uni-bremen.deelib.suub.uni-bremen.de/diss/docs/00010494.pdf · Contents 1 Introduction 7 2 Construction of the Classical Limit 11 2.1 The Classical Limit in the Simple Case .

Representation TheoreticalConstruction of the Classical Limit and

Spectral Statistics of GenericHamiltonian Operators

von Ingolf Schäfer

November 21, 2006

Dissertation

zur Erlangung des Grades eines Doktors derNaturwissenschaften

– Dr. rer. nat. –

Datum des Promotionskolloquiums: 09.11.2006

Gutachter: Prof. Dr. E. Oeljeklaus (Universität Bremen)Prof. Dr. A. T. Huckleberry (Ruhr-Universität Bochum)

Contents

1 Introduction 7

2 Construction of the Classical Limit 112.1 The Classical Limit in the Simple Case . . . . . . . . . . . . . . . . . 112.2 The Classical Limit in the General Case . . . . . . . . . . . . . . . . 122.3 Realizing the Classical Limit as an Analytical Limit . . . . . . . . . . 14

3 Spectral Statistics of Simple Operators 213.1 A Convergence Theorem for Simple Operators . . . . . . . . . . . . . 213.2 Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Rescaling and Spectral Statistics . . . . . . . . . . . . . . . . 253.2.2 Rescaling and exp . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Spectral Statistics of Generic Operators 294.1 Topology and Completion of U(g) . . . . . . . . . . . . . . . . . . . . 294.2 A Notion of Hermitian Operators for O(Cn) . . . . . . . . . . . . . . 314.3 Examples of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Rational Independence of the Spectra in Representations . . . . . . . 354.5 Ergodic Properties of Hgen . . . . . . . . . . . . . . . . . . . . . . . . 364.6 The Sets BN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.7 Convergence to µPoisson . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 The Poisson Spectral Statistics for Tori 435.1 Some Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 The Random Variable Z(n, F, T (N)) . . . . . . . . . . . . . . . . . . 455.3 Moving the Estimates to TCor(k, a, f, T (N)) . . . . . . . . . . . . . . 505.4 The Weak Convergence of µ(naive, U(N), 1) to the Poisson Distribution 515.5 The M -grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.6 The Key Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.7 The Final Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Appendix 596.1 Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.1.1 Representations of Compact Lie Groups . . . . . . . . . . . . 596.1.2 The Universal Enveloping Algebra . . . . . . . . . . . . . . . . 626.1.3 The Laplace Operator . . . . . . . . . . . . . . . . . . . . . . 636.1.4 The Theorem of Borel-Weil and the Embedding Of Line Bundles 63

3

Contents

6.2 Symplectic geometry and momentum maps . . . . . . . . . . . . . . . 666.3 Generalities on Level Spacings . . . . . . . . . . . . . . . . . . . . . . 67

6.3.1 The Nearest Neighbor Distribution . . . . . . . . . . . . . . . 676.3.2 The Kolmogorov-Smirnov Distance . . . . . . . . . . . . . . . 696.3.3 Approximating N -tuples . . . . . . . . . . . . . . . . . . . . . 716.3.4 The Nearest Neighbor Statistics under exp . . . . . . . . . . . 736.3.5 The Nearest Neighbor Statistics and the CUE Measure . . . . 75

4

List of Figures

1.1 A sample histogram of the nearest neighbor statistics . . . . . . . . . 8

2.1 A picture of the U−-section. . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 A picture of B3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Pictures of B3 and B4 intersected with the hyperplane normal to the

diagonal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.1 Approximation of µPoisson. . . . . . . . . . . . . . . . . . . . . . . . . 72

5

List of Figures

6

1 Introduction

The theory of spectral statistics is concerned with the spectral properties of en-sembles of linear operators. Typically, these depend on a parameter N which issupposed to be very large or even approaching infinity. The origin of this field isquantum physics, where such ensembles arose as models for the energy spectra oflarge atoms.

Another branch of physics, namely semiclassical physics, is also concerned withsuch ensembles and their spectral statistics. In semiclassical physics large values ofN should correspond to a quantum mechanical system which approaches classicalmechanics. Details about these relations can be found in [Meh91] and [Haa99].

Finally, spectral statistics have been studied in the context of number theory, withthe most famous example being the distribution of zeros of the Riemann ζ-functionon the critical line. An introduction to this field is given in [Sna00].

Under the assumption of genericity one might hope that there exist natural se-quences of operators taken from these ensembles such that the spectral propertiesof the individual operators reflect those of the ensembles.

We are concerned here with two examples, in which spectral statistics appear.The first being the theory of Random Matrices. In this theory natural sequencesof symmetric spaces with invariant measures on them are given. These spaces havenatural representations as matrices and one is interested in the limit of the spectralstatistics as N → ∞. An example is the sequence of unitary groups U(N) withthe Haar measure. In [KS99] it is proven that a limit measure of a special kind ofspectral statistics exists for this example.

The second example, in which spectral statistics appear, is given by the approachsuggested in [GHK00]. In this article the authors consider two fixed operators in theuniversal enveloping algebra of SL(3,C) in a sequence of irreducible representationsof SL(3,C) and study the spectral statistics by numerical methods. The motivationfrom the approach stems from a previous paper (cf. [GK98]) of two of the authors:Such a sequence of irreducible representations occurs in the construction of the clas-sical mechanical system in the limit of a quantum mechanical system with SL(3,C)symmetry. We will follow this approach in the following chapters.

Our main device in the study of spectral statistics is the nearest neighbor statistics,i.e. the normalized distribution of distances of neighboring eigenvalues (counted withmultiplicity) of such linear operators. It is frequently drawn as a histogram (seeFigure 1.1). A detailed explanation of this plot can be found in the Appendix.

The nearest neighbor statistics lead to Borel measures on the positive real lineby putting a Dirac measure for every occurring distance of neighboring eigenvalueswith proper normalization. Out of the wealth of notions of convergence for such

7

1 Introduction

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 1.1: A sample histogram of the nearest neighbor statistics

measures we choose the weak convergence (in probability theory: convergence indistribution) and the Kolmogorov-Smirnov convergence. The Kolmogorov-Smirnovdistance of two measures µ, ν is given by

dKS(µ, ν) = supt∈R

∣∣∣∣∫ t

−∞dµ−

∫ t

−∞dν

∣∣∣∣ , (1.1)

i.e., Kolmogorov-Smirnov convergence is uniform convergence of the cumulative dis-tribution functions. We will examine dKS for sequences of individual operatorsrelative to a fixed measure ν, but also average dKS with respect to a fixed probabil-ity measure ν over the full ensemble. Here sequences of irreducible representationswill arise.

This text is structured into six chapters. Following the approach in [GK98] wegive a general construction of the classical limit for semi-simple compact Lie groupsin Chapter 2. This can be done in a functorial way, but the objective of Chapter 2is to give an interpretation as a mathematical limit as a parameter n converges to∞.

Chapter 3 deals with the spectral statistics of operators in the Lie algebra alongsequences of irreducible representations. It is necessary to discuss possible scalingsof these operators in this context.

The goal of Chapter 4 is to study the spectral statistics of exponentiated operators,which satisfy certain conditions of genericity, in a certain completion of the universalenveloping algebra of a semi-simple complex Lie group. The main tools are Birkhoff’sErgodic Theorem and an estimation on dKS for maximal tori of U(N).

Chapter 5 is devoted to the proof of this estimation, where we follow the structureof [KS99] for the proof.

In the Appendix we collect the necessary background facts of representation theory

8

and symplectic geometry for the readers’ convenience. The Appendix closes withsome general observations about nearest neighbor statistics.

Acknowledgments

I would like to mention all the people who helped me while writing this thesis.First and foremost, I would like to thank my supervisors Alan T. Huckleberry andEberhard Oeljeklaus for their support and guidance. Furthermore, I am indebtedto Marek Kuś, who gave me insight on the physical motivation for the topic of thisthesis and who invited me to Warsaw for joint research.

Special thanks go to my colleagues in Bremen and Bochum, where I would liketo mention Kristina Frantzen, Daniel Greb, Christian Miebach, Elmar Plischke,Patrick Schützdeller, Monika Winklmeier and Christian Wyss, who always foundtime to discuss and helped me in various ways.

Last but not least, I would like to thank my wife Silke Schäfer for her supportand patience.

Support by the Sonderforschungbereich TR 12 “Symmetries and Universality inMesoscopic Systems” is gratefully acknowledged.

9

1 Introduction

10

2 Representation TheoreticalConstruction of the Classical Limit

In this chapter we will give a construction of the classical limit of Hamiltonianmechanics by a representation theoretical approach. Our method is an abstractgeneralization of the method given in [GK98] and [Gnu00] and covers systems withcompact semi-simple Lie groups as symmetry groups.

The following notation will be used without further notice (for details cf. Ap-pendix): K is a compact semi-simple Lie group with complexification G and thecorresponding Lie algebras are k and g. Every representation of K will be assumedto be continuous, finite-dimensional and unitary, where the scalar product is de-noted by 〈·, ·〉. By convention the scalar product is complex linear in the second,and anti-linear in the first variable.

Furthermore we assume that we have fixed a Borel subgroup B ⊂ G and obtain anotion of positivity of roots and weights. Recall that the choice of B also determinesa maximal tours T ⊂ K.

2.1 The Classical Limit in the Simple Case

A guiding principle in quantum mechanics is that of correspondence. It statesthat quantum mechanical systems whose size is large compared to microscopicallength scales can be described by classical physics. The classical system attached tothe quantum mechanical system is called the classical limit (cf. [GK98]). So thereshould be some kind of functor from Hilbert spaces with Hamiltonian operators tosymplectic manifolds with Hamiltonian functions. Actually, one might require thatthis functor is inverse to so-called geometric quantization. At least it should satisfythe Dirac correspondence, i.e., if ξH1 and ξH2 are two Hamiltonian operators withcorresponding Hamiltonian functions h1 and h2, then the Lie bracket of ξH1 and ξH2

should correspond to the Poisson bracket of h1 and h2:

[ξH1 , ξH2 ] 7→ ch1, h2, (2.1)

where c is a constant, usually i~.More often, one discusses the opposite direction, i.e., quantization (cf. [Woo97]

Chapter 9.2). Therefore one may call the procedure presented here dequantization.Let ρ : K → U(V ) be an irreducible representation. Let ρ∗ : k → End(V ) be

the induced representation of the Lie algebra. Both ρ and ρ∗ extend to holomorphic

11

2 Construction of the Classical Limit

resp. linear representations of the corresponding complexifications G and g. To keepnotation as simple as possible we will also denote these by ρ and ρ∗

The map µ : P(V ) → k∗ given by

µξ([v]) = −2i〈v, ρ∗(ξ).v〉〈v, v〉

∀ξ ∈ k, v ∈ P(V ) (2.2)

is the momentum map with respect to the symplectic structure on P(V ) inducedby the Fubini-Study metric (cf. Appendix for details). Moreover, if λ ∈ t∗ is thehighest weight of ρ, then

µ([vmax]) = λ (2.3)

for any vector vmax of highest weight.Since µ is an K-equivariant map and the stabilizers of λ and vmax agree, this map

is a symplectic diffeomorphism of the orbit K.[vmax] onto the coadjoint orbit K.λwith the Kostant-Kirillov form.

In the literature, this coadjoint orbit is called the set of coherent states (cf.[Per86], [Woo97]). To simplify notation we write Z = K.λ for this set.

Equivariance implies that the map µ : k → C∞(Z), ξ 7→ µξ(·), satisfies

µ([ξ1, ξ2]) = µ(ξ1), µ(ξ2). (2.4)

If we compare this equation with the Dirac condition (2.1), then, up to constants,this is exactly what we are looking for. But the Lie algebra k acts by skew self-adjoint operators on V . Thus we define cl : ik → C∞(Z) for an element ξH ∈ ikby

cl(ξH)([x]) =1

2µ(iξH)(x) =

〈x, ρ∗(ξH).x〉〈x, x〉

, (2.5)

where the factor 12

will become clear in the following. First note that while iξH isrepresented as a skew self-adjoint operator, ξH is self-adjoint. Now, we have thefollowing version of the Dirac correspondence for the classical limit cl:

cl(i[ξH1 , ξH2 ]) =1

2µ(ii[ξH1 , ξH2 ]) =

1

2µ([iξH1 , iξH2 ])

= 2 · 1

2µ(iξH1),

1

2µ(iξH2) = 2 · cl(ξH1), cl(ξH2).

(2.6)

2.2 The Classical Limit in the General Case

So far our classical limit has been defined for those self-adjoint operators which canbe expressed as the image of an element of ik under ρ∗. But we want to define theclassical limit for every self-adjoint linear operator on V . In fact, it will be defined forall linear operators on V , although in general we do not obtain real-valued functionson Z if we take the classical limit of an operator which is not self-adjoint.

Let T (g) denote the full tensor algebra of g. The Lie algebra representation ρ∗extends uniquely to a representation ρ∗ : T (g) → End(V ). This map is surjective by

12

2.2 The Classical Limit in the General Case

the lemma of Burnside. Thus, in particular every self-adjoint operator is containedin the image of ρ∗.

We fix an R-basis ξ1, . . . , ξk of ik for the rest of this chapter. Note that thisis a C-basis of g. Thus, an element ξH of T (g) has a unique decomposition intohomogeneous terms consisting of sums of “monomials” ξα1 ⊗ · · · ⊗ ξαp for someindices αj ∈ 1, . . . , k. (These are not monomials in the usual sense because of thenon-commutativity.)

Definition 2.1. The classical limit of such a “monomial” is

cl(ξα1 ⊗ · · · ⊗ ξαp) := cl(ξα1) · · · · · cl(ξαp). (2.7)

The classical limit of

ξH =∑

αIξα1 ⊗ · · · ⊗ ξαpI∈ T (g) (2.8)

is the sum of all classical limits of each “monomial” multiplied by the correspondingcoefficient.

We call the resulting map cl : T (g) → C∞(Z,C) the classical limit map.

Let us discuss this definition. First note that if ξH is abstractly self-adjoint, thencl(ξH) is real-valued. To see this, we calculate

cl(ξH) = cl(ξ†H) = cl(ξH), (2.9)

where the last step is due to (6.12) and (2.7). The converse is false since, in general,T (g) contains nilpotent elements.Remark 2.2. The map cl : T (g) → C∞(Z,C) has a natural factorization clS : S(g) →C∞(Z,C) to the full algebra of symmetric tensors S(g).

In this way the classical limit map is a link between the non-commutative algebraT (g) and a certain commutative subalgebra of C∞(Z,C). But since C∞(Z,C) iscommutative, we have to work with the tensor algebra and cannot pass to theuniversal enveloping algebra U(g) in the definition of the classical limit, otherwisethe quotient will not be well-defined. To see this, take any operators ξand ξb suchthat [ξa, ξb] 6= 0. Then it follows that cl(ξaξb − ξbξa − [ξa, ξb]) is not equal to zero.

Let x ∈ V be a vector of unit length. Reading cl as a map to C∞(V \0,C) wesee that

cl(ρ∗(ξaξb))(x) = cl(ξa)(x) cl(ξb)(x) = 〈x, ρ∗(ξa)x〉 · 〈x, ρ∗(ξb)x〉, (2.10)

which has a meaningful physical interpretation. Namely, if we think of ξa and ξb asobservables, then in the classical limit the expectation value of the operator ξaξb isgiven by the product of the expectation values of ξa and ξb

1. But this means thatthe operators ξa and ξb are stochastically independent in the classical limit.

1This remark has to be taken cum grano salis, because of the possible complex phases on theright-hand side. For probabilities one has to take the absolute value squared, which is animplicit convention in theoretical physics.

13


The main point of this chapter is to give an analytical realization of this purelyalgebraic construction, i.e., there will be a parameter and we will obtain the aboveclassical limit as an analytical limit when this parameter goes to infinity. Thiswill make the notion of ~ → 0 precise in our context. Here the theme of non-commutativity vs. commutativity will appear again.

2.3 Realizing the Classical Limit as an Analytical Limit

The Lie algebra g can be decomposed as

g = u− ⊕ tC ⊕ u+, (2.11)

where tC is the Lie algebra of the complexified maximal torus and u− and u+ areunipotent Lie subalgebras corresponding to the positive and negative roots. Wedefine the groups

U+ = exp(u+), U− = exp(u−), and TC = exp(tC). (2.12)

Recall that the decomposition of the Lie algebra g almost yields a decomposition ofG. “Almost” in this context means that it is a decomposition of G\S, where S is aZariski-closed set,

G = Zarsiki closure of U−TCU+, (2.13)

and even strongerG\S ' U− × TC × U+. (2.14)

Let us again consider the representation ρ∗ : U(g) → End(V) and choose a vectorof highest weight vmax ∈ V . By the definition of vmax we see that U+ ⊂ StabG(vmax)and ρ(T ) ⊂ C∗ · vmax. Moreover, the K-orbit through [vmax] agrees with the G-orbitthrough this point, i.e. K.[vmax] = G.[vmax].

Thus, there exists a Zariski-closed set A in K.[vmax] such that K.[vmax]\A is iso-morphic to the orbit of U− through vmax in V . Therefore, the U−-orbit is isomorphicto a dense, Zariski-open subset of Z if we identify Z = K.λ with K.[vmax] via themomentum map.

We will write cl as composition of two maps r and s:

r : ik → Vect(V \0), ξ 7→ −1

2Xξ, with (Xξf)(x) =

d

dt

∣∣∣∣t=0

f(exp(−ξt).x) (2.15)

and

s : Vect(V \0) → C∞(V \0,C), X 7→ 1

N(XN), (2.16)

where N(x) = ‖x‖2 is the norm function squared.

14


Slightly changing the definition of cl to a map to C∞(V \0,C) the definition ofthe momentum map (2.2) yields the following commutative diagram:

ik r//

cl ((RRRRRRRRRRRRRRRR Vect(V \0)s

C∞(V \0,C)

(2.17)

Let us explicitly calculate the map s on the U−-orbit through vmax:

(XξN)(x) =d

dt

∣∣∣∣t=0

N(exp(−tξ).x). (2.18)

Since x lies on the U−-orbit, there exists a u ∈ U− such that

x = u.vmax. (2.19)

Now we can decompose exp(−tξ)u uniquely as

exp(−ξt)u = u−(t)l(t)u+(t) (2.20)

for t in a neighborhood of 0, where u−(t) ∈ U−, l(t) ∈ TC and u+(t) ∈ U+. To seethis note that we can decompose the identity and the set of decomposable elementsis a Zariski open set by (2.13). Using the chain rule and self-adjointness of ξ, weobtain

(XξN)(x) = 2

⟨x,

d

dt

∣∣∣∣t=0

exp(−ξt).x⟩

= 2

⟨x,

d

dt

∣∣∣∣t=0

u−(t)l(t)u+(t).vmax

⟩.

(2.21)But since u+(t) ∈ U+ ⊂ StabG(vmax) for all t we have

(XξN)(x) = 2

⟨x,

d

dt

∣∣∣∣t=0

u−(t)l(t).vmax

⟩. (2.22)

According to the product rule and using l(0) = Id, u−(0) = u we find

(XξN)(x) = 2〈x, u d

dt

∣∣∣∣t=0

l(t).vmax〉+ 2

⟨x,

d

dt

∣∣∣∣t=0

u−(t).vmax

⟩. (2.23)

Due to the fact that l(t) ∈ T acts as scalar on vmax this can be simplified as follows

(XξN)(x) = 2l(0)〈x, x〉+ 2

⟨x,

d

dt

∣∣∣∣t=0

u−(t).vmax

⟩. (2.24)

Thus, we can read the right hand side as a differential operator applied to thenorm function. This operator consists of a multiplication part with 2l(0) and a vectorfield part which is tangential to the U−-orbit. Let D(U−.vmax) denote the algebra of

15


linear differential operators on U−.vmax. We claim that the above procedure affordsa map

r : ik → D(U−.vmax), ξ 7→ mξ + ξtan, (2.25)

where ξtan is the vector field tangent to the U− orbit whose one parameter groupat x is given by 2 d

dt

∣∣t=0

u−(t) with respect to the above decomposition, and mξ isa smooth function on the U−-orbit with mξ(x) = 2l(0). The only thing we have toshow is that the construction is independent of the choice of u in (2.19). But if wechoose u′ with

x = u.vmax = u′.vmax (2.26)

then u′u−1 ∈ StabG(vmax). So, u′ = ug, where g ∈ StabG(vmax). But as g actstrivially on vmax the calculation does not change.

The map r will be the crucial point in the following. We will discuss it from anabstract point of view later on, but first we extend r to T (g) in the following manner

r(ξα1 ⊗ · · · ⊗ ξαp) = r(ξα1) · · · r(ξαp). (2.27)

This is well-defined because the ξj are linear differential operators, so they respectscalar multiplication and addition.

Before we go into the details of the convergence, we need a fact about the norm.

Theorem 2.3. Let λ be the highest weight of the representation ρ with decompositioninto fundamental weights fj as follows

λ =r∑

j=1

λjfj. (2.28)

Then the squared norm function N on the U−-orbit decomposes as

N(u.vmax) = c ·N1(u.vmax)λ1 · · · · ·Nr(u.vmax)

λr , (2.29)

where r is the rank of g and N1, . . . , Nr are the squared norms of the fundamentalunitary representations corresponding to the fundamental weights f1, . . . , fr.

Proof. For every fundamental representation ρ(j) we have a holomorphic line bundleLj → G/B− such that the representation of G on Γhol(G/B−, L) is equivalent to ρ(j)

(cf. Appendix Theorem 6.14).By induction and Lemma 6.16, we find that the representation with highest weight

λ =∑λjfj is given by the action on the sections of

L = Lλ1

(1) ⊗ . . .⊗ Lλr

(r). (2.30)

Let hj denote the induced K-invariant, hermitian bundle metric on Lj, which isgiven in Lemma 6.15, and h the induced metric for L.

Choose a common open covering Wk of G/B−, such that L and all Lj aretrivializable over each Wk. Without loss of generality we may assume that W1 =

16


U− · [vmax]. Each hermitian bundle metric hj is given by a family mk,j : Wk → R+,h by the family mk : Wk → R+.

A direct calculation shows that the family m′k : Wk → R+ given by

m′k := mλ1

k,1 · . . . ·mλrk,r (2.31)

represents a hermitian, K-invariant bundle metric h′ on L. Thus, h′ = ch for somepositive constant c. Using (6.23) we see that the norm on W1 is defined by thebundle metric up to this scalar.

This completes the proof of Theorem 2.3.

In the following we will consider a highest weight λ =∑

j λjfj. If we are given afunction like

u+ u

1 + ‖u‖2λ1 + 17λ2 (2.32)

then we can think of the function as a polynomial in λ1, λ2 where the coefficientsare smooth functions. It is even a homogeneous polynomial of degree 1.Notation 2.4. The ring of smooth functions on U−.vmax is denoted by the symbolR, i.e. R := C∞(U−.vmax,C), and the ring of polynomials in the λj with coefficientsin R by R[λ].

The key result of this chapter is the following:

Theorem 2.5. Let λ =∑

j λjfj be the highest weight of ρ and assume that at leastone λj > p for a fixed natural number p. Furthermore, let α = ξα1 ⊗ · · · ⊗ ξαp be a“monomial” element of degree p in the generators ξj of g as chosen above.

Then f(λ) := 1Nr(α)(N) ∈ R[λ] and deg f = p. The homogeneous part of degree

p of f is, up to a real, multiplicative constant, given by cl(ξα1) · · · · · cl(ξαp), wherewe view the cl(ξαj

) as elements of R[λ]. Moreover, the constant does not depend onα.

Proof. By definition, every r(ξαj) is a first order partial differential operator. Hence

the summands in the derivative of N = Nλ11 · · · · · Nλr

r , after dividing by N , arepolynomials in λ of degree at most p. On the other hand, at least one such summandmust be a polynomial of degree at least p. If all were of lesser degree, one of the ξαj

would be multiplication by a constant, which is not the case, or the partial derivativeswould lower every exponent λj to 0, which yields a contradiction because at leastone λj is larger than p. This proves the first part of the theorem.

For the second part, we consider the case p = 1 first.Then there is no degree zeroterm in the polynomial 1

Nr(α)(N) since

1

Nr(α)(N) =

1

Nr(α)(N) (2.33)

in the above construction. But r(α) is a vector field and contains no multiplicativepart, so we have only partial derivatives turning N into a homogeneous polynomialof degree 1 after dividing by N . This proves the second statement for p = 1.

17


Let p ≥ 2 and ξα1 ⊗ . . .⊗ ξαp be given. We have r(ξa) = c+∑aj

∂∂zj

for some aj

and c in some coordinate system zj on U−.vmax. By the induction hypothesis

cl(ξα2 ⊗ · · · ⊗ ξαp) = cl(ξα2) · . . . · cl(ξαp) + q, (2.34)

where q is a polynomial of degree less than p− 1. Using the product rule of differ-entiation we calculate explicitly

cl(ξα1 ⊗ . . .⊗ ξαp) =1

Nr(ξα1 ⊗ . . .⊗ ξαp)(N) =

1

Nr(ξα1)(N cl(ξα2) . . . cl(ξαp)+Nq)

=1

N

(c+

∑aj

∂

∂zj

)(N cl(ξα2) . . . cl(ξαp) +Nq)

= c · cl(ξα2) . . . cl(ξαp) + q +1

Ncl(ξα2) . . . cl(ξαp)

(∑aj

∂

∂zj

)(N)

+1

N

(∑aj

∂

∂zj

)(cl(ξα2) . . . cl(ξαp)) +

1

N

(∑aj

∂

∂zj

)(Nq)

=1

N(cl(ξα2) · . . . · cl(ξαp))

(∑aj

∂

∂zj

+ c

)(N) + terms of degree less than p.

= cl(ξα1) + (cl(ξα2) · . . . · cl(ξαp)) + terms of degree less than p. (2.35)

Here the first summand is a homogeneous polynomial of degree p, as claimed. Theremaining summands are certainly of lower degree, because each cl(ξb) is of degreeone and taking the partial derivatives can only lower the degree.

After these preparations we define the classical limit along a ray in the followingway.

Definition 2.6. Let ρ : K → U(V ) be a non-trivial, irreducible, unitary represen-tation of a semisimple, compact Lie group K on a finite-dimensional vector space Vcorresponding to the highest weight λ.

We call a sequence (ρn : K → U(Vn))n∈N∗ of irreducible, unitary representations,each ρn corresponding to the highest weight n ·λ, the ray through ρ. For simplicity,we shall always assume that ρ1 = ρ.

Let ξH ∈ T (g) be an abstractly hermitian operator and ξ1, . . . , ξk be a basis of ik.We have a unique decomposition into “monomials” of ξH =

∑j ajξj1 ⊗ · · · ⊗ ξjd(j)

,where each aj is a complex number. (Keep in mind that these are not monomialsin the usual sense because of the non-commutativity.)

Definition 2.7. The n-th approximation of the classical limit is

cln(ξH) =∑

j

aj1

nd(j)

1

Nrn(ξj1 ⊗ · · · ⊗ ξjd(j)

)(N). (2.36)

Here rn is defined as in (2.25) and (2.27) with respect to the representation ρn, i.e.we substitute every λj in the resulting polynomials by n · λj.

18


Theorem 2.8. Along a ray through the non-trivial, irreducible representation ρ then-th approximations of the classical limit converge to the classical limit uniformlyon compact subsets of U−.vmax for every fixed ξH ∈ T (g), i.e.

cln(ξH) → cl(ξH) uniformly on compact subsets, as n→∞. (2.37)

Proof. Decompose ξH into its homogeneous parts:

ξH =∑

j

ξj (2.38)

where each xj is homogeneous of degree j. Since ρ is a non-trivial representation,at least one λj in the decomposition λ =

∑j λjfj is not zero. Because ξH has only

a finite degree, the conditions of Theorem 2.5 are satisfied for all n sufficiently big.Applying this theorem to each “monomial” in every ξj implies

cln(ξj) = cl(ξj) +1

n(terms of lower degree). (2.39)

It follows that for any compact set M

cln(ξH)(x) → cl(ξH)(x) as n→∞ (2.40)

for all x ∈M uniformly.

This completes the construction of the classical limit as a mathematical limit. Thereader might wonder whether the convergence on a dense, open subset of Z suffices.Note that cl is defined on the whole of Z, but our U− chart is not. Unfortunately,it is not clear that every approximation can be extended to Z, but nevertheless thelimit does extend continuously.

Let us now discuss the procedure a more abstractly. The main step is the sub-stitution of r for r in the definition of the classical limit. After this, the othertheorems follow from Theorem 2.3. But what are these deformed vector fields r(ξ)?In a way this is at least in a formal sense similar to a connection in a line bundleplus multiplicative function, like in geometric quantization. Indeed, we have a linebundle here. It is the tautological bundle V \0 → P(V ) restricted to K.[vmax].Furthermore, the U−-orbit can be thought of as a section of this bundle over thedense open set U−.[vmax]. Since U− is biholomorphic to some Cp, we get a chart forthe bundle here. In this chart r is in fact just a connection plus a multiplicativepart.

A visualization of the situation is provided by Figure 2.1. Here we see the originin V and vmax. Since K acts unitarily, the K-orbit preserves the metric and isdrawn as a circular arc. The U−-orbit is non-compact and drawn as a very flatparabola. If we look at this in P(V ), we see that the U−-orbit is not a global sectionof the tautological bundle because the horizontal axis has no intersection with theU−-orbit.

19


K-orbit

0

U--orbit

vmax

Figure 2.1: A picture of the U−-section.

20

3 Spectral Statistics of SimpleHamiltonian Operators

The spectral statistics of simple Hamiltonian operators, i.e., the nearest neighborstatistics for elements of some semi-simple Lie algebra, are discussed in this chapter.The main interest is in the behavior of the spectral statistics in irreducible repre-sentations as the dimension goes to infinity. Thereafter, the notion of rescaling isintroduced and some consequences of the choice of rescaling are given.

3.1 A Convergence Theorem for Simple Operators

In this section we give an estimation on the number of weights of irreducible rep-resentations and in certain cases deduce from it the convergence of the spectralstatistics for simple operators.

Here K always denotes a semi-simple, compact Lie group with a fixed maximaltorus T and a fixed notion of positivity of roots. We write W for the Weyl group ofK with respect to T . Further, let G be the complexification of K and denote thecorresponding Lie algebras by g and k. For any hermitian matrix A we write µA forthe nearest neighbor statistics of A, i.e.,

µA := µ(X(A)) (3.1)

as defined in (6.30). If U is a unitary matrix we will write µU for the nearest neighborstatistics of unitary matrices (6.37), i.e.

µU := µc(X(A)). (3.2)

It is clear by the subscript which kind of statistics is meant, so we use the sameabbreviation.

We start with a lemma.

Lemma 3.1. Let ρλ : K → U(Vλ) be an irreducible, unitary representation withhighest weight λ. Let λ =

∑λjfj be the decomposition of λ into the basis of fun-

damental weights fj. Then the number nλ of possible weights of ρλ is bounded asfollows

nλ ≤ ord(W ) ·∏

j

(λj + 1). (3.3)

21

3 Spectral Statistics of Simple Operators

Proof. Starting from λ we get all other weights by subtracting multiples of the roots.The lattice of roots is a sublattice of the lattice of weights, so we can reach everyweight by subtracting multiples of the fundamental weights fj.

There are at most∏

j(λj +1) of the such possible substractions that give positiveweights and every weight is in the W -orbit of a positive weight, which has at most|W | elements.

Now we give a rough estimate for the dimension of an irreducible representation.

Lemma 3.2. Under the assumptions of Lemma 3.1 we have the following inequalityfor the dimension of ρλ:

dim ρλ ≥∏

α∈Π+,〈λ,α〉>0

〈λ, α〉〈δ, α〉

, (3.4)

where Π+ denotes the set of positive roots and δ = 12

∑α∈Π+ α.

Proof. Weyl’s dimension formula reads

dim ρλ =∏

α∈Π+

〈δ + λ, α〉〈δ, α〉

=∏

α∈Π+

(1 +

〈λ, α〉〈δ, α〉

). (3.5)

Now, 〈λ, α〉 ≥ 0 and 〈δ, α〉 > 0 for all positive roots α. Thus, the inequality isclear.

We write δDirac for the Dirac measure with mass 1 at 0 and apply these lemmasto the situation of Chapter 2 where we looked at rays to infinity.

Theorem 3.3. Let ρ : K → U(V ) be an irreducible representation with highestweight λ =

∑λjfj and the sequence (ρn : K → U(Vn))n∈N∗ be a ray through ρ.

If r := rank(K) ≥ 2 and

r < #α ∈ Π+ : 〈α, λ〉 > 0 (3.6)

then for every ξ ∈ ik\0

µρ∗,mλ(ξ) → δDirac in dKS, as m→∞. (3.7)

Proof. Let ξ ∈ ik be given. The element iξ ∈ k is conjugated to an element η ∈ t =Lie(T ). We will show that

pm :=number of (different) eigenvalues of ρ∗,mλ(η)

dim ρmλ

→ 0 (3.8)

as m→∞. This implies the convergence to δDirac since the value of lims→0

∫ s

0dµA

is1− number of (different) eigenvalues

number of rows of A(3.9)

22

3.1 A Convergence Theorem for Simple Operators

for any hermitian matrix A by the definition of the nearest neighbor statistics. Thus,µρ∗,mλ(ξ) has mass 1− pm at zero, which proves the convergence.

It remains to show the claim about pm. To do so, note that the eigenvalues ofρmλ(ξ) are just the values of the weights of the representation evaluated at ξ. So,it is sufficient to prove that the ratio of the different weights and the dimension ofρmλ converges to zero.

To show this we combine the inequalities of Lemma 3.1 and 3.2, but first wesimplify the notation a bit. We denote by Q the set of α ∈ Π+, such that 〈α, λ〉 > 0and by q the cardinality of Q. Finally, the number of different weights in ρmλ isnmλ.

We obtain

nmλ

dimρmλ

≤

(ord(W ) ·

∏rj=1(λj + 1)

)mr(∏

α∈Q〈λ,α〉〈δ,α〉

)mq

= c(λ)mr−q. (3.10)

Here c(λ) is a constant, depending only on λ, and, since r < q by the hypothesis,the ratio converges to zero as promised. This proves the theorem.

Remark 3.4. The number q in the above proof is the complex dimension of thecoadjoint orbit through λ, i.e., the complex dimension of the classical phase spacein the classical limit of Chapter 2.

Corollary 3.5. The conditions of the above theorem will be automatically satisfiedif K is simple, rankK ≥ 2, and λ lies in the interior of the Weyl chamber.

Proof. First, we remark that r equals the number of positive roots for any represen-tation whose highest weight is in the interior of the Weyl chamber, since the interioris defined by the condition 〈λ, α〉 > 0 for every simple root α. But positive rootsare positive integer combinations of simple roots 〈λ, α〉 > 0 for all positive roots α.This completes the proof.

We now give another corollary.

Corollary 3.6. Under the assumptions of the theorem let t1, . . . , tp ∈ g be givensuch that ξ = t1 ⊗ · · · ⊗ tp ∈ T (g) is abstractly hermitian in the sense of definition6.10. Furthermore, let ρ∗,mλ be the induced Lie algebra representation with highestweight mλ extended to the full tensor algebra. Then

µρ∗,mλ(ξ) → δDirac weakly as m→∞, (3.11)

if p · r < #α ∈ Π : 〈α, λ〉 > 0.

Proof. We can assume without loss of generality that all tj are always representedas diagonal matrices and we proceed by induction. From (3.10) it follows that foreach tj the number of its eigenvalues nj,mλ divided by the dimension is smaller thanc(λ)mr−q. But the maximal number of eigenvalues in a product of diagonal matrices

23


is just the product of the number of eigenvalues of each matrix. Thus, we have anumerator mrp here instead of mr in (3.10). But by assumption rp < q, i.e. thenumber of eigenvalues of the product divided by the dimension is decreasing fasterthan 1/m.

This proves the corollary.

3.2 Rescaling

In this section we discuss the notion of rescaling. This concept appeared alreadyin Chapter 2. There the classical limit along rays (ρm : K → U(Vm))m∈N∗ througha given representation ρ was considered and the scaling was given by substituting1mξj for ξj. Since we are interested in the problem of scaling in general, we define

the notion of a rescaling map abstractly.Let U(g) denote the universal enveloping algebra of g and † the formal adjoint

(cf. Appendix). We choose a fixed basis ξ1, . . . , ξn of g and write the elements ofU(g) as ordered polynomials in the ξj. Furthermore the multiindex notation ΞI willbe used for ξi1

1 . . . ξinn .

The basic problem can be seen if one considers the hermitian operators ξ and ξη ina sequence of irreducible representations. As the dimensions of the representationsincrease the maximal eigenvalues of ξη will in general grow faster than those of ξ.In principle, we would like the rate of growth to be the same, including the optionof no growth at all. This motivates the following definition.

Definition 3.7. A rescaling map rρ for the irreducible representation ρ : K →U(V ) is given by a map

rρ : U(g) → U(g),∑

aIΞI 7→

∑ 1

s|I|aIΞ

I , (3.12)

where s a positive integer number.

Lemma 3.8. Every rescaling map rρ is linear, injective and compatible with †.

Proof. This follows directly from the definition of rρ.

Of all possible scalings the most natural one is the scaling by inverse dimensionsince we have no other natural quantity associated to arbitrary sequences of irre-ducible representations.

Definition 3.9. Let Irr(K) denote the set of equivalence classes of irreducible, uni-tary representations of K. The rescaling by inverse dimension is the family ofrescaling maps (iρ)ρ∈Irr(K) given by

iρ : U(g) → U(g),∑

aIXI 7→

∑ 1

(dim ρ)|I|aIX

I (3.13)

for each ρ ∈ Irr(K).

24

3.2 Rescaling

If we are considering rays through a fixed irreducible representation with highestweight λ, then we have another natural quantity: the parameter m for each ρmλ.

Definition 3.10. Let ρ : K → U(V ) be an irreducible representation with highestweight λ =

∑λjfj and the sequence (ρm : K → U(Vm))m∈N∗ be a ray through ρ.

The rescaling by inverse parameter is the family of rescaling maps (pρmλ)

given by

pρm : U(g) → U(g),∑

aIXI 7→

∑ 1

m|I|aIXI . (3.14)

3.2.1 Rescaling and Spectral Statistics

In the first section we considered simple operators only, i.e. Lie algebra elements.Rescaling has no effect in this case since for any self-adjoint matrix A and any c > 0

µA = µc·A. (3.15)

But rescaling has an effect if we consider operators whose monomial parts havedifferent degrees, e.g.

ξ + η2 ∈ U(g). (3.16)

Recall that for a highest weight λ the set Q is defined as Q = α ∈ Π+ : 〈α, λ〉and q = #Q. We state the following lemma:

Lemma 3.11. Let ξH =∑

I aIΞI ∈ U(g) be given with ξ†H = ξH and consider the

ray (ρm : K → U(Vm))m∈N∗ through an irreducible representation ρ : K → U(V ) ofhighest weight λ.

Then‖ρ∗,m(iρm(ξH))‖End(Vm) ≤ c1(λ)

∑I

|aI |c2(λ)|I| ·m|I|−q|I| (3.17)

where the cj(λ) are constants depending only on λ and ‖·‖End(Vm) denotes the operatornorm on End(Vm).

Proof. We use the explicit construction of irreducible representations by Borel-Weil.For this let

Sj = (s(j)1 , . . . , s

(j)d(j)), j = 1, . . . , r (3.18)

denote a basis of the j-th fundamental representation. These are holomorphic sec-tions in a holomorphic line bundle

Lj → G/B (3.19)

where B is a Borel subgroup of G and L = G×χjC, such that χj : B → C is the ex-

ponentiated character of the fundamental weight λj. The irreducible representationwith highest weight λ is then given by the action on sections of the line bundle

L = L⊗λ11 ⊗ . . .⊗ L

⊗λj

j → G/B. (3.20)

25


By the theorem of Borel-Weil the tensors of the form

SI11 ⊗ . . .⊗ SIr

r , (3.21)

with I1, . . . , Ir multiindices of degree |Ij| = λj constitute a generating system of thespace of sections.

Without loss of generality we may take a basis ξ1, . . . , ξn of g, such that ξ1 isrepresented by a diagonal hermitian matrix of spectral norm 1 in every fundamentalrepresentation. Since the operator norm is equal to the spectral norm, we wish togive an estimate for the maximal absolute value of an eigenvalue of ξ1 in ρ∗,λ.

But on the generating system of vectors given by (3.21) the action is on eachfactor separately, so we have

‖ρ∗,λ(ξ1)‖ 6 λ1 + . . .+ λr =: |λ|. (3.22)

Clearly, the same argument can be carried out for ξ2, . . . , ξn. So we have the followingestimate

‖ρ∗,m(ξj)‖ 6 m(λ1 + . . .+ λr) = m|λ| (3.23)

for all j = 1, . . . , n.Now, consider γ =

∑I aIX

I . Then

‖ρ∗,m(iρm(γ)t)‖End(Vm) 6∑

I

1

(dim ρk)|I|‖ρ∗,m(ξ1)‖i1

End(Vm) · . . . · ‖ρ∗,m(ξn)‖inEnd(Vm).

(3.24)Using the estimates given by (3.23) and Lemma 3.2, we see that

‖ρ∗,m(iρm(γ)t)‖End(Vm) ≤∑

I

|aI |C ·mq|I|m

|I| · |λ||I| = C ′∑

I

|aI ||λ||I|m|I|−q|I|, (3.25)

where C and C ′ are constants depending only on λ, which completes the proof.

We use this lemma to prove the following theorem.

Theorem 3.12. Consider the ray (ρm : K → U(Vm))m∈N∗ through an irreduciblerepresentation ρ : K → U(V ) of highest weight λ and assume q > 2.

Then for all ξH = η +∑

|I|≥2 aIΞI ∈ U(g) with η ∈ g\0 and ξ†H = ξH

dKS(µρ∗,m(iρm (ξH)), µρ∗,m(η)) → 0 as m→∞. (3.26)

Proof. We claim, that

limm→∞

(dimVm) ·

∥∥∥∥∥∥ρ∗,miρm

∑|I|≥2

aIΞI

∥∥∥∥∥∥End(Vm)

= 0. (3.27)

26

3.2 Rescaling

This implies the theorem, because the nearest neighbor statistics for hermitian ma-trices are scaling invariant, i.e.

µ(dim Vn)ρ∗,m(ξH) = µρ∗,m(ξH) (3.28)

and (dimVm)ρ∗,m(iρmη) = ρ∗,m(η). Thus,

limm→∞

‖(dimVn)ρ∗,m(iρm(ξH))− ρ∗,m(η)‖End(Vm) = 0. (3.29)

It remains to proof (3.27). But by (3.25) we obtain

(dimVm)∑|I|≥2

aI

∥∥ρ∗,m (iρm

(ΞI))∥∥

End(Vm)≤ C

∑I

|aI ||λ||I|m|I|−(q−1)|I|, (3.30)

where C is a constant. Since λ is fixed and q > 2 the right hand side converges tozero.

So, we only have to study the convergence of µρ∗,m(η) to gain information aboutthe convergence of the nearest neighbor distribution of the whole operator underrescaling by inverse dimension. For example, we may use Theorem 3.3.

3.2.2 Rescaling and exp

Rescaling can affect the limit measure of exponentiated operators as shown in thefollowing lemma.

Lemma 3.13. Let ρk : K → U(Vk), k ∈ N, be a sequence of irreducible, unitaryrepresentations and γ ∈ U(g) with γ† = γ.

Let us assume that

limn→∞

‖ρ∗,k(rρk(γ)t)‖End(Vk) = 0 for all t > 0, (3.31)

where ‖ · ‖End(Vk) denotes the operator norm on End(Vk).Then µexp(ρ∗,k(rρk

(γ))t) does not converge to any Borel measure µ on the positivereal line with ∫ 1

2π

0

dµ < 1 (3.32)

as n goes to infinity for any t > 0. In particular it does not converge to µPoisson orµCUE.

Proof. For simplicity set γk = rρk(γ) and let t > 0 be fixed. Now by (3.31) we see

that starting from a sufficiently large k0 the spectrum of ρ∗,k(γk)t is in the interval[−π,−π].

Now we may consider a subsequence of ρkjsuch that the spectrum of ρ∗,k(γk)t is

in the interval ] − 12j,− 1

2j[. Analogously to the counterexample in Remark 6.28 in

Chapter 6, one proves that a limit measure must necessarily have the whole massbetween 0 and 1/2π.

27


The following theorem states that rescaling by inverse dimension will destroyconvergence to µPoisson in many cases.

Theorem 3.14. Choose a fixed irreducible, unitary representation ρλ : K → U(Vλ)with highest weight λ, where λ = λ1f1 + . . .+ λrfr is the decomposition into funda-mental weights with every λj > 0.

Let rank(g) > 2 and assume that at least two fj are positive. Then for everyγ ∈ U(g) without constant term

limk→∞

‖ρ∗,k(iρk(γ)t)‖End(Vk) = 0 for all t > 0, (3.33)

where ρk : K → U(Vk) is an irreducible representation with highest weight k · λ and‖ · ‖End(Vk) is the usual operator norm in End(Vk).

Proof. Apply Lemma 3.11 and note that the right hand side of (3.17) converges tozero.

Corollary 3.15. Under the above assumptions µexp(ρ∗,k(rρk(γ))t) does not converge to

the measures µPoisson or µCUE.

Proof. This follows from Lemma 3.13, since we proved that (3.31) is fulfilled.

Remark 3.16. Note that there is an obvious counterexample to Theorem 3.14 if therank of g is 1. Namely, the irreducible representation of sl(2,C) on the homogeneouspolynomials in two indeterminates.

Take ξ = diag(1,−1). Then ‖ρk(ξ)‖Vk= k where Vk is the vector space of

homogeneous polynomials of degree k. Therefore dim ρk(ξ) = k + 1. We see that

‖ρk(rk(ξ))‖Vk=

k

k + 1→ 1. (3.34)

The reader may wonder what happens in the case of the rescaling by inverse pa-rameter as in Chapter 2. There is no analogue of Theorem 3.14 in this case, becausethe denominator in (3.24) scales like the numerator, so there is no convergence tozero.

In fact, the statements of this chapter can be made more general by allowingrescaling maps which decrease operators faster than the rescaling by inverse param-eter. The theorems will still be true in this case, although some corrections to theconstants will be required.

28

4 Spectral Statistics of GenericHamiltonian Operators

Having studied the spectral statistics of simple Hamiltonian operators, i.e., simple“polynomials” of Lie algebra elements in irreducible representations, we are nowinterested in more complicated operators.

In Chapter 2 “polynomials” in some basis of the Lie algebra were considered,which gave rise to Hamiltonians. But for a more analytic treatment of the matter,we investigate the spectral statistics in a completion of the polynomial algebra. Notethat such a completion was already implicitly used in [GHK00], where the authorsused the sine of a Lie algebra element.

Thereafter we will define the notion of a generic Hamiltonian operator and provethat the irreducible representations of the flows through the generic operators havespectral statistics converging to µPoisson under special assumptions on the dimensionsof the representation spaces.

We will use the following notation throughout this chapter. Let K denote a com-pact semi-simple Lie group with complexification G. The corresponding Lie algebrasare called k and g. Every representation of K will be assumed to be continuous,finite-dimensional and unitary. The K-invariant inner product will be denoted by〈·, ·〉 without putting the representation space into the notation. It will be clear bythe arguments or by the context which representation space is meant.

4.1 Topology and Completion of U(g)

In this section we introduce a topology on the universal enveloping algebra U(g)and complete it to a Fréchet space. To do so, choose a basis ξ1, . . . , ξn of g. By thePoincaré-Birkhoff-Witt Theorem we have a vector space isomorphism

ψ : C[X1, . . . , Xn] → U(g) (4.1)

given by substituting ξi for Xi in every polynomial p in which we have ordered theindeterminates in each monomial lexicographically. Note that this ordering is nec-essary since ψ is only a vector space isomorphism, but not an algebra isomorphism.

We use ψ to give a topology to U(g) by the natural embedding of C[X1, . . . , Xn]into the algebra of holomorphic functions O(Cn).

It is a well-known fact that O(Cn) is a Fréchet space with respect to the topologyof uniform convergence on compact subsets of Cn. If we change the basis of g toη1, . . . , ηn we obtain a priori another completion of C[X1, . . . Xn]. But changing the

29

4 Spectral Statistics of Generic Operators

basis is nothing more than a linear change of coordinates, yielding an induced linearhomeomorphism of Fréchet spaces. So, a different choice of basis does not changethe topology.

Remark 4.1. If a sequence of holomorphic functions on Cn converges to zero inthe Fréchet topology, then the suprema of the coefficients in the Taylor expansionaround the origin also converge to zero.

Proof. Let (fj)j∈N be a sequence of holomorphic functions with Taylor expansionfj =

∑I a

(j)I XI , where I is a multiindex with the usual conventions.

By the general Cauchy integral formula in several variables we see that

a(j)I =

1

(2πi)n

∮ζ∈∂P

fj(ζ)

ζI+(1,...,1)dζ, (4.2)

where P is the unit polycylinder in Cn and ∂P its distinguished boundary. Fromthis we obtain

|a(j)I | 6 sup

ζ∈∂P|fj(ζ)|. (4.3)

The right hand side does not depend on I, so the inequality holds for the supremumof the |a(j)

I | for a fixed j, but the fj converge uniformly on compact sets, especiallyon ∂P .

Let ρ∗ : g → End(V ) be an irreducible representation on a finite-dimensionalcomplex vector space V . This map extends to an irreducible representation of U(g),which we will again call ρ∗.

Proposition 4.2. The map ρ∗ : U(g) → End(V ) extends to a continuous, surjective,linear map

ρ∗ : O(Cn) → End(V ) (4.4)

with respect to the above completion of U(g), where the topology on End(V ) is givenby the operator norm with respect to some norm on V .

Proof. Let f =∑aIX

I ∈ O(Cn) be given. We define

ρ∗(f) =∑

aIρ∗(ξ1)i1 . . . ρ∗(ξn)in . (4.5)

By the basic inequality for the operator norm

‖AB‖ 6 ‖A‖ · ‖B‖ ∀A,B ∈ End(V ) (4.6)

it follows that‖aIρ∗(ξ1)

i1 . . . ρ∗(ξn)in‖ 6 |aI |bi11 . . . binn (4.7)

for bi := ‖ρ∗(ξi)‖. This series is convergent since f ∈ O(Cn). Moreover, ρ∗ is linear.To show the continuity, it suffices to show that ρ∗ is continuous at zero. So let

30

4.2 A Notion of Hermitian Operators for O(Cn)

(fj)j∈N be a sequence of holomorphic functions on Cn converging to zero uniformlyon compact subsets. We must show that

limj→∞

ρ∗(fj) = 0, (4.8)

but this is the claim that

‖∑

a(j)I ρ∗(ξ1)

i1 . . . ρ∗(ξn)in‖ → 0. (4.9)

Note that‖∑

a(j)I ρ∗(ξ1)

i1 . . . ρ∗(ξn)in‖ 6∑

|a(j)I |bi11 . . . binn . (4.10)

Again the right-hand side converges to zero because the ξi can be chosen such that|bi| 6 1

2for all i ∈ 1, ., , n, and the right-hand side is less or equal to

sup |a(j)I |∑ 1

2|I|, (4.11)

which converges to zero according to Remark 4.1. We can then scale back to theoriginal ξi, which is just an isomorphism of Frechét spaces.

To see that ρ∗ is surjective we use the Lemma of Burnside which states thatρ∗ : U(g) → End(V ) is already surjective.

4.2 A Notion of Hermitian Operators for O(Cn)

In the following a notion of self-adjointness or hermitian operators for O(Cn) willbe required. For this we will extend the definition of † on U(g) by continuity.

Lemma 4.3. The map † extends to a continuous involution of O(Cn).

Proof. We choose a basis of g in the following way. First, fix a maximal torus t ing. Let τ1, . . . , τr be a basis of the torus such that τ †i = τi for all i. Then choose asystem Π of positive roots and a basis ξα of the root spaces gα for α ∈ Π such that

ξ†α = ξ−α. (4.12)

With this basis, † operates on the basis elements just by permutation.Let f =

∑I aIX

I be in O(Cn). We define f † :=∑

I aI(XI)†. Clearly, f † is again

everywhere convergent because we just changed the order of the summation andconjugated each coefficient.

Let (fj)j∈N be a sequence of holomorphic functions on Cn converging to zerouniformly on compact subsets. To show that † is continuous, we must show that

limj→∞

(f †j ) = 0. (4.13)

But since in each f †j we have only changed the order of the summands and conjugatedto coefficients, this is also a series of holomorphic functions converging uniformly oncompact subsets.

As stated before, the choice of basis has no effect on the topology.

31


We define the notion of an abstractly hermitian operator as follows.

Definition 4.4. f ∈ O(Cn) is called an abstractly hermitian operator if f † = f .The set of all abstract hermitian operators is denoted by H.

Note that this definition is compatible with the one given for the tensor algebrain the Appendix.Remark 4.5. H is a closed subspace of O(Cn) and as such is a Fréchet space.

Proof. The linear map † − idO(Cn) is continuous and H is its kernel.

Lemma 4.6. Let ρ : K → U(V ) be an irreducible unitary representation and ρ∗ :U(g) → End(V ) the induced representation with extension ρ∗ : O(Cn) → End(V ).Then the restriction of ρ∗ to H is surjective onto the subspace of self-adjoint linearoperators of V .

Proof. For A ∈ End(V ) we denote by A† the conjugate transpose of A. We remarkthat by the definition of † we have

ρ∗(ξ)† = ρ∗(ξ

†) ∀ξ ∈ g. (4.14)

Thereforeρ∗(H ∩ U(g)) ⊂ self-adjoint operators in End(V). (4.15)

To show that the restriction is surjective, consider a self-adjoint operator A ∈End(V ). Since ρ∗ is surjective, we find an α ∈ H ∩ U(g), such that ρ∗(α) = A. By(4.14) it follows that

ρ∗(α†) = ρ∗(α)† = A† = A. (4.16)

Therefore we see that

ρ∗

(1

2(α+ α†)

)=

1

2ρ∗(α) +

1

2ρ∗(α

†) =1

2A+

1

2A = A. (4.17)

But1

2(α+ α†) ∈ H ∩ U(g), (4.18)

so the restriction of ρ∗ to H is surjective.

4.3 Examples of Convergence

In this section we will give a class of examples for the convergence of nearest neigh-bor statistics of abstractly hermitian operators in suitable sequences of irreduciblerepresentations.

Before these examples are considered we briefly discuss the effect of holomorphicmaps on operators. Consider a holomorphic map f : C → C. It induces a map

f : O(Cn) → O(Cn), g 7→ f g. (4.19)

Let ρ : K → U(V ) be an irreducible representation and ξ ∈ O(Cn) be a fixedoperator. We are interested in the spectrum of ρ∗(f(ξ)).

32

4.3 Examples of Convergence

Remark 4.7.Spec(ρ∗(f(ξ))) = f( Spec(ρ∗(ξ)) ). (4.20)

Proof. Let∑

j bjzj be the power series expansion for f at zero. Since ρ∗ is continu-

ous, it follows thatρ∗(f(ξ)) =

∑j

bj ρ∗(ξ)j. (4.21)

Conjugating ρ∗(ξ) to a diagonal matrix and inserting in the above equation givesthen the desired result.

Theorem 4.8. Let (ρm : K → U(Vm))m∈N be a sequence of irreducible represen-tations with strictly increasing dimension. Assume that ξ ∈ H has the followingproperties:

1. Every eigenvalue of ρ∗,m(ξ) has multiplicity one.

2. S :=⋃

m∈N Spec(ρ∗,m(ξ)) is a discrete subset of R.

Then for every absolutely continuous measure µ on R+ with∫∞

0xdµ ∈ [0, 1] there

exists a function f ∈ Hol(C) and a subsequence (ρmk: K → U(Vmk

))k∈N such thatη := f(ξ) satisfies

dKS(µρ∗,mk(η), µ) → 0 as k →∞. (4.22)

Proof. We begin by choosing a subsequence ρmkin the following way. First, we set

rm1 = ρ1 and proceed inductively by requiring that

Nk+1 := dim ρmk+1 ≥ k(dim ρmk+ 2). (4.23)

Without loss of generality we assume that N1 ≥ 3 and find an N1-tuple X1 suchthat

dKS(µ(X1), µ) ≤ 2

N1

. (4.24)

We now proceed inductively again, i.e. by Corollary 6.25 in the Appendix, thereis an Nk+1-tuple Xk+1 that contains the Nk-tuple Xk as subset such that

dKS(µ(Xk+1), µ) ≤ Nk + 2

Nk+1

≤ 1

k, (4.25)

where the last inequality follows from (4.23).Elementary complex analysis yields that there exists a holomorphic function f :

C → C, such thatf( Spec(ρ∗,mk

(ξ)) ) = Xk ∀k ∈ N, (4.26)

since S is a discrete subset in R and each Xk ⊂ Xk+1. By (4.20) it follows that

Spec(ρ∗,mk(f(ξ))) = Xk ∀k ∈ N. (4.27)

Thus, η = f(ξ) has the property

dKS(µρ∗,mk(η), µ) → 0 as k →∞. (4.28)

33


Operators ξ with the above properties will in general exist for every ray of irre-ducible representations. One strategy of producing them goes as follows:

Start with an operator ξ of degree 2 that fulfills condition 1. Such operators can befound for every simple group K and should exist in general. We now force condition2 to hold by adding Casimir operators to ξ. Recall that Casimir operators act byscalar multiplication so they just add these scalars to the eigenvalues. If these scalarsincrease quickly enough, the spectra of ξ along the irreducible representations willlie in disjoint intervals and consequently condition 2 is satisfied.

The problem is that the operator ξ depends on the group K and we do not know ifthere is an abstract way of giving examples. So we will give here an example for K =SUn for the ray of irreducible representations through the standard representation.

Proposition 4.9. Let (ρm : SUn → U(Vm))m∈N be the sequence of irreducible rep-resentations on the homogeneous polynomials of degree m in C[x1, . . . , xn].

Then there exists an operator ξ ∈ U(g) that satisfies the conditions of Theorem4.8.

Proof. Let αj denote the n × n-matrix with 1 in the j-th diagonal component and−1 in the (j+1)-th diagonal component. Every other component should be equal tozero. These matrices form a basis for the standard maximal torus in SLn(C) = SUC

n .They also define a system of simple roots (cf. the tables in Appendix C of [Kna02]).

The operation of αj on the homogeneous polynomial xa11 . . . xan

n of degree m isgiven by

ρ∗,n(αj).xa11 . . . xan

n = (aj − aj+1)xa11 . . . xan

n . (4.29)

Therefore, the largest eigenvalue of ρ∗,nαj is m and the smallest −m and every othereigenvalue is an integer number in-between these extremes. Now, we consider theoperator ξ =

∑cjαj, where the cj are real constants with 0 < cj <

1n

and whichare linearly independent over Q. Thus, ρ∗,m(ξ) is represented as diagonal matrixand has eigenvalues with multiplicity greater than 1, since otherwise there wouldexist a linear relation between the cj over Q. Note that by the choice of the cj theeigenvalues of ξ are still in the interval [−m,m].

By now ξ satisfies condition 1 of Theorem 4.8 and we will now add the Laplaceoperator to ξ to guarantee that condition 2 holds. For this, let Ω ∈ U(sln(C)) be theLaplace operator associated to sln(C). It acts on the homogeneous polynomials ofdegree m by rΩ,m := 〈mλ,mλ+2δ〉Kil, where λ is the highest weight of the standardrepresentation of SUk, δ denotes the half sum of positive weights and 〈·, ·〉Kil denotesthe Killing form. It follows that

rΩ,m+1 − rΩ,m = 〈λ, λ+ 2δ〉Kil +m〈λ, λ〉Kil +m〈λ, λ+ 2δ〉Kil. (4.30)

Choosing a constant b such that

b(rΩ,m+1 − rΩ,m) ≥ 2m ∀ m ∈ N∗ (4.31)

yields thatξ′ := bΩ +

∑j

cjαj (4.32)

34

4.4 Rational Independence of the Spectra in Representations

fulfills conditions 1 and 2 of Theorem 4.8.

4.4 Rational Independence of the Spectra inRepresentations

In this section we give a notion of generic operators in H.

Definition 4.10. An abstract hermitian operator α ∈ H is called generic if forevery irreducible representation ρ the eigenvalues of ρ are linearly independent overQ. We denote the set of generic operators in H by Hgen.

We start with the following theorem.

Theorem 4.11. The set of generic operators Hgen is dense in H.

Before the prove is given, we need to fix the notation. The ordered tuple ofeigenvalues with multiplicity of a hermitian matrix A will be denoted by X(A) andthe set of ordered n-tuples by Rn

ord.

Lemma 4.12. Let V be a unitary vector space of dimension n and Herm(V ) be thereal subspace of hermitian endomorphisms of V . For every λ ∈ (Qn)∗ the set

Sλ := A ∈ Herm(V ) : λ(X(A)) = 0 (4.33)

is nowhere dense in Herm(V ).

Proof. Let λ ∈ (Qn)∗ be a non-zero linear form. The set λ−1(0) is a hyperplane inRn, thus nowhere dense. In follows that the intersection of Rn

ord ∩ λ−1(0) is nowheredense in Rord.

Now, let us fix a given point x ∈ Rnord. From linear algebra we know that the

set of hermitian operators with spectrum x1, . . . , xn is just the U(n) orbit undermatrix conjugation through the diagonal matrix

X = diag(x1, . . . , xn). (4.34)

Therefore, the set Rnord can be identified with Herm(V )/U(n) and the projection

map p : Herm(V ) → Herm(V )/U(n) = Rnord is an open map.

Because preimages of nowhere dense sets under open maps are nowhere dense,the lemma is proved.

Proof. (Theorem 4.11) Since ρ∗ : U(g) → End(V ) is an irreducible, finite-dimen-sional representation, the induced mapping ρ∗ : H → Herm(V ) is a real linear,surjective mapping between Fréchet spaces. Therefore it is an open mapping by theopen mapping theorem.

So for any given non-zero linear from λ ∈ (Qdim V )∗, the set

Mλ,ρ := α ∈ H : X(ρ∗(α)) ∈ λ−1(0) (4.35)

35


is nowhere dense in H. Otherwise, we could find an inner point in this set, butbecause ρ is an open mapping this would contradict Lemma 4.12.

Thus, the setM :=

⋃ρ irrep. ,λ∈(Qdim V )∗

Mλ,ρ (4.36)

contains no inner point by Baire’s category theorem, i.e. its complement is dense.It follows that Hgen is dense.

4.5 Ergodic Properties of Hgen

Before we come to the main point of this section, we have to recall some terminologyfrom ergodic theory. All details can be found in [Sin94] or [CFS82]. We follow thelatter in terminology.

Let (X,µ) be a measure space, where µ denotes the measure on some σ-algebrain the power set of X. A measurable map f : X → X is called an automorphismof the measure space (X,µ), if f is bijective, f−1 is measurable again, and for allmeasurable sets A ⊂ X, we have

µ(f(A)) = µ(f−1(A)) = µ(A). (4.37)

By a flow (ϕt)t∈R of the measure space (X,µ), we mean a 1-parameter group ofautomorphisms of (X,µ), i.e., a group homomorphism of R into the group of allautomorphisms of the measure space (X,µ) such that ϕ : R×X → X is measurable.

For us X will be an N -dimensional torus, i.e., X = [0, 1]N mod 1 and the measureµ is the Haar measure on X, which is equal to the Lebesgue measure here. Weconsider some N -tuple x = (x1, .., xn) such that 0 < xi < 1 for all i ∈ 1, . . . , Nand the xi’s are linearly independent over the rational numbers. The map ϕt :X → X, z 7→ z + t · xmod 1 defines a group homomorphism R → Diff(X), t 7→ ϕt,where Diff(X) denotes the group of diffeomorphisms of X. It is a standard factfrom ergodic theory that (ϕt)t∈R is a flow of the measure space (X,µ) (cf. [CFS82])Chapter 3, §1, Theorem 1).

A flow is called ergodic if for every t 6= 0, the only invariant sets of ϕt have measureeither 0 or 1. We make use of the following

Theorem 4.13. (Birkhoff) Let (X,µ) be a measure space with µ(X) = 1 and(ϕt)t∈R be a flow of the measure space (X,µ). Then for every integrable functionf : X → R,

f(y) := limt→∞

1

2t

∫ t

−t

f(ϕτ (y))dτ =

∫X

f(x)dx (4.38)

for almost all y ∈ T with respect to µ.

It is a standard result of ergodic theory that (ϕt)t∈R is a uniquely ergodic flow,i.e., f is constant, (cf. [CFS82]) Chapter 3, §1, Theorem 2).

36

4.6 The Sets BN

In this case, we obtain the formula for the characteristic function χA of a measur-able set A:

limt→∞

1

2t

∫ t

−t

χA(ϕτ (y))dτ = µ(A) ∀y ∈ X. (4.39)

Let us now consider an element α ∈ Hgen and the induced irreducible, finite-dimen-sional representation ρ∗ : U(g) → End(V ). Since ρ∗(α) is a self-adjoint operator, itfollows that (exp(2πiρ(α)t))t∈R is a uniquely ergodic flow on the torus

T (V ) = closure(exp(2πiρ∗(α)t)|t ∈ R). (4.40)

This torus depends on the starting direction ρ∗(α), but we will in the followingalways assume that we have conjugated it into a diagonal matrix. There is no loss ofgenerality because we are only interested in the eigenvalues and they do not changeunder conjugation. Thus, we will just write TN for the N -dimensional torus, i.e.,

TN = diag(e2πiφ1 , . . . , e2πiφN ) : φj ∈ [0, 1]. (4.41)

4.6 The Sets BN

In this section we will use the ergodic properties of Hgen in combination with atheorem of Chapter 5 to connect the spectral properties of an abstract hermitianoperator with the Poisson-statistics. For this we first need to fix some notation.

For a unitary automorphism A ∈ U(V ) of a finite-dimensional unitary vectorspace V of dimension N we have the nearest neighbor statistics µc(X(A)) as definedin Definition 6.26 of the Appendix. By µPoisson we denote the absolutely contin-uous probability measure on the positive real line with density function exp(−x)with respect to the Lebesgue measure. Finally, let us write dKS(µ1, µ2) for theKolmogorov-Smirnoff distance (cf. (6.39) in the Appendix).

The following theorem is analogous to the second main theorem of [KS99] and isthe main result of Chapter 5.

Theorem 4.14. Let α > 0 be given. Then there exists an natural number N0 suchthat for every N ≥ N0∫

TN

dKS(µc(X(A)), µPoisson)dA <1

eα√

log N. (4.42)

The rather technical proof is given in Chapter 5, cf. Theorem 5.20.

Corollary 4.15. For all α ∈ R with α > 0 and any N ≥ N0 = N0(α) we have

dKS(µA, µPoisson) 6 e−12α√

log(N) (4.43)

for all A in a set in TN of measure at least 1− e−12α√

log(N).

37


0 0.2

0.4 0.6

0.8 1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Figure 4.1: A picture of B3.

Proof. Let us assume the contrary, i.e., we assume that

dKS(µA, µPoisson) > e−12α√

log(N) (4.44)

on a set M of measure at least e−12α√

log(N). Then∫M

dKS(µA, µPoisson)dHaar(A) > e−12α√

log(N)e−12α√

log(N) = e−α√

log(N). (4.45)

Since the integrand is always positive, this is a contradiction to Theorem 4.14.

This motivates the following definition.

Definition 4.16. Let α > 0 be given. The set BN is given by 1

BN :=B ∈ TN : dKS(µB, µPoisson) ≥ e−

12α√

log(N). (4.46)

It is clear that BN depends on the choice of α. However, for reasons of simplicitywe suppress this fact in the notation. In the following we will always assume thatthe N are so large that Theorem 4.14 is valid, i.e. N ≥ N0 ≥ 2.

Let us now collect some properties of BN . First of all, BN is not empty becausethe identity matrix EN is in BN . For this just recall that

∫ c

0µPoisson is close to zero

for small c and that∫ c

0dµEN

= 1 for every non-negative c, so dKS(µEn , µPoisson) = 1.Due to the fact that the map A 7→ dKS(µA, µPoisson) is continuous (cf. Lemma

6.29), BN is closed and the identity matrix is an inner point as a consequence ofcontinuity.

Moreover, BN is invariant under scalar multiplication with z = eiλ, where λ ∈ R,cf. Chapter 6.

The set B3 for α = 43

is visualized by Figure 4.1. For the drawing, we havediscretized the torus T3 into a cubical lattice with 20×20×20 points and calculated a

1The letter B in BN is not an abbreviation for big. In fact these sets are small.

38

4.6 The Sets BN

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

0 0.2

0.4 0.6

0.8 1

0 0.2

0.4 0.6

0.8 1

Figure 4.2: Pictures of B3 and B4 intersected with the hyperplane normal to thediagonal.

discretized version of dKS for a grid size of 20 points. The axises show the coordinatesφ1, φ2 and φ3. The intersection of B3 and the cubical grid is the drawn set of points.The definition of the discretized version is given as Definition 5.7. One can see theinvariance under multiplication with eiφ here as invariance under diagonal shifts.

Thus, it is enough to know the sets BN only on that hyperplane which is normalto the diagonal and contains the point 1

2(1, . . . , 1), i.e., the hyperplane given by

a1φ1 + · · ·+ aNφN = N/2. (4.47)

Figure 4.2 shows these hyperplanes for N = 3, 4 parametrized by φ1, . . . , φN−1.We now use the ergodic properties of Hgen to formulate our key lemma.

Lemma 4.17. Let γ ∈ Hgen and ρ : K → U(V ) be an irreducible, finite-dimensional,unitary representation with dimV = N and denote the characteristic function of theset BN by χ. Then

limt→∞

1

2t

∫ t

−t

χ(exp(2πiρ∗(γ)τ))dτ = volTN(BN), (4.48)

where volTN(BN) denotes the measure of BN with respect to the Haar measure on

TN .

Proof. This is just the ergodic property of equation (4.39).

We would like to emphasize the role of t in the above lemma. Consider the setR(N) defined by

R(N) = t ∈ R : exp(2πiρ∗(γ)t ∈ BN. (4.49)

Corollary 4.18. Under the assumptions of the above lemma

dKS(µexp(2πiρ∗(γ)t, µPoisson) < e−12α√

log(N) (4.50)

for every t 6∈ R(N).

39


Moreover R\R(N) has infinite measure and he have the following estimation onthe size of R(N)

0 < limt→∞

1

2t

∫ t

−t

χR(N)(τ)dτ < e−12α√

log(N), (4.51)

where χR(N) denotes the characteristic function of R(N).

Proof. By virtue of equation (4.48) we obtain the corollary.

4.7 Convergence to µPoisson

From now on, consider a sequence (ρk)k∈N of irreducible, unitary representationsρk : K → U(Vk) such that dk := dim(Vk) is increasing. Before the main result canbe stated, it is necessary to introduce two rather technical conditions.

Definition 4.19. A sequence (ρk)k∈N is said to be of admissible growth, if thereexists an α > 0 such that

∞∑k=0

e−12α√

log(dk) <∞. (4.52)

Definition 4.20. A generic hermitian operator γ ∈ Hgen is said to be admissibleof width ε for the sequence (ρk)k∈N, where 0 < ε < 1 if there exists a k0 and a t0such that for all t ≥ t0 and all k ≥ k0 the inequality∣∣∣∣vol(Bdk

)− 1

2t

∫ t

−t

χBdk(exp(2πiρ∗,k(γ)τ)dτ

∣∣∣∣ < ε (4.53)

holds. Here χBdkdenotes the characteristic function of the set Bdk

as defined above.

Let us briefly discuss these definitions. As will become clear in the followingtheorem the first describes a condition on the growth of the dimensions dk. By adirect calculation we see that the condition requires dk to grow faster than e(

2 log(k)α )

2

.We will come back to this later.

The second definition guarantees that we are outside the sets BN in each repre-sentation. For fixed k the condition can be fulfilled for every ε by Birkhoff’s ergodictheorem. But we require here that t0 as a function of k is bounded. So, condition(4.53) only fails, if ∣∣∣∣vol(Bdk

)− 1

2t

∫ t

−t

χBdk(. . . )dτ

∣∣∣∣→ 1 (4.54)

is true. This will happen if the leaving time, i.e., the supremum of all t, such thatexp(2πiρ∗,k(γ)τ) ∈ Bdk

, converges too rapidly to infinity as function of k. In Lemma3.13 we saw this kind of behavior. The reader may wonder if operators of width εdo exist at all. But in Section 4.3 we saw examples of operators γ whose nearestneighbor statistics converge to a given measure µ. Although the situation is a little

40

4.7 Convergence to µPoisson

different here, because of the exponentiation, we could use the proof of Theorem4.8 to construct operators γ such that exp(2πiρ∗,k) has nearest neighbor statisticswhich converge to µPoisson. These γ have a leaving time less than 1 by construction.

Now we state our key theorem in this chapter.

Theorem 4.21. Let γ ∈ Hgen be admissible of width ε for a sequence (ρk : K →U(Vk))k∈N of irreducible, unitary representations which is of admissible growth. Thenfor every ε′ > 0 there exists a set R = R(ε′) in R, such that

limr→∞

1

2r

∫ r

−r

χR(x)dx ≤ ε+ ε′ (4.55)

andµexp(2πiρ∗,k(γ)t) → µPoisson as k →∞ (4.56)

for all t 6∈ R.

Before we prove the theorem, let us discuss the claim about the measure of R.Any bounded set R is of this type, or any set of measure 0. But from the point ofview of percentage of real numbers, we prove that a fraction of (1 − ε − ε′) of thereal numbers yields convergence to µPoisson for the subsequence.

Proof. According to the condition of (4.53), we find a t0 such that for all t ≥ t0 andall k ≥ k0 ∣∣∣∣vol(Bdk

)− 1

2t

∫ t

−t

χBdk(exp(2πiρ∗,k(γ)τ)dτ

∣∣∣∣ < ε. (4.57)

By the definition of admissible growth it follows that

∞∑k=1

e−12α√

log(dk) <∞. (4.58)

Thus for every ε1 > 0 we find a natural number N0 = N0(ε1) such that

∞∑k=N0

e−12α√

log(dk) < ε1. (4.59)

Now set

Rε1 =∞⋃

k=N0

R(dk), (4.60)

where R(dk) = t ∈ R : exp(2πiρ(γ)t ∈ Bdk. We set Qε1 = R\Rε1 and note that

for all t ∈ Qε1

µexp(2πiρ∗,k(γ)t) → µPoisson as k →∞. (4.61)

Now we have to show that Qε1 6= ∅.

41


By enlarging N0 if necessary, we may also assume that k0 ≤ N0. We fix an interval[−t, t], where t ≥ t0, and obtain

2t(vol(Bdk)− ε) ≤ vol(R(dk) ∩ [−t, t]) ≤ 2t(vol(Bdk

) + ε). (4.62)

Summing over all k ≥ N0 and applying (4.59) it follows that

vol(Rε1 ∩ [−t, t]) = vol(⋃

R(dk) ∩ [−t, t]) ≤ 2t(ε1 + ε). (4.63)

We can choose ε1 so small that ε1 + ε < 1. This yields

Qε1 ∩ [−t, t] 6= ∅. (4.64)

It remains to show (4.55). But since (4.63) holds for all t ≥ t0:

vol(Rε1 ∩ [−t, t]) =

∫ t

−t

χRε1(s)ds ≤ 2t(ε1 + ε). (4.65)

This completes the proof of the theorem.

Let us briefly discuss this theorem. For every generic, admissible operator onehas convergence of the nearest neighbor distributions for all t 6∈ R. But the readermay wonder how restrictive the condition of admissible width is. This will dependon the geometric structure of the sets BN . If they are regular enough, the conditionof admissible width should be automatically fulfilled for most generic operators.Unfortunately, we do not know enough about this structure yet, although in lowdimensions the sets BN are very regular (cf. Figures 4.1 and 4.2).

42

5 The Poisson Spectral Statistics forTori

In this chapter we give a proof for the convergence of the nearest neighbor statisticsof a real torus T (N) to the Poisson spectral statistics in the sense of the Kolmogorov-Smirnov distance.

We follow the structure of the proof in [KS99] for the CUE case but will tryto make this chapter as self-contained as possible, citing only some combinatoriallemmas and some facts about measures.

5.1 Some Combinatorics

We give here the basic definitions of Sep, Cor, Clump and so on from [KS99] again.To do this let f : R → R be a function, a be a non-negative integer called theseparation and X be an N -tuple of real numbers in increasing order.

We defineClump(a, f,N,X) =

∑1≤t1≤...≤ta+2≤N

f(xta+2 − xt1) (5.1)

andSep(a, f,N,X) =

∑1≤t1≤...≤ta+2≤N,tj+1−tj=1 for all j

f(xta+2 − xt1). (5.2)

Let us briefly discuss what these definitions signify, first taking a closer look atClump. We sum over all (a + 2)-tuples (t1, . . . , ta+2) with increasing entries suchthat the last entry is smaller or equal than N , thereby evaluating the function f atthe differences between xta+2 − xt1 . If a + 2 > N then there are no tuples to sumover, so Sep and Clump vanish identically.

Formally we can think of this as integrating the function f over a sum of Diracmeasures at the points xta+2 − xt1 . The same applies to the function Sep with therestriction that we sum only about the (a+2)-tuples of the form (t1, t1 +1, . . . , t1 +a+ 1).

If we consider a = 0, then we evaluate f exactly at the nearest neighbor spacings.This may give a clear motivation why we are interested in Sep. The point in thedefinition of Clump will become clear later on. For the moment, let us just indicatethat there will be a combinatorial identity expressing Sep as alternating sum oversome versions of Clump.

By now, Sep and Clump are defined over increasing N -tuples X. We extend thisdefinition to all N -tuples by first ordering the tuple X.

43

5 The Poisson Spectral Statistics for Tori

Clump(a, f,N, ·) : RN → R, X → Clump(a, f,N,X ordered) (5.3)

andSep(a, f,N, ·) : RN → R, X 7→ Sep(a, f,N,X ordered). (5.4)

Sep and Clump are special cases of a certain class of functions which we will dealwith in the following. We define this class in the following way:

Definition 5.1. Let N ≥ 2 be an integer.A function f : RN → R is called a function of class T (N) if f is Borel mea-

surable, SN -invariant and invariant under additive diagonal translations

(x1, . . . , xN) 7→ (x1 + t, . . . , xN + t), (5.5)

with t ∈ R.A function f : RN → R is called a function of class T0(N) if f is a function of

class T (N) and f vanishes outside the set (x1, . . . , xN) ∈ RN : maxi,j |xi−xj| ≤ αfor some α > 0. We abbreviate this condition by

supp f ≤ α. (5.6)

The following lemma lists some basic properties of Sep and Clump. This is Lemma2.5.11 of [KS99].

Lemma 5.2. For a ∈ N and f : R → R Borel measurable and N ≥ 2.Then Sep(a, f,N, ·) and Clump(a, f,N, ·) are functions of class T (N). If f is

continuous, then Sep(a, f,N, ·) and Clump(a, f,N, ·) are also continuous.If f vanishes outside the interval [−α, α], Sep(a, f,N, ·) and Clump(a, f,N, ·) are

of class T0(N) and

supp Sep(a, f,N, ·) ≤ α and supp Clump(a, f,N, ·) ≤ α. (5.7)

Proof. See [KS99] p.52.

Using Clump we define a third function for an integer k, k ≥ a.

TClump(k, a, f,N, ·) : RN → R, X 7→(k

a

)Clump(k, f,N,X). (5.8)

Note that this definition may seem a bit superfluous, but it is added here to showthe parallels to [KS99]. If we were working with multiple neighbor statistics, i.e.r > 1 in terms of [KS99], then TClump would be a more complicated sum.

We now relate this functions on RN to functions on the torus T (N). Againfollowing [KS99], we name these functions Int for “integral”, Cor for “correlation”and TCor for “total correlation”.

44

5.2 The Random Variable Z(n, F, T (N))

These are defined as functions from T (N) to R which map A ∈ T (N) as follows

Int(a, f, T (N), A) :=1

NSep

(a, f,N,

N

2πX(A)

)Cor(a, f, T (N), A) :=

1

NClump

(a, f,N,

N

2πX(A)

)TCor(k, a, f, T (N), A) :=

1

NTClump

(k, a, f,N,

N

2πX(A)

),

where X(A) is −i times the component-wise logarithm of A, i.e. for the matrixA = diag(eiϕ1 , . . . , eiϕN ) with 0 ≤ ϕj < 2π for all j, we have X(A) = (ϕ1, . . . , ϕN).It is now, obvious why we study these objects because

Int(a, f, T (N), A) =

∫Rfdµ(naive, A, T (N), a). (5.9)

It is exactly this µ(naive, A, T (N), a) we want to study for a=0. For a ≥ 1 we maytake the above equation as definition of µ(naive, A, T (N), a). In the notation ofChapter 1 this measure is given as

µ(naive, A, T (N), 0) =1

N

∫A

N−1∑j=1

δ

(y − N

2π· (ϕj+1 − ϕj)

)dy (5.10)

if a = 0, which is almost identical to µc(X)(A) but the wrapped eigenangle betweenxN and x1 is missing. Therefore it is called “naive” in [KS99].

If we think of Int, Cor and TCor as random variables, we may calculate their ex-pectation value. But instead of writing E(Int(a, f, T (N), A)) we use capital letters:

INT(a, f, T (N), A) :=

∫T (N)

Int(a, f, T (N), A)dA,

COR(a, f, T (N), A) :=

∫T (N)

Cor(a, f, T (N), A)dA,

TCOR(k, a, f, T (N), A) :=

∫T (N)

TCor(k, a, f, T (N), A)dA.

There are numerous relations between these functions, but we will stop the combi-natorics here, coming back when we need it.

5.2 The Random Variable Z[n, F, T (N)]

Define the random variable Z[n, F, T (N)] by

Z[n, F, T (N)](A) =1

N

∑#T=n

F

(N

2πpr(T )X(A)

), (5.11)

45


where pr(T ) is the projection from T (N) to T (n), (x1, . . . , xN) 7→ (xt1 , . . . , xtn) fora subset T ⊂ 1, . . . , N of cardinality n and X(A) is the vector of angles for A.

We will later use this random variable with F = TCor, but for the start weformulate our version of Theorem 4.2.2 of [KS99].

The following theorem should be thought of as a very special limit theorem formeasures on the tori T (N) as N goes to infinity. We fix a small torus of dimensionn and sum over all projections of T (N) to T (n). In doing so we obtain inducedmeasures on T (n) and the statement of the following theorem can be interpreted asstating that these induced measures on T (n) have a converging expectation valueand decreasing variance.

Theorem 5.3. Consider n ∈ N, n ≥ 2 and F ∈ T0(n) with suppF < α for α > 0.Assume furthermore F ≥ 0.

1. The sequence E(Z[n, F, T (N)]) converges for N →∞ to a limit E(n, F, univ)and the estimation

|E(Z[n, F, T (N)])− E(n, F, univ)| ≤ ‖F‖sup1

N

αn−1

(n− 2)!. (5.12)

is true for all N ≥ 2.

2. For all N ≥ 2 the expectation is bounded as follows:

|E(Z[n, F, T (N)])| ≤ ‖F‖supαn−1

(n− 1)!. (5.13)

3. For all N ≥ 2 the variance is bounded as follows:

Var(Z[n, F, T (N)]) ≤‖F‖2

sup

Nmax1, (2α)2n−2 2n2(

floor(

n2

)!)2 , (5.14)

where floor denotes the function rounding a real number down to the nextinteger.

Proof. We start with the proof of statement 2.By a direct calculation we see that

E(Z[n, F, T (N)]) =1

N

∫[0,N ]n

(N

n

)1

NnF (x)dx1 . . . dxn. (5.15)

Since suppF < α, we consider the set ∆(n, α) = x ∈ Rn : supi,j |xi − xj| < α. ByLemma 5.8.3 of [KS99], we know

1

NVol(∆(n, α) ∩ [0, N ]n) ≤ nαn−1. (5.16)

46


Applying this to the above, it follows that

|E(Z[n, F, T (N)])| ≤ 1

N‖F‖sup

1

Nn

(N

n

)Nαn−1n

≤ ‖F‖supαn−1

(n− 1)!

N

N· N − 1

N· . . . · N − n+ 1

N.

(5.17)

The following inequalityk∏

ν=1

(1− ν

N

)≤ 1− k

N, (5.18)

gives

|E(Z[n, F, T (N)])| ≤ ‖F‖supαn−1

(n− 1)!

(1− n− 1

N

)≤ ‖F‖sup

αn−1

(n− 1)!. (5.19)

Thus, statement 2 has been proven.Now we wish to prove the first statement. For this, recall that F ∈ T0(n) means,

that F is Sn-invariant and invariant under diagonal addition. So

E(Z[n, F, T (N)]) =1

N

(N

n

)1

Nnn!

∫[0,N ]n(ordered)

F (x)dx1 . . . dxn. (5.20)

This is true since the tuples with two or more equal components are a zero set andcan be neglected. Substituting

y1 = x1, y2 = x2 − x1, . . . , yn = xn − x1 (5.21)

yields

E(Z[n, F, T (N)])

=1

N

(N

n

)n!

Nn

∫ N

0

(∫[0,N−y1]n−1(ordered)

F (0, y2, . . . , yn)dy2 . . . dyn

)dy1.

(5.22)We call the inner integral g(y1) and assume that α < N . Note that

g(y1) ≤ ‖F‖sup

∫[0,α]n−1(ordered)

dy2 . . . dyn = ‖F‖supαn−1

(n− 1)!(5.23)

since suppF < α. Therefore the integral extends from 0 to min(α,N −y1). We nowset

E(n, F, univ) := gα :=

∫[0,α]n−1(orderhned)

F (0, y2, . . . , yn)dy2 . . . dyn (5.24)

and consider the difference

D = |E(Z[n, F, T (N)])− E(n, F, univ)| =∣∣∣∣ 1

N

(N

n

)n!

Nn

∫ N

0

g(y1)dy1 − gα

∣∣∣∣ . (5.25)

47


By splitting the integral into two parts, i.e., integrating from 0 to N − α and fromN − α to N , it follows that in the first case g(y1) = gα because of the suppF < αcondition, and thus

D =

∣∣∣∣ 1

N

(N

n

)n!

Nn

∫ N−α

0

gαdy1 +1

N

(N

n

)n!

Nn

∫ N

N−α

g(y1)dy1 − gα

∣∣∣∣ . (5.26)

Therefore we have

D ≤((

N

n

)n!

Nn

N − α

N− 1

)gα +

α

N

(N

n

)n!

Nngα (5.27)

which leads to

D ≤((

N

n

)n!

Nn− 1

)gα ≤

n− 1

Ngα ≤ ‖F‖sup

αn−1

(n− 2)!

1

N. (5.28)

This proves statement 1, if N > α.If N ≤ α, we define gα as above, but immediately see that

E(n, F, univ) = gα =

∫[0,N ]n−1(ordered)

F (0, y2, . . . , yn)dy2 . . . dyn. (5.29)

Inserting this into (5.25) it follows that statement 1 is fulfilled in this case as well.For the proof of the last statement let us first look at

E(Z[. . . ]2) =1

N2

∫T (N)

∑#T=n,#S=n

F

(N

2πpr(T )X(A)

)F

(N

2πpr(S)X(A)

)dA,

(5.30)where the sum extends over all subsets S and T of cardinality n of the set 1, . . . , Nand we write E(Z[. . . ]2) for E(Z[n, F, T (N)]2). This can be written as

E(Z[. . . ]2) =1

N2

2n∑l=n

(N

l

)1

N l

(l

n

)(n

l − n

)×

×∫

[0,N ]lF (x1, . . . , xn)F (xl−n+1, . . . , xl)dx1 . . . dxl,

(5.31)

as can seen by writing the double sum over T and S as a sum over the cardinalityof S ∪ T and an inner sum. Using the Sn-invariance of F one obtains the aboveformula. Now we consider the summands with l < 2n. This means that

suppF (x1, . . . , xn)F (xl−n+1, . . . , xl) ≤ 2α (5.32)

because |xj − xi| ≤ |xj − xn| + |xn − xi| and suppF ≤ α. Using again (5.16) and(5.18) we obtain

E(Z[. . . ]2) ≤ 1

N

2n−1∑l=n

(1− l − 1

N

)(2α)l−1 l

(l − n)!(l − n)!(2n− l)!‖F‖2

sup

+1

N2

(N

2n

)1

N2n

(2n

n

)(∫[0,N ]n

F (x1, . . . , xn)dx1 . . . dxn

)2

.

(5.33)

48


Now, compute the variance:

Var(E(Z[n, F, T (N)])) = E(Z[n, F, T (N)]2)− E(Z[n, F, T (N)])2

≤‖F‖2

sup

N

2n−1∑l=n

(2α)l−1 l

((l − n)!)2 (2n− l)!

+1

N2

(∫[0,N ]n

F (x1, . . . , xn)dx1 . . . dxn

)2((

2n

n

)1

N2n

(N

2n

)−((

N

n

)1

Nn

)2)

≤‖F‖2

sup

N

2n−1∑l=n

(2α)l−1 l

((l − n)!)2 (2n− l)!

+(αn−1

)2n2

((2n

n

)1

N2n

(N

2n

)−((

N

n

)1

Nn

)2)‖F‖2

sup.

(5.34)

But the last summand is negative:(2n

n

)1

N2n

(N

2n

)−((

N

n

)1

Nn

)2

=1

(n!)2

(2n−1∏ν=0

(1− ν

N

)−

n−1∏ν=0

(1− ν

N

)2)

≤ 0

(5.35)

which gives the result

Var(E(Z[n, F, T (N)])) ≤‖F‖2

sup

Nmax(2α)2n−2, 1

n−1∑p=0

n+ p

(p!)2(n− p)!. (5.36)

For simplicity we estimate furthern−1∑p=0

n+ p

(p!)2(n− p)!≤ 2n

n−1∑p=0

1

n!

n!

p!p!(n− p)!≤ 2n

n−1∑p=0

1

n!p!

(n

p

). (5.37)

For n even this yields (n

p

)≤ n!(

n2

)!(

n2

)!. (5.38)

and for n odd (n

p

)≤ n!(

n+12

)!(

n−12

)!. (5.39)

So we have the following estimation for the variance:

Var(Z[n, F, T (N)]) ≤‖F‖2

sup

Nmax(2α)2n−2, 1 2n2(

floor(

n2

)!)2 . (5.40)

Combining everything finishes the proof of the last statement.

49


5.3 Moving the Estimates to TCor(k, a, f, T (N))

The reader may wonder how the above theorem is related to spectral statistics. Theanswer is given by the following theorem which transfers the above estimation onZ[n, F, T (N)] to estimations about TCor.

Theorem 5.4. Let f : R → R≥0 be a bounded, non-negative, Borel-measurablefunction with upper bound α and a, k ∈ N with k ≥ a.

1. The sequence TCOR(k, a, f, T (N)) converges for N → ∞ to a limit which isdenoted by TCOR(k, a, f, univ), and the following estimation

|TCOR(k, a, f, T (N))− TCOR(k, a, f, univ)| ≤(k

a

)‖f‖sup

1

N

αk+1

k!(5.41)

holds for all N ≥ 2.

2. For all N ≥ 2 the expectation is bounded as follows:

TCOR(k, a, f, T (N)) ≤(k

a

)‖f‖sup

αk+1

(k + 1)!. (5.42)

3. For all N ≥ 2 the variance is bounded as follows:

Var(A 7→ TCOR(k, a, f, T (N), A) on T (N))

≤(k

a

)2 ‖f‖2sup

Nmax(2α)2k+2, 1 2(k + 2)2(

floor(

k2

+ 1)!)2 . (5.43)

Proof. This is Proposition 4.2.3 of [KS99]. For self-containtedness we give the proofhere again.

The idea is to use Theorem 5.3 for the function F (X) = TClump(k, a, f, k+2, X),where k = n+ 2. We claim that

Z[k + 2, F, T (N)](A) = TCor(k, a, f, T (N), A) (5.44)

then. This can be seen by unwinding the definitions and using a combinatorialidentity for equation (5.45):

Z[k + 2, F, T (N)](A) =1

N

∑#T=k+2

F

(N

2πpr(T )X(A)

)=

1

N

∑#T=k+2

TClump

(k, a, f, k + 2,

N

2πpr(T )X(A)

)=

1

NTClump

(k, a, f,N,

N

2πX(A)

)(5.45)

= TCor(k, a, f, T (N), A).

The theorem now follows from the fact that proved F ∈ T0(n) and ‖f‖sup

(ka

)≥

‖F‖sup. But these are direct consequences of the definition of TClump as(

ka

)Clump

and Lemma 5.2.

50

5.4 The Weak Convergence of µ(naive, U(N), 1) to the Poisson Distribution

5.4 The Weak Convergence of µ(naive, U(N), 1) to thePoisson Distribution

We only cite a part of Proposition 2.9.1 of [KS99] here without repeating the proof.

Theorem 5.5. (Katz, Sarnak) Assume that a ∈ N is fixed. If for every k ∈ N,k ≥ a and every f : R → R which is bounded, Borel measurable, non-negative andof compact support,

limN→∞

TCOR(k, a, f, T (N)) =: TCOR(k, a, f, univ) (5.46)

exists and moreover ∑k≥a

TCOR(k, a, f, univ) <∞, (5.47)

then the limit measure µ(naive, a) exists and∫Rfdµ(naive, a) =

∑k≥a

(−1)k−a TCOR(k, a, f, univ). (5.48)

Proof. [KS99], p. 58 and following.

Since we are dealing with T (N), it is possible to give an explicit formula for theLebesgue density of µ(naive, 0).

Theorem 5.6. The limit measure µ(naive, a) exists and for a = 0 it has the proba-bility density e−x.

Proof. Here Theorem 5.5 will be applied to prove the convergence result. By state-ment 1 of Theorem 5.4, we know the existence of TCOR(k, a, f, univ) and by state-ment 2 we see that

TCOR(k, a, f, univ) ≤(k

a

)‖f‖sup

αk+1

(k + 1)!≤ ‖f‖sup

(2α)k+1

(k + 1)!. (5.49)

Thus ∑k≥a

TCOR(k, a, f, univ) ≤ ‖f‖sup

∞∑k=0

(2α)k+1

(k + 1)!≤ ‖f‖supe

2α <∞. (5.50)

It remains to prove the explicit form for a = 0. For this it suffices to calculate to∫Rfdµ(naive, 0) (5.51)

for the characteristic functions of intervals of the form [0, p]. But this integral canbe calculated directly∫ p

0

dµ(naive, 0) = limN→∞

∫ p

0

dµ(naive, T (N), 0), (5.52)

51


where∫ p

0

dµ(naive, T (N), 0) =1

N

N−1∑j=1

N !

∫T (N)(ordered)

f

(N

2π(xj+1 − xj)

)dA

= (N − 1)!N−1∑j=1

∫ 1

0

∫ xN

0

. . .

∫ xj+2

0

××∫ xj+1

xj+1−p/N

∫ xj

0

. . .

∫ x2

0

dx1 . . . dxN . (5.53)

The desired result follows by evaluating the right-hand side. For this let us definethe integrands Ij by ∫ p

0

dµ(naive, T (N), 0) = (N − 1)!N−1∑j=1

IJ , (5.54)

By a direct calculation we derive the recursion formula

Ij+1 = Ij − (−1)j 1

N !

(N

j + 1

)( pN

)j+1

(5.55)

and thus the explicit formula for the Ij

Ij =1

N !

j∑k=1

(N

k

)( pN

)k

(−1)k+1. (5.56)

Now, we insert this into (5.54) and compare it to the power series for 1− exp(−p)

∫ p

0

dµ(naive, T (N), 0) = (N − 1)!N−1∑j=1

1

N !

j∑k=1

(N

k

)( pN

)k

(−1)k+1

=1

N

N−1∑j=1

j∑k=1

(N

k

)( pN

)k

(−1)k+1 = − 1

N

N−1∑j=1

j∑k=1

(−p)k

k!

k−1∏ν=1

(1− ν

N

)= − 1

N

N−1∑k=1

N−1∑j=k

(−p)k

k!

k−1∏ν=1

(1− ν

N

)=

N−1∑k=1

(−p)k

k!

N − k

N

k−1∏ν=1

(1− ν

N

)=

N−1∑k=1

ak(N)(−p)k

k!,

(5.57)

where the ak(N) are the coefficients defined above. For fixed k

ak(N) =k∏

ν=1

(1− ν

N

)→ 1 as N →∞, (5.58)

which completes the proof.

52

5.5 The M -grid

5.5 The M -grid

We would like to study the Kolmogorov-Smirnov distance dKS for the nearest neigh-bor measures µ(naive, A, T (N), a). This is a quite complicated matter and thereforewe discretize on the so-called M -grid.

For this let M be a (big) positive natural number. Divide the interval [0, 1] intopieces of length 1

M. This defines a grid

−∞ = s(0) < s(1) < . . . < s(M − 1) < s(M) = +∞, (5.59)

where ∫ s(j)

−∞µ(naive, a) =

j

Mfor all 1 ≤ j ≤M − 1. (5.60)

Definition 5.7. Define the M-grid version of the Kolmogorov-Smirnov distance tobe

dM,KS(µ, ν) = maxi=1,...,M−1

|∫ s(i)

s(1)

dµ−∫ s(i)

s(1)

dν|. (5.61)

Lemma 5.8. For any Borel measure of total mass ≤ 1 we have the inequality

dKS(ν, µ(naive, a)) ≤ 5

M+ 2 · dM,KS(ν, µ(naive, a)). (5.62)


5.6 The Key Lemma

For simplicity we cite here Lemma 3.2.16 of [KS99].

Lemma 5.9. Let f ≥ 0 be a bounded, Borel measurable function with compactsupport and L ≥ a be an integer, then the following basic inequality

| INT(a, f, univ)− Int(a, f, T (N), A)|

≤∑

L≥k≥a

|TCOR(k, a, f, T (N))− TCor(k, a, f, T (N), A)|

+∑

L≥k≥a

|TCOR(k, a, f, T (N))− TCOR(k, a, f, univ)|

+ TCOR(L, a, f, univ) + TCOR(L+ 1, a, f, univ)

(5.63)

holds.


53


Notation 5.10. Since it is too cumbersome to write the measure µ(naive, A, T (N), a),we abbreviate in the following

µ = µ(naive, a) and µA = µ(naive, A, T (N), a) (5.64)

for fixed a.

Corollary 5.11. Let R ⊂ [s(1), s(M − 1)] be a Borel measurable. Then

|µ(R)− µA(R)| ≤∑

L≥k≥a

|TCOR(k, a, χ, T (N))− TCor(k, a, χ, T (N), A)|

+

(L

a

)αL+1

(L+ 1)!+

(L+ 1

a

)αL+2

(L+ 2)!+

1

N

∑L≥k≥a

(k

a

)αk+1

k!

<∑

L≥k≥a

|TCOR(k, a, χ, T (N))− TCor(k, a, χ, T (N), A)|

+(2α)L+1

(L+ 1)!+

(2α)L+2

(L+ 2)!+

1

N

∑L≥k≥a

(2α)k+1

k!

where χ is the characteristic function of R and α = diam(R).

Proof. Apply the above Lemma and the TCOR estimations.

Corollary 5.12. Set β = s(M − 1)− s(1). The following estimation holds:

dM,KS(µ, µA) < maxi∑

L≥k≥a

|TCOR(k, a, χ[s(1),s(i)], T (N))

− TCor(k, a, χ[s(1),s(i)], T (N), A)|

+(2β)L+1

(L+ 1)!+

(2β)L+2

(L+ 2)!+

1

N

∑L≥k≥a

(2β)k+1

k!,

(5.65)

where χR denotes the characteristic function of the interval R.

Proof. This is clear from the definition of dM,KS.

Lemma 5.13.∫T (N)

|TCOR(k, a, χ[s(1),s(i)], T (N))− TCor(k, a, χ[s(1),s(i)], T (N), A)|dA

≤(k

a

)√2

Nmax(2s(i)− 2s(1))k+1, 1 k + 2

floor(k2

+ 1)!. (5.66)

Proof. This is just the Cauchy-Schwarz inequality∫T (N)

|h(A)|dA ≤√∫

T (N)

|h(A)|2dA , (5.67)

where h(A) = TCOR(k, a, χ[s(1),s(i)], T (N))− TCor(k, a, χ[s(1),s(i)], T (N), A)combined with the statement about the variance of the TCOR estimations.

54

5.6 The Key Lemma

Theorem 5.14. For N →∞∫T (N)

dKS(µ, µA)dA→ 0. (5.68)

Proof. Putting everything together, we obtain∫T (N)

dKS(µ, µA)dA <5

M+ 2 ·

((2β)L+1

(L+ 1)!+

(2β)L+2

(L+ 2)!+

1

N

∑L≥k≥0

(2β)k+1

k!

)

+

√2

Nmax

i

∑L≥k≥a

(k

a

)k + 2

floor(k2

+ 1)!max1, (2s(i)− 2s(1))k+1

(5.69)

where β = s(M − 1)− s(1) as above.Now we combine the summands to make more explicit estimations∑

L≥k≥0

(2β)k+1

k!< (2β)e2β (5.70)

and if s(i)− s(1) ≥ 12

we have the following estimation for the second sum∑L≥k≥0

(4(s(i)− s(1)))k+1 k + 2

floor(k2

+ 1)!≤ 8(s(i)− s(1))

∑L≥k≥0

(64(s(i)− s(1))2)k/2

floor(k2

+ 1)!

< 8(s(i)− s(1))e64(s(i)−s(1))2 .(5.71)

If s(i)− s(1) < 12

we may estimate the sum as∑L≥k≥0

2k k + 2

floor(k2

+ 1)!≤∑

L≥k≥0

3 · 2k ≤ 3L · 2L. (5.72)

Applying this to the above it follows that∫T (N)

dKS(µ, µA)dA <5

M+ 2

((2β)L+1

(L+ 1)!+

(2β)L+2

(L+ 2)!

)+

1√N

(1√N

(2β)e2β + 8β√

2e64β2

+ 6L · 2L

).

(5.73)

It is clear that β depends only on M . So given ε > 0, we first choose M so large,that

5

M<ε

3, (5.74)

then we can choose L so large, that(2β)L+1

(L+ 1)!+

(2β)L+2

(L+ 2)!<ε

6(5.75)

and finally N so large that1√N

(1√N

(2β)e2β + 8β√

2e64β2

+ 6L · 2L

)<ε

3. (5.76)

55


5.7 The Final Estimation

In this last section we will give the final form of the estimation. But before we doso we state a series of lemmas which we will combine to give the main estimation.

We start by fixing two positive constants α, γ ∈ R>0. Set the grid size M to bethe largest integer smaller than eα

√log N and the cut-off L to be the largest integer

such that(L− 1)! ≤ Nγ2 ≤ L! . (5.77)

ThenlogM ≤ α

√logN ≤ log(M + 1) (5.78)

andlog(L− 1)! ≤ γ2 logN ≤ logL! . (5.79)

Thus, we see thatlogM ≤ α

γ

√logL! . (5.80)

The following lemma is a useful corollary of Stirling’s formula.

Lemma 5.15. Given ε > 0 and c > 0, there exists a k0 such that for all k ≥ k0:

1. (log k!)k+2 ≤ (k!)1+ε.

2. ck+2 ≤ (k!)ε/2.

Proof. The proof can be found in [KS99] p.93.

Next, note that β = s(M − 1)− s(0)) < logM by construction. We will now giveestimations for each summand in (5.89).

Lemma 5.16. The following estimation holds:

(2β)L+1

(L+ 1)!+

(2β)L+2

(L+ 2)!≤ 1

Nγ2−εγ2 . (5.81)

Proof. By lemma 5.15 we see that

(2β)L+1

(L+ 1)!+

(2β)L+2

(L+ 2)!≤ 2(2β)L+2

(L+ 1)!≤ 2

L+ 1

(2 logM)L+2

L!

≤ 2

L+ 1

√L!

1+ε

L!

(2α

γ

)L+2

≤ (L!)ε−1 ≤ 1

Nγ2−εγ2 ,

(5.82)

which is the desired result.

56

5.7 The Final Estimation

Lemma 5.17.

1

N

∑L≥k≥0

(2β)k+1

k!≤ 1√

N

∑L≥k≥0

(4 logM)k+1(k + 1)

floor(

k2

+ 1)!

. (5.83)

Proof. This follows by direct calculation.

Lemma 5.18.

1√N

∑L≥k≥0

(4 logM)k+1(k + 2)

floor(

k2

+ 1)!

≤ 1√NNγ2+2γ2ε (5.84)

Proof.

1√N

∑L≥k≥0

(4 logM)k+1(k + 2)

floor(

k2

+ 1)!

≤ 1√N

∑L≥k≥0

(8 logM)k+1

≤ 1√NL(8 logM)L+1 ≤ 1√

N(16 logM)L+1

≤ 1√N

(16α

β

)L+1√logL!

L+1≤ 1√

N(L!)

12+ε

≤

(N2γ2

)( 12+ε)

√N

,

(5.85)where in the last line we used that

L! ≤ LNγ2 ≤ N2γ2

. (5.86)

Lemma 5.19.

1√N

∑L≥k≥0

(k

a

)k + 2

floor(k2

+ 1)!≤ 3N2γ2− 1

2 for sufficiently large L. (5.87)

Proof.

∑L≥k≥0

(k

a

)k + 2

floor(k2

+ 1)!≤ 3L · 2L ≤ 3L! for sufficiently large L. (5.88)

Now, we want to combine these estimations. Starting with equation (5.89)

57


∫T (N)

dKS(µ, µA)dA <5

M+ 2 ·

((2β)L+1

(L+ 1)!+

(2β)L+2

(L+ 2)!+

1

N

∑L≥k≥0

(2β)k+1

k!

)

+

√2

Nmax

i

∑L≥k≥a

(k

a

)k + 2

floor(k2

+ 1)!max1, (2s(i)− 2s(1))k+1

(5.89)

the following intermediary result is a consequence of the above lemmas:∫T (N)

dKS(µ, µA)dA ≤ 5

M+

2

Nγ2−εγ2 +2√

2

N12−γ2−2γ2ε

+3√

2

N12−2γ2

. (5.90)

The summand 5M

decreases like exp(−α√

logN) as N goes to infinity. The othersummands decrease much faster. Therefore we may neglect them, i.e. for N suf-ficiently large, the left-hand side is smaller than 6

M. If we substitute α from the

beginning by α/2 the constant 6 can also be neglected. Thus, we have proved themain theorem of this chapter.

Theorem 5.20. Let α be a positive constant. Then the following estimation∫T (N)

dKS(µ, µA)dA <1

eα√

log N(5.91)

holds for N sufficiently large.

58

6 Appendix

In this Appendix the fundamental results from representation theory and momentumgeometry which are used in the main body of the text are stated in detail. Withfew exceptions, for the proofs only references to the literature are given. We closethis Appendix with elementary observations about nearest neighbor statistics.

6.1 Representation Theory

Throughout this text we are concerned with the representation theory of compactLie groups. For the standard facts we refer the reader to [BtD85] and [Kna02].

We will always use the following conventions: K denotes a semi-simple, compactLie group with Lie algebra k. Further, let G denote the complexification of K andg be the Lie algebra of G. Furthermore for any unitary vector space V the symbolU(V ) is used for the set of unitary automorphisms.

6.1.1 Representations of Compact Lie Groups

Fix a maximal torus T in K with Lie algebra t, i.e. T is a maximal, connected,commutative subgroup of K, and every irreducible representation can be decom-posed into one dimensional representations of T . On each of these T acts by scalarmultiplication, i.e., we are given a group homomorphism f : T → S1 ⊂ C∗. Wemake the following definition.

Definition 6.1. Let ρ : K → U(V ) be an irreducible, unitary representation of Kon some finite dimensional vector space V . Then a weight of ρ is an element λ ∈ t∗

such that there exists a non-trivial subspace Vλ of V with

deρ(t).x = 2πiλ(t)x ∀ x ∈ Vλ, t ∈ t. (6.1)

Note that these weights are sometimes called real infinitesimal weights.

Proposition 6.2. The set of weights (with multiplicity) of an irreducible represen-tation determines the representation uniquely.

Proof. This is a very weak form of the Theorem 5.110 in [Kna02].

Moreover, one can order the set of all weights such that every irreducible repre-sentation has a unique highest weight. We will define such an ordering here, but wehave to elaborate on the weights first.

59

6 Appendix

Recall that the adjoint representation Ad : K → GL(k) is given by k 7→ de int(k),where int : K → Aut(K), k 7→ (g 7→ kgk−1). The complexified weights of theadjoint representation are called roots.

Of all Ad-invariant scalar products on g the most important one is the so calledKilling form 〈·, ·〉Kil, which is defined by

〈ξ, η〉Kil = trace(ad(ξ) ad(η)), (6.2)

where ξ, η ∈ g and ad : g → End(g), ξ 7→ [ξ, ·].Let us denote the set of roots by ∆. The following lemma summarizes some

properties of the roots.

Lemma 6.3. The set ∆ has the following properties:

1. α ∈ ∆ generates t∗.

2. α ∈ ∆ if and only if −α ∈ ∆.

3. There exists a set of simple roots, i.e. a smallest subset ∆′ of ∆, such thatevery α ∈ ∆ is an integer combination of simple roots.

4. In the integer combination either all coefficients are non-negative or all arenon-positive.

5. The simple roots form a basis for t∗.

6. The non-negative linear combinations over R of the simple roots give a closedconvex cone in t∗.

Proof. Cf. [Kna02] Chapter II.5.

The cone in the lemma above is usually called the Weyl chamber with respect tothe system of simple roots. Identifying t∗ with t via an Ad-invariant scalar productwe can think of this cone as a subset of t.

Since every root is an integer combination of the simple roots, where all coefficientsare either non-negative or non-positive, we divide the set ∆ into the set of positiveroots

Π+ = α ∈ ∆ : α is non-negative combination of simple roots (6.3)

and negative roots

Π− = α ∈ ∆ : α is non-positive combination of simple roots. (6.4)

The simultaneous eigenspace of a root α is denoted by gα, i.e.

gα = ξ ∈ g : α(τ)ξ = [τ, ξ]. (6.5)

60


This yields a direct sum decomposition of the Lie algebra g

g = u+ ⊕ tC ⊕ u−, (6.6)

whereu− :=

⊕α∈Π−

gα and u+ :=⊕

α∈Π+

gα. (6.7)

Definition 6.4. The group generated by the reflections on the faces of the Weylchamber is called the Weyl group.

We denote the Weyl group by Wand remark that it is a finite group.

Definition 6.5. The ordering of weights is given by

λ ≤ µ :⇔ Conv(W.λ) ⊂ Conv(W.µ), (6.8)

where λ, µ are weights.

Lemma 6.6. Every weight is equivalent to a weight in the Weyl chamber under theaction of the Weyl group.

Proof. Cf. [Kna02] Corollary 2.68.

The main statement about weights is called the Theorem of the Highest Weight.

Theorem 6.7. Every irreducible representation has a unique highest weight in theWeyl chamber. Moreover, two irreducible representations are equivalent if and onlyif the highest weights are equal.

Proof. Cf. [Kna02] Theorem 5.110.

Connected to the above definitions are special complex subgroups of G, which areintroduced subsequently.

Definition 6.8. A Borel subgroup of G is a maximal, connected, solvable, complexsubgroup of G. A parabolic subgroup is a complex subgroup which contains a Borelsubgroup.

Given a fixed torus and a notion of positivity of roots, we have two natural Borelsubgroups, which are called B+ and B−. These can be obtained as follows:

B− := exp(u− ⊕ tC) and B+ := exp(u+ ⊕ tC). (6.9)

61

6 Appendix

6.1.2 The Universal Enveloping Algebra

Let T (g) denote the full tensor algebra of g, .i.e. T (g) = ⊕j∈N(⊗jg). The universalenveloping algebra U(g) of g is given by the quotient algebra

U(g) = T (g)/I, (6.10)

where I is the ideal generated by all 〈ξ ⊗ η − η ⊗ ξ − [ξ, η]〉 for ξ, η ∈ g.One directly checks that U(g) is an associative algebra.

Theorem 6.9. The universal enveloping algebra U(g) has the following properties:

1. g is embedded in U(g) by X 7→ X + I.

2. Every Lie algebra representation ρ∗ : g → End(V ) has a continuation as ahomomorphism of associative algebras ρ∗ : T (g) → End(V ). The kernel ofρ∗ contains I so this yields an induced homomorphism of associative algebrasρ∗ : U(g) → End(V ).

3. (Lemma of Burnside) Let ρ∗ : g → End(V ) be an irreducible Lie algebrarepresentation on a finite dimensional vector space. Then ρ∗ : U(g) → End(V )is surjective.

4. (Theorem of Poincare-Birkhoff-Witt) Let ξ1, . . . , ξn be a basis of g. Then themap

ψ : C[X1, .., Xn] → U(g),∑

I

aIXI →

∑I

aIξI (6.11)

is an isomorphism of vector spaces, where it is assumed that every monomialin C[X1, .., Xn] is ordered lexicographically.

Proof. The proof of the Lemma of Burnside can be found in [Far01] Chapter 3.3.The rest is proved in [Kna02] Chap. III.

Note that ψ is not an isomorphism of algebras since C[X1, .., Xn] is commutativeand U(g) is not.

In the text a notion of hermitian operators on the tensor algebra and on theuniversal enveloping algebra is needed.

Definition 6.10. The R-linear map † : T (g) → T (g) defined by

1. (zα1 ⊗ · · · ⊗ αn)† = zα†n . . . α†1 ∀α1, . . . , αn ∈ g, z ∈ C

2. ξ† = −ξ ∀ξ ∈ k

and, extended by R-linearity to T (g), is called the formal adjoint. An operatorα ∈ T (g) is called abstractly self-adjoint or abstractly hermitian, if α† = α.

Note that the formal adjoint is not complex linear because of the conjugationinvolved in condition 1.

62


Remark 6.11. The map † is compatible with ρ in the following sense:

ρ∗(ξ†) = ρ∗(ξ)

†. (6.12)

Lemma 6.12. The map † induces a R-linear map U(g) → U(g), which we also call†.Proof. We have to show that the ideal I in T (g) is fixed by †. For this let ξ = ξ1+iξ2and η = η1 + iη2 with ξ1, ξ2, η1 and η2 ∈ k be given. We calculate

((ξ1 + iξ2)(η1 + iη2)− (η1 + iη2)(ξ1 + iξ2)− [ξ1 + iξ2, η1 + iη2])†

= (ξ1η1 − η1ξ1 − [ξ1, η1])† + (i(ξ2η1 − η1ξ2 − [ξ2, η1]))

†

+ (i(ξ1η2 − η2ξ1 − [ξ1, η2]))† − (ξ2η2 − η2ξ2 − [ξ2, η2])

†

= (η†1ξ†1 − ξ†1η

†1 − [ξ1, η1]

†)− i(η†1ξ†2 − ξ†2η

†1 − [ξ2, η1]

†)− i(η†2ξ†1 − ξ†1η

†2 − [ξ1, η2]

†)

− (η†2ξ†2 − ξ†2η

†2 − [ξ2, η2]

†)

= (η1ξ1 − ξ1η1 − [η1, ξ1])− i(η1ξ2 − ξ2η1 − [η1, ξ2])− i(η2ξ1 − ξ1η2 − [η2, ξ1])

− (η2ξ2 − ξ2η2 − [ξ2, η2]).(6.13)

This proves the lemma.

6.1.3 The Laplace Operator

A Casimir operator is by definition an element of the center Z(g) of U(g). If weconsider an irreducible representation ρ∗ : U(g) → End(V ), then due to Schur’sLemma every Casimir operator has to act by scalar multiplication.

The most important example of a Casimir operator is the Laplace operator Ω.Sometimes it is even called the Casimir element, e.g. in [Kna02]. We do not give anexplicit formula for the Laplace operator here, but just state that it is an operatorof degree two in the basis elements of g.

Let δ denote half the sum of positive roots.

Lemma 6.13. The Laplace operator Ω operates by the scalar 〈λ, λ + 2δ〉Kil in anirreducible representation of g of highest weight λ.

6.1.4 The Theorem of Borel-Weil and the Embedding Of Line Bundles

Let H ⊂ G be a closed complex subgroup and ρ : H → End(V ) be a holomorphicrepresentation. The fiber product F := G×H V is the quotient space of G× V bythe equivalence relation

(g1, v1) ∼ (g2, v2), if g1 = g2h−1, v1 = ρ(h)v2 for some h ∈ H. (6.14)

The projection p : F → G/H, [(g, v)] 7→ gH is holomorphic and it can be shownby a direct calculation that p : F → G/H is a vector bundle with typical fiber V .We define a G-action on F by

x.[(g, v)] := [(xg, v)]. (6.15)

63

6 Appendix

This action induces a representation of G on the vector space of holomorphic sec-tions1 Γ(G/H,F ). For our purpose it is useful to give this representation in thecontext of H-invariant functions. Therefore, we identify the sections of F → G/Hwith the H-invariant functions f : G→ V , i.e.,

f(gh−1) = ρ(h)f(g) ∀ h ∈ H, g ∈ G. (6.16)

The G-action on these functions is given by

x.f(g) := f(x−1g) ∀ g, x ∈ G. (6.17)

In our context H will be a Borel subgroup of G.After this preparation, we can formulate a weak version of the Borel-Weil Theo-

rem. For a more complete version we refer to [Huc91] and [Akh91] for a treatmentfrom the complex analytic point of view. An algebraic approach can be found in[WG99].

Theorem 6.14. (Borel-Weil) Let ρ : G→ End(V ) be an irreducible representa-tion with highest weight λ and B− the Borel subgroup of the negative roots. ThenB− acts by multiplication on Vλ with character χ : B− → C∗, where deχ|t = 2πiλand the representation on Γ(G/B−, G×B− C) is isomorphic to ρ.

Proof. Cf. [Akh91] Chap. 4.3.

We now follow the classical construction of embedding a G-line bundle into thedual of the vector space of its sections. For this, set L = G ×B− C and fix a basiss0, . . . , sN of Γ(G/B−, L) and the corresponding dual basis s∗0, . . . , s∗N .

Let Z be the zero section of L. In the view of L = G × C/∼, the zero sectionis exactly given by the elements of the form (g, 0) for g ∈ G. We claim that weobtain an equivariant, holomorphic map of L\Z into Γ(G/B−, L)∗ by the followingconstruction. We think of the si’s as B−-equivariant functions G→ C and define

ϕ : L\Z → Γ(G/B−, L)∗, [(g, z)] 7→ 1

z

N∑j=0

sj(g)s∗j . (6.18)

This is reasonable because z is not 0, otherwise we would have [(g, 0)] ∈ Z. More-over, ϕ is well-defined. Indeed, if we take another representative (gb−1, χ(b)z), weget

N∑j=0

sj(gb−1)

χ(b)zs∗j =

N∑j=0

χ(b)

χ(b)

sj(g)

zs∗j (6.19)

because the sj are equivariant under B−, i.e.

sj(gb) = χ(b)−1sj(g). (6.20)1Since we only deal with holomorphic sections, we write Γ(G/H, F ) instead of Γhol(G/H, F ) for

the rest of this chapter.

64


Next, we have to show the equivariance of ϕ with respect to the left action of Gon L and the dual representation on Γ(G/B−, L)∗. For this let x−1.sj =

∑Ni=0 aisi

for a fixed x ∈ G and we calculate

x.ϕ([g, z])(sj) =1

z

(x.

N∑i=0

si(g)s∗i

)(sj)

=1

z

N∑i=0

si(g)s∗i (x

−1.sj)

=1

z

N∑i=0

aisi(g)

=1

z(x−1.sj)(g)

=1

zsj(xg)

=1

z

N∑i=0

si(xg)s∗i (sj)

= ϕ([xg, z])(sj).

Now, we claim that a vector of maximal weight is in the image of ϕ. For this,consider the mapping

j : G/B− → P(Γ(G/B−, L)∗), x 7→ [s0(x) : . . . : sN(x)] (6.21)

where the coordinates on the right hand side are the s∗j . It is an equivariant, holo-morphic map of G/B− into the projective space of Γ(G/B−, L)∗. Thus, the imageis a closed orbit in P(Γ(G/B−, L)∗). But the orbit of the projection of a maximalweight vector is the only such orbit (cf. [Huc91]). By comparison of (6.18) and (6.21)we obtain that a vector vmax of maximal weight is in the image of ϕ. Actually, everyc · vmax, c 6= 0, is in the image then. By equivariance, we conclude that the wholeU−-orbit through every vector of maximal weight is contained in the image of ϕ.

We state the following lemma.

Lemma 6.15. Let ϕ : L\Z → Γ(G/B−, L)∗ be the equivariant embedding de-scribed above. Then any K-invariant unitary structure on Γ(G/B−, L)∗ inducesa K-invariant hermitian bundle metric which is unique up to multiplication by aconstant.

Proof. First, we recall that the K-action on G/B− is transitive (cf. [Huc91]), soevery K-invariant bundle metric is the same up to a constant factor and we havecompleted the proof once we find the induced bundle metric is indeed K-invariant.

For [g, z1], [g, z2] ∈ L we define

hg(z1, z2) =

1〈ϕ([g,z1]),ϕ([g,z2])〉 if z1, z2 6= 0

0 otherwise.(6.22)

65

6 Appendix

By the relation

1

〈ϕ([g, z1]), ϕ([g, z2])〉= z1z2

1

〈∑n

i=0 fi(g)f ∗i ,∑n

i=0 fi(g)f ∗i 〉= z1z2

1

‖ϕ([g, 1])‖2

(6.23)we obtain a hermitian inner product at every point, since ϕ is well-defined and hasonly values different from zero. We claim that hg is a smooth bundle metric. Wesee that hg is continuous and smooth outside the zero section. Recall the standardfact that such a bundle metric is then smooth everywhere (cf. [Lan87] p.96). Thismetric is K-invariant because 〈·, ·〉 is K-invariant and ϕ is equivariant.

Lemma 6.16. Let L1 → G/B− and L2 → G/B− be homogeneous complex linebundles that realize the representations corresponding to the highest weights λ1 andλ2.

The representation of highest weight λ1 +λ2 is then realized by Γ(G/B−, L1⊗L2).

Proof. This is a corollary to the Theorem of the Highest Weight as written in [Huc91]Chap. 7.1.

6.2 Symplectic geometry and momentum maps

In this section the basic definitions of symplectic manifolds and momentums mapsare given.

By definition a symplectic manifold (M,ω) is a real manifold M with a non-degenerate two-form ω.

An action of a Lie group H on M is said to be symplectic if

h∗ω = ω ∀ h ∈ H. (6.24)

Before we define the notion of a momentum map, let us fix the notation.The induced vector field of the flow exp(−ξt) on M is denoted by Xξ and the

Lie derivative along Xξ by LXξ. For a smooth map µ : M → Lie(H)∗ we obtain an

induced map µξ : Lie(H) → C∞(M) by

µξ(x) := µ(x)(ξ) ∀ ξ ∈ Lie(H). (6.25)

Definition 6.17. Let (M,ω) be a symplectic manifold on which H acts by symplectictransformations.

A momentum map is an equivariant, smooth map µ : M → Lie(H)∗ such that

d(µξ) = ω(Xξ, ·), (6.26)

where the action on Lie(H)∗ is the coadjoint action.

66

6.3 Generalities on Level Spacings

We will use the momentum map only in the context of representations of a compactLie group. Let ρ : K → U(V ) be a unitary representation of the compact Lie groupK on a finite-dimension vector space V . This representation induces an action of Kon P(V ) which is symplectic with respect to the Fubini-Study metric on C. Recallthat the Fubini-Study metric is given by the imaginary part of the form i

2∂∂ log || · ||2

pushed down from V \0 to P(V ).

Theorem 6.18. Let ρ : K → U(V ) be an irreducible representation of highest weightλ.

The map µ : P(V ) → k∗ given by

µξ([v]) = −2i〈v, ρ∗(ξ).v〉〈v, v〉

∀ξ ∈ k, v ∈ P(V ) (6.27)

is the unique momentum map and

µ([vmax]) = λ (6.28)

for any vector vmax of highest weight.

Proof. Cf. [Huc91] Chap. IV.7.


In this section we summarize the foundational facts on level spacings.

6.3.1 The Nearest Neighbor Distribution

The material in this subsection applies to arbitrary N -tuples of real numbers, N > 1.Later on it will be used only for eigenvalues of hermitian matrices.

Definition 6.19. Let X = (x1, . . . , xN) ∈ RN be an N-tuple of real numbers, orderedby increasing value

x1 ≤ x2 ≤ · · · ≤ xN . (6.29)

The nearest neighbor distribution of X is the Borel measure on R given by

µ(X)(A) =

∫A

1

N

N−1∑i=1

δ

(y − N

xN − x1

· (xj+1 − xj)

)dy, (6.30)

if x1 6= xN , and

µ(X)(A) =N − 1

N

∫A

δ(y)dy, (6.31)

if x1 = . . . = xN , where A is a Borel set in R and δ(y−p) denotes the Dirac measurewith mass one at the point p.

67

6 Appendix

Thus, if x1 6= xN , µ(X) is a measure of total mass2 1− 1N

with expectation value

E(µ(X)) =

∫Ry dµ(X)(y)

=1

N

N−1∑j=1

N

xN − x1

· (xj+1 − xj)

=1

xN − x1

· (xN − x1) = 1.

(6.32)

Remark 6.20. Note, that µ(X) does not change under scalar multiplication, i.e.,

µ(aX) = µ(X) ∀a ∈ R, a 6= 0 (6.33)

nor under diagonal addition

µ((x1 + a, . . . , xn + a)) = µ((x1, . . . , xn)). (6.34)

If we know a priori that our N -tuple X is contained in [a, b]N mod 1, it is custom-ary to measure the wrapped around distance between xN and x1:

b− a− xN + x1 (6.35)

and to replace xN − x1 in the denominator by b− a:

µw(X)(A) =1

N

∫A

δ

(y − N

b− a(b− a− xN + x1)

)+

N−1∑i=1

δ

(y − N

b− a· (xj+1 − xj)

)dy.

(6.36)

Note that the total mass of this measure is 1 and the expectation value is also 1.Our main example for the above measure on the torus is given by the logarithms

of eigenvalues of a unitary matrix U . Here a = 0 and b = 2π and we obtain thefollowing definition

µc(X)(A) =1

N

∫A

δ

(y − N

2π(2π − xN + x1)

)+

N−1∑i=1

δ

(y − N

2π· (xj+1 − xj)

)dy

(6.37)where X = (x1, . . . , xN) is the set of ordered logarithms of the eigenvalues withmultiplicities, i.e. spec(U) = eix1 , . . . , eixN. Here the differences xj+1 − xj are theangles between the eigenvalues and 2π − xN + x1 is the angle between the first andthe last eigenvalue.

In physical models such a wrapping occurs naturally because the only physicaldata is encoded in the difference of the arguments of the eixj . Thus, the choice ofthe branch of the logarithm is artificial, i.e. the position of zero cannot be measured.

2Note that for this reason it is common to use the factor 1N−1 in front of the sum and N − 1

instead of N inside the δ measures, but we will see that this is of no importance for questionsof convergence.

68


Note. The measures µc(XN) and µw(XN) are no longer invariant under scalar mul-tiplication.

6.3.2 The Kolmogorov-Smirnov Distance

Since we want to discuss convergence of measures on the real line, we need a precisenotion of the type of convergence we are dealing with. For us only two types ofconvergence are important: the weak convergence of distribution functions and thesup-norm convergence of distribution functions.

Recall that a sequence of measure µn is said to converge weakly to a measure µ iffor every bounded, continuous function f the following holds:

limn→∞

∫fdµn =

∫fdµ. (6.38)

Definition 6.21. Let µ, ν be Borel measures on R of finite mass. The Kolmogorov-Smirnov distance dKS of µ and ν is

dKS(µ, ν) = supt∈R

∣∣∣∣∫ t

−∞dµ−

∫ t

−∞dν

∣∣∣∣ , (6.39)

which is the sup-norm for the difference of the cumulative distribution functions.We say a sequence of Borel measures µN converges to µ if dKS(µN , µ) converges

to zero.

Remark 6.22. Convergence with respect to the Kolmogorov-Smirnov distance impliesweak convergence.

Proof. The convergence in the Kolmogorov-Smirnov distance implies the pointwiseconvergence of the cumulative distribution functions. But this implies weak con-vergence by a standard result of measure theory (cf. [Els04] chap. 8 Theorem 4.12).

We now show that the scaling, with N − 1 instead of N which is common in theliterature (cf. [Meh91]), gives the same results.

Lemma 6.23. Let (XN)N∈N be a sequence of N-tuples such that XN ∈ RN and letν be a Borel measure on R+ with continuous density function p(x) with respect tothe Lesbesgue measure. Then the following are equivalent:

1. limN→∞ µ(XN) = ν.

2. limN→∞ µ1(XN) = ν, where

µ1(X)(A) =1

N − 1

∫A

N−1∑i=1

δ

(y − N

xN − x1

· (xj+1 − xj)

)dy. (6.40)

69

6 Appendix

3. limN→∞ µ2(XN) = ν, where

µ2(X)(A) =1

N − 1

∫A

N−1∑i=1

δ

(y − N − 1

xN − x1

· (xj+1 − xj)

)dy. (6.41)

Proof. The equivalence of 1. and 2. is clear, since dKS(µ1(X)(A), µ2(X)(A)) = 1N

.For the proof of the equivalence of 2. and 3. we note that

µ1(X)([0, y]) =1

N − 1card

j :

xj+1 − xj

xN − x1

·N ≤ y

(6.42)

andµ2(X)([0, y]) =

1

N − 1card

j :

xj+1 − xj

xN − x1

· (N − 1) ≤ y

. (6.43)

Therefore we see that

µ2(X)([0, y]) = µ1(X)

([0,N − 1

N· y])

. (6.44)

Now suppose 2. is true. Then

|ν([0, y])− µ2(X)([0, y])| ≤∣∣∣∣µ1(X)

([0,N − 1

N· y])

− ν([0, y])

∣∣∣∣≤∣∣∣∣µ1(X)

([0,N − 1

N· y])

− ν

([0,N − 1

N· y])∣∣∣∣

+

∣∣∣∣ν ([0, N − 1

N· y])

− ν([0, y])

∣∣∣∣ .(6.45)

Since p(x) is continuous, the cumulative density function of ν is uniformly continuousand the lemma follows from the estimation by a direct ε

2proof. Therefore, 2. implies

3. and, analogously, we see that the converse is true.

In the literature one often comes across histograms with densities plotted intothem for the nearest neighbor statistics. Compare Figure 1.1 in the introduction,where we see a histogram containing two curves.

Let us briefly discuss how the histogram in Figure 1.1 was built. We start withan N -tuple X = (x1, . . . , xN) of non-decreasing real numbers as input and considerthe N − 1 rescaled nearest neighbor distances

φj =N − 1

xN − x1

· (xj+1 − xj) ∀j = 1, . . . , N − 1. (6.46)

Now, we divide the real line into bins of some fixed width w and count the numberof φj in each bin. At last we scale the height of the boxes with a common factorsuch that the total area of the histogram is one.

70


One usually has some measures with a continuous density function with which tocompare the histogram. In Figure 1.1 two such densities are plotted.

This can be thought of as a visualization of the dKS-convergence in the followingsense: As the width w becomes smaller and the N -tuples become larger, the his-togram should approach the density of the limit measure. This can be made precisein the following way. Fix p ≥ 0 and think of the histogram restricted to [0, p] as aRiemannian sum, which should converge to the integral of the density over [0, p].

Unfortunately, this depends on the ratio of N and w. Being a bit sloppy wecan say that at the locus φj we obtain a contribution of mass 1/(N − 1) if thewidth is small enough. This is exactly the point of the definition of µ(X). Thus,a visualization of the convergence is obtained, although it is not without problemsbecause of the new dependence on the parameter w.

6.3.3 Approximating N -tuples

The following lemma shows how to construct approximating N -tuples for any abso-lutely continuous measure.

Lemma 6.24. Let µ be a measure on R≥0 with continuous density f such that∫ ∞

0

xf(x)dx ∈ [0, 1]. (6.47)

For every N ≥ 3 there exists an N-tuple X = (x1, . . . , xN), x1 ≤ . . . ≤ xn such that

dKS(µ(X), µ) ≤ 2

N − 1. (6.48)

Moreover, x1 can be chosen to be 0.

Proof. First, define yj by the requirement

j

N=

∫ yj

0

dµ ∀ j = 1, . . . , N − 1. (6.49)

If we could choose X in such a way that µ(X) has mass 1N

exactly at the yj, i.e.,

yj!=

N

xN − x1

(xj+1 − xj) ∀ j = 1, . . . , N − 1, (6.50)

then|∫ y

0

dµ−∫ y

0

dµ(X)| ≤ 1

N − 1(6.51)

since the cumulative distribution functions agree at the yj by construction and differonly by at most 1

N−1as indicated in the following picture for a certain measure.

Unfortunately, the system (6.50) might have no solution since

N−1∑j=1

yj 6= N =N−1∑j=1

N

xN − x1

(xj+1 − xj). (6.52)

71

6 Appendix

0 0,5 1 1,5 2 2,5 3

0,5

1

Figure 6.1: Approximation of µPoisson.

Thus, we redefine yN−1 in the following way

yN−1 = N −N−2∑j=1

yj. (6.53)

We claim that yN−1 is non-negative, i.e.,

N ≥N−2∑j=1

yj. (6.54)

This follows at once from the inequality∫ ∞

0

xf(x)dx =

∫ y1

0

xf(x)dx+ . . .+

∫ ∞

yN−2

xf(x)dx

≥ 0 ·∫ y1

0

f(x)dx+ y1 ·∫ y2

y1

f(x)dx . . .+ yN−2

∫ ∞

yN−2

f(x)dx (6.55)

= y11

N+ . . . yN−3

1

N+ yN−2

2

N≥ 1

N

N−2∑j=1

yj,

because∫∞

0xf(x)dx ∈ [0, 1] by assumption.

Writing (6.50) as a linear system

(xN − x1)yj −N(xj+1 − xj) = 0 (6.56)

and calculating the space of solutions, we see that the solutions depend on realparameters a and b:

xj = a+b

N

j−1∑1

yj ∀ j = 1, . . . , N. (6.57)

Note that no solution with b = 0 solves the original problem (6.50).For any solution X with b 6= 0 the estimate

dKS(µ(X), µ) ≤ 2

N − 1(6.58)

holds since yN−1 is not in the optimal position any more. Hence, we have to adjustby the factor 2

N−1.

72


Corollary 6.25. Let µ be an absolutely continuous measure on R≥0 with∫∞

0xdµ ∈

[0, 1] and p > 0 be a fixed integer. Then for any p-tuple (z1, . . . , zp) and any N ≥ p+2there is an N-tuple X = (x1, . . . , xN) such that every zj occurs as one of the xk and

dKS(µ(X), µ) ≤ 2 + p

N − 1. (6.59)

Proof. By Lemma 6.24 we find an N -tuple X such that

dKS(µ(X), µ) ≤ 2

N − 1. (6.60)

Due to the invariance of µ(X) under scalar multiplication and diagonal addition, wemay assume that

x2 ≤ zj ≤ xN − 1 ∀ j = 1, . . . , p. (6.61)

Now, insert the zj into the ordered sequence x1 ≤ . . . ≤ xN at the correspondingpositions and remove the closest xk for each zj inserted, as long as xk is neither x1

nor xN . In this case take the closest xk in the middle. The resulting sequence iscalled X.

In the picture of Figure 6.1 we have changed p points of the jump loci of theapproximating staircase function. Thus, we have to add an extra p

N−1to the esti-

mation.

6.3.4 The Nearest Neighbor Statistics under exp

In this work the most important examples of sequences (XN) of non-decreasingN -tuples are given by the spectra of sequences of Hamiltonian operators on finite-dimensional Hilbert spaces or, equally important, by the restrictions of Hamiltonianoperators to finite dimensional subspaces of some infinite-dimensional Hilbert spacesuch that the dimension of the finite-dimensional parts is approaching infinity.

In the setting of general finite-dimensional Hilbert spaces the Hamiltonians arejust skew self-adjoint operators. The space of these operators is again a finite-dimensional vector space. If A is hermitian, the one-parameter group

exp(iAt) : t ∈ R (6.62)

is a subgroup of the unitary group of this Hilbert space and the exponential mappingA 7→ exp(iA) is surjective but not injective.

Now, we consider the spectrum of a unitary operator exp(iA) and take the nearestneighbor statistics µc of the eigenangles, i.e., the aj in the eigenvalue eiaj , where0 ≤ aj < 2π.

Definition 6.26. Let U ∈ U(N) be a unitary matrix, whose eigenvalues are givenas e2iπφ1 , . . . , e2iπφN , and X(U) = (φ1, . . . , φN). The nearest neighbor statistics ofthe unitary matrix U is µc(X(U)).

73

6 Appendix

Frequently, we will write µA as an abbreviation for µ(X(A)) and µU as abbrevia-tion for µc(X(U)).

The nearest neighbor statistics of exp(iA) will not agree with the nearest neighborstatistics of A for two reasons, the first being the wrapping discussed above and thesecond and more important is the problem of reordering.

The eigenvalues x1 ≤ · · · ≤ xN of A give the φj only modulo 2π, i.e.

aj = φj mod 2π (6.63)

and it may happen that there are j1 and j2, such that φj1 < φj2 but aj1 > aj2 .If, however, all eigenvalues are sufficiently close to each other, meaning that they

all lie in an interval of width 2π, one does not have to reorder, if choosing a differentbranch of the logarithm or just by adding a constant to all eigenvalues such that thesmallest eigenvalue is 0.

Lemma 6.27. Let (XN)N∈N be a sequence of non-decreasing N-tuples such thatXN ∈ [0, 2π[N and let ν be a Borel measure on R+ with continuous density withrespect to the Lebesgue measure. Assume that the difference between the largest andthe smallest eigenvalue converges to 2π. Then the following are equivalent:

1. limN→∞ µ(XN) = ν.

2. limN→∞ µc(XN) = ν.

Proof. Since the differences between the largest and the smallest eigenvalue con-verge to 2π, the µ(XN) come arbitrarily close to the µc(XN) as is evident by theirdefinitions.

Remark 6.28. The above lemma is false if we drop the assumption on the largest andsmallest eigenvalues. Indeed, assume that every XN is contained in the subinterval[0, 1

N[ with smallest eigenvalue 0 and largest eigenvalue 1

N, then µC(XN) has the

wrapping eigenangle given by

aN =N

2π(2π − 1

N) = N − 1

2π, (6.64)

all other eigenangles are less than or equal to 1/2π. Therefore µc can only convergeto a measure whose cumulative distribution function is 1 for all t ∈ R, t ≥ 1/2π.

To summarize, care has to be taken if considering the nearest neighbor statisticsunder exp. It is not enough to ensure that the eigenvalues of a hermitian operatorare in an interval [0, 2π] but one must also ensure that the difference between thesmallest and the largest eigenvalue approaches 2π.

74


6.3.5 The Nearest Neighbor Statistics and the CUE Measure

As discussed above we are mainly interested in the nearest neighbor statistics asso-ciated to unitary matrices. We give some more details about these statistics here.In this section µc(X(A)) will be abbreviated by µA.

The following lemma is necessary in certain of our applications.

Lemma 6.29. If ν is an absolutely continuous probability measure on R, then themap

U(N) → [0, 1], A 7→ dKS( ν, µA ) (6.65)

is continuous.

Proof. Cf. [KS99] where the proof is given in lemma 1.0.11. and 1.0.12.

Since U(N) is a compact group, functions on U(N) can be averaged. It is alsopossible to average the map A 7→ µA. This can be done in the following way. Letµ(U(N)) denote the Borel measure given by

µ(U(N))(X) :=

∫U(N)

(µA(X)) dHaar(A) (6.66)

for any Borel-measurable set X.We now state Lemma 1.2.1 of [KS99].

Lemma 6.30. There exists an absolutely continuous probability measure ν on Rwith real analytic cumulative distribution function such that

µ(U(N)) → ν weakly, as n→∞. (6.67)

We call this measure µCUE. In [KS99] the following theorem is given in a moregeneral form as Lemma 1.2.6.

Theorem 6.31. For every ε > 0 there is a natural number N0 such that∫U(N)

dKS(µCUE, µA)dHaar ≤ N ε−1/6 (6.68)

for all N ≥ N0.

The complete proof of the lemma and the theorem is given in all detail in [KS99],where it takes the first half of the book, so it cannot be given here.

More details on µCUE can be found in [Meh91] and again in [KS99].

75

6 Appendix

76

List of Symbols

† The formal adjoint on T (g) or U(g), page 62

∆ The set of roots of K, page 60

〈·, ·〉Kil The Killing-Form on k or g, page 60

C[X1, .., Xn] The Ring of polynomials in n indeterminates with coefficients in C,page 62

O(Cn) The algebra of holomorphic functions on Cn, page 29

T (g) The full tensor algebra of g, page 62

U(g) The universal enveloping algebra of g, page 62

g The Lie algebra of G, page 59

k The Lie algebra of K, page 59

t The Lie algebra of T , page 59

u+ The Lie algebra of positive roots, page 61

u− The Lie algebra of negative roots, page 61

µ(X) The nearest neighbor statistics of the tuple X, page 67

µA The nearest neighbor statistics of the eigenvalues of hermitian matrix A,page 74

µc(X) The nearest neighbor statistics of the angles X with wrapping at 2π, page 68

µU The nearest neighbor statistics of the eigenangles of unitary matrix U , page 74

µCUE The limit measure of the nearest neighborhood statistics of the unitary group,page 75

Ω The Laplace operator in U(g), page 63

Π+ The set of positive roots of K, page 60

Π− The set of negative roots of K, page 60

77

6 Appendix

ρ∗ The Lie algebra representation associated to a Lie group representation ρ,page 62

cl The classical limit map, page 13

cln n-approximation of the classical limit map, page 18

U(V ) The set of unitary operators on the unitary vector space V , page 59

B+ The Borel subgroup of positive roots of G, page 61

B− The Borel subgroup of negative roots of G, page 61

BN The set of unitary N ×N -matrices with nearest neighbor statistics not closeto µPoisson, page 38

dKS The Kolmogorov-Smirnov distance, page 69

G The complexification of K, page 59

K A semi-simple, compact Lie group, page 59

T A maximal torus of K, page 59

W The Weyl group of K with respect to fixed Π+, page 61

78

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[CFS82] I. P. Cornfeld, S.V. Fomin, and Ya. G. Sinai. Ergodic Theory. Springer,1982.

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[Gnu00] Sven Gnutzmann. Klassischer Grenzfall, Semiklassik und Quantenchaosbei kollektiv gekoppelten n-Niveau-Atomen. PhD thesis, Universität Essen,2000.

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Documents

uni-bremen.deelib.suub.uni-bremen.de/diss/docs/00010494.pdf · Contents 1 Introduction 7 2 Construction of the Classical Limit 11 2.1 The Classical Limit in the Simple Case .