Asymptotic windings of the block determinants of a unitary Brownian motion and related diffusions*

We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.


Introduction
The Brownian motion on the Lie group of complex unitary matrices, in short unitary Brownian motion, is an extensively studied object in random matrix theory (see for instance [8,23] and references therein). In this paper we study several diffusion processes naturally associated to this Brownian motion and show how they can be used to compute the exact or asymptotic distribution of some functionals. In particular, we emphasize the role of the Stiefel fibration over the complex Grassmannian space as an effective computational tool. As an application of our methods we obtain for instance the following result.
be a Brownian motion on the unitary group U(n) with Z t ∈ C k×k , 1 ≤ k ≤ n − 1. Assume that det Z 0 = 0. One has then the polar decomposition det(Z t ) = t e iθt where 0 < t ≤ 1 is a continuous semimartingale such that, in distribution, when t → +∞, where B a,b are independent beta random variables with parameters (a, b) and θ t is a realvalued continuous martingale such that the following convergence holds in distribution where C k(n−k) is a Cauchy distribution of parameter k(n − k).
The study of the limit law of t can be deduced from known results about the complex Jacobi ensemble, see [26]. Therefore, our main contribution lies in the study of the winding process θ t . We note that for k = 1, the theorem therefore yields a Spitzer's type theorem (see [27]) for the windings of each of the entries of a unitary Brownian motion (the choice of bottom left corner block for Z is arbitrary). Overall, the proof of Theorem 1.1 is rather long. An important step, interesting in itself, is to obtain the representation Asymptotic windings of the block determinants of a unitary Brownian motion of [6], we prove that w[0,t] α is a time-changed Brownian motion with independent clock t 0 tr(w * s w s )ds and finally study the distribution of this latter additive functional using a Girsanov transform in the spirit of [6,13,30]. The paper is structured as follows. Section 2 mostly collects preliminary results. We present the geometry of the Stiefel fibration over the complex Grassmannian space, introduce a convenient set of coordinates and show how to obtain representations of the Brownian motions on the Stiefel and complex Grassmannian spaces from a unitary Brownian motion, see Theorem 2.1. We then study some related eigenvalue processes. In Section 3, we take advantage of the Stiefel fibration over the complex Grassmannian space to explicitly write a skew-product decomposition of the Stiefel Brownian motion. Section 4 is devoted to the study of limit laws of some functionals of the unitary Brownian motion and culminates with the proof of Theorem 1.1

Notations:
• If M ∈ C n×n is a n × n matrix with complex entries, we will denote M * = M T its adjoint.
• If z i = x i + iy i is a complex coordinate system, let • Throughout the paper we work on a filtered probability space (Ω, (F t ) t≥0 , P ) that satisfies the usual conditions.
• If X and Y are semimartingales, we denote XdY the Itô integral, X • dY the Stratonovich integral and dXdY or X, Y the quadratic covariation. • For matrix-valued semimartingales M and N , the quadratic variation dM dN is a matrix such that • If M is a semimartingale and η a one-form, then M [0,t] η denotes the Stratonovich line integral of η along the paths of M .

Stiefel fibration
Let n ∈ N, n ≥ 2, and k ∈ {1, . . . , n}. The complex Stiefel manifold V n,k is the set of unitary k-frames in C n . In matrix notation we have As such V n,k is therefore an algebraic compact embedded submanifold of C n×k and inherits from C n×k a Riemannian structure. We note that V n,1 is isometric to the unit sphere S 2n−1 . There is a right isometric action of the unitary group U(k) on V n,k , which is simply given by the right matrix multiplication: M g, M ∈ V n,k , g ∈ U(k). The quotient space by this action G n,k := V n,k /U(k) is the complex Grassmannian manifold. It is a compact manifold of complex dimension k(n−k). We note that G n,k can be identified with the set of k-dimensional subspaces of C n . In particular G n,1 is the complex projective space CP n−1 . Since G n,k and G n,n−k can be identified with each other via orthogonal complement, without loss of generality we will therefore assume throughout the paper that k ≤ n − k , except when explicitly mentioned otherwise (see section 4.4).
Let us quickly comment on the Riemannian structure of G n,k that we will be using and that is induced from the one of V n,k . From Example 2.3 in [4], there exists a unique Riemannian metric on G n,k such that the projection map π : V n,k → G n,k is a EJP 26 (2021), paper 38. Riemannian submersion. From Example 2.5 in [4] and Theorem 9.80 in [7] the fibers of this submersion are totally geodesic submanifolds of V n,k which are isometric to U(k).
This therefore yields a fibration: which is often referred to as the Stiefel fibration, see [3,22]. We note that for k = 1 it is nothing else but the classical Hopf fibration considered from the probabilistic viewpoint in [6]: For further details on the Riemannian geometry of the complex Grassmannian manifolds we also refer to [28,29], see in particular Theorem 4 in [28].

Inhomogeneous coordinates on G n,k
We consider the open set V n,k ⊂ V n,k given by and the smooth map p : V n,k → C (n−k)×k given by p X Z = XZ −1 . It is clear that for every g ∈ U(k) and M ∈ V n,k , p(M g) = p(M ). Since p is a submersion from V n,k onto its image p( V n,k ) = C (n−k)×k we deduce that there exists a unique diffeomorphism Ψ from an open set of G n,k onto C (n−k)×k such that The map Ψ induces a (local) coordinate chart on G n,k that we call inhomogeneous by analogy with the case k = 1 which corresponds to the complex projective space. The Riemannian metric of G n,k is then transported to C (n−k)×k using the map Ψ. In the sequel, we will denote C (n−k)×k endowed with this Riemannian metric by G n,k in order to emphasize the Riemannian structure that is used. Note that by construction G n,k is isometric to an open subset of G n,k and that it differs from G n,k by the sub-manifold at ∞, det(Z) = 0. In the case n = 2, k = 1, G n,k = CP 1 and the above description corresponds to the classical one-point compactification description CP 1 = C ∪ {∞}. We note that the Stiefel fibration that we still refer to as the Stiefel fibration. The projection map p : V n,k → G n,k , X Z → XZ −1 is then a Riemannian submersion with totally geodesic fibers isometric to U(k).

Brownian motions on G n,k and V n,k
In this section, we show how the Brownian motions on G n,k and V n,k can be constructed from a Brownian motion on the unitary group U(n). In the following, we will use the block notations as below: For any U ∈ U(n) and A ∈ u(n) we will write where X, γ ∈ C (n−k)×k , Y, ∈ C (n−k)×(n−k) , Z, α ∈ C k×k , V, β ∈ C k×(n−k) . We recall that the Lie algebra u(n) consists of all skew-Hermitian matrices u(n) = {X ∈ C n×n |X = −X * }, which we equip with the inner product X, Y u(n) = − 1 2 tr(XY ). This induces a Riemannian metric on U(n). Consider now on u(n) a Brownian motion (A t ) t≥0 and the matrix-valued process (U t ) t≥0 that satisfy the Stratonovich stochastic differential equation: The process (U t ) t≥0 is a Brownian motion on U(n) (which is not started from the identity).
If we write the block decomposition then, it is known that both of the processes X * X and Z * Z = I k − X * X belong to the wellknown family of (complex) matrix Jacobi processes that have already been extensively studied in the literature, see for instance [9], [11], [13], [19] and [20]. In particular, one can see that Z * Z satisfies where B is a C k×k Brownian motion.

The process
X t Z t t≥0 is a Brownian motion on V n,k ; 2. We have P (inf{t > 0, det Z t = 0} < +∞) = 0 and the process (w t ) t≥0 := (X t Z −1 t ) t≥0 is a Brownian motion on G n,k with generator 1 2 ∆ G n,k , where Proof. The first part of the theorem is straightforward to prove. Indeed, the map is a Riemannian submersion with totally geodesic fibers, therefore the process (Π(U t )) t≥0 is a Brownian motion on V n,k . We now turn to the proof of the second part of the theorem. We first note that, as noticed above, (Z * t Z t ) t≥0 is a matrix Jacobi process and therefore, from known properties of those processes, P (inf{t > 0, det(Z * t Z t ) = 0} < +∞) = 0. We then turn to the study of w = XZ −1 which is therefore well defined for all times. We need to introduce some notations. Let us consider the block decomposition EJP 26 (2021), paper 38. with α t ∈ C k×k . Note that α t , β t = −γ * t and t are independent. From (2.2) we obtain the following system of stochastic differential equations: Using then Itô's formula, long but routine computations yield From this expression, after further computations, we deduce Therefore (w t ) t≥0 is a diffusion with generator 1 2 ∆ G n,k . Then, to conclude, we note that it is indeed the Brownian motion on G n,k because p is a Riemannian submersion with totally geodesic fibers.

Remark 2.2.
If we consider on C (n−k)×k the probability measure dµ := c n,k det(I k + w * w) −n dw with normalizing constant c n,k , then a direct computation shows that µ is the symmetric and invariant measure for the diffusion (w t ) t≥0 , i.e. if f, g are smooth and compactly supported functions on C (n−k)×k , then The complex Grassmannian G n,k is a compact irreducible symmetric space of rank k and the complex Stiefel manifold is a Riemannian homogeneous space (but is not symmetric). As such, the Brownian motions on G n,k and V n,k and their distributions and pathwise properties can be studied using representation theory and stochastic differential geometry. The literature on those topics is nowadays quite substantial. We for instance refer to the early works by Eugene Dynkin [14,15] and Paul & Marie-Paule Malliavin [24] or more recent presentations like [25] and the book [16] (see in particular Chapter 8: Riemannian submersions and Symmetric spaces). In some sense, Theorem 2.1 provides a more pedestrian approach: We work in a specific choice of coordinates within the algebra of complex matrices and describe the G n,k and V n,k Brownian motions in those coordinates taking advantage of the additional structure given by the matrix multiplication.

Eigenvalue process
In this section, for later use, we collect some properties of the eigenvalues of the We explicitly note that the proof of P(inf{t > 0, det(J t ) = 0} < +∞) = 0 uses the fact that k ≤ n − k.
Remark 2.5. The symmetric and invariant probability measure of the diffusion process J can be easily obtained. Indeed, let w be a random variable on C (n−k)×k whose law is the probability measure with density c n,k det(I k + w * w) −n dw. From Proposition 1 in [17] one has for every bounded Borel function g and some normalization constant c n,k that Thus, from Remark 2.2, the probability measure on the coneĤ k of positive definite Hermitian matrices with density c n,k det(I k + S) −n det(S) n−2k dS is the invariant and symmetric probability measure for the diffusion process J.

Moreover, we have
Under the assumptions of the previous theorem, let us denote ρ i = 1−λi 1+λi , i = 1, . . . , k. Note that ρ is the eigenvalue process of 2ZZ * − I k . Then, as an application of Itô's formula, we have where (B t ) t≥0 is the same Brownian motion as in (2.6). Therefore, ρ is a diffusion process with generator given by EJP 26 (2021), paper 38.
If we consider the Vandermonde function 1]. Therefore (ρ(t)) t≥0 is a Karlin-McGregor diffusion associated to a k-dimensional Jacobi process with independent components and conditioned by its ground state. Using then the well-known Karlin-McGregor formula, see [2], one deduces that for every t > 0, ρ(t) has a density with respect to the Lebesgue measure dx given by where p n−2k,0 t is the transition density of a one-dimensional Jacobi diffusion (see (4.4) for a precise formula) and Moreover, when t → +∞, ρ(t) converges in distribution to the invariant probability where c n,k is again a normalization constant and it might be explicitly computed using the well-known Selberg integral formula (see [1]).

Connection form and horizontal Brownian motion on V n,k
Let us consider the Stiefel fibration that was described in Sections 2.1 and 2.2. According to this fibration, one can see V n,k as a U(k)-principal bundle over G n,k . The next lemma gives a formula for the connection form of this bundle.
Then, ω is the connection form of the bundle U(k) → V n,k → G n,k . Proof. We first observe that if v = X Z ∈ V n,k , then the tangent space to V n,k at v is given by Then, if θ ∈ u k , one easily computes that the generator of the one-parameter group {q → qe tθ } t∈R is given by the vector field on V n,k whose value at v is Xθ Zθ . Applying ω to this vector field yields θ. To show that ω is the connection form it remains therefore to prove that the kernel of ω is the horizontal space of the Riemannian submersion X Z → XZ −1 . This horizontal space at v, say H v , is the orthogonal complement of the vertical space at v, which is the subspace V v of T v V n,k tangent to the fiber of the submersion. The previous argument shows that Therefore we have We deduce from this that from which it is clear that ω |H = 0.
Our next goal is to describe the horizontal lift to V n,k of a Brownian motion on G n,k . We still denote by p : V n,k → G n,k the Riemannian submersion. A continuous semimartingale (M t ) t≥0 on V n,k is called horizontal if for every t ≥ 0, M [0,t] ω = 0, where M [0,t] ω denotes the Stratonovich line integral of ω along the paths of M . If (N t ) t≥0 is a continuous semimartingale on G n,k with N 0 ∈ G n,k , then if N 0 ∈ V n,k is such that p( N 0 ) = N 0 , there exists a unique horizontal continuous semimartingale ( N t ) t≥0 on V n,k such that p( N t ) = N t for every t ≥ 0. The semimartingale ( N t ) t≥0 is then called the horizontal lift at N 0 of (N t ) t≥0 to V n,k . We refer to [5] or [16] for a more general description of the horizontal lift of a semimartingale in the context of foliations.
We then consider on G n,k the u(k)-valued one-form η given by be a Brownian motion on G n,k started at w 0 ∈ G n,k as in is the horizontal lift at X 0 Z 0 of (w t ) t≥0 to V n,k , where (Θ t ) t≥0 is the U(k)-valued solution of the Stratonovich stochastic differential equation Proof. As before we denote by p the submersion X Z → XZ −1 . It is easy to check that for every t ≥ 0, p( w t ) = w t and that w 0 = X 0 Z 0 . It is therefore enough to prove that w is A long, but routine, computation shows that and thus w[0,t] ω = 0.

Skew-product decomposition of the Stiefel Brownian motion
We now turn to the description of the Brownian motion on V n,k as a skew-product. Theorem 3.3. Let (w t ) t≥0 be a Brownian motion on G n,k started at w 0 = X 0 Z −1 0 ∈ G n,k as in Theorem 2.1 and let (Ω t ) t≥0 be a Brownian motion on the unitary group U(k) independent from (w t ) t≥0 . Let (Θ t ) t≥0 be the U(k)-valued solution of the Stratonovich stochastic differential equation is a Brownian motion on V n,k started at X 0 Z 0 . EJP 26 (2021), paper 38.
Proof. We denote by ∆ H the horizontal Laplacian and by ∆ V the vertical Laplacian of the Stiefel fibration; see [4] for the definitions of horizontal and vertical Laplacians. Since the submersion V n,k → G n,k is totally geodesic, the operators ∆ H and ∆ V commute (see [4]). We note that the Laplace-Beltrami operator of V n,k is given by ∆ V n,k = ∆ H + ∆ V and that the horizontal lift of the Brownian motion on G n,k is a diffusion with generator 1 2 ∆ H , see [5]. The fibers of the submersion V n,k → G n,k are isometric to U(k), thus if f is a bounded Borel function on V n,k , one has

Limit theorems
Throughout the section, let (w t ) t≥0 = (X t Z −1 t ) t≥0 be a Brownian motion on G n,k where X t Z t t≥0 is a Brownian motion on V n,k . Our goal is to prove Theorem 1.1. Without loss of generality we will assume throughout the section that the eigenvalues of Z * 0 Z 0 are distinct; Even if the eigenvalues of the complex Jacobi process Z * t Z t are not distinct for t = 0, they will be distinct for any t > 0, see [9,11], and thus from the Markov property, the limit Theorem 1.1 still holds.

Main limit theorem
We first give a limit theorem for the process t 0 tr (w * s w s )ds) t≥0 that shall be used in the next subsections. The method we use, a Girsanov transform, takes its root in the paper by M. Yor [30] and was further developed in the situation of matrix Wishart diffusions in [12] and in the situation of the real Jacobi matrix processes in Section 9.4.2 of the thesis [13]. Our result is the following: (therefore X is the inverse of a gamma distributed random variable).
The proof is rather long and will be decomposed in several steps. We first recall that from Lemma 2.4, there exists a Brownian motion (B t ) t≥0 in C k×k such that:
Hence the first order term writes det(I k + J)tr((I k + J) −1 dJ). Next, we will use the following formula to compute the cross second order derivatives: Moreover, from (4.1) we know that Hence we have d(det(I k + J)) = det(I k + J)tr((I k + J) −1 dJ) As a direct consequence of d det(I k + J), det(I k + J) = 4 det(I k + J) 2 tr(J)dt, we obtain and thus From this expression, it is clear that there exists a constant C > 0 such that we almost surely have |M α t | ≤ Ce 2kα(n−k)t and thus the process (M α t ) t≥0 is a martingale.
In the next lemma, we provide a formula for the Laplace transform of the functional t 0 tr(J)ds using a Girsanov transform. In this computation, the transition kernel of one-dimensional Jacobi diffusions naturally appears. We recall the formula for this transition kernel. If we denote by p a,b t (x, y) the transition density, with respect to the Lebesgue measure, of the diffusion with generator is the normalization constant, λ 1 (0), · · · , λ k (0) are the ordered eigenvalues of J 0 , p n−2k,2α t is given by the formula (4.4) and Proof. Let α ≥ 0 and consider the probability measure P α defined by We first note that From Girsanov theorem, the process is under P α a k × k-matrix-valued Brownian motion and we have We now denote by λ(t) = (λ i (t)) 1≤i≤k the eigenvalues of J t , t ≥ 0. From the previous equation satisfied by J we deduce that there exists a Brownian motion (B t ) t≥0 in R k for the probability measure P α such that Using then the formula for the density of non-colliding Jacobi processes, see [2], we deduce that the process (ρ(t)) t≥0 has, under P α , a density with respect to the Lebesgue measure dx given by We conclude then with (4.5).
We are now ready for the proof of Theorem 4.1.
Proof of Theorem 4.1. From Lemma 4.4, we have for every α ≥ 0 and t > 0 In order to analyze the large time behavior of this Laplace transform we we will use the formula (4.4). We can write p n−2k,2α t (x, y) Similarly to [11], or Section 3.9.1 in [2], denoting as before ρ i (0) = 1−λi(0) 1+λi(0) we now compute det p n−2k,2α By skew-symmetrization, we can rewrite the previous sum as Let us note that when t → +∞, the term of leading order in this sum corresponds to (m 1 , · · · , m k ) = (0, 1, · · · , k − 1) and is given by On the other hand, from (4.7) one has for every λ ≥ 0 and t > 0 that and thereforeC = 1. One now concludes using an inverse Laplace transform that the following convergence takes place in distribution when t → +∞ . We incidentally note that X is therefore distributed as the hitting time of k(n − k) by a one-dimensional Brownian motion, even though this does not seem to be readily explainable.

Asymptotics of a generalized stochastic area
By the definition of the one-form η in (3.3), we note that where as before J = w * w. From simple computations one can verify that tr(dη) = ∂∂ log det(I k + w * w), which implies that itr(dη) is the Kähler form on G n,k . Therefore i w[0,t] tr(η) can be, in some sense, considered as a generalized stochastic area process on G n,k ; we refer to [6] for further explanation on the terminology of generalized stochastic area. In the proposition below we deduce large time limit distributions of such functionals.
Proposition 4.5. The following convergence holds in distribution when t → +∞ where C k(n−k) is a Cauchy distribution of parameter k(n − k).
We obtain Therefore from (4.10), we have in distribution that where B is a one-dimensional Brownian motion independent from the process tr(J). Therefore, for every λ > 0, We conclude then from (4.9) after straightforward computations

Asymptotic windings
We are now interested in the windings of the complex valued process det(Z t ). We first note that from Theorem 3.3, we have identity in law We shall then use the following result.
Lemma 4.6. Let G be a connected compact Lie group of m×m matrices with Lie algebra g. Let (M t ) t≥0 be a g-valued continuous semimartingale such that M 0 = 0 and let (C t ) t≥0 be the G-valued solution of the Stratonovich stochastic differential equation Then, for t ≥ 0, det C t = (det C 0 ) exp (tr(M t )).
Proof. Let T > 0. Consider on the time interval [0, T ] the sequence of G-valued semimartingales (C n t ) 0≤t≤T inductively defined by where t k = kT 2 n , k = 0, . . . , 2 n . From Theorem 2 in [21], the sequence of semimartingales (C n t ) 0≤t≤T converges in probability to (C t ) 0≤t≤T uniformly on [0, T ]. However, det(C n t ) = det(C n t k ) exp We deduce therefore by induction that det(C n T ) = (det C 0 ) exp (tr(M T )) .
Letting then n → +∞ yields the conclusion.
Using the previous lemma, we deduce the following: Thus from Lemma 4.6 we have det Θ t = det Z0 | det Z0| exp tr w[0,t] η .
We are now finally in position to prove one of our main results.  where 0 < t ≤ 1 is a continuous semimartingale and θ t is a continuous martingale such that the following convergence holds in distribution when t → +∞ where C k(n−k) is a Cauchy distribution of parameter k(n − k).
Proof. From the decomposition det(Z t ) = det(I k + w * t w t ) −1/2 det Θ t det Ω t one deduces from Lemmas 4.6 and 4.7 that where D t is a Brownian motion on u(k) independent from w and θ 0 is such that e iθ0 = det Z0 | det Z0| . The conclusion follows then from Proposition 4.5.
Let us remark that it is also possible to compute the asymptotic law of the radial part t . Indeed, from the previous proof, we know that t = det(I k + J t ) −1/2 and the limit distribution of the ordered eigenvalues of (I k − J)(I k + J) −1 is computed explicitly in (2.7) to be a distribution with density Using then the Selberg's integral formula, one obtains that in distribution one has t → ∞ , where ρ ∞ is a random variable such that for every s ≥ 0 wherec n,k is a normalization constant. Thus, using uniqueness of the Mellin transform, one concludes that where B j,n−k are independent beta random variables with parameters (j, n − k). This recovers a result by A. Rouault (Proposition 2.4 in [26]).

The case k ≥ n − k
In this section, we prove Theorem 1.1 in the case of k ≥ n − k. Thus, unlike the rest of the paper, we assume in this section that k ≥ n − k. This is essentially a duality argument equivalent to the isomorphism G n,k G n,n−k . Let EJP 26 (2021), paper 38.
with Z ∈ C k×k , det(Z) = 0. Using X * X + Z * Z = I k and XX * + Y Y * = I n−k we deduce the following equality of spectrum sp(Z * Z) = sp(Y Y * ) ∪ {1} and that the eigenvalue 1 of Z * Z has multiplicity at least 2k − n. Consider now a Brownian motion with Z 0 ∈ C k×k , det(Z 0 ) = 0. The process (U * t ) t≥0 is also a Brownian motion on U(n). Therefore, when t → +∞, For the study of the winding process of det(Z t ), we notice that in the case k ≤ n − k the only part of the proof of Theorem 4.8 that actually uses the fact that k ≤ n − k is the proof of Theorem 4.1 (in the case k > n − k the stochastic differential equation for J = w * w is only defined up to the hitting time inf{t ≥ 0, det J t = 0} < +∞ and the Girsanov transform method fails). To handle the case k ≥ n − k, we note that trJ = tr[(ZZ * ) −1 −I k ] = tr[(Y Y * ) −1 −I n−k ]. Since the process (U * t ) t≥0 is also a Brownian motion on U(n), one can use the argument of the proof of Theorem 4.1 to deduce that . Therefore, the conclusion of Theorem 4.8 still holds in the case of k ≥ n − k.

On the moments of det Z t
To conclude the paper with a possible view toward free probability that concerns the limit n → ∞ with k = αn (see [10]) we show that it is possible to give a formula for the mixed moments of the process det Z t that can be deduced from the previous computations. Let be a Brownian motion on the unitary group U(n) with Z t ∈ C k×k , 1 ≤ k ≤ n − 1.
Assume that det Z 0 = 0. Let now p, q ∈ N. As before, from the decomposition det(Z t ) = det(I k + w * t w t ) −1/2 det Θ t det Ω t one has from Lemmas 4.6 and 4.7 that det(Z t ) = t e iθt with t = det(I k + J t ) −1/2 , iθ t = iθ 0 + tr(D t ) + where D t is a Brownian motion on u(k) independent from w and θ 0 is such that e iθ0 = det Z0 | det Z0| . Therefore, using the martingale of Lemma 4.3 and the Girsanov transform as in Lemma 4.4, we obtain: where C is a normalization constant depending only on p, q and the ordered eigenvalues of J 0 , and p n−2k,|p−q| t is given by the formula (4.4). We note that if q = 0 the formula simplifies considerably and yields E ((det Z t ) p ) = (det Z 0 ) p e −kp 2 t−k(n−k)pt .