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Quantum ergodicity of random orthonormal bases of spaces of high dimension Steve Zelditch rsta.royalsocietypublishing.org
Research Cite this article: Zelditch S. 2014 Quantum ergodicity of random orthonormal bases of spaces of high dimension. Phil. Trans. R. Soc. A 372: 20120511. http://dx.doi.org/10.1098/rsta.2012.0511
One contribution of 13 to a Theo Murphy Meeting Issue ‘Complex patterns in wave functions: drums, graphs and disorder’.
Subject Areas: analysis, statistics Keywords: quantum ergodcity, laplace eigenfunctions, random orthonormal basis Author for correspondence: Steve Zelditch e-mail: [email protected]
Department of Mathematics, Northwestern University, Evanston, IL 60208-2370, USA We consider a sequence HN of finite-dimensional Hilbert spaces of dimensions dN → ∞. Motivating examples are eigenspaces, or spaces of quasimodes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of HN may be identified with U(dN ), and a random orthonormal basis of N HN is a choice of a random sequence UN ∈ U(dN ) from the product of normalized Haar measures. We prove that if dN → ∞ and if (1/dN )Tr A|HN tends to a unique limit state ω(A), then almost surely an orthonormal basis is quantum ergodic with limit state ω(A). This generalizes an earlier result of the author in the case where HN is the space of spherical harmonics on S2 . In particular, it holds on the flat torus Rd /Zd if d ≥ 5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.
1. Introduction The purpose of this article is to prove a general result on the quantum ergodicity of random orthonormal N of finite-dimensional Hilbert spaces HN ⊂ bases {ψN,j }dj=1 L2 (M) of dimensions dN → ∞ of a compact Riemannian manifold (M, g). The proof is based on a ‘moment polytope’ interpretation of quantum ergodicity from [1]: the quantum variances of a Hermitian observable A ∈ Ψ 0 (M) (where Ψ 0 (M)) is the class of pseudo-differential operators of degree zero) are identified with moments of inertia of the convex polytopes Pλ defined as the convex hull of the vectors λ = (λ1 , . . . , λdN ) of eigenvalues (in all possible orders) of ΠN AΠN where ΠN : L2 (M) → HN is the orthogonal projection. Equivalently, Pλ is the image of the coadjoint orbit Oλ of the diagonal matrix D(λ) under the moment map for the Hamiltonian action of the
2013 The Author(s) Published by the Royal Society. All rights reserved.
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(indexed by A ∈ Ψ 0 (M)) by A ({ψNk }) := VN
dN 1 |AψN,j , ψN,j − ω(A)|2 . dN
(1.2)
j=1
Here, ω(A) = S∗ M σA dμL , where dμL is normalized Liouville measure (of mass one), and σA is the principal symbol of A. Definition 1.1. A sequence {ψNj }N of ONBs of HN is a quantum ergodic ONB of L2 (M) if (EP )
N 1 A n Vn ({ψn,j }dj=1 ) = 0, N→∞ N
lim
∀A ∈ Ψ 0 (M).
(1.3)
n=1
We say sequence {ψN,j }N is fine quantum ergodic if (F EP )
n lim VA ({ψN,j }dj=1 )) = 0,
N→∞
∀A ∈ Ψ 0 (M).
(1.4)
By a standard diagonal argument, this implies a subsequence of density of one of the individual elements AψN,j , ψN,j tends to ω(A). The fine notion of density is of course stronger, because there
.........................................................
to HN of a pseudo-differential operator A ∈ Ψ 0 (M); here ΠN is the orthogonal projection to HN and Ψ 0 (M) is the space of pseudo-differential operators of order zero. The same methods and results apply to other contexts such as semiclassical pseudo-differential operators or to Toeplitz operators on holomorphic sections of powers of a positive line bundle [3]. Given an orthonormal N of HN we define the (normalized) quantum variances of the ONB basis (ONB) {ψN,j }dj=1
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maximal torus TdN ⊂ U(dN ) of diagonal matrices acting by conjugation on Oλ . In particular, the main estimates of quantum ergodicity can be formulated in terms of estimates of the first four moments of inertia of Pλ . The main result, theorem 1.3, states that random orthonormal bases are almost surely quantum ergodic as long as dN → ∞ and (1/dN )Tr ΠN AΠN → ω(A) for all A ∈ Ψ 0 (M), where ω(A) is the Liouville state. More generally, if these traces have any unique limit state, then almost surely it is the quantum limit of a random orthonormal basis. The proof is essentially implicit in [1], but we bring it out explicitly here and also give detailed calculations of the moments of inertia, which seem of independent interest. Quantum ergodicity of random orthonormal bases is a rigorous result on the ‘random wave model’ in quantum chaos, according to which eigenfunctions of quantum chaotic systems should behave like random waves. It also has implications for the approximation of modes by quasimodes. As eigenfunctions of the Laplacian of a compact Riemannian manifold (M, g) form an orthonormal basis, it is natural to compare the orthonormal basis of eigenfunctions to a ‘random orthonormal basis’. In [1], the result of this article was proved for the special case, where HN is the space of degree N spherical harmonics on the standard S2 . In [2], the quantum ergodic property was generalized to any compact Riemannian manifold, with HN the span of √ the eigenfunctions in a spectral interval [N, N + 1] for ; in [3], essentially the same result was proved for holomorphic sections of line bundles over Kähler manifolds. Related results for eigenfunctions have recently been proved in [4,5]. The dimension of such HN grows at the rate Nm−1 , where m = dim M, and thus a random element of HN is a superposition of Nm−1 states. The results of this article show that the same quantum ergodicity property holds for sequences of eigenspaces (or linear combinations) whose dimensions dN tend to infinity at any rate. For instance, the results show that random orthonormal bases of eigenfunctions on a flat torus of dimension ≥ 5 are quantum ergodic (for the precise statement, see §4a, and for further discussion, see §1c.) To explain the moment map interpretation and the variance formula, recall that quantum ergodicity is concerned with quantum variances, i.e. with the dispersion from the mean of the diagonal part of a Hermitian matrix on a large-dimensional vector space HN . The matrix is the restriction A := ΠN AΠN (1.1) TN
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where dU is the unit mass Haar measure on U(dN ). We denote by E the expectation with respect to dν. Here, we are working with Hermitian orthonormal bases and Hermitian pseudo-differential operators. We could also work with real self-adjoint operators and real orthonormal bases, which are then related by the orthogonal group. The results in that setting are essentially the same but the proofs are somewhat more complicated; for expository simplicity, we stick to the unitary Hermitian framework. A by λ , . . . , λ . The empirical measure of Let A ∈ Ψ 0 and denote the eigenvalues of TN 1 dN A eigenvalues of TN is defined by dN 1 δλj . (1.6) νλN := dN j=1
Its moments are given by pk (λ1 , . . . , λdN ) =
dN
A k λkj = Tr(TN ) .
(1.7)
j=1
To obtain quantum ergodicity, we put the following constraint on the sequence {HN }: Definition 1.2. We say that HN has local Weyl asymptotics if, for all A ∈ Ψ 0 (M), 1 A Tr TN = ω(A) + o(1). dN i.e.
(1.8)
In fact, the results generalize to the case where ω(A) is replaced by any other limit state, S∗ M σA dμ, where dμ is another invariant probability measure for the geodesic flow. Our main result is:
Theorem 1.3. Let HN be a sequence of subspaces of L2 (M) of dimensions dN = dim HN → ∞. Assume that (1/dN )Tr ΠN AΠN = ω(A) + o(1) for all A ∈ Ψ 0 (M). Then with probability one in (ON B, dν), a random orthonormal basis of N HN is quantum ergodic. A natural question is whether a random orthonormal basis is quantum uniquely ergodic (QUE), i.e. whether max{|AψN,j , ψN,j − ω(A)|2 , j = 1, . . . , dN } → 0 (a.s.) dν? As a tail event, the probability of a random orthonormal basis being QUE is either 0 or 1. Recently, this has been proved in [4,5] when the spectral intervals satisfy certain growth assumptions.
(a) Outline of the proof The first step is to reformulate quantum ergodicity properties (1.3) and (1.4) of a random ONB in terms of moment maps and polytopes. As mentioned above, quantum ergodicity of a random A , which orthonormal basis concerns the dispersion from the mean of the diagonal part of TN A can depends on the choice of an orthonormal basis of HN . Once an orthonormal basis is fixed, iTN be identified with an element HN of the Lie algebra u(dN ) of U(dN ), and a unitary change of the
.........................................................
N=1
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could exist a sparse subsequence of N for which the individual normalized variances do not tend to zero. As this aspect of quantum ergodicity is the same as in [1,3] (e.g.), we do not discuss it here. To define random orthonormal bases, we introduce the probability space (ON B, dν), where ON B is the infinite product of the sets ON BN of orthonormal bases of the spaces HN , and ν= ∞ N=1 νN , where νN is the Haar probability measure on ON BN . A point of ON B is thus a sequence Ψ = {(ψN,1 , . . . , ψN,dN )}N≥1 of orthonormal basis. Given one orthonormal basis {eN,j } of HN any other is related to it by a unique unitary matrix. So the probability space is equivalent to the product ∞ (U(dN ), dU), (1.5) (ON B, dν)
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j=1
of the conjugation action of the Cartan subgroup TdN of diagonal matrices (see [1] for background and references). Finally, let
1 1 Tr H IddN and D0 (λN ) = D(λN ) − Tr H IddN , J¯dN (H) = dN dN for Hermitian matrices H ∈ iu(dN ). We also introduce notation for the diagonal of D0 (λ) D0 (λ) = D(Λ),
with Λj := λj −
dN 1 λj . dN
(1.10)
j=1
Thus, H = H0 + J¯d (H),
resp., D(λN ) = D0 (λN ) +
1 Tr H IddN , dN
with H0 traceless, corresponds to the decomposition u(dN ) = su(dN ) ⊕ R. We now rewrite quantum variances (1.2) in terms of the polytopes PλN . Define A (Ψ ) := XN
1
Jd (U∗ D(λ)UN ) − ω(A)I 2 . dN N N
(1.11)
The following is immediate from the definitions: Lemma 1.4. The ergodic property of an ONB Ψ (EP ) is equivalent to N 1 1 A X (Ψ ) = 0, dn n N→∞ N
lim
∀A ∈ Ψ 0 (M).
(1.12)
n=1
A (Ψ ) → 0 almost surely. Similarly, (1.4) is equivalent to (1/dN )XN
As lemma 1.4 indicates, quantum ergodicity of random orthonormal bases is essentially a A of result about the asymptotic geometry of the polytopes PλN corresponding to a sequence TN Toeplitz operators satisfying trace condition (1.8). Under assumption (1.8), the point ω(A)I is A slightly to replace almost the centre of mass of PλN . We now modify the random variables XN ω(A)I by the exact centre of mass: Definition 1.5. A : ON BN → [0, +∞), YN
Ψ = (Ud1 , Ud2 , . . .)
⎫ ⎬
A ∗ ∗ (Ψ ) := JdN (UN D(λ)UN ) − D(λ) 2 = JdN (UN D0 (λ)UN ) 2 .⎭ YN
In lemma 2.1, we will show that the ergodic properties of an orthonormal basis are equivalent A replaced by YA . to the statements in lemma 1.4 with XN N A and The main step in the proof of theorem 1.3 is to determine the asymptotic mean EYN variance A A 2 A 2 ) := E((YN ) ) − (E((YN )) ) Var(YN of these random variables with respect to dν. For the proof of theorem 1.3, it suffices to prove that the mean tends to zero and the variance is bounded.
.........................................................
denote the orthogonal projection (extracting the diagonal). Extracting the diagonal from each element of the orbit is precisely the moment map ⎛ ⎞ dN ∗ 2 λj |Uij | , . . .⎠ (1.9) JdN : OλN → PλN ⊂ it(dN ), JdN (UD(λ)U ) = ⎝. . . ,
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∗ H U of H . If the vector of eigenvalues orthonormal basis results in the conjugation HN → UN N N N of HN is denoted λN , then the conjugates sweep out the orbit OλN . Let t(dN ) denote the Cartan subalgebra of diagonal elements in u(dN ), and let · 2 denote the Euclidean inner product on t(dN ). Also let JdN : iu(dN ) → it(dN )
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SN :=
1 (YA − EYnA ) dn n
(1.13)
1 1 A Y → 0 almost surely. N dn n
(1.14)
n≤N
have the property 1 SN → 0, N
hence
n≤N
Moreover, if dN grows at a sufficiently fast rate we obtain stronger results such as (1.4) from the A Borel–Cantelli lemma: e.g. if ∞ n=1 (1/dn ) < ∞, one obtains almost sure convergence (1/dn )Yn → 0 A (a.s.). By lemma 2.1, the same results then hold for XN . By lemma 1.4, this will conclude the proof of theorem 1.3. A . To do so, we rewrite them in Thus, the main step is to compute the mean and variance of YN terms of moments of inertia of PλN . The pushforward of the U(dN )-invariant normalized measure on Oλ to Pλ is the so-called Duistermaat–Heckman measure dLDH λ , a piecewise polynomial measure on Pλ . We denote the centre of mass of the polytope Pλ ⊂ RN by 1 D(λ) = x dLDH λ . Vol(Pλ ) Pλ By volume we mean the integral of the function 1 with respect to Duistermaat–Heckman measure, which is simply the volume of the orbit Oλ . It is normalized to equal to 1, but we include the volume normalization to emphasize that the measure on the polytope is a probability measure. We further denote the moments of the polytope by 1 |x − D(λ)|2k dLDH (1.15) m2k (Pλ ) = λ . Vol(Pλ ) Pλ Unravelling the definitions gives A m2 (PλN ) := E JdN (U∗ D0 (λ)U) 2 = EYN
and
⎫ ⎬
A ⎭ m4 (PλN ) := E JdN (U∗ D0 (λ)U) 4 = Var YN .
(1.16)
A obtained from A ∈ Ψ 0 (M) with spectra {λ }, Thus, it suffices to show that for all sequences TN N are bounded. Note that the second and fourth moments of inertia of Pλ with respect to dLDH λ the concentration of mass of these convex bodies in high-dimensional spaces is evidently quite different from that of isotropic convex bodies [6]. We asymptotically evaluate the moments of inertia of PλN using the Fourier transform eiX,diag(Y) dμλ (Y) (1.17) μˆ λ (X) := Oλ
of the δ-function on Oλ . Here, we assume X ∈ RdN . We may identify X with a diagonal matrix, so that X, diag(Y) = Tr XY, so that μˆ λ (X) is the standard Fourier transform. We translate λ λj = 0, i.e. we by its centre of mass to make the centre of mass of Pλ equal to 0, i.e. replace λ by Λ (1.10). Then, differentiating under the integral sign gives
.........................................................
i.e. the mean of the quantum variances (1.2) tends to zero. In corollary 1.8 and lemma 1.7, we A ) which shows that it too is bounded. As in [1,3], we then have given an exact formula for Var(YN A } of independent apply the Kolmogorov SLLN (strong law of large numbers) to the sequence {YN random variables. The SLLN implies that the partial sums
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In lemma 3.1, we have given an exact formula for E JdN (U∗ D0 (λ)U) 2 for all A ∈ Ψ 0 (M). From the formula, it is obvious that the mean is bounded, so that
1
JdN (U∗ D0 (λ)U) 2 → 0 as long as dN → ∞, E dN
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Lemma 1.6. Let be the Euclidean Laplacian of RdN acting in the X variable. Then, ⎫ m2 (PΛ ) = −μˆ Λ (X)|X=0 ⎬
6
Lemma 1.7. Let pk be power functions (1.7). Then for any λ such that p1 (λ) = 0, μˆ λ (0) =
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
p2 (λ) , dN + 1
2 μˆ λ (0) = β4 (dN ) p22 (λ), 4dN (dN − 1) 4dN (dN − 1) ⎪ − with β4 (dN ) = ⎪ 2 (d + 2)(d (dN + 1)dN (dN − 1) N N + 1)dN (dN − 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ (12dN + 4dN (dN − 1)) ⎪ ⎪ ⎪ . + ⎭ (dN + 3)(dN + 2)(dN + 1)dN A are bounded, and Combining (1.16) and lemmas 1.6 and 1.7, we find that the variances of YN indeed have the asymptotics given in
Corollary 1.8. We have
A Var(YN ) = β4 (dN ) −
1 (dN + 1)2
p22 (ΛN )
3 d2N
p22 (ΛN ).
A ) ≤ 3 A 2 . Hence, Var(YN
Boundedness holds because A ∈ Ψ 0 (M) is a bounded operator, hence the numerator is bounded by 3 A 2 times the number dN of terms.
(b) More on limit shapes of PλN As mentioned above, the calculations of lemma 1.7 go beyond what is necessary for almost sure quantum ergodicity. They have their own intrinsic interest in the asymptotic geometry of the polytopes PλN . This sequence of polytopes has nice asymptotic properties as long as empirical measures (1.6) tend to a weak limit ν. It seems of interest to explore the limit shapes of any sequence of polytopes with this property. Independently of any connection to quantum ergodicity, the results of this article show: Proposition 1.9. Let λN ∈ RdN be a sequence of vectors with the property that empirical measures (1.6) tend to a weak limit ν. Let PλN be the associated polytopes. Then, ⎫ ⎪ ⎪ m2 (PλN ) → (t − t¯)2 dν ⎪ ⎬ R
2 ⎪ ⎪ ⎭ (t − t¯)2 dν .⎪ and m4 (PλN ) → 4 R
This proposition is closely related to the ‘Weingarten theorem’ that the matrix elements dN Uij are asymptotically complex normal random variables, where Uij are the matrix elements of U ∈ U(dN ) [7]. Perhaps this explains why the fourth moment is a constant multiple of the square of the second moment. It would be interesting to see whether the pattern continues to higher moments.
.........................................................
These formulae are useful because we can calculate μˆ Λ (X) in terms of Schur polynomials. The definition of the Schur polynomials is recalled in §1d and the calculation of μˆ Λ (X) in terms of Schur polynomials is recalled in lemma 2.3. It leads to the following:
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m4 (PΛ ) = 2 μˆ Λ (X)|X=0 . ⎭
and
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It seems of some interest in spectral asymptotics to determine when the condition in this definition holds. For instance, we do not know if it holds for the sequence of eigenspaces of flat tori in dimensions ≥ 5.
(c) Discussion The motivation for proving quantum ergodicity of random orthonormal bases for HN of any dimensions tending to infinity was prompted by the general question: how many diffuse states (modes or quasi-modes) does it take to synthesize localized modes or quasi-modes? Vice versa, how many localized states does it take to synthesize diffuse states? We would like to synthesize entire orthonormal bases rather than individual states and measure the dimensions of the space of states in terms of the Planck constant h¯ . Let us consider some examples. In the case of the standard S2 , the eigenspaces HN of are the spaces of spherical harmonics N of joint eigenfunctions of of degree N. They have the well-known highly localized basis Ym N } with m/N → and of rotations around the x3 -axis. By localized we mean that a sequence {Ym α microlocally concentrates on the invariant tori in S∗ S2 where pθ = α. Here, pθ (x, ξ ) = ξ (∂/∂θ ) where ∂/∂θ generates the x3 -axis rotations. On the other hand, it is proved in [1] that independent ‘random’ orthonormal bases of HN are quantum ergodic, i.e. are highly diffusive in S∗ S2 . As dim HN = 2N + 1, it is perhaps not surprising that the same eigenspace can have both highly localized and highly diffuse orthonormal bases when its dimension is so large. The question is, how large must it be for such incoherently related bases to exist? A setting where the eigenvalues have high multiplicity but of a lower order of magnitude than on S2 is that of flat rational tori Rn /L, such as Rn /Zn . Of course it has an orthonormal basis of localized eigenfunctions, eik,x . The key feature of such rational tori is the high multiplicity of eigenvalues of the Laplacian of the flat metric. It is well known and easy to see that the multiplicity is the number of lattice points of the dual lattice L∗ lying on the surface of a Euclidean sphere. We denote the distinct multiple -eigenvalues by μN , the corresponding eigenspace by HN and the multiplicity of μ2N by dN = dim HN . In dimensions n ≥ 5, dN ∼ μn−2 N , one degree lower than the maximum possible multiplicity of a -eigenvalue on any compact Riemannian manifold, achieved on the standard Sn . Furthermore, (1/dN )Tr ΠN AΠN → ω(A). Hence, the results of this article show that despite the relatively slow growth of dN on a flat rational torus, orthonormal bases of HN in dimensions ≥ 5 are almost surely quantum ergodic. The statement for dimensions 2, 3, 4 is more complicated (see §4a). An interesting setting where the behaviour of eigenfunctions is largely unknown is that of Kolmogorov–Arnold–Moser (KAM) systems. For these, one may construct a ‘nearly’ complete and orthonormal basis for L2 (M) by highly localized quasi-modes associated with the Cantor set of invariant tori. It seems unlikely that the actual eigenfunctions are quantum ergodic; but the results of this article show that if they resemble random combinations of the quasi-modes, then it is possible that they are. Further discussion is in §4b.
2. Background In this section, we review the definition of random orthonormal basis and relate it to properties of the moment map for the diagonal action of the maximal torus TdN on co-adjoint orbits of U(dN ).
.........................................................
Definition 1.10. We say that the sequence {HN } has Szegö asymptotics if, for all A ∈ Ψ 0 (M), there exists a unique weak* limit, νλN → νA ∈ M(R) as N → ∞. Here, M(R) is the set of probability measures on R.
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In the context of quantum ergodicity and pseudo-differential operators, one may ask when the sequence of polytopes associated with A ∈ Ψ 0 (M) and the Hilbert spaces HN has the property in proposition 1.9. It is a much stronger condition than the one in definition 1.8.
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(a) Random orthonormal bases of eigenspaces
8
∞
HN
N=1
and orthogonal projections ΠN : H → HN .
(2.1)
We then consider orthonormal bases (1.5) of H which arise from sequences of orthonormal bases of HN .
(b) The basic random variables Let A ∈ Ψ 0 (M) be a zeroth-order pseudo-differential operator. By a Toeplitz operator, we mean the A (1.1) of A to H . compression TN N A can be identified with a Hermitian d × d matrix. N of HN , TN Given one ONB {eN,j }dj=1 N N
N N is related to {eN,j }dj=1 by a unitary matrix UN . We thus Moreover, any other ONB {ψN,j }dj=1 introduce the random variables on (1.5)
A ANj (Ψ ) = |AψN,j , ψN,j − ω(A)|2 = |(TN ψN,j , ψN,j ) − ω(A)|2 ∗ A = |(UN TN UN eN,j , eN,j ) − ω(A)|2 ,
where Ψ = {UN }, UN ∈ U(dN ) ≡ OBNN . We also define 2 ∗ A 1 A TN UN eN,j eN,j ) − Tr TN Aˆ Nj (Ψ ) = (UN . dN
(2.2)
(2.3)
Evidently, dN dN 1 A 1 1 YN (Ψ ) = ANj (Ψ ) + o(1) Aˆ Nj (Ψ ) = dN dN dN j=1
(2.4)
j=1
(where the o(1) term is independent of Ψ ). It follows that A 1 A sup XN − YN = o(1). dN ON BN Combining with lemma 1.4, Lemma 2.1 ([1,3]). The ergodic property of an ONB Ψ (EP ) is equivalent to N 1 1 A Y (Ψ ) = 0, dn n N→∞ N
lim
∀A ∈ Ψ 0 (M).
(2.5)
n=1
Similarly, (1.4) is equivalent to (1/dn )YnA (Ψ ) → 0 almost surely. As mentioned in the Introduction, it follows by a standard diagonal argument that almost all the individual elements AψN,j , ψN,j tend to ω(A) for all A. We do not discuss this step because it is nothing new.
(c) Moment map interpretation In the case where the components of λN are distinct, the convex polytope PλN is the permutahedron determined by λ, that is, the simple convex polytope defined as the convex hull
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H=
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Suppose that we have a sequence of Hilbert spaces HN N = 1, 2, . . . of dimensions dN = dim HN → ∞. We define the large Hilbert space
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9
Xσ (λN ) = 0
⇐⇒
X=
σ ∈SdN
1 σ (λN ), (dN )! σ ∈SdN
where XY = X − Y is the vector from X to Y. The centre of mass is evidently invariant under N λj . In effect, we want SdN , hence has the form (a, a, . . . , a) for some a and clearly a = (1/dN ) dj=1 to asymptotically calculate the moments of inertia of the sequence of permutahedra (figure 1) associated with a Toeplitz operator.
(d) Symmetric polynomials and Schur polynomials The elementary symmetric polynomial of degree k in d variables is defined by ek (X1 , . . . , Xd ) = Xi1 · · · Xik . i1
Quantum ergodicity of random orthonormal bases of spaces of high dimension Steve Zelditch rsta.royalsocietypublishing.org
Research Cite this article: Zelditch S. 2014 Quantum ergodicity of random orthonormal bases of spaces of high dimension. Phil. Trans. R. Soc. A 372: 20120511. http://dx.doi.org/10.1098/rsta.2012.0511
One contribution of 13 to a Theo Murphy Meeting Issue ‘Complex patterns in wave functions: drums, graphs and disorder’.
Subject Areas: analysis, statistics Keywords: quantum ergodcity, laplace eigenfunctions, random orthonormal basis Author for correspondence: Steve Zelditch e-mail: [email protected]
Department of Mathematics, Northwestern University, Evanston, IL 60208-2370, USA We consider a sequence HN of finite-dimensional Hilbert spaces of dimensions dN → ∞. Motivating examples are eigenspaces, or spaces of quasimodes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of HN may be identified with U(dN ), and a random orthonormal basis of N HN is a choice of a random sequence UN ∈ U(dN ) from the product of normalized Haar measures. We prove that if dN → ∞ and if (1/dN )Tr A|HN tends to a unique limit state ω(A), then almost surely an orthonormal basis is quantum ergodic with limit state ω(A). This generalizes an earlier result of the author in the case where HN is the space of spherical harmonics on S2 . In particular, it holds on the flat torus Rd /Zd if d ≥ 5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.
1. Introduction The purpose of this article is to prove a general result on the quantum ergodicity of random orthonormal N of finite-dimensional Hilbert spaces HN ⊂ bases {ψN,j }dj=1 L2 (M) of dimensions dN → ∞ of a compact Riemannian manifold (M, g). The proof is based on a ‘moment polytope’ interpretation of quantum ergodicity from [1]: the quantum variances of a Hermitian observable A ∈ Ψ 0 (M) (where Ψ 0 (M)) is the class of pseudo-differential operators of degree zero) are identified with moments of inertia of the convex polytopes Pλ defined as the convex hull of the vectors λ = (λ1 , . . . , λdN ) of eigenvalues (in all possible orders) of ΠN AΠN where ΠN : L2 (M) → HN is the orthogonal projection. Equivalently, Pλ is the image of the coadjoint orbit Oλ of the diagonal matrix D(λ) under the moment map for the Hamiltonian action of the
2013 The Author(s) Published by the Royal Society. All rights reserved.
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(indexed by A ∈ Ψ 0 (M)) by A ({ψNk }) := VN
dN 1 |AψN,j , ψN,j − ω(A)|2 . dN
(1.2)
j=1
Here, ω(A) = S∗ M σA dμL , where dμL is normalized Liouville measure (of mass one), and σA is the principal symbol of A. Definition 1.1. A sequence {ψNj }N of ONBs of HN is a quantum ergodic ONB of L2 (M) if (EP )
N 1 A n Vn ({ψn,j }dj=1 ) = 0, N→∞ N
lim
∀A ∈ Ψ 0 (M).
(1.3)
n=1
We say sequence {ψN,j }N is fine quantum ergodic if (F EP )
n lim VA ({ψN,j }dj=1 )) = 0,
N→∞
∀A ∈ Ψ 0 (M).
(1.4)
By a standard diagonal argument, this implies a subsequence of density of one of the individual elements AψN,j , ψN,j tends to ω(A). The fine notion of density is of course stronger, because there
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to HN of a pseudo-differential operator A ∈ Ψ 0 (M); here ΠN is the orthogonal projection to HN and Ψ 0 (M) is the space of pseudo-differential operators of order zero. The same methods and results apply to other contexts such as semiclassical pseudo-differential operators or to Toeplitz operators on holomorphic sections of powers of a positive line bundle [3]. Given an orthonormal N of HN we define the (normalized) quantum variances of the ONB basis (ONB) {ψN,j }dj=1
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maximal torus TdN ⊂ U(dN ) of diagonal matrices acting by conjugation on Oλ . In particular, the main estimates of quantum ergodicity can be formulated in terms of estimates of the first four moments of inertia of Pλ . The main result, theorem 1.3, states that random orthonormal bases are almost surely quantum ergodic as long as dN → ∞ and (1/dN )Tr ΠN AΠN → ω(A) for all A ∈ Ψ 0 (M), where ω(A) is the Liouville state. More generally, if these traces have any unique limit state, then almost surely it is the quantum limit of a random orthonormal basis. The proof is essentially implicit in [1], but we bring it out explicitly here and also give detailed calculations of the moments of inertia, which seem of independent interest. Quantum ergodicity of random orthonormal bases is a rigorous result on the ‘random wave model’ in quantum chaos, according to which eigenfunctions of quantum chaotic systems should behave like random waves. It also has implications for the approximation of modes by quasimodes. As eigenfunctions of the Laplacian of a compact Riemannian manifold (M, g) form an orthonormal basis, it is natural to compare the orthonormal basis of eigenfunctions to a ‘random orthonormal basis’. In [1], the result of this article was proved for the special case, where HN is the space of degree N spherical harmonics on the standard S2 . In [2], the quantum ergodic property was generalized to any compact Riemannian manifold, with HN the span of √ the eigenfunctions in a spectral interval [N, N + 1] for ; in [3], essentially the same result was proved for holomorphic sections of line bundles over Kähler manifolds. Related results for eigenfunctions have recently been proved in [4,5]. The dimension of such HN grows at the rate Nm−1 , where m = dim M, and thus a random element of HN is a superposition of Nm−1 states. The results of this article show that the same quantum ergodicity property holds for sequences of eigenspaces (or linear combinations) whose dimensions dN tend to infinity at any rate. For instance, the results show that random orthonormal bases of eigenfunctions on a flat torus of dimension ≥ 5 are quantum ergodic (for the precise statement, see §4a, and for further discussion, see §1c.) To explain the moment map interpretation and the variance formula, recall that quantum ergodicity is concerned with quantum variances, i.e. with the dispersion from the mean of the diagonal part of a Hermitian matrix on a large-dimensional vector space HN . The matrix is the restriction A := ΠN AΠN (1.1) TN
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where dU is the unit mass Haar measure on U(dN ). We denote by E the expectation with respect to dν. Here, we are working with Hermitian orthonormal bases and Hermitian pseudo-differential operators. We could also work with real self-adjoint operators and real orthonormal bases, which are then related by the orthogonal group. The results in that setting are essentially the same but the proofs are somewhat more complicated; for expository simplicity, we stick to the unitary Hermitian framework. A by λ , . . . , λ . The empirical measure of Let A ∈ Ψ 0 and denote the eigenvalues of TN 1 dN A eigenvalues of TN is defined by dN 1 δλj . (1.6) νλN := dN j=1
Its moments are given by pk (λ1 , . . . , λdN ) =
dN
A k λkj = Tr(TN ) .
(1.7)
j=1
To obtain quantum ergodicity, we put the following constraint on the sequence {HN }: Definition 1.2. We say that HN has local Weyl asymptotics if, for all A ∈ Ψ 0 (M), 1 A Tr TN = ω(A) + o(1). dN i.e.
(1.8)
In fact, the results generalize to the case where ω(A) is replaced by any other limit state, S∗ M σA dμ, where dμ is another invariant probability measure for the geodesic flow. Our main result is:
Theorem 1.3. Let HN be a sequence of subspaces of L2 (M) of dimensions dN = dim HN → ∞. Assume that (1/dN )Tr ΠN AΠN = ω(A) + o(1) for all A ∈ Ψ 0 (M). Then with probability one in (ON B, dν), a random orthonormal basis of N HN is quantum ergodic. A natural question is whether a random orthonormal basis is quantum uniquely ergodic (QUE), i.e. whether max{|AψN,j , ψN,j − ω(A)|2 , j = 1, . . . , dN } → 0 (a.s.) dν? As a tail event, the probability of a random orthonormal basis being QUE is either 0 or 1. Recently, this has been proved in [4,5] when the spectral intervals satisfy certain growth assumptions.
(a) Outline of the proof The first step is to reformulate quantum ergodicity properties (1.3) and (1.4) of a random ONB in terms of moment maps and polytopes. As mentioned above, quantum ergodicity of a random A , which orthonormal basis concerns the dispersion from the mean of the diagonal part of TN A can depends on the choice of an orthonormal basis of HN . Once an orthonormal basis is fixed, iTN be identified with an element HN of the Lie algebra u(dN ) of U(dN ), and a unitary change of the
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N=1
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could exist a sparse subsequence of N for which the individual normalized variances do not tend to zero. As this aspect of quantum ergodicity is the same as in [1,3] (e.g.), we do not discuss it here. To define random orthonormal bases, we introduce the probability space (ON B, dν), where ON B is the infinite product of the sets ON BN of orthonormal bases of the spaces HN , and ν= ∞ N=1 νN , where νN is the Haar probability measure on ON BN . A point of ON B is thus a sequence Ψ = {(ψN,1 , . . . , ψN,dN )}N≥1 of orthonormal basis. Given one orthonormal basis {eN,j } of HN any other is related to it by a unique unitary matrix. So the probability space is equivalent to the product ∞ (U(dN ), dU), (1.5) (ON B, dν)
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j=1
of the conjugation action of the Cartan subgroup TdN of diagonal matrices (see [1] for background and references). Finally, let
1 1 Tr H IddN and D0 (λN ) = D(λN ) − Tr H IddN , J¯dN (H) = dN dN for Hermitian matrices H ∈ iu(dN ). We also introduce notation for the diagonal of D0 (λ) D0 (λ) = D(Λ),
with Λj := λj −
dN 1 λj . dN
(1.10)
j=1
Thus, H = H0 + J¯d (H),
resp., D(λN ) = D0 (λN ) +
1 Tr H IddN , dN
with H0 traceless, corresponds to the decomposition u(dN ) = su(dN ) ⊕ R. We now rewrite quantum variances (1.2) in terms of the polytopes PλN . Define A (Ψ ) := XN
1
Jd (U∗ D(λ)UN ) − ω(A)I 2 . dN N N
(1.11)
The following is immediate from the definitions: Lemma 1.4. The ergodic property of an ONB Ψ (EP ) is equivalent to N 1 1 A X (Ψ ) = 0, dn n N→∞ N
lim
∀A ∈ Ψ 0 (M).
(1.12)
n=1
A (Ψ ) → 0 almost surely. Similarly, (1.4) is equivalent to (1/dN )XN
As lemma 1.4 indicates, quantum ergodicity of random orthonormal bases is essentially a A of result about the asymptotic geometry of the polytopes PλN corresponding to a sequence TN Toeplitz operators satisfying trace condition (1.8). Under assumption (1.8), the point ω(A)I is A slightly to replace almost the centre of mass of PλN . We now modify the random variables XN ω(A)I by the exact centre of mass: Definition 1.5. A : ON BN → [0, +∞), YN
Ψ = (Ud1 , Ud2 , . . .)
⎫ ⎬
A ∗ ∗ (Ψ ) := JdN (UN D(λ)UN ) − D(λ) 2 = JdN (UN D0 (λ)UN ) 2 .⎭ YN
In lemma 2.1, we will show that the ergodic properties of an orthonormal basis are equivalent A replaced by YA . to the statements in lemma 1.4 with XN N A and The main step in the proof of theorem 1.3 is to determine the asymptotic mean EYN variance A A 2 A 2 ) := E((YN ) ) − (E((YN )) ) Var(YN of these random variables with respect to dν. For the proof of theorem 1.3, it suffices to prove that the mean tends to zero and the variance is bounded.
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denote the orthogonal projection (extracting the diagonal). Extracting the diagonal from each element of the orbit is precisely the moment map ⎛ ⎞ dN ∗ 2 λj |Uij | , . . .⎠ (1.9) JdN : OλN → PλN ⊂ it(dN ), JdN (UD(λ)U ) = ⎝. . . ,
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∗ H U of H . If the vector of eigenvalues orthonormal basis results in the conjugation HN → UN N N N of HN is denoted λN , then the conjugates sweep out the orbit OλN . Let t(dN ) denote the Cartan subalgebra of diagonal elements in u(dN ), and let · 2 denote the Euclidean inner product on t(dN ). Also let JdN : iu(dN ) → it(dN )
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SN :=
1 (YA − EYnA ) dn n
(1.13)
1 1 A Y → 0 almost surely. N dn n
(1.14)
n≤N
have the property 1 SN → 0, N
hence
n≤N
Moreover, if dN grows at a sufficiently fast rate we obtain stronger results such as (1.4) from the A Borel–Cantelli lemma: e.g. if ∞ n=1 (1/dn ) < ∞, one obtains almost sure convergence (1/dn )Yn → 0 A (a.s.). By lemma 2.1, the same results then hold for XN . By lemma 1.4, this will conclude the proof of theorem 1.3. A . To do so, we rewrite them in Thus, the main step is to compute the mean and variance of YN terms of moments of inertia of PλN . The pushforward of the U(dN )-invariant normalized measure on Oλ to Pλ is the so-called Duistermaat–Heckman measure dLDH λ , a piecewise polynomial measure on Pλ . We denote the centre of mass of the polytope Pλ ⊂ RN by 1 D(λ) = x dLDH λ . Vol(Pλ ) Pλ By volume we mean the integral of the function 1 with respect to Duistermaat–Heckman measure, which is simply the volume of the orbit Oλ . It is normalized to equal to 1, but we include the volume normalization to emphasize that the measure on the polytope is a probability measure. We further denote the moments of the polytope by 1 |x − D(λ)|2k dLDH (1.15) m2k (Pλ ) = λ . Vol(Pλ ) Pλ Unravelling the definitions gives A m2 (PλN ) := E JdN (U∗ D0 (λ)U) 2 = EYN
and
⎫ ⎬
A ⎭ m4 (PλN ) := E JdN (U∗ D0 (λ)U) 4 = Var YN .
(1.16)
A obtained from A ∈ Ψ 0 (M) with spectra {λ }, Thus, it suffices to show that for all sequences TN N are bounded. Note that the second and fourth moments of inertia of Pλ with respect to dLDH λ the concentration of mass of these convex bodies in high-dimensional spaces is evidently quite different from that of isotropic convex bodies [6]. We asymptotically evaluate the moments of inertia of PλN using the Fourier transform eiX,diag(Y) dμλ (Y) (1.17) μˆ λ (X) := Oλ
of the δ-function on Oλ . Here, we assume X ∈ RdN . We may identify X with a diagonal matrix, so that X, diag(Y) = Tr XY, so that μˆ λ (X) is the standard Fourier transform. We translate λ λj = 0, i.e. we by its centre of mass to make the centre of mass of Pλ equal to 0, i.e. replace λ by Λ (1.10). Then, differentiating under the integral sign gives
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i.e. the mean of the quantum variances (1.2) tends to zero. In corollary 1.8 and lemma 1.7, we A ) which shows that it too is bounded. As in [1,3], we then have given an exact formula for Var(YN A } of independent apply the Kolmogorov SLLN (strong law of large numbers) to the sequence {YN random variables. The SLLN implies that the partial sums
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In lemma 3.1, we have given an exact formula for E JdN (U∗ D0 (λ)U) 2 for all A ∈ Ψ 0 (M). From the formula, it is obvious that the mean is bounded, so that
1
JdN (U∗ D0 (λ)U) 2 → 0 as long as dN → ∞, E dN
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Lemma 1.6. Let be the Euclidean Laplacian of RdN acting in the X variable. Then, ⎫ m2 (PΛ ) = −μˆ Λ (X)|X=0 ⎬
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Lemma 1.7. Let pk be power functions (1.7). Then for any λ such that p1 (λ) = 0, μˆ λ (0) =
⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
p2 (λ) , dN + 1
2 μˆ λ (0) = β4 (dN ) p22 (λ), 4dN (dN − 1) 4dN (dN − 1) ⎪ − with β4 (dN ) = ⎪ 2 (d + 2)(d (dN + 1)dN (dN − 1) N N + 1)dN (dN − 2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ (12dN + 4dN (dN − 1)) ⎪ ⎪ ⎪ . + ⎭ (dN + 3)(dN + 2)(dN + 1)dN A are bounded, and Combining (1.16) and lemmas 1.6 and 1.7, we find that the variances of YN indeed have the asymptotics given in
Corollary 1.8. We have
A Var(YN ) = β4 (dN ) −
1 (dN + 1)2
p22 (ΛN )
3 d2N
p22 (ΛN ).
A ) ≤ 3 A 2 . Hence, Var(YN
Boundedness holds because A ∈ Ψ 0 (M) is a bounded operator, hence the numerator is bounded by 3 A 2 times the number dN of terms.
(b) More on limit shapes of PλN As mentioned above, the calculations of lemma 1.7 go beyond what is necessary for almost sure quantum ergodicity. They have their own intrinsic interest in the asymptotic geometry of the polytopes PλN . This sequence of polytopes has nice asymptotic properties as long as empirical measures (1.6) tend to a weak limit ν. It seems of interest to explore the limit shapes of any sequence of polytopes with this property. Independently of any connection to quantum ergodicity, the results of this article show: Proposition 1.9. Let λN ∈ RdN be a sequence of vectors with the property that empirical measures (1.6) tend to a weak limit ν. Let PλN be the associated polytopes. Then, ⎫ ⎪ ⎪ m2 (PλN ) → (t − t¯)2 dν ⎪ ⎬ R
2 ⎪ ⎪ ⎭ (t − t¯)2 dν .⎪ and m4 (PλN ) → 4 R
This proposition is closely related to the ‘Weingarten theorem’ that the matrix elements dN Uij are asymptotically complex normal random variables, where Uij are the matrix elements of U ∈ U(dN ) [7]. Perhaps this explains why the fourth moment is a constant multiple of the square of the second moment. It would be interesting to see whether the pattern continues to higher moments.
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These formulae are useful because we can calculate μˆ Λ (X) in terms of Schur polynomials. The definition of the Schur polynomials is recalled in §1d and the calculation of μˆ Λ (X) in terms of Schur polynomials is recalled in lemma 2.3. It leads to the following:
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m4 (PΛ ) = 2 μˆ Λ (X)|X=0 . ⎭
and
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It seems of some interest in spectral asymptotics to determine when the condition in this definition holds. For instance, we do not know if it holds for the sequence of eigenspaces of flat tori in dimensions ≥ 5.
(c) Discussion The motivation for proving quantum ergodicity of random orthonormal bases for HN of any dimensions tending to infinity was prompted by the general question: how many diffuse states (modes or quasi-modes) does it take to synthesize localized modes or quasi-modes? Vice versa, how many localized states does it take to synthesize diffuse states? We would like to synthesize entire orthonormal bases rather than individual states and measure the dimensions of the space of states in terms of the Planck constant h¯ . Let us consider some examples. In the case of the standard S2 , the eigenspaces HN of are the spaces of spherical harmonics N of joint eigenfunctions of of degree N. They have the well-known highly localized basis Ym N } with m/N → and of rotations around the x3 -axis. By localized we mean that a sequence {Ym α microlocally concentrates on the invariant tori in S∗ S2 where pθ = α. Here, pθ (x, ξ ) = ξ (∂/∂θ ) where ∂/∂θ generates the x3 -axis rotations. On the other hand, it is proved in [1] that independent ‘random’ orthonormal bases of HN are quantum ergodic, i.e. are highly diffusive in S∗ S2 . As dim HN = 2N + 1, it is perhaps not surprising that the same eigenspace can have both highly localized and highly diffuse orthonormal bases when its dimension is so large. The question is, how large must it be for such incoherently related bases to exist? A setting where the eigenvalues have high multiplicity but of a lower order of magnitude than on S2 is that of flat rational tori Rn /L, such as Rn /Zn . Of course it has an orthonormal basis of localized eigenfunctions, eik,x . The key feature of such rational tori is the high multiplicity of eigenvalues of the Laplacian of the flat metric. It is well known and easy to see that the multiplicity is the number of lattice points of the dual lattice L∗ lying on the surface of a Euclidean sphere. We denote the distinct multiple -eigenvalues by μN , the corresponding eigenspace by HN and the multiplicity of μ2N by dN = dim HN . In dimensions n ≥ 5, dN ∼ μn−2 N , one degree lower than the maximum possible multiplicity of a -eigenvalue on any compact Riemannian manifold, achieved on the standard Sn . Furthermore, (1/dN )Tr ΠN AΠN → ω(A). Hence, the results of this article show that despite the relatively slow growth of dN on a flat rational torus, orthonormal bases of HN in dimensions ≥ 5 are almost surely quantum ergodic. The statement for dimensions 2, 3, 4 is more complicated (see §4a). An interesting setting where the behaviour of eigenfunctions is largely unknown is that of Kolmogorov–Arnold–Moser (KAM) systems. For these, one may construct a ‘nearly’ complete and orthonormal basis for L2 (M) by highly localized quasi-modes associated with the Cantor set of invariant tori. It seems unlikely that the actual eigenfunctions are quantum ergodic; but the results of this article show that if they resemble random combinations of the quasi-modes, then it is possible that they are. Further discussion is in §4b.
2. Background In this section, we review the definition of random orthonormal basis and relate it to properties of the moment map for the diagonal action of the maximal torus TdN on co-adjoint orbits of U(dN ).
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Definition 1.10. We say that the sequence {HN } has Szegö asymptotics if, for all A ∈ Ψ 0 (M), there exists a unique weak* limit, νλN → νA ∈ M(R) as N → ∞. Here, M(R) is the set of probability measures on R.
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In the context of quantum ergodicity and pseudo-differential operators, one may ask when the sequence of polytopes associated with A ∈ Ψ 0 (M) and the Hilbert spaces HN has the property in proposition 1.9. It is a much stronger condition than the one in definition 1.8.
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(a) Random orthonormal bases of eigenspaces
8
∞
HN
N=1
and orthogonal projections ΠN : H → HN .
(2.1)
We then consider orthonormal bases (1.5) of H which arise from sequences of orthonormal bases of HN .
(b) The basic random variables Let A ∈ Ψ 0 (M) be a zeroth-order pseudo-differential operator. By a Toeplitz operator, we mean the A (1.1) of A to H . compression TN N A can be identified with a Hermitian d × d matrix. N of HN , TN Given one ONB {eN,j }dj=1 N N
N N is related to {eN,j }dj=1 by a unitary matrix UN . We thus Moreover, any other ONB {ψN,j }dj=1 introduce the random variables on (1.5)
A ANj (Ψ ) = |AψN,j , ψN,j − ω(A)|2 = |(TN ψN,j , ψN,j ) − ω(A)|2 ∗ A = |(UN TN UN eN,j , eN,j ) − ω(A)|2 ,
where Ψ = {UN }, UN ∈ U(dN ) ≡ OBNN . We also define 2 ∗ A 1 A TN UN eN,j eN,j ) − Tr TN Aˆ Nj (Ψ ) = (UN . dN
(2.2)
(2.3)
Evidently, dN dN 1 A 1 1 YN (Ψ ) = ANj (Ψ ) + o(1) Aˆ Nj (Ψ ) = dN dN dN j=1
(2.4)
j=1
(where the o(1) term is independent of Ψ ). It follows that A 1 A sup XN − YN = o(1). dN ON BN Combining with lemma 1.4, Lemma 2.1 ([1,3]). The ergodic property of an ONB Ψ (EP ) is equivalent to N 1 1 A Y (Ψ ) = 0, dn n N→∞ N
lim
∀A ∈ Ψ 0 (M).
(2.5)
n=1
Similarly, (1.4) is equivalent to (1/dn )YnA (Ψ ) → 0 almost surely. As mentioned in the Introduction, it follows by a standard diagonal argument that almost all the individual elements AψN,j , ψN,j tend to ω(A) for all A. We do not discuss this step because it is nothing new.
(c) Moment map interpretation In the case where the components of λN are distinct, the convex polytope PλN is the permutahedron determined by λ, that is, the simple convex polytope defined as the convex hull
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H=
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Suppose that we have a sequence of Hilbert spaces HN N = 1, 2, . . . of dimensions dN = dim HN → ∞. We define the large Hilbert space
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9
Xσ (λN ) = 0
⇐⇒
X=
σ ∈SdN
1 σ (λN ), (dN )! σ ∈SdN
where XY = X − Y is the vector from X to Y. The centre of mass is evidently invariant under N λj . In effect, we want SdN , hence has the form (a, a, . . . , a) for some a and clearly a = (1/dN ) dj=1 to asymptotically calculate the moments of inertia of the sequence of permutahedra (figure 1) associated with a Toeplitz operator.
(d) Symmetric polynomials and Schur polynomials The elementary symmetric polynomial of degree k in d variables is defined by ek (X1 , . . . , Xd ) = Xi1 · · · Xik . i1
