MOMENTS OF THE HERMITIAN MATRIX JACOBI PROCESS

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1 MOMENTS OF THE HERMITIAN MATRIX JACOBI PROCESS LUC DELEAVAL AND NIZAR DEMNI Abstract In this paper, we compute the expectation of traces of powers of the hermitian matrix Jacobi process for a large enough but fixed size To proceed, we first derive the semi-group density of its eigenvalues process as a bilinear series of symmetric Jacobi polynomials Next, we use the expansion of power sums in the Schur polynomial basis and the integral Cauchy-Binet formula in order to determine the partitions having non zero contributions after integration It turns out that these are hooks of bounded weight and the sought expectation results from the integral of a product of two Schur functions with respect to a generalized Beta distribution For special values of the parameters on which the matrix Jacobi process depends, the last integral reduces to the Cauchy determinant and we close the paper with the investigation of the asymptotic behavior of the resulting formula as the matrix size tends to infinity Reminder and motivation Given three integers d, p, m such that p, m < d, the hermitian matrix Jacobi process J t ) t 0 of parameters p, q := d p) was defined in [6, page 4] as the product of the m p upper-left corner of a d d Brownian motion Y t ) t 0 on the unitary group Ud, C) [24]) and of its Hermitian conjugate Equivalently, if P m and Q p are two d d diagonal projections of ranks m and p respectively, then ) J t 0 m,d m := P 0 d m,m 0 m Y t Q p )P m Y t Q p ) = P m Y t Q p Yt P m, d m where 0 d m,m, 0 m,d m, 0 d m are the null matrices of shapes d m m, m d m, and d m d m respectively 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 = dm) and p = pm) depend on m such that pm) ) := θ ]0, [, dm) m := η > 0, with ηθ ]0, [, pm) exist, then the expectation of the normalized trace of any finite-tuple of matrices drawn from J t/dm) ) t 0 converge as m [], see also the recent paper [8] where the convergence is shown to hold in the strong sense) 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 [ ) n ]) J t/dm) 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 []) Furthermore, the it M n, η, θ) := t M n t, η, θ), n 0, is the moment sequence of the spectral distribution of the large m-it of P m UQ pm) U P m, where U is a dm) dm) 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 [5], [7], []) Besides, an explicit expression of M n, η, θ) obtained from large m-asymptotics of the moments of the multivariate Beta distribution figures in [6, Theorem 44] However, the situation becomes rather considerably more complicated when dealing with M n t, η, θ) for 200 Mathematics Subject Classification 5B52, 33C45, 60H5 Key words and phrases Hermitian matrix Jacobi process, Schur polynomial, symmetric Jacobi polynomial, hook

2 fixed time t > 0, as witnessed by the series of papers [4], [5] and [3] For instance, it was proved in [4] that 3) M n t,, /2) = ) 2n 2 2n + n ) 2n n 2 2n n k k L k 2kt)e kt, 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 [4], [23], [28]), 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 [5], 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 In the recent paper [3], a complicated expression of M n t,, θ) is obtained using sophisticated tools from complex analysis Motivated by these findings, we tackle here the problem of computing the large m-it 2) by deriving an explicit expression of Etr[J t/d ) n ]) for fixed t > 0, 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 this respect, it was noticed in [6] that the latter process is realized as m independent real Jacobi processes of parameters 2p m + ) > 0, 2q 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 [2] for the details) and is given by a bilinear series of symmetric Jacobi polynomials indexed by partitions see for instance [9], [22]) We shall also prove the absolute-convergence of the series defining this density, so that Fubini Theorem applies when computing Etr[J t/d ) n ]) Next, with the help of the expansion of the n-th power sum in the Schur polynomial basis [25]) and of the integral Cauchy-Binet formula [0, 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 [2] and [22] 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, p386 in [25]) 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 Uτ p m,q m,m of parameters p m, q m), 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 = αn, 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) Etr[J t/d ) n ]) = n k=0 ) k τ α where a r,s,m τ, b r,s,m µ,τ a r,s,m τ e Kr,s,m τ k= t/d) Uτ r,s,m m ) b r,s,m µ,τ det βα i + µ j + 2m i j + r +, s + )) m i,j=, µ τ R are given in 3) and 4) respectively and where m Kτ r,s,m = τ i τ i + r + s + + 2m 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 Etr[J t/d ) n ]) = k=0 ) k τ α 5) If further r = s = 0, then e Kr,0,m τ t/d) [s τ m )] 2 Uτ r,0,m m ) µ τ n Etr[J t/d ) n ]) = k=0 ) k τ α m [2τ i + m i) + r + ] i<j m τ i + τ j + 2m i j + r + ) 2 m i,j= α i + µ j + 2m i j + r + ) e K0,0,m τ t/d) [s τ m )] 2 Uτ 0,0,m m ) µ τ m [2τ i + m i) + ] Let us point out that, for s =, the determinant i<j m τ i + τ j + 2m i j + ) 2 m i,j= α i + µ j + 2m i j + ) { } 2 Γm i + )Γr + τi + m i + ) Γτ i + m i + )Γr + m i + ) b r,0,m µ,τ s µ m )s α m ) b 0,0,m µ,τ s µ m )s α m ) det βα 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 [9], 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, as we shall see below, the term corresponding to the null partition τ = 0) may be computed using Kadell s integral see Exercise 7, p385 in [25]) and as such, we retrieve the moments derived in [6] see Proposition 22 and Corollary 23 there) of the multivariate Beta distribution arising from the Jacobi unitary ensemble This is by no means a surprise since all but this term cancel when we let t in 4) and the distribution of J t converges weakly as t to that of the Jacobi unitary ensemble also known as the matrix-variate Beta distribution) Back to formula 5), some of the products involved there terminate after cancellations, since the lengths of µ, τ, α satisfy lµ) lτ) lα) n < m This observation allows to take the it as m there, assuming p = pm) and d = dm) are such that ) holds Moreover, we show that bµ,τ rm),sm),m s µ m ) has finite large m-it which, together with the generalized binomial formula for Schur functions [20]), entail U τ rm),sm),m m ) = θ ) τ Here, we write r = rm) = pm) m, s = sm) = dm) pm) m and the assumption sm) = 0 corresponds in the large m-it to the set {θ ]0, [, θ + η) = } Since s α m ) = Om α ) for any partition α [6], p4), 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 [6] is based on the inverse binomial transform The paper is organized as follows In the next section, we recall the definition of the Brownian motion on the unitary group Ud, C) and derive the stochastic differential equation satisfied by the hermitian matrix Jacobi process which was announced in [6] without proof In the same section, we recall also the stochastic differential system satisfied by the corresponding eigenvalues process and prove the absolute convergence of the semi-group density of the latter 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 investigate the asymptotic behavior of all the terms appearing in the right-hand side of 5) 3 ) m i,j=

4 2 The Hermitian matrix Jacobi process and its eigenvalues process 2 From the unitary Brownian motion to the Hermitian matrix Jacobi 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 Brownian motion [4]) The time normalization t/d is equivalent to the normalization of the Laplace-Beltrami operator on Ud) by a factor /d, which in this case corresponds to the Killing form d trxy ), where X, Y are skew-hermitian matrices With this normalization, the unitary Brownian motion solves the following stochastic differential equation see [24]): 6) 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 while its off diagonal entries are complex Brownian motions, all of them being independent and have common variance t/d Besides, the process Y t/d ) t 0 is a left Brownian motion in the sense that the semi-group operator f E [fzy t )], Z Ud, C), defined on the space of continuous functions f on Ud, C) is left-invariant Equivalently, the right-increments Y s/d Y t/d, 0 s < t, of Y t/d ) t 0 are invariant under left multiplication by any complex unitary matrix This choice is by no means a loss of generality since the process Y t/d ) t 0 has the same distribution as Y t/d ) t 0 and is a right Brownian motion on Ud, C) Now, one can use in order to derive a stochastic differential equation satisfied by J t To this end, 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, 2) readily gives and Itô formula yields dx t = ix 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 [29]) Since < db t, db t >= t, < db t, db t >= 0, for any complex Brownian motion B t ) t 0 of variance t, 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 the semi-martingale decomposition of dj 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 ], whose bracket coincides with that 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 whose entries are independent and have common 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 4

5 as long as J t and I m J t remain so According to Bru s Theorem see [8, 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)) 7) 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 [6, page 50] Consequently, λ i) m is realized as a Doob transform of m independent real Jacobi processes of parameters 2p m + ), 2q m + )) killed when they first collide, the sub-harmonic function being the Vandermonde polynomial On the other hand, the main result proved in [2] shows that if p q > m /2), then 7) admits, for any starting point λ0) = 0 λ m 0) λ 0) ), a unique strong solution defined on the whole positive half-line Altogether, we deduce from the last section of [2] that the semi-group density of λ i ) m, say Gr,s,m t, is given at time t by: 8) G r,s,m t λ0), λ) = τ=τ τ m 0) where we recall r = p m, s = q m, where we have set r,s det[p e Kr,s,m τ t/d) τ λ i+m i j0))] m i,j= det[p r,s τ λ i+m i j)] m i,j= W r,s,m λ), V λ0)) V λ) K r,s,m τ := m τ i τ i + r + s + + 2m i)), V λ) := i<jλ 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, ] Actually, 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! Γr + n + )Γs + n + ) 2n + r + s + Γn + )Γn + + r + s), r + ) n = Γr + + n), Γr + ) and 2 F is the Gauss hypergeometric function see [, chapters 2 and 6] for more details) 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], [9], [22], [26] and [27] For instance, since p r,s τ i+m i 0) = r + ) τ i+m i τ i + m i)!, With respect to Lebesgue measure dλ = m dλ i 5

6 then G r,s,m t λ0), λ) may be written as 9) G r,s,m t λ0), λ) = τ=τ τ m 0) e Kr,s,m τ t/d) V τ) i<j m 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 λ0))uτ r,s,m λ)w r,s,m λ), where Uτ r,s,m denotes the polynomial considered in [22], normalized to be equal to at 0,, 0), see [22, }{{} m times Theorem 0] More explicitely Uτ r,s,m λ) := )mm )/2 V τ) with V τ) = i<j m i<j m 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 9) 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 [22] 22 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) see Table II in [26]) It has the merit to be well-suited for proving that the series given in 8) 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 72 in [26] 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 for any φ [, ] m, Q r,s,m τ φ) Q r,s,m τ m ), r s 0, while the special value Q r,sm τ m ) is given by [26, Proposition 7]): m m ) = V τ) Since Q r,s,m τ P r,s,m τ Γτ i + m i + r + )2 m i) Γτ i + m i + )Γm i + r + )Γm i + ) x) = 2) mm )/2 m then the absolute convergence of 8) amounts to that of [ By the virtue of the bound and from the expression 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 Γr + τ i + m i + )Γs + τ i + m i + ) τ i+m i 2 2 = 2τ i + m i) + r + s + Γτ i + m i + )Γτ i + m i + + r + s), 6

7 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 ) Γτ i + m i + r + )Γτ i + m i + r + s + ) Γτ i + m i + )Γτ 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 proving the absolute convergence of the series e jj+r+s++2m 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 q r,s n x) = ) n q s,r n x), it follows that Q r,s,m τ φ) = ) τ Q r,s,m τ φ) whence the absolute-convergence of the series 8) may be proved for 0 r s along the previous lines As a matter of fact, if the hermitian matrix Jacobi process starts at the identity matrix J 0 = I m, then Fubini Theorem yields 0) m Etr[J t/d ) n ]) = = λ n i τ τ m 0 ) G r,s,m t m, λ)dλ 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 the former relies mainly on the lemma below, where we determine the partitions having non zero contributions to the integral displayed in the right hand side of 0) reduces to the orthonormal one- 3 Partitions When m =, τ is a nonnegative integer and Pτ r,s, dimensional Jacobi polynomial Pτ r,s of degree τ In this case, the integral 0 x j Pτ r,s x)x r x) s dx 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 [25, page 68, exercise 0] the following expansion of the n-th power sum: m n λ n i = ) k s α λ), where α = αk, n) = n k, k ) := n k,,,, 0,, 0 ), 0 k n, }{{}}{{} k times m k times are hooks of common weight m α = α i = n, and k=0 s α λ) = s α λ,, λ m ) = detλαi+m i j ) m i,j= detλ m i j ) m i,j= 7

8 are the corresponding Schur polynomials Recall also from [0, 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, m ) m detψ i x j )) m i,j=detφ i x j )) m i,j= κdx i ) = m!det ψ i x)φ j x)κdx) i,j= We can now state the lemma alluded to above, where we use the ordering τ α meaning that τ i α i for all i m 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 detλ αj+m j i )detp τj+m jλ i )) λ r i λ i ) s dλ m! [0,] m ) m = det x αj+m j P τi+m ix)x r x) s dx Set A = A ij ) m i,j= := 0 0 i,j= ) m x αj+m j P τi+m ix)x r x) s dx i,j= and note that deta) = 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, deta) = 0 unless τ i = 0 for all k + 2 i m If k = 0, then A is a lower triangular matrix and deta) = 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 one-dimensional 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, deta) = 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 deta) = 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 8

9 32 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 = Γz i + + µ i ) Γz i + ) the generalized Pochhammer symbol From [2] and [22], we dispose of an explicit expansion of Uτ r,s,m in the Schur polynomial basis More precisely, by specializing [22, Theorem 3] to α =, we claim that ) Uτ r,s,m λ) = ) µ ) τ C r + m) µ µ µr τ + s + 2m) s µλ) s µ m ) µ τ where if then ) τ = µ µ = n k γ, k l ), δ γ n k, g l k, n k δ γ δ ) ) k g n δ l)n g γ) γ δ)l g) l g n γ l) 2 is the generalized binomial coefficient specialize [2, Theorem 4] to α = ), and where for any real X specialize [22, Theorem 6] to α = ) 2) CµX) τ = n k δ)n k δ ) k g)k g + ) X + n δ g ) n k γ 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 9), 0) and Lemma, i=2 k l X + n k δ + i 2) X k + g i) 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 µr + s + 2m) s µ m W r,s,m λ)dλ ) µ τ = ) µ ) τ ) m r + m) µ s µ m C τ ) µ µr + s + 2m) det x αi+µj+2m i j+r x) s dx µ τ 0 i,j= = ) µ ) τ r + m) µ s µ m C ) µ µr τ + s + 2m) det βα i + µ j + 2m i j + r +, s + )) m i,j= µ τ The formula displayed in Theorem follows after setting a r,s,m τ := V τ) 3) r + j i)i 4) b r,s,m µ,τ := i<j m m p r,s 2 τ 0) i+m i p r,s, ) µ ) τ r + m) µ s µ m C τ ) µ µr + s + 2m) τ i+m i 2 Remark The product s α s µ is linearized via the Littlewood-Richardson coefficients [25], p42) as: s α λ)s µ λ) = κ c κ αµs κ λ), 9

10 where the summation is over the set of partitions {κ α, κ µ, α + µ = κ } Thus 5) 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, p385 in [25]): s κ λ)w r,s,m λ)dλ = i<j m κ i κ j + j i) m Γκ i + r + m i + )Γs + m i + ) Γκ 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 5) Nonetheless, if τ = 0) is the null partition then µ = 0) and the left-hand side of 5) reduces to Kadell s integral Moreover, b r,s,m 0,0 =, U r,s,m 0) = and a r,s,m 0) is exactly the normalizing constant of W r,s,m whose multiplicative inverse is a special instance of the value of the Selberg integral see eg [6]) Consequently, if we let t in ), then the only non-vanishing term corresponds to τ = 0) and as such, we retrieve the moments of W r,s,m which is the stationary distribution of the eigenvalues process λt)) t 0 ) derived in [6] 33 The case s = 0: proof of Corollary Specializing Theorem with s = 0, then the Cauchydeterminant yields: ) m det βα i + µ j + 2m i j + r +, )) m i,j= = det α i + µ j + 2m i j + r + i,j= i<j m = 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 and the equality entail i<j m i<j m s µ m ) = [r + j i)i] = i<j m m α i α j + j i)µ i µ j + j i) [r + j i)i] 2 = Formula 5) in Corollary follows then from the equality together with i<j m j i) = µ i µ j + j i) j i Γr + m i + ) Γr + ) i<j m j i) m { } 2 Γr + ) s α m )s µ m ) Γr + m i + ) m Γm i + ) p r,0 τ i+m i 0) = Γr + + τ i + m i) Γr + )Γτ i + m i + ), The second formula in the corollary is obvious pr,0 τ i+m i 2 2 = 2τ i + m i) + r + 2 This is referred to as Pieri formula 0

11 4 Asymptotics The purpose of this section is to determine the its of various terms appearing in 5) under the assumption that the its ) exist Doing so is the crucial step in our future investigations aiming in particular to derive the moments 3) as its of their matrix analogues and more generally to derive an expression for M n t, η, θ) We start with dm) Krm),sm),m τ = lτ) dm) τ i τ i + dm) + 2i) = τ which holds for any hook τ of weight τ n Next, we prove the following lemma: Lemma 2 Let τ be a hook of weight τ n and let µ τ Then In particular, brm),sm),m µ,τ s µ m ) = ) µ θ µ ) τ µ U τ rm),sm),m m ) = ) τ θ Proof: Since lµ) n < m, then the generalized Pochammer symbol splits as lµ) lµ) rm) + m) µ = pm) i + ) µi = pm)) µ pm) i + ) Thus rm) + m) µ pm) µ as m On the other hand, it is obvious from 2) that Hence, we get from 4): brm),sm),m µ,τ i=2 C τ µrm) + sm) + 2m) = C τ µdm)) dm) µ as m s µ m ) = ) µ rm) + m) µ τ µ )C τµrm) + sm) + 2m), = ) µ and from ): U τ rm),sm),m m ) = ) ) µ τ ) µ = s τ /θ),, /θ)), µ θ s µ τ τ lτ) ) where the last equality follows from the generalized binomial Theorem [20]) The lemma follows from the homogeneity of the Schur polynomials Now, assume sm) = 0 and note that this assumption yields in the large m-it the relation θ + η) = η = θ θ If lµ) lτ) lα) n < m are the lengths of the partitions µ τ α respectively, then the following cancellations occur: m lτ) Γr + τ i + m i + )Γm i + ) Γτ i + m i + )Γr + m i + ) = m θ µ Γr + τ i + m i + )Γm i + ) Γτ i + m i + )Γr + m i + ), lα) [2τ i + m i) + r + ] α i + µ i + 2m 2i + r + ) = [2τ i + m i) + r + ] α i + µ i + 2m 2i + r + ), and lα)+ i<j m τ i + τ j + 2m i j + r + ) 2 lα)+ i j m α i + µ j + 2m i j + r + ) = lα)+ i j m ) τ µ τ i + τ j + 2m i j + r + ) α i + µ j + 2m i j + r + ) =

12 As a result, m and similarly lτ) Γrm) + τ i + m i + )Γm i + ) Γτ i + m i + )Γrm) + m i + ) = m [2τ i + m i) + rm) + ] α i + µ i + 2m 2i + rm) + ) =, i<j lα) τ i + τ j + 2m i j + rm) + ) 2 i j lα) α i + µ j + 2m i j + rm) + ) = Finally, consider the product i lα) lα)+ j m It can be rewritten as i lα) lα)+ j m α j + µ i + 2m i j + rm) + ) i lα) lα)+ j m i lα) lα)+ j m i j lα) τ i + 2m i j + rm) + ) 2 j lα) lα)+ i m ) τi pm) = θ m η = τ θ ) τ, τ i + τ j + 2m i j + rm) + ) α i + µ j + 2m i j + rm) + ) = α i + µ j + 2m i j + rm) + ) τ i + 2m i j + rm) + ) 2 α i + 2m i j + rm) + )µ i + 2m i j + rm) + ) which shows that it is equivalent to [dm)] 2 τ α µ as m Indeed, recall rm)+2m = pm)+m = dm) and consider τ i + dm) i j + ) = τ i + dm) i j + ) dm) i j + ) dm) i j + ) Then the terms corresponding to i = are i lτ) lα)+ j m dm) lα) + τ )dm) lα) + τ 2) dm) lα))dm) lα) ) dm) m) dm) lα) ) dm) m) which reduces to Consequently dm) lα) + j) dm) τ, m τ j=0 i lα) lα)+ j m The same reasoning shows that i lα) lα)+ j m i lα) lα)+ j m τ i + dm) i j + ) dm) i j + ) α i + dm) i j + ) dm) i j + ) µ i + dm) i j + ) dm) i j + ) dm) τ, m dm) α, m, dm) µ, m, whence the claimed equivalence follows Summing up, all the terms of the finite sum in the right hand side of formula 5) admit finite its except s α m ) and s µ ) m Since the latter are equivalent to dm) α and to dm) µ respectively as m and due to the presence of alternating signs, taking the it as m in formula 5) leads to an indeterminate it To solve this problem, one needs to seek some cancellations in a similar fashion this was done for the unitary Brownian motion [4]) 2

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