises in the large d-it of (Y t/d ) t 0. This observation led to a beautiful, yet striking, representation of the spectral distribution of the free Jacobi process associated with the couple of values η =, θ = /2. In [2], partial results on the spectrum of the free Jacobi process associated with η =, θ ]0, ] were obtained. There, a unitary process related to the free Jacobi process was considered and a detailed analysis of the dynamics of its spectrum was performed. The connection between both spectra is then ensured by a non commutative binomial-type expansion. Motivated by these findings, we tackle the problem of computing the large m-it (2) by deriving an explicit expression of E(tr[(J t/d ) n ]). To this end, we shall assume that m is large enough so that m > n and make use of the semi-group density of the eigenvalues process of (J t ) t 0. In [3], it was noticed that the latter is realized as m independent real Jacobi processes of parameters (2(p m + ) > 0, 2(q m + ) > 0) and conditioned never to collide. As a matter of fact, its semi-group density follows readily from the Karlin and McGregor formula (see [0] for the details) and is given by a bilinear series of symmetric Jacobi polynomials indexed by partitions (see for instance [7], [8]). However, this realization was announced in [3] with few details. For that reason and for the reader s convenience, we shall perform below the stochastic analysis of (J t/d ) t 0 leading to the semi-group density of its eigenvalues process. We shall also prove the absolute-convergence of the series defining this density, so that Fubini Theorem applies when computing E(tr[(J t/d ) n ]). Nest, with the help of the expansion of the n-th power sum in the Schur polynomial basis ([2]) and of the integral Cauchy-Binet formula ([8, page 37]), we determine the partitions having non zero contributions after integration. These are exactly the hooks of weights less than n, and both papers [7] and [8] provide an explicit expansion of the corresponding symmetric Jacobi polynomial in the Schur polynomial basis. The sought expectation follows from the integral of a product of Schur functions with respect to a multivariate Beta weight. The Cauchy-Binet formula allows once more to express this integral as a determinant of a matrix whose entries are Beta functions (see Exercise 8, p.386 in [2]). Summarizing, we obtain the following result, where we denote by Uτ p m,q m,m ( m ) the value at the point m = (,..., ) }{{} m times of the symmetric Jacobi polynomial, by µ τ the ordering induced by the Young diagrams associated with the partitions µ = (µ... µ m 0), τ = (τ... τ m 0), by α = α(n, k) := (n k, k ) a hook of weight α = n and by β(, ) the Beta function (see the next sections for more details on both Jacobi polynomials and partitions). Theorem. Let p q m and set r := p m 0, s := q m 0. Then (4) E(tr[(J t/d ) n ]) = n k=0( ) k τ α a r,s,m τ e Kr,s,m τ (t/d) Uτ r,s,m ( m ) b r,s,m µ,τ det (β(α i + µ j + 2m i j + r +, s + )) m i,j, where a r,s,m τ, b r,s,m µ,τ R are given in (2) and (3) respectively and where K r,s,m τ = m τ i (τ i + r + s + + 2(m i)). When s = 0, the determinant of Beta functions reduces to the well-known Cauchy determinant. Together with Weyl dimension formula, we get the following corollary where, for a partition τ, s τ denotes the associated Schur polynomial (see Section 3 for more details on Schur polynomial). 2

3 Corollary. If s = 0, then we have n E(tr[(J t/d ) n ]) = k=0( ) k τ α e Kr,0,m τ (t/d) [s τ ( m )] 2 Uτ r,0,m ( m ) b r,0,m µ,τ m [2(τ i + m i) + r + ] Moreover, in the particular case where r = s = 0, then we have n E(tr[(J t/d ) n ]) = k=0( ) k τ α e K0,0,m τ (t/d) [s τ ( m )] 2 Uτ 0,0,m ( m ) b 0,0,m µ,τ { } 2 Γ(m i + )Γ(r + τi + m i + ) Γ(τ i + m i + )Γ(r + m i + ) (τ i + τ j + 2m i j + r + ) 2 m i,j= (α s α ( m )s µ ( m ). i + µ j + 2m i j + r + ) m [2(τ i + m i) + ] Let us point out that, for s =, the determinant (τ i + τ j + 2m i j + ) 2 m i,j= (α s α ( m )s µ ( m ). i + µ j + 2m i j + ) det (β(α i + µ j + 2m i j + r +, 2)) m i,j ( = det (α i + µ j + 2m i j + r + )(α i + µ j + 2m i j + r + 2) was already considered in [5], where it is expanded in some basis of symmetric functions. Up to our best knowledge, there is no general explicit expression of the above determinant for arbitrary s 0. Nonetheless, when τ = (0) is the null partition then we can rather appeal to an instance of Kadell s integral (see Exercise 7, p.385 in [2]) and retrieve the moments derived in [5] (see Proposition 2.2 and Corollary 2.3 there) of the multivariate Beta distribution arising from the Jacobi unitary ensemble. This is by no means a surprise since the distribution of J t converges as t to the unitary-invariant matrix-variate Beta distribution. Back to the first formula displayed in the corollary, some of the products involved there terminate after cancellations, since the lengths of µ, τ, α satisfy l(µ) l(τ) l(α) n < m. This observation allows to take the it as m there, assuming p = p(m) and d = d(m) are such that () holds. Moreover, we show that b r(m),s(m),m µ,τ s µ ( m ) has finite large m-it which, together with the generalized binomial formula for Schur functions ([6]), entail ( U τ r(m),s(m),m ( m ) = ) τ. θ Here, we write r = r(m) = p(m) m, s = s(m) = d(m) p(m) m and the assumption s(m) = 0 corresponds in the large m-it to the set {θ ]0, [, θ( + η) = }. Since s α ( m ) = O(m α ) for any partition α ([5], p.4), then we are led after normalizing by the factor (/m) to an indeterminate it and as such, the computation of (2) seems to be out of reach for the moment. Note in this respect that the derivation of the moments M n (, η, θ) performed in [5] is based on the inverse binomial transform. The paper is organized as follows. In the next section, we perform the stochastic analysis of the hermitian Jacobi process and we prove the absolute convergence of the semi-group density of its eigenvalues process. In section 3, we prove our main results, that is Theorem, and his corollary. For that purpose, we recall some facts on both Schur polynomials and symmetric Jacobi polynomials associated with hooks then generalize an orthogonality relation for the real Jacobi polynomial to its multivariate analogue. In the last section, we study the asymptotic behavior of all the terms appearing in the first formula of the corollary. 2. The Hermitian Jacobi process, its eigenvalues process and symmetric Jacobi polynomials 2.. Stochastic analysis and semi-group density of the eigenvalues process. The existence of the it (2) relies to a large extent on the convergence of the moments of (Y t/d ) d 0 to those of the free unitary 3 ) m i,j

4 Brownian motion ([3]). The time normalization t/d is equivalent to the normalization of the Laplace operator on U(d) by a factor /d, which in this case corresponds to the Killing form d tr(xy ), where X, Y are skew-hermitian matrices. With this normalization, the unitary Brownian motion solves the stochastic differential equation (see [20]) (5) dy t = iy t dh t 2 Y tdt, Y 0 = I d, where I d is the d d identity matrix and (H t ) t 0 is a d d matrix-valued Hermitian process whose diagonal entries are real Brownian motions with variance t/d and its off diagonal entries are complex Brownian motions of variance t/d, all of them being independent. In order to derive a stochastic differential equation satisfied by J t, let ( ) ( ) Xt U Y t = t Rt S, H V t W t = t, t M t N t be the block decompositions of Y t and H t. Here, X t is the m p upper-left corner of Y t so that J t = X t X t, while U t, V t, W t, R t, S t, M t, N t are m q, d m p, d m q, p p, p q, q p, q q matrices respectively. Hence, (5) readily gives and Itô formula yields dx t = i(x t dr t + U t dm t ) X t 2 dt dj t = X t (dx t ) + (dx t )X t + < (dx t ), (dx t ) > where <, > denotes the bracket of continuous semi-martingales. Since for any complex Brownian motion (B t ) t 0 of variance t < db t, db t >= t, < db t, db t >= 0, since (R t ) t 0 and (M t ) t 0 are independent and since X t Xt + U t Ut = I m, then the finite-variation part of J t is given by ( p ) d I m J t dt. Again, since R t is Hermitian, then the local-martingale part of dj t is given by i(u t dm t X t X t dm t U t ), and its bracket is equal to the bracket of the local martingale Jt df t Im J t + I m J t dft Jt, where (F t ) t 0 is a complex Brownian matrix the entries of each have variance t/d. Hence, if J 0 and I m J 0 are positive-definite, the following stochastic differential equation holds dj t = J t df t Im J t + ( p ) I m J t dft Jt + d I m J t dt as long as J t and I m J t are positive-definite. According to Bru s Theorem (see [4, page 306]), there exist m real Brownian motions (ν i ) m with common variance t such that the eigenvalues process, say (λ i) m, satisfies the stochastic differential system dλ i (t) = (2/d)(λ i (t)( λ i (t))dν i (t) + (p d λ i (t)) + λ i (t)( λ j (t)) + λ j (t)( λ i (t)) (6) dt d λ i (t) λ j (t) j i as long as 0 < λ m (t) < λ m (t) < < λ (t) <. Recalling q = d p, then the infinitesimal generator of (λ i (2td), t 0) m coincides with the one displayed in [3, page 50]. Consequently, (λ i) m is realized as a Doob transform of m independent real Jacobi processes of parameters (2(p m + ), 2(q m + )) killed when they first collide, the sub-harmonic function being the Vandermonde polynomial. On the other hand, the main result proved in [0] shows that if p q > m (/2), then (6) admits, for any starting point λ(0) = (0 λ m (0) λ (0) ), a unique strong solution defined on the whole positive half-line. 4

5 Altogether, we deduce from the last section of [0] that the semi-group density of (λ i ) m, say Gr,s,m t, is given at time t by: (7) G r,s,m t (λ(0), λ) = τ=(τ τ m 0) r,s det[p e Kr,s,m τ (t/d) τ (λ i+m i j(0))] m i,j= det[p r,s τ (λ i+m i j)] m i,j= W r,s,m (λ), V (λ(0)) V (λ) where we recall that r = p m, s = q m and that m Kτ r,s,m := τ i (τ i + r + s + + 2(m i)), where we have set V (λ) := i<j(λ i λ j ), W r,s,m (λ) := m λ r i ( λ i ) s V (λ) 2 {0<λm< <λ <}, and where P r,s n with stands for the n-th orthonormal Jacobi polynomial on [0, ]. More precisely, p r,s n 2 2 := Pn r,s := pr,s n p r,s (r + ) n = n 2 p r,s 2F ( n, n + r + s +, r +, ) n 2 n! Γ(r + n + )Γ(s + n + ) 2n + r + s + Γ(n + )Γ(n + + r + s), (r + ) n = Γ(r + + n), Γ(r + ) and 2 F being the Gauss hypergeometric function (see [, chapters 2 and 6] for more details) Symmetric Jacobi polynomials. Set r,s Pτ r,s,m det[pτ (x) := (x i+m i j)] m i,j= = V (x) m det[p r,s τ (x i+m i j)] m i,j= p r,s τ, i+m i 2 V (x) then Pτ r,s,m is known as the symmetric (orthonormal) Jacobi polynomial associated with the partition τ. Under different normalizations, the family (Pτ r,s,m ) τ appeared independently in [2], [7], [8], [22] and [23]. For instance, since p r,s τ (0) = (r + ) i+m i τ i+m i/(τ i + m i)! then G r,s,m t (λ(0), λ) may be written as (8) G r,s,m t (λ(0), λ) = τ=(τ τ m 0) e Kr,s,m τ (t/d) V ( τ) (r + j i)i m p r,s τ i+m i (0) p r,s τ i+m i 2 2 U r,sm τ (λ(0))uτ r,s,m (λ)w r,s,m (λ), where Uτ r,s,m denotes the polynomial considered in [8], normalized to be equal to at (0,..., 0), see [8, }{{} m times Theorem 0]. More explicitely Uτ r,s,m (λ) := ( )m(m )/2 V ( τ) with V ( τ) = (r+j i)i det( 2F ( (τ i + m i), τ i + m i + r + s +, r +, λ j )) m i,j= V (λ) (τ i τ j + j i)(τ i + τ j + 2m i j + r + s + ). The representation (8) is convenient for our purposes since when τ is a hook, an explicit expansion of U r,s,m τ in the Schur polynomial basis is given in [8]. With respect to Lebesgue measure dλ = m dλ i. 5

6 2.3. Absolute convergence of the semi-group density. Another normalization of the symmetric Jacobi polynomial is related to the spherical function property they satisfy for special parameters (r, s) ([22]). It has the merit to be well-suited for proving that the series given in (7) is absolutely convergent. Indeed, let φ [, ] m and let qn r,s (x) = p r,s n (( x)/2), Q r,s,m τ (φ) = det[qr,s τ (φ i+m i j)] m i,j= V (φ) be the Jacobi polynomial in [, ] and the symmetric Jacobi polynomial in [, ] m respectively. Then Proposition 7.2 in [22] shows that Q r,s,m τ coincides up to a constant with the symmetric Jacobi polynomials considered there. Moreover, Proposition. in the same paper shows that Q r,s,m τ (φ) Q r,s,m τ ( m ), m = (,..., ), r s 0, }{{} m times and the special value Q r,sm τ ( m ) is given by ([22, Proposition 7.]) m ( m ) = V ( τ) Since Q r,s,m τ P r,s,m τ Γ(τ i + m i + r + )2 (m i) Γ(τ i + m i + )Γ(m i + r + )Γ(m i + ). (x) = ( 2) m(m )/2 m then the absolute convergence of (7) amounts to that of [ Using the bound and from τ τ m 0 V ( τ) e Kr,s,m τ (t/d) p r,s τ Q r,s,m τ ( 2x), i+m i 2 Q r,s,m τ ( m ) m ] 2 p r,s τ. i+m i 2 m [(τ i + m)(2τ i + 2m + r + s + )] m p r,s Γ(r + τ i + m i + )Γ(s + τ i + m i + ) τ i+m i 2 2 = 2(τ i + m i) + r + s + Γ(τ i + m i + )Γ(τ i + m i + + r + s) it then suffices to prove the absolute convergence of the series ( m e Kr,s,m τ (t/d) [(τ i + m)(2τ i + 2m + r + s + )] 2m [2τ i + 2m + r + s + ] τ τ m 0 ) Γ(τ i + m i + r + )Γ(τ i + m i + r + s + ). Γ(τ i + m i + )Γ(τ i + m i + s + ) Since this is a series of positive numbers then we can bound it from above by the series over all the m-tuples (τ,..., τ m ) N m. Doing so leads to the absolute convergence of the series e j(j+r+s++2(m i))(t/d) [(j + m)(2j + 2m + r + s + )] 2m j 0 for any i m. But this holds true since [2j + 2m + r + s + ](j + m i + ) r (j + m i + s + ) r, (j + m i + ) r (j + m i + s + ) r (j + m i + ) r (j + m i + s + ) r, j. From the mirror symmetry qn r,s ( x) = ( ) n qn s,r (x), it follows that Q r,s,m τ ( φ) = ( ) τ Q r,s,m τ (φ) whence the absolute-convergence of the series (7) may be proved for 0 r s along the same lines. As a matter of 6

7 fact, if the hermitian Jacobi process starts at the identity matrix J 0 = I m then Fubini Theorem yields ( ) m E(tr[(J t/d ) n ]) = G r,s,m t ( m, λ)dλ (9) = λ n i τ τ m 0 ( m e Kr,s,m τ (t/d) Pτ r,s,m ( m ) 3. Proof of Theorem λ n i ) Pτ r,s,m (λ)w r,s,m (λ)dλ. In this section, we prove both Theorem and Corollary. The proof of Theorem mainly relies on the lemma below, where we determine the partitions having non zero contributions to the integral displayed in the right hand side of (9). 3.. Partitions. When m =, τ is a nonnegative integer and Pτ r,s, polynomial Pτ r,s of degree τ. In this case, the integral 0 x j Pτ r,s (x)x r ( x) s dx reduces to the orthonormal Jacobi vanishes unless j τ, since x j may be written as a linear combination of P τ, τ j. For general m 2, the situation is quite similar. More precisely, fix n < m and recall from [2, page 68, exercise 0] the following expansion m n λ n i = ( ) k s α (λ), where α = α(k, n) = (n k, k ) := (n k,,...,, 0,..., 0 ), 0 k n, }{{}}{{} k times m k times are hooks of weight m α = α i = n, and k=0 s α (λ) = s α (λ,..., λ m ) = det(λαi+m i j ) m i,j= det(λ m i j are the corresponding Schur polynomials. Recall also from [8, page 37] the integral form of the Cauchy-Binet formula: for any probability measure κ and any sequences (ψ i ) i, (φ i ) i of real-valued bounded functions, det(ψ i (x j )) m i,j=det(φ i (x j )) m i,j= m ( κ(dx i ) = m!det ) m i,j= ) m ψ i (x)φ j (x)κ(dx). i,j= We can now state the lemma, where τ α means that, for any i m, τ i α i. Lemma. For any k n, the integral s α (λ)pτ r,s,m (λ)w r,s,m (λ)dλ vanishes unless τ α. Proof: For sake of simplicity, let us omit in this proof the super-scripts and write simply P τ, P n, W instead of Pτ r,s,m, Pn r,s, W r,s,m respectively. From the Cauchy-Binet formula, it follows that s α (λ)p τ (λ)w (λ)dλ = m det(λ αj+m j i )det(p τj+m j(λ i )) λ r i ( λ i ) s dλ m! [0,] m ( ) m = det x αj+m j P τi+m i(x)x r ( x) s dx 0 7. i,j=

8 Set ( A = (A ij ) m i,j= := 0 ) m x αj+m j P τi+m i(x)x r ( x) s dx i,j= and note that det(a) = 0 if τ m since the last column is the null vector. Assuming τ m = 0, τ m and expanding the determinant along the last column, then the same conclusion holds for the principal minor (A ij ) m i,j= and so on up to the principal minor of size k +. Thus, det(a) = 0 unless τ i = 0 for all k + 2 i m. If k = 0, then A is a lower triangular matrix and det(a) = 0 unless τ n. Otherwise k n, and if τ i 2 for some 2 i k +, then τ τ 2 2 so that for any j 2 τ + m τ 2 + m 2 m > α j + m j. From the orthogonality of the real Jacobi polynomials, it follows that A j = A 2j = 0 for all j 2 so that the first and the second row are proportional. Thus, det(a) = 0 and we are left with the hooks τ = (τ τ 2 τ k+ 0,..., 0 ) }{{}}{{} {0,} m k times But if τ > n k then the first row is the null vector and det(a) = 0 as well. The lemma is proved. Remark. We shall see below that the symmetric Jacobi polynomial has a lower-triangular expansion in the basis of Schur polynomials with respect to the ordering. It is very likely that the inverse expansion of the Schur polynomial in the basis of symmetric Jacobi polynomials is also lower-triangular. In this case, the lemma would follow from the fact that symmetric Jacobi polynomials are mutually orthogonal with respect to W r,s,m : Pτ r,s,m (x)pκ r,s,m (x)w r,s,m (x)dx = 0 whenever the partitions τ and κ are different. Now we proceed to the end of the proof of Theorem Symmetric Jacobi polynomials associated with hooks. Let 0 k n and τ α be a hook τ = (n k δ, k g ), 0 δ n k, 0 g k. For a partition µ, we denote by m m (z) µ = (z i + ) µi = Γ(z i + + µ i ) Γ(z i + ) the generalized Pochhammer symbol. From [7] and [8], we dispose of an explicit expansion of Uτ r,s,m in the Schur polynomial basis. More precisely, by specializing [8, Theorem 3] to α =, we claim that (0) Uτ r,s,m (λ) = ( ) µ ( ) τ C (r + m) µ µ µ(r τ + s + 2m) s µ(λ) s µ ( m ) where if µ = (n k γ, k l ), δ γ n k, g l k, then ( ) ( )( ) τ n k δ k g (n δ l)(n g γ) (γ δ)(l g) = µ γ δ l g (n γ l) 2 is the generalized binomial coefficient (specialize [7, Theorem 4] to α = ), and where for any real X (specialize [8, Theorem 6] to α = ) () Cµ(X) τ = ( (n k δ)(n k δ ) (k g)(k g + ) X + n δ g 8 ) n k γ i=2 k l (X + n k δ + i 2) (X k + g i).

9 In order to prove Theorem, we need to compute s α (λ)uτ r,s,m (λ)w r,s,m (λ)dλ. With regard to (8), (9) and Lemma, s α (λ)uτ r,s,m (λ)w r,s,m (λ)dλ = ( ) µ ( ) τ C τ sα (λ)s µ (λ) (r + m) µ µ µ(r + s + 2m) s µ ( m W r,s,m (λ)dλ ) = ( ) µ ( ) ( τ ) m (r + m) µ s µ ( m C τ ) µ µ(r + s + 2m) det x αi+µj+2m i j+r ( x) s dx 0 i,j= = ( ) µ ( ) τ (r + m) µ s µ ( m C ) µ µ(r τ + s + 2m) det (β(α i + µ j + 2m i j + r +, s + )) m i,j=. The formula displayed in Theorem follows after setting a r,s,m τ := V ( τ) (2) (r + j i)i (3) b r,s,m µ,τ := m p r,s 2 τ (0) i+m i p r,s, ( ) µ ( ) τ (r + m) µ s µ ( m C τ ) µ µ(r + s + 2m). τ i+m i 2 Remark. The product s α s µ is linearized via the Littlewood-Richardson coefficients ([2], p.42) as: (4) s α (λ)s µ (λ) = c κ αµs κ (λ), κ where the summation is over the set of partitions {κ α, κ µ, α + µ = κ }. Thus s α (λ)s µ (λ)w r,s,m (λ)dλ = c κ αµ s κ (λ)w r,s,m (λ)dλ κ and the value of the integral in the right hand side is an instance of Kadell s integral (see Exercice 7, p.385 in [2]): s κ (λ)w r,s,m (λ)dλ = m Γ(κ i + r + m i + )Γ(s + m i + ) (κ i κ j + j i). Γ(κ i + r + s + 2m i + ) However, up to our best knowledge, there is no simple formula for c κ αµ except when µ is a partition with one row or one column 2. For that reason, we preferred the use of the Cauchy-Binet formula when evaluating (4). Nonetheless, if τ = (0) is the null partition then µ = (0) and we are only left with Kadell s integral. In particular, b r,s,m 0,0 =, U r,s,m (0) = and we retrieve the moments of the normalized multivariate distribution Beta derived in [5]. Indeed, a r,s,m (0) is exactly the normalizing constant of W r,s,m whose inverse is a special instance of the value of the Selberg integral (see e.g. [5]) The case s = 0. Now, we specialize Theorem to s = 0. Then the Cauchy-determinant yields ( ) m det (β(α i + µ j + 2m i j + r +, )) m i,j = det α i + µ j + 2m i j + r + i,j = (α i α j + j i)(µ i µ j + j i) m i,j= (α. i + µ j + 2m i j + r + ) Besides, the Weyl dimension formula s µ ( m ) = 2 This is referred to as Pieri formula. 9 (µ i µ j + j i) j i

10 and the equality entail Corollary follows from the equality together with [(r + j i)i] = m (α i α j + j i)(µ i µ j + j i) [(r + j i)i] 2 = (j i) = p r,0 τ i+m i (0) = Γ(r + + τ i + m i) Γ(r + )Γ(τ i + m i + ), Γ(r + m i + ) Γ(r + ) (j i) m { } 2 Γ(r + ) s α ( m )s µ ( m ). Γ(r + m i + ) m Γ(m i + ) 4. Asymptotics pr,0 τ i+m i 2 2 = 2(τ i + m i) + r +. Below, we determine the iting behavior of the terms appearing in the first formula displayed in the corollary. As claimed in the introduction (after the statement of the corollary), we first notice that the following cancellations occur: if l(µ) l(τ) l(α) n < m are the lengths of the partitions µ τ α respectively then m l(τ) Γ(r + τ i + m i + )Γ(m i + ) Γ(τ i + m i + )Γ(r + m i + ) = Γ(r + τ i + m i + )Γ(m i + ) Γ(τ i + m i + )Γ(r + m i + ), m l(α) [2(τ i + m i) + r + ] (α i + µ i + 2m 2i + r + ) = [2(τ i + m i) + r + ] (α i + µ i + 2m 2i + r + ), and l(α)+ (τ i + τ j + 2m i j + r + ) 2 l(α)+ i j m (α i + µ j + 2m i j + r + ) = As a result, if s = s(m) = q(m) m = 0 and () holds then m l(α)+ i j m l(τ) Γ(r(m) + τ i + m i + )Γ(m i + ) Γ(τ i + m i + )Γ(r(m) + m i + ) = (τ i + τ j + 2m i j + r + ) (α i + µ j + 2m i j + r + ) =. ( p(m) m ) τi = η τ. Note that the assumption s(m) = 0 yields in the large m-it the relation θ( + η) = whence Now, it is obvious that and that m Γ(r(m) + τ i + m i + )Γ(m i + ) ( θ Γ(τ i + m i + )Γ(r(m) + m i + ) = θ m [2(τ i + m i) + r(m) + ] (α i + µ i + 2m 2i + r(m) + ) = i<j l(α) (τ i + τ j + 2m i j + r(m) + ) 2 i j l(α) (α i + µ j + 2m i j + r(m) + ) = 0 i j l(α) ) τ. (τ i + τ j + 2m i j + r(m) + ) (α i + µ j + 2m i j + r(m) + ) =.

11 As to i l(α) l(α)+ j m we can rewrite it as i l(α) l(α)+ j m (α j + µ i + 2m i j + r(m) + ) i l(α) l(α)+ j m i l(α) l(α)+ j m (τ i + 2m i j + r(m) + ) 2 j l(α) l(α)+ i m (α i + µ j + 2m i j + r(m) + ), (τ i + 2m i j + r(m) + ) 2 (α i + 2m i j + r(m) + )(µ i + 2m i j + r(m) + ) and deduce that it is equivalent to [d(m)] 2 τ α µ as m. Indeed, recall r(m)+2m = p(m)+m = d(m) and consider (τ i + d(m) i j + ) = (τ i + d(m) i j + ). (d(m) i j + ) (d(m) i j + ) Then the terms corresponding to i = are i l(τ) l(α)+ j m (d(m) l(α) + τ )(d(m) l(α) + τ 2) (d(m) l(α))(d(m) l(α) ) (d(m) m) (d(m) l(α) ) (d(m) m) which reduces to Consequently (d(m) l(α) + j) d(m) τ, m. τ j=0 i l(α) l(α)+ j m (τ i + d(m) i j + ) (d(m) i j + ) d(m) τ, m. Applying this reasoning to the partitions α and µ, the claimed equivalence follows. Next, without assuming s(m) = 0 then d(m) Kr(m),s(m),m τ = holds for any hook τ of weight τ n, while Lemma 2. l(τ) d(m) ( U τ r(m),s(m),m ( m ) = ) τ. θ τ i (τ i + d(m) + 2i) = τ Proof: Pick µ τ. Then we have l(µ) n < m so that the generalized Pochammer symbol splits l(µ) l(µ) (r(m) + m) µ = (p(m) i + ) µi = (p(m)) µ (p(m) i + ). Thus (r(m) + m) µ p(m) µ as m. On the other hand, it is obvious from () that Hence br(m),s(m),m µ,τ i=2 C τ µ(r(m) + s(m) + 2m) = C τ µ(d(m)) d(m) µ as m. s µ ( m ) = ( ) µ (r(m) + m) µ ( τ µ )C τµ(r(m) + s(m) + 2m) = ( ) µ and from (0), we get U τ r(m),s(m),m ( m ) = ( ) ( ) µ τ ( ) µ = s τ ( (/θ),..., (/θ)), µ θ s τ ( l(τ) ) θ µ ( ) τ µ

12 where the last equality follows from the generalized binomial Theorem ([6]). The lemma follows from the homogeneity of the Schur polynomials. Collecting all these its and since s τ ( m ) = O(m τ ), s α ( m ) = O(m α ) ([5]), then we are led to an indeterminate it when computing (2). References [] G. E. Andrews, R. Askey, R. Roy. Special functions. Cambridge University Press [2] R. J. Beerends, E. M. Opdam. Certain hypergeometric series related to the root system BC. Trans. Amer. Math. Soc. 339, no , [3] P. Biane. Free Brownian motion, free stochastic calculus and random matrices. Fields. Inst. Commun., 2, Amer. Math. Soc. Providence, RI, [4] M. Capitaine, M. Casalis. Asymptotic freeness by generalized moments for Gaussian and Wishart Matrices. Application to Beta random matrices. Ind. Univ. Math. J. 53, no. 2, 2004, [5] C. Carré, M. Deneufchatel, J.G. Luque, P. Vivo. Asymptotics of Selberg-like integrals: the unitary case and Newton s interpolation formula. J. Math. Phys. 5 (200), no. 2, 9p. [6] B. Collins. Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theor. Rel. Fields. 33, no. 3, 2005, [7] A. Débiard. Système différentiel hypergéométrique et parties radiales des opérateurs invariants des espaces symétriques de type BC p. Lecture Notes in Math., 296, Springer, Berlin, 987, [8] P. Deift. D. Gioev. Random matrix theory: invariant ensembles and universality. Courant Lecture Notes in Mathematics, 8. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, [9] N. Demni. Free Jacobi process. J. Theo. Probab. 2, no.. (2008), [0] N. Demni. β-jacobi processes. Adv. Pure Appl. Math,, no [] N. Demni, T. Hamdi, T. Hmidi. Spectral distribution of the free Jacobi process. Indiana Univ. J. 6, no [2] N. Demni, T. Hmidi. Spectral distribution of the free Jacobi process associated with one projection. To appear in Colloquium Math. [3] Y. Doumerc. Matrices aléatoires, processus stochastiques et groupes de réflexions. Ph.D. Thesis, Paul Sabatier Univ. Available at [4] M. Katori, H. Tanemura. Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems. J. Math. Phys. 45, 2004, no. 8, [5] A. Lascoux. Square-ice enumeration. Sém. Lothar. Combin. 42, 999, Art. B42p, 5 pp. [6] M. Lassalle. Une formule du binôme généralisée pour les polynômes de Jack. C. R. Acad. Sci. Paris. t. 30. Série I [7] M. Lassalle. Coefficients du binôme généralisés. C. R. Acad. Sci. Paris. t. 30. Série I [8] M. Lassalle. Polynômes de Jacobi. C. R. Acad. Sci. Paris. t. 32, Série I. 99. p [9] T. Lévy. Schur-Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 28, 2008, no. 2, [20] M. Liao. Lévy processes in Lie groups. Cambridge University press [2] I. G. MacDonald. Symmetric Functions and Hall Polynomials. Second edition, Mathematical Monographs, Oxford [22] G. Olshanski, A. Okounkov. Limits of BC-type orthogonal polynomials as the number of variables goes too infinity. Jack, Hall-Littlewood and Macdonald polynomials, 28-38, Contemp. Math. 47, Amer. Math. Soc., Providence, RI, [23] G. I. Olshanski, A. A. Osinenko. Multivariate Jacobi polynomial and the Selberg integral. Functional Analysis and Its Applications. Vol. 46. No , 202. [24] Rains, E. M. Combinatorial properties of Brownian motion on the compact classical groups. J. Theoret. Probab. 0, 997, no. 3, LAMA, Université Marne la Vallée, Champs sur Marne, Marne la Valle Cedex 2, France address: luc.deleaval@u-pem.fr IRMAR, Université de Rennes, Campus de Beaulieu, Rennes cedex, France address: nizar.demni@univ-rennes.fr 2