1 Introduction

A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

\ArticleName

A Central Limit Theorem for Random Walks
on the Dual of a Compact Grassmannian

\Author

Margit RÖSLER  and Michael VOIT

M. Rösler and M. Voit

Warburger Str. 100, D-33098 Paderborn, Germany \EmailDroesler@math.upb.de

Fakultät für Mathematik, Technische Universität Dortmund,
Vogelpothsweg 87, D-44221 Dortmund, Germany \EmailDmichael.voit@math.uni-dortmund.de

\ArticleDates

Received October 14, 2014, in final form February 03, 2015; Published online February 10, 2015

\Abstract

We consider compact Grassmann manifolds over the real, complex or quaternionic numbers whose spherical functions are Heckman–Opdam polynomials of type . From an explicit integral representation of these polynomials we deduce a sharp Mehler–Heine formula, that is an approximation of the Heckman–Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of , which are constructed by successive decompositions of tensor powers of spherical representations of . The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.

\Keywords

Mehler–Heine formula; Heckman–Opdam polynomials; Grassmann manifolds; spherical functions; central limit theorem; asymptotic representation theory

\Classification

33C52; 43A90; 60F05; 60B15; 43A62; 33C80; 33C67

1 Introduction

For Riemannian symmetric spaces of the compact or non-compact type, there is a well-known contraction principle which states that under suitable scaling, the spherical functions of tend to the spherical functions of the tangent space of in the base point, which is a symmetric space of the flat type:

 limn→∞φnλ(exp(x/n))=ψλ(x).

See [4] and, for a more recent account, [2]. This curvature limit, also known as Mehler–Heine formula, extends to the more general setting of hypergeometric functions associated with root systems, which converge under rescaling to generalized Bessel functions. This is proven in [6] by a limit transition in the Cherednik operators; see also [2] for a different approach. In the compact rank one case, the contraction principle is a weak version of the classical Hilb formula for Jacobi polynomials (see [27, Theorem 8.21.12]), which provides in addition a precise estimate on the rate of convergence. In this paper, we prove a Mehler–Heine formula with a precise estimate on the error term for a certain class of orthogonal polynomials associated with root systems, which in particular encompasses the spherical functions of compact Grassmannians. This result is a “compact” analogue of Theorem 5.4 in [26], which gives a scaling limit with error bounds for hypergeometric functions in the dual, non-compact setting. In the second part of the paper, we shall use the Mehler–Heine formula 2.4 in order to establish a central limit theorem for random walks on the semi-lattice of dominant weights parametrizing the unitary dual of a compact Grassmannian.

To become more precise, we consider the compact Grassmannians over one of the (skew-) fields , with and , where . Via polar decomposition of , the double coset space may be topologically identified with the fundamental alcove

 A0:={x=(x1,…,xq)∈Rq:π2≥x1≥x2≥⋯≥xq≥0},

with being identified with the matrix

 ax=⎛⎜⎝cosx––−sinx––0sinx––cosx––000Ip−q⎞⎟⎠.

Here we use the diagonal matrix notation , and the functions , are understood component-wise. For details, see [23, Theorem 4.1]. The spherical functions of  can be viewed as Heckman–Opdam polynomials of type , which are also known as multivariable Jacobi polynomials. They may be described as follows: denote by  the Heckman–Opdam hypergeometric function associated with the root system

 R=2BCq={±2ei,±4ei:1≤i≤q}∪{±2ei±2ej:1≤i

with spectral variable and multiplicity parameter corresponding to the roots , and . Fix the positive subsystem

 R+={2ei,4ei,1≤i≤q}∪{2ei±2ej,1≤i

and the associated semi-lattice of dominant weights,

 P+={λ∈(2Z)q:λ1≥λ2≥⋯≥λq≥0}.

Then the set of spherical functions of is parametrized by and consists of the functions

 φpλ(ax)=FBC(λ+ρp,k(p),ix)=:Rpλ(x),λ∈P+ (1.1)

with multiplicity parameter

 k(p)=(d(p−q)/2,(d−1)/2,d/2), (1.2)

where and

 ρp=12∑α∈R+kαα=q∑i=1(d2(p+q+2−2i)−1)ei.

The functions are the Heckman–Opdam polynomials associated with the root system  (called Jacobi polynomials in the following) and with multiplicity , normalized according to . We refer to [12, 13, 22] for Heckman–Opdam theory in general, and to [23] and the references cited there for the connection with spherical functions in the compact case. Notice that our notion of  coincides with that of Heckman, Opdam and [25, 26], while it differs from the geometric notion in [23]. Theorem 4.3 of [23] corresponds to (1.1).

In Theorem 4.2 of [23], the product formula for spherical functions of was written as a formula on  and analytically extended to a product formula for the Jacobi polynomials  with multiplicity  corresponding to arbitrary real parameters . This led to three continuous series of positive product formulas for Jacobi polynomials corresponding to and to associated commutative hypergroup structures on ; see [15] and [3] for the notion of hypergroups. Using a Harish-Chandra-type integral representation for the , we shall derive a Mehler–Heine formula with a precise asymptotic estimate for the Jacobi polynomials  in terms of Bessel functions associated with root system on the Weyl chamber

 C={x=(x1,…,xq)∈Rq:x1≥⋯≥xq≥0}.

This Mehler–Heine formula will be the key ingredient for the main result of the present paper, a central limit theorem for random walks on the semi-lattice , which parametrizes the spherical unitary dual of . To explain this CLT, let us first recall that via the GNS representation, the spherical functions of , which are necessarily positive definite, are in a one-to-one correspondence with the (equivalence classes of) spherical representations of , that is those irreducible unitary representations of  whose restriction to  contains the trivial representation with multiplicity one, see [9] or [14, Chapter IV]. The decomposition of tensor products of spherical representations into their irreducible components leads to a probability preserving convolution  and finally a Hermitian hypergroup structure on the discrete set ; see [7] and [18]. Following, e.g., [3, 29, 32], we introduce random walks on associated with and derive some limit theorems for . The main result of this paper will be the Central Limit Theorem 3.12. This CLT implies the following result for :

Theorem 1.1.

Let be a non-trivial spherical representation of  associated with . Let be -invariant with . For each , decompose the -fold tensor power into its finitely many irreducible unitary components

 (π⊗,nλ,H⊗,nλ)=(⨁τnπτn,⨁τnHτn),

where the components are counted with multiplicities. Consider the orthogonal projections and a -valued random variable with the finitely supported distribution

 ∑τn∥∥pτn(u⊗,nλ)∥∥2δτn∈M1(P+)

with the point measures at . Then, for ,

 Xn,λm(λ)√n

tends in distribution to

 dρd,p(x)=c−1d,pq∏j=1xd(p−q+1)−1j∏1≤i

with a suitable normalization . Notice that the probability measure is the distribution of a Laguerre ensemble on . The modified variance parameter is a second order polynomial in  and given explicitly in Lemma 3.3 below.

For , the Central Limit Theorem 3.12 has a long history as a CLT for random walks on  whose transition probabilities are related to product linearizations of Jacobi polynomials. This includes random walks on the duals of and in [8] and [11]. See also [29] for further one-dimensional cases. For our results are very closely related to the work [5] of Clerc and Roynette on duals of compact symmetric spaces. For a survey on limits for spherical functions and CLTs in the non-compact case for we refer to [31].

2 A Mehler–Heine formula

In this section we derive a Mehler–Heine formula for the Jacobi polynomials , describing the approximation of these polynomials in terms of Bessel functions with a precise error bound. Our result will be based on Laplace-type integrals for the Jacobi polynomials and the associated Bessel functions, where we treat the group cases with integers as well as the case with beyond the group case. The integral representation for below is a special case of a more general Harish-Chandra integral representation for hypergeometric functions in [26]. To start with, let us introduce some notation:

Let denote the space of Hermitian matrices over , and denote by the determinant of , which may be defined as the product of (right) eigenvalues of . We mention that for , this is just the Moore determinant, which coincides with the Dieudonné determinant if  is positive semi-definite, see, e.g., [1]. On , we consider the power functions

 Δλ(a):=Δ1(a)λ1−λ2⋯Δq−1(a)λq−1−λqΔq(a)λq,λ∈Cq,

with the principal minors of the matrix , see [10]. We introduce the matrix ball , where means for matrices that is (strictly) positive definite. On , we define the probability measures

 dmp(w)=1κpd/2Δ(I−w∗w)pd/2−γdw∈M1(Bq),

with , . Here is the Lebesgue measure on the ball ,

 γ:=d(q−12)+1

and

 κpd/2=∫BqΔ(I−w∗w)pd/2−γdw.

According to Theorem 2.4 of [26], the Heckman–Opdam hypergeometric function with , and as in (1.2) has the following integral representation for with :

 FBC(λ,k(p),x)=∫Bq×U0(q,F)Δ(λ−ρp)/2(gx(u,w))dmp(w)du,

where denotes the identity component of and

 gx(u,w)=u−1(coshx––+w∗sinhx––)(coshx––+sinhx––w)u.

It is easily checked that may be replaced by in the domain of integration. Notice further that extends to a holomorphic function on . As the principal minors  are polynomial in the entries of , it follows that extends to a holomorphic function on  for each . In view of relation (1.1), this leads to the following integral representation for the Jacobi polynomials :

Proposition 2.1.

Let with and with . Then the Jacobi polynomials , , have the integral representation

 Rpλ(x)=∫Bq×U(q,F)Δλ/2(gix(u,w))dmp(w)duforx∈A0 (2.1)

with

 gix(u,w)=u−1(cosx––+w∗isinx––)(cosx––+isinx––w)u.

We next turn to the Bessel functions which will show up in the Mehler–Heine formula. They are given in terms of Bessel functions of Dunkl type which generalize the spherical functions of Cartan motion groups; see [6] and [21] for a general background. We denote by the Bessel function which is associated with the rational Dunkl operators for the root system and multiplicity corresponding to the roots and . We shall be concerned with multiplicities which are connected as follows to the multiplicities from (1.2):

 k=(k1,k2)withk1=k(p)1+k(p)2=d(p−q+1)/2−1/2,k2=k(p)3=d/2.

For such  on , we use the notion

 ˜φpλ(x):=JBk(x,iλ),x∈C,λ∈Cq.

It is well-known that for integers , the are the spherical functions of the Euclidean symmetric spaces , where and is the Cartan motion group associated with the Grassmannian . Hereby the double coset space is identified with the Weyl chamber  such that corresponds to the double coset of , and in this way, -biinvariant functions on may be considered as functions on . Two functions and coincide if and only if  and  are in the same Weyl group orbit. Finally, the bounded spherical functions are exactly those with . The Bessel functions with and not necessarily integral parameter  are closely related to Bessel functions on the symmetric cone of positive definite -matrices over , see Section 4 of [24]. It has been shown there that for , they have a positive product formula which generalizes the product formula in the Cartan motion group cases and leads to a commutative hypergroup structure on the Weyl chamber .

Lemma 2.2.

For with , the Bessel functions with have the following integral representation:

 ˜φpλ(x)=∫Bq∫U(q,F)eiRetr(wx––uλ––)dmp(w)du. (2.2)
Proof.

This follows readily from equations (3.12) and (4.4) in [24]; see also Proposition 5.3 of [26]. ∎

Remark 2.3.

There are finitely many geometric cases which are not covered by the range , namely the indices . In these cases, the Jacobi polynomials  and the Bessel functions both admit interpretations as spherical functions and have an integral representation similar to that above, by the following reasoning: According to Lemma 2.1 of [25], the measure with , is just the pushforward measure of the normalized Haar measure on under the mapping

 v↦σ∗0vσ0,withσ0=(Iq0(p−q)×q)∈Mp,q(F).

For , we now define the measure in the same way as a pushforward measure of the Haar measure on . (But in contrast to the case , it will not have a Lebesgue density in these cases). From the integral representations (3.3) and (4.4) of [24] for the Bessel functions, as well as Theorem 2.1 of [26] and relation (1.1) between Jacobi polynomials and hypergeometric functions, one obtains that the integral representations of Proposition 2.1 and Lemma 2.2 extend to the case .

We shall now compare the integral representations of Proposition 2.1 and Lemma 2.2, which will lead to the following quantitative Mehler–Heine (or Hilb-type) formula.

Theorem 2.4.

There exist constants such that for all , all , and ,

 ∣∣Rpλ(x)−˜φpλ(x)∣∣≤C1x21λ1eC2x21λ1.

Thus in particular,

 ∣∣Rpnλ(xn)−˜φpλ(x)∣∣≤C1nx21λ1eC2x21λ1/n→0forn→∞.

Notice that the estimate of Theorem 2.4 is uniform in , a fact which was to our knowledge so far not even noticed in the rank-one case. We conjecture that the statement of this theorem remains correct for .

Proof.

We only consider the case where the proof is based on Proposition 2.1 and Lemma 2.2. By the previous remark, the cases can be treated in the same way. Notice that it suffices to check uniformity in  for .

We substitute in the integral (2.2) and obtain

 ˜φpλ(x)=∫Bq×U(q,F)eiRetr(u∗w∗x––uλ––)dmp(w)du.

Denoting the trace of the upper left -block of a -matrix by , we have

 Retr(u∗w∗x––uλ––)=12q∑r=1(u∗((x––w)∗+x––w)u)rrλr Retr(u∗w∗x––uλ––)=q∑r=1[trr(u∗((x––w)∗+x––w)u)−trr−1(u∗((x––w)∗+x––w)u)]λr/2 Retr(u∗w∗x––uλ––)=q∑r=1trr(u∗((x––w)∗+x––w)u)(λr−λr+1)/2

with . Hence

 ˜φpλ(x)=∫Bq×U(q,F)q∏r=1eitrr(u∗((xw)∗+xw)u)(λr−λr+1)/2dmp(w)du.

Furthermore, by Proposition 2.1,

 Rpλ(x)=∫Bq×U(q,F)q∏r=1Δr(gix(u,w))(λr−λr+1)/2dmp(w)du.

Telescope summation yields the well-known estimate

 ∣∣ ∣∣q∏r=1ar−q∏r=1br∣∣ ∣∣≤(max(|a1|,…,|aq|,|b1|,…,|bq|))q−1q∑r=1|ar−br|

for . We thus obtain

 ∣∣Rpλ(x)−˜φpλ(t)∣∣≤q∑r=1∫Bq×U(q,F)M(x,u,w,λ) (2.3) ∣∣Rpλ(x)−˜φpλ(t)∣∣≤×∣∣Δr(gx(u,w))(λr−λr+1)/2−eitrr(u∗((t–w)∗+t–w)u)(λr−λr+1)/2∣∣dmp(w)du

with

 M(x,u,w,λ):=max(1,maxr=1,…,q∣∣Δr(gx(u,w))(λr−λr+1)/2∣∣q−1).

We now investigate more closely. As , , run through compacta, we obtain that uniformly in , , ,

 gix(u,w)=u−1(cosx––+w∗isinx––)(cosx––+isinx––w)u gix(u,w)=u−1(Iq+w∗ix––+O(x2))(Iq+ix––w+O(x2))u gix(u,w)=Iq+u−1(ix––w+w∗ix––)u+O(x2),

and thus

 Δr(gx(u,w))=1+trr(u−1(ix––w+w∗ix––)u)+O(x2). (2.4)

Using the power series for , we further have

Notice that is skew-Hermitian, that is . Therefore , which implies that . It follows that

 ∣∣Δr(gix(u,w))(λr−λr+1)/2∣∣=exp[12(λr−λr+1)Re(trr(y)+O(x2))]=e(λr−λr+1)O(x2).

Note that these considerations apply for all fields . It follows that there exists a constant (independent of , , , ) such that

 M(x,u,w,λ)≤eC3x21λ1% for allx∈A0, λ∈P+, u∈U(q,F), w∈Bq.

From this inequality we obtain by the mean value theorem that for all and ,

 ∣∣Δr(gx(u,w))(λr−λr+1)/2−eitrr(u∗((x––w)∗+x––w)u)(λr−λr+1)/2∣∣ ≤eC3x21λ1−1≤C3x21λ1eC3x21λ1.

These estimates together with (2.3) imply the assertion. ∎

Example 2.5 (the rank one case).

For the Jacobi polynomials  associated with root system are classical one-variable Jacobi polynomials in trigonometric parametrization. For integers , the associated Grassmannians are the projective spaces . For the details, recall that the classical Jacobi polynomials with the normalization are given by

 R(α,β)n(x)=2F1(α+β+n+1,−n;α+1;(1−x)/2), (2.5)

where , . It is easily derived from the example on p. 17 of [22] that

 Rpλ(x)=R(α,β)λ/2(cos2x) (2.6)

for , with

 α=(dp−2)/2,β=(d−2)/2;

see also [23, Section 5]. In the rank one case, the integral in representation (2.1) cancels by invariance of  under unitary conjugation. Thus (2.1) reduces to

 Rpλ(x)=1κpd/2∫B1((cosx+¯¯¯¯wisinx)(cosx+isinxw))λ/2(1−|w|2)d(p−1)/2−1dw

for , . In particular, if , then and . Thus

 R(p/2−1,−1/2)n(cos2x)=1κp/2∫1−1(cosx+itsinx)2n(1−t2)(p−3)/2dt.

If , then and . Using polar coordinates , one obtains

The quaternionic case can be treated in a similar way. These formulas are just special cases of a well-known Laplace-type integral representation for Jacobi polynomials with general indices ; see, e.g., [20, Section 18.10].

Let us finally mention that the Mehler–Heine formula 2.4 corresponds to [27, Theorem 8.21.12] and that in the case of rank two (), the Jacobi polynomials of type BC were first studied by Koornwinder [16, 17].

3 Random walks on the dual of a compact Grassmannian and on P+

Recall that for integers the functions , form the spherical functions of the compact Grassmannians . As functions on , they are positive-definite. In other words, the Jacobi polynomials are just the hypergroup characters of the compact double coset hypergroups . We now recapitulate the associated dual hypergroup structures on .

3.1 Dual hypergroup structures on P+

As mentioned in the introduction, there is a one-to-one correspondence between the set of (positive definite) spherical functions of , which is parametrized by , and the spherical unitary dual of , i.e., the set of all equivalence classes of irreducible unitary representations  of  whose restriction to  contains the trivial representation with multiplicity one. Here a representation and its spherical function are related by

 φπ(x)=⟨u,π(x)u⟩forx∈G

with some -invariant vector with , which is determined uniquely up to a complex constant of absolute value 1.

Now consider with associated spherical functions , and the corresponding representations with -invariant vectors , . The tensor product is a finite-dimensional unitary representation of  which decomposes into a finite orthogonal sum

 (⊕kτk=πλ⊗πμ,⊕k˜Hk=Hλ⊗Hμ)

of irreducible unitary representations where some of them may appear several times. Consider the orthogonal projections . Then the vectors are -invariant, and for , we obtain , i.e., is equal to some , . Moreover, for ,

 φλ(g)φμ(g)=⟨uλ⊗uμ,(πλ⊗πμ)(g)uλ⊗uμ⟩ φλ(g)φμ(g)=∑k⟨pk(uλ⊗uμ),τk(g)pk(uλ⊗uμ)⟩=∑k∥pk(uλ⊗uμ)∥2φτk(g)

with . For we now define as , whenever appears above with a positive part, and otherwise. As for all  in our case, these nonnegative linearization coefficients also satisfy

 cλ,μ,τ=dimHτ∫Gφλ(g)φμ(g)φτ(g)dg.

For we define the probability measure

 δλ∗d,pδμ:=∑τ∈P+cλ,μ,τδτ∈M1(P+) (3.1)

with finite support. By its very construction, this convolution can be extended uniquely in a weakly continuous, bilinear way to a probability preserving, commutative, and associative convolution on the Banach space of all bounded, signed measures on . Moreover, as all spherical functions are -valued in our specific examples, the contragredient representation of any element in is the same representation, i.e., the canonical involution on is the identity. In summary, is a commutative Banach--algebra with the complex conjugation as involution. Moreover, becomes a so-called Hermitian hypergroup in the sense of Dunkl, Jewett and Spector; see [3, 7, 15].

The Haar measure on this hypergroup, which is unique up to a multiplicative constant, is the positive measure with

 h(λ)=c−1λ,λ,0=∫Gφ2λ(g)dg=1dimHλ,

where the first two equations follow from general hypergroup theory (see [15]) and the last one from the theory of Gelfand pairs (see, e.g., [9]).

The coefficients of the convolution on are related to the unique product linearization

 RpλRpμ=∑τ∈P+cλ,μ,τRpτ

of the Jacobi polynomials . It is clear by our construction that for integers , all are nonnegative with .

Clearly, as for all , the normalization also holds for all real . We conjecture that actually all are nonnegative for all or at least for all .

Suppose that for fixed the linearization coefficients are all nonnegative. Then equation (3.1) defines a commutative discrete hypergroup structure with the convolution

 δλ∗d,pδμ:=∑τ∈P+cλ,μ,τδτ∈M1(P+)

of point measures. For instance, in the rank one case of Example 2.5 the linearization coefficients  are explicitly known and nonnegative for all as the product linearization coefficients of the associated one-dimensional Jacobi polynomials.

3.2 Random walks on P+

We next introduce certain random walks on , i.e., time-homogeneous Markov chains on whose transition probabilities are given in terms of the product linearizations coefficients for some fixed . This concept even works, under a suitable restriction, in the case where some of the are negative. To describe the restriction, we fix