Spherical Matrix Ensembles

0.27 MB·English

by  Gene S. Kopp

SPHERICAL MATRIX ENSEMBLES
GENES.KOPPANDSTEVENJ.MILLER
ABSTRACT. Thesphericalorthogonal,unitary,andsymplecticensembles(SOE/SUE/SSE)Sβ(N,r)
consist of N N real symmetric, complex hermitian, and quaternionic self-adjoint matrices of
5 ×
Frobeniusnormr,madeintoaprobabilityspacewiththeuniformmeasureonthesphere. Foreach
1
of these ensembles, we determine the joint eigenvalue distribution for each N, and we prove the
0
2 empirical spectral measures rapidly convergeto the semicircular distribution as N . In the
→ ∞
unitarycase(β =2),wealsofindanexplicitformulafortheempiricalspectraldensityforeachN.
b
e
F

2

CONTENTS
]
R
1. Introduction 1
P
. 1.1. Definitions 3
h
1.2. Results 4
t
a
2. PreliminaryLemmas 5
m
3. ProofofTheorem 1.3 6
[
4. ProofofTheorem 1.4 8

2
5. ProofofTheorem 1.5 9
v
8 6. Spacing Statistics 12
4
References 13
8
1
0
.
1
0 1. INTRODUCTION
5
1 After writing this paper, it was brought to our attention that the
v: our spherical ensembles have been studied previously by physicists
i Caër and Delannay, under the name of fixed trace ensembles [CD, DC].
X
Delannay and Caër prove results equivalent to our Theorem 1.4 (on
r
a moments of spherical ensembles) for β = 1,2 in Appendix B of [DC],
and our Theorem 1.5 (on empirical spectral density) in Section 3.1
of [DC]. Our methods differ from those of Delannay and Caër: We use
the method of moments, and they use the Laplace transform method.
Date:February3,2015.
2010MathematicsSubjectClassification. 15B52(primary),60F05(secondary).
Keywordsandphrases. Randommatrixensembles,Wigner’ssemicirclelaw,empiricalspectralmeasure,spherical
ensembles,Besselfunctions.
Portions of this work were completed at the 2010 SMALL REU at Williams College; we thank our colleagues
there,especiallyMuratKolog˘lu,andparticipantsattheICMSatelliteMeetinginProbability&StochasticProcesses
inBangalore,India,especiallyArupBose,forhelpfulconversations. Thefirstnamedauthorwaspartiallysupported
byNSFGrantDMS0850577,andthesecondnamedauthorwaspartiallysupportedbyNSFgrantsDMS0970067and
DMS1265673.
Random matrix theory has grown enormously in results and scope from Wishart’s [Wis] 1928
paper. Through the work of Wigner [Wig1, Wig2, Wig3, Wig4, Wig5], Dyson [Dy1, Dy2], and
others,itnowsuccessfullymodelsavarietyofsystemsfrom theenergy levelsofheavy nucleiand
zeros of L-functions [Con, Por] to bus routes [BBDS, KrSe]; see for example [AGZ, For, Meh,
MT-B] for detailed expositions of the general theory, and [FM, Hay] for a review of some of the
historicaldevelopmentand applications.
Much of the subject revolves around determining properties of the spectra of self-adjoint ma-
trices. In this paper we concentrate on the density of states, though there has been significant
progressinrecentyearsonthespacingsbetweennormalizedeigenvalues(seeforexample[ERSY,
ESY, TV1, TV2]). The limiting spectral measure has been computed for many ensembles of pat-
terned or structured matrices, including band, circulant, Hankel and Toeplitz matrices, as well as
adjacency matrices of d-regular graphs [BBDS, BasBo1, BasBo2, BanBo, BCG, BHS1, BHS2,
BM, BDJ, GKMN, HM, JMP, Kar, KKMSX, LW, McK, Me, Sch]. Many of these papers show
thatwhen aclassicalensembleismodifiedby enforcingparticular, usuallylinear, relationsamong
the entries, new limiting behavior emerges. We study spherical ensembles introduced in [MT-B]
(seeResearch Project 16.4.7). Whileingeneral itisdifficulttoobtainclosed-formexpressionsfor
spectralstatisticsofstructuredensembles,weareabletoexploitthesymmetryofthesphericalen-
sembles to give explicit formulas for the joint spectral distribution and empirical spectral density.
Weproveasemicircularlaw and givenumericalevidenceofGUEspacing statistics.
As a set, the N N spherical unitary ensemble of radius r consists of N N Hermitian
× ×
matrices
a a +b √ 1 a +b √ 1
11 12 12 1N 1N
− ··· −
a b √ 1 a a +b √ 1
A =  12 − ...12 − 2...2 ·.·..· 2N ...2N −  (1.1)
 
 a b √ 1 a b √ 1 a 
1N 1N 2N 2N NN
 − − − − ··· 
 
satisfyingtherelation
a2 +2 a2 +b2 = r2, (1.2)
ii ij ij
i i<j
X X(cid:0) (cid:1)
ormoresuccinctly,Tr(A2) = r2. Thissetofmatricesbecomesaprobabilityspacewiththeuniform
measureonthe(compact)ellipsoiddefinedbyTr(A2) = r2. Oneinterpretationofthenameisthat
Tr(A2) = r2 isaspherein theFrobenius norm(seebelow).
The spherical ensemble may be understood to arise naturally from the problem of normaliza-
tion. ConsideraHermitianmatrixAchosenfromaGaussianunitaryensemble,withvolumeform
e Tr(A2)/2dA(uptoaconstant). TostudythelimitingbehaviorasthedimensionN ,it’sstan-
−
→ ∞
dard to normalize the eigenvalues by a factor of 1/√N. With this normalization, the eigenvalue
densityconvergesto Wigner’ssemicircle,
1 √4 x2, if x 1,
f (x) := 2π − | | ≤ (1.3)
Wig
0, otherwise.
(
2
One could instead normalize the matrices themselves before looking at the eigenvalues. We can
normalizeaHermitianmatrixAbysendingitto √NA,where A istheFrobeniusnorm(see(1.5)).
A | |
| |
This“pre-normalized GUE”isjustthesphericalunitaryensembleofradius √N.
We also consider spherical orthogonal and symplectic ensembles. In all cases, we prove that
the eigenvalue density converges to Wigner’s semicircle as the dimension N . In the fixed-
→ ∞
dimensionalcase,however,theempiricalandjointspectraldistributionsexhibitnewbehavior. We
provideexplicitformulasand investigatetheiranalyticproperties.
1.1. Definitions. Let V be a finite-dimensionalreal, complex, or quaternionic1vector space with
a Hermitian inner product , . For any A End(V), let A End(V) be the adjoint, so
∗
h− −i ∈ ∈
Av,w = v,A w . TheFrobenius inner product onEnd(V) is defined by
∗
h i h i
A,B := Tr(AB ), (1.4)
∗
h i
and inducestheFrobenius norm
A := A,A = Tr(AA ). (1.5)
∗
| | h i
When Aiswrittenas amatrix(withresppect to anorthopnormalbasisonV), theFrobenius normof
A equals the square root of the sum of the norm squares of the entries of A. If A is self-adjoint,
then A = Tr(A2)alsoequalsthesquarerootofthesumofthesquaresoftheeigenvaluesofA.
Fo|ro|urpurposes,anN N matrixensembleissimplyasubsetofM (K),whereK = R,Cor
N
H, alongwipthprobability×measure. Foramatrixensemble ,theunderlyingsetisalsodenotedby
E
, and the associated volume form is denoted by d A. The ensembles we examineare all ensem-
E E
bles of self-adjoint matrices and thus have real eigenvalues, so we assume that in the discussion
thatfollows.
Ifitexists,theempiricalspectraldensityoftheensemble istheunique(uptoameasurezero
E
set ofvalues)real-valuedfunctionf(x, )such that
E
1
E g(λ (A)) = ∞ g(x)f(x, )dx (1.6)
A i
∈E N ! E
Xi Z−∞
for all g L1(R), where λ (A) 6 λ (A) 6 6 λ (A) are the eigenvalues of A. The kth
1 2 N
∈ ···
momentoff(x, )is
E
1 1
m ( ) := E λ (A)k = Tr(Ak)d A, (1.7)
k A i
E ∈E N N E
(cid:18) (cid:19) Z
E
and itscorrespondingcharacteristic function is
∞ m ( )
φ(t, ) = k E (it)k (1.8)
E k!
k=0
X
and isrelated to thedensityby
1
f(x, ) = ∞ e itxφ(t, )dt. (1.9)
−
E 2π E
Z
−∞
Let S be the symmetricgroup on 1,2,...,N . The joint spectral distribution is the measure
N
on theset RN/S ofN unordered p{ointsinduce}d by pushingforward themeasure on alongthe
N
E
1A quaternionic vector space is a free (H,H)-bimodule, where H denotes the Hamilton quaternions. A finite-
dimensionalquaternionicvectorspaceisnecessarilyisomorphictosomeHn.
3
mapA (λ (A),λ (A),...,λ (A)). WeconsideritasameasureonRN bysymmetrizingitand
1 2 N
7→
normalizingit tobeaprobabilitymeasure.
Wecan nowcarefully statetheensemblesofinterestin thispaper.
Definition 1.1(Gaussian ensembles). The Gaussianorthogonal/ unitary/symplecticensembles
(GOE/GUE/GSE)aredenotedG (N,q)andconsistofN N realsymmetric,complexHermitian,
β
×
andquaternionicself-adjointmatrices,madeintoaprobabilityspacewiththedensity
d A := C (N,q)e βqA2/4dA. (1.10)
Gβ(N,q) β − | |
Here β = 1,2,4 for the GOE, GUE, and GSE, respectively, q is a positive real number, and
C (N,q)is chosentogivea probabilitydensity.
β
Definition 1.2(Spherical ensembles). The sphericalorthogonal/ unitary/symplecticensembles
(SOE/SUE/SSE)S (N,r)consistofN N realsymmetric,complexhermitian,andquaternionic
β
×
self-adjointmatricesofFrobeniusnormr,madeintoaprobabilityspacewiththeuniformmeasure
on thesphere.
1.2. Results. Forallspherical ensembles,wedescribethejointeigenvaluedistribution.
Theorem 1.3. Let β 1,2,4 . The joint eigenvalue distribution for S (N,r) (on vectors of
β
eigenvaluesλ RN)i∈s g{ivenby}
∈
ρ(λ,S (N,r))dλ = C (N,r) ∆(λ) βδ(r2 λ 2)dλ. (1.11)
β β′′ | | −| |
Here, C (N,r)isa computableconstant,δ istheDiracdelta,and
β′′
∆(λ) = (λ λ ). (1.12)
j i
−
16i<j6N
Y
The formula in (1.11) can be used—at least theoretically—to describe any statistic for these
ensembles,such as spacingdensity,largest spacing,n-levelcorrelation, n-leveldensity,et. cetera.
However,thisis easiersaidthan done. Theanalysiseventocomputethemarginals
ρ(m)((λ ,...,λ ),S (N,r)) := ∞ ∞ ρ(λ,S (N,r)) dλ dλ (1.13)
1 m β β m+1 N
··· ···
Z Z
−∞ −∞
was toocumbersome. Theremainderofourresultsconcern theempiricalspectral density
f (x,S (N,r)) = ρ(1)((x),S (N,r)) (1.14)
β β
and itsmoments.
For all spherical ensembles, we are able to express the moments in terms of the moments of
the corresponding Gaussian ensembles, which allows us to prove a semicircular law in the limit.
Explicitly,
Theorem 1.4. Let β 1,2,4 , and let q,r > 0. The kth moment of a spherical ensemble is
∈ { }
relatedto thekth moment ofa Gaussianensembleby
β k2 Γ n
m (S (N,r)) = qr2 2 m (G (N,q)), (1.15)
k β 4 Γ k+n k β
(cid:18) (cid:19) (cid:0)2(cid:1)
where n = dimRGβ(N,q) = β N2 + N. Specializ(cid:0)ing to(cid:1) r = √N, the eigenvalue density of
S (N,√N)hasmeanzeroandvarianceone. Thedensityfunctionsf(x,S (N,√N))convergein
β (cid:0) (cid:1) β
measuretof (x).
Wig
4
In the Hermitian case, we are able to give explicit formulas for the characteristic function and
densityfunction.
Theorem 1.5. Let J bea Bessel functionof thefirstkind (seechapter9 of [AS]). Thecharacter-
α
isticfunctionequals
2
Γ(N2) 2 N2 −1N−1 1 N
φ(t,S (N,r)) = 2 J (rt)( rt)j, (1.16)
2 N rt j! j +1 N2+j 1 −
(cid:18) (cid:19) j=0 (cid:18) (cid:19) 2 −
X
andthespectraldensityis
f (x,S (N,r)) = 1 N−1 N Γ N22 d 2j r2 x2 N22−3+j
2 rN2 2N j +1 2j√πΓ (cid:16)N2 1(cid:17)+j j! dx −
− j=0 (cid:18) (cid:19) 2− (cid:18) (cid:19)
X (cid:0) (cid:1)
1 x x 2 12(N2−2(cid:0)N−1) (cid:1)
= p 1 (1.17)
N
r r − r
(cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
fora particularpolynomialp .
N
Thepaperisorganizedasfollows. Wederivesomenecessarylemmasin§2. WeproveTheorem
1.3in§3,Theorem1.4in§4,andTheorem1.5in§5. Weconcludebydiscussingspacingstatistics
in§6.
2. PRELIMINARY LEMMAS
We calculate the joint and empirical spectral distributions of the spherical ensembles by relating
themto theGaussianensembles,whosespectraldistributionsareknownexplicitly.
Lemma 2.1. The map2 of measure spaces A r A defines a measure-preserving map from
7→ A
G (N,q) toS (N,r). In otherwords, iff L1(S (|N|,r)),then
β β β
∈
r
f(A)d A = f A d A. (2.1)
Sβ(N,r) A Gβ(N,q)
ZSβ(N,r) ZGβ(N,q) (cid:18)| | (cid:19)
Proof. BecausethedensityfunctionofaGaussianensembledependsonlyontheFrobeniusnorm,
the mass of each ray from the origin is the same. Collapsing each such ray to the point on the
Frobeniussphereit intersectsgivestheuniformdistributionon thesphere. (cid:3)
Thenextlemmawillbeneeded in theproofofTheorem 1.4.
Lemma 2.2. Let G be a degree k homogeneous polynomial in n variables with complex coeffi-
cients, B a positive definite symmetric matrix, the standard 2-norm on Rn, √~x B~x the norm
⊤
|·|
inducedbyB, and α > 0. Then
G(~x) α k Γ n
e α(~x⊤B~x)/4d~x = 2 2 G(~x)e α~x⊤B~x/4d~x. (2.2)
− −
(~x B~x)k/2 4 Γ k+n
ZRn ⊤ (cid:16) (cid:17) (cid:0)2(cid:1) ZRn
(cid:0) (cid:1)
2While this map is not defined at the zero matrix, this is immaterial for our investigationsbelow since the zero
matrixisasinglepointofmass0.
5
Proof. We first do the special case when B = I. Denote by d~θ the volume element on Sn 1. For
−
anyfunctionf : R R so thatG(~x)f( ~x ) isintegrable,
→ | |
G(~x)f( ~x )d~x = ∞G(r~θ)f(r)rn 1drd~θ
−
| |
ZRn ZSn−1Z0
= ∞G(~θ)f(r)rk+n 1drd~θ
−
ZSn−1Z0
= ∞f(r)rk+n 1dr G(~θ)d~θ. (2.3)
−
(cid:18)Z0 (cid:19)ZSn−1
Iff(r) = e αr2/4rγ forα > 0, wehavether-integralis
−
∞f(r)rk+n 1dr = ∞e αr2/4rγ+k+n 1dr
− − −
Z0 Z0
γ+k+n 1
4u − 1
= ∞e u du
−
α αu
Z0 r r
γ+k+n
1 4 2 γ +k +n
= Γ . (2.4)
2 α 2
(cid:18) (cid:19) (cid:18) (cid:19)
Thecases γ = k andγ = 0give
−
n
G(~x) 1 4 2 n
e α~x2/4d~x = Γ G(~θ)d~θ,
− | |
~x k 2 α 2
ZRn | | (cid:18) (cid:19) (cid:16) (cid:17)ZSn−1
k+n
1 4 2 k +n
G(~x)e α~x2/4d~x = Γ G(~θ)d~θ. (2.5)
− | |
2 α 2
ZRn (cid:18) (cid:19) (cid:18) (cid:19)ZSn−1
Dividingthesetwo equationsgivesthedesired formulafortheB = I case.
Thegeneral case followsimmediatelyfrom (2.5). As B is positivedefinite, wehave B = A A
⊤
forsomefull rank matrixA. Thelinearsubstitution~y = A~x yieldsthemoregeneral formula,as
G(~x) e α(~x⊤B~x)/4d~x = G(~x)e αA~x2/4d~x
− − | |
ZRn (~x⊤B~x)k/2 ZRn |A~x|k
G(A 1~y)
= − e α~y2/4 detA 1d~y
− | | −
~y k | |
Rn
Z | |
α k Γ n
= 2 2 G A 1~y e α~y2/4 detA 1d~y
− − | | −
4 Γ k+n | |
(cid:0)2(cid:1) ZRn
(cid:16) (cid:17)
(cid:0) (cid:1)
= α k2 Γ(cid:0) n2 (cid:1) G(~x)e α(~x⊤B~x)/4d~x. (2.6)
−
(cid:16)4(cid:17) Γ (cid:0)k+2n(cid:1) ZRn
(cid:3)
(cid:0) (cid:1)
3. PROOF OF THEOREM 1.3
The (order-symmetric) joint eigenvalue density for the N N GOE / GUE / GSE ensembles are
×
givenby theformula
ρ(λ,G (N,q)) = C (N,q)e βqλ2/4 ∆(λ) β (3.1)
β β′ − | | | |
6
(see Theorem 3.1 of [Rez]). Here λ = (λ ,...,λ ) is the vector of eigenvalues, C (N,q) is a
1 N β′
computableconstant,and
∆(λ) = (λ λ ). (3.2)
j i
−
i<j
Y
In otherwords,forasymmetricintegrablefunctionf ofN real variables,
f(λ(A))d A = f(λ)ρ (λ,G (N,q))dλ, (3.3)
Gβ(N,q) N β
ZGβ(N,q) ZRN
with ρ(λ,G (N,q)) as above. As by Lemma 2.1 the spherical ensemble measure is the pushfor-
β
ward oftheGaussianensemblemeasureunderthemap A √rA, wehave
→ A
| |
r
f(λ(A))d A = f λ A d A
Sβ(N,r) A Gβ(N,q)
ZSβ(N,r) ZGβ(N,q) (cid:18) (cid:18)| | (cid:19)(cid:19)
r
= f λ(A) d A
A Gβ(N,q)
ZGβ(N,q) (cid:18)| | (cid:19)
r
= f λ(A) d A
λ(A) Gβ(N,q)
ZGβ(N,q) (cid:18)| | (cid:19)
r
= f λ ρ(λ,G (N,q))dλ. (3.4)
β
λ
ZRN (cid:18)| | (cid:19)
Changingto sphericalcoordinates,weobtain
f(λ(A))d A = ∞f (θ)ρ(tθ,G (N,q))rtN 1dtdθ, (3.5)
Sβ(N,r) β −
ZSβ(N,r) ZrSN−1Z0
where dθ denotes the volume form of rSN 1 (called d~θ earlier, in the r = 1 case). As ∆
−
is a homogeneous polynomial in the eigenvalues of degree N , we have ρ(tθ,G (N,q)) =
2 β
Cβ′(N,q)e−βq|tθ|2/4|∆(tθ)|β = Cβ′(N,q)e−βqr2t2/4tβ(N2)|∆(θ)|β.(cid:0)S(cid:1)ubstitutinginto(3.5),
f(λ(A))dSβ(N,r)A = ∞f (θ) Cβ′(N,q)e−βqr2t2/4tβ(N2)|∆(θ)|β rtN−1dtdθ
ZSβ(N,r) ZrSN−1Z0 (cid:16) (cid:17)
= C (N,q)r ∞tn 1e βqr2t2/4dt f (θ) ∆(θ) βdθ ,
β′ − − | |
(cid:18)Z0 (cid:19)(cid:18)ZrSN−1 (cid:19)
(3.6)
where (as usual) n = β N + N. The substitution u = βqr2t2/4 immediately gives the value
2
2n−1 Γ n forthefirst integral. Thus,
(βqr2)n/2 2 (cid:0) (cid:1)
(cid:0) (cid:1)
f(λ(A))d A = C (N,r) f (θ) ∆(θ) βdθ, (3.7)
Sβ(N,r) β′′ | |
ZSβ(N,r) ZrSN−1
wherethenormalizingconstant
2n 1 n
−
C (N,r) = C (N,q)r Γ (3.8)
β′′ β′ (βqr2)n/2 2
(cid:16) (cid:17)
is independent of q. This equality (3.7) is equivalentto Theorem 1.3: The Dirac delta in Theorem
1.3specifies that themass isconcentrated onrSN 1. ✷
−
7
4. PROOF OF THEOREM 1.4
Proofof Theorem 1.4. Westartbyprovingthemomentexpansion,equation(1.15). Appliedtothe
momentsoftheempiricalspectral density,Lemma2.1 implies
1
m (S (N,r)) = Tr(Ak)d A
k β N Sβ(N,r)
ZSβ(N,r)
k
1 r
= Tr A d A
N A Gβ(N,q)
ZGβ(N,q) (cid:18)| | (cid:19) !
1 Tr Ak
= C (N,q)rk e βqA2/4dA. (4.1)
N β A k − | |
ZGβ(N,q) |(cid:0)| (cid:1)
By Lemma2.2, wemayrewritetheaboveas
βq k2 Γ n 1
m (S (N,r)) = 2 C (N,q)rk Tr Ak e βqA2/4dA, (4.2)
k β 4 Γ k+n N β − | |
(cid:18) (cid:19) (cid:0)2(cid:1) ZGβ(N,q)
(cid:0) (cid:1)
where n = dimRGβ(N) = β N2 +(cid:0)N ((cid:1)and we are using the fact that the norm of the Frobenius
innerproductisareal innerproduct). Thus,
(cid:0) (cid:1)
β k2 Γ n
m (S (N,r)) = qr2 2 m (G (N,q)), (4.3)
k β 4 Γ k+n k β
(cid:18) (cid:19) (cid:0)2(cid:1)
as desired. (cid:0) (cid:1)
We now turn to the second part of the theorem, the convergence to the semi-circle. The eigen-
value density of G (N,N) has mean zero and variance 1+ 2 1 1 = 1 +o(1). The correct
β β − N
scalingforthesphericalensembleistotaker = √N,asthisw(cid:16)illlead(cid:17)toeigenvaluesofsizeonthe
orderof1. By (1.15)wefind
β 2 k2 Γ n
m (S (N,√N)) = N√N 2 m (G (N,N))
k β 4 Γ k+n k β
(cid:18) (cid:19) (cid:0)2(cid:1)
β k2 Γ n (cid:0) (cid:1)
= N2 2 m (G (N,N)), (4.4)
4 Γ k+n k β
(cid:18) (cid:19) (cid:0)2(cid:1)
wheren = β N2 +N = N(βN+2(2−β)). In particular, S(cid:0)β(N,(cid:1)√N) has mean zero and variance
(cid:0) (cid:1) β Γ n 2 1
m (S (N,√N)) = N2 2 1+ 1
2 β 4 Γ 1+ n β − N
(cid:18) (cid:19) (cid:0) (cid:1)2 (cid:18) (cid:18) (cid:19) (cid:19)
β 1 βN +(2 β)
= N2 (cid:0) (cid:1) −
4 n/2 βN
(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
βN2 4 βN +(2 β)
= −
4 N(βN +(2 β)) βN
(cid:18) (cid:19)(cid:18) − (cid:19)(cid:18) (cid:19)
= 1. (4.5)
8
As N , the asymptotic relation Γ(z + c)/Γ(z) zc for z implies the asymptotic
→ ∞ ∼ | | → ∞
equality
k
β 2 n k
m (S (N,√N)) N2 −2 m (G (N,N))
k β k β
∼ 4 2
(cid:18) (cid:19)
(cid:16) (cid:17)
k k
β 2 βN2 −2
N2 m (G (N,N))
k β
∼ 4 4
(cid:18) (cid:19) (cid:18) (cid:19)
m (G (N,N)). (4.6)
k β
∼
In otherwords,ifC denotes theℓth Catalan number,then
ℓ
C ifk iseven,
lim m (S (N,√N)) = lim m (G (N,N)) = k/2 (4.7)
k β k β
N N (0 ifk isodd,
→∞ →∞
which by the Method of Moments (see [Bi, Ta]) implies that, in the large N limit, the empirical
spectral distribution for a spherical ensemble converges in measure to the semicircle. (See [Rez],
Remark 2.2 for the fact that the limiting moments of Gaussian ensembles—and, more generally,
Wignerensembles—areCatalan numbers.) (cid:3)
5. PROOF OF THEOREM 1.5
WefirstprovetheexpansionforthecharacteristicfunctioninTheorem1.5,andthenwederivethe
claimedformulafortheempirical spectraldensity.
ThemomentsoftheGUEarewell-known. Themomentm (S (N,r))vanishesifk isodd. For
2 β
k = 2ℓ wehavethefollowingexpressionfromHarer and Zagier[HZ]:
1
m (G (N,q)) = q ℓ(2ℓ 1)!!c(ℓ,N), (5.1)
2ℓ 2 −
N −
wherec(ℓ,N) are powerseries coefficients foraparticularrational function,
N
∞ 1 1+z
c(ℓ,N)zℓ = 1 , (5.2)
2z 1 z −
!
ℓ=0 (cid:18) − (cid:19)
X
and(2ℓ 1)!! = (2ℓ 1)(2ℓ 3) 3 1isthe2ℓth momentofthestandardnormal. Substituting
− − − ··· ·
(5.1)into(1.15)yields
1 r2 ℓ Γ n
m (S (N,r)) = 2 (2ℓ 1)!!c(ℓ,N), (5.3)
2ℓ 2 N 2 Γ n +ℓ −
(cid:18) (cid:19) 2(cid:0) (cid:1)
and wemaytherefore writethecharacteristicfunctio(cid:0)nas (cid:1)
∞ 1 r2 ℓ Γ n (it)2ℓ
φ(t,S (N,r)) = 2 (2ℓ 1)!!c(ℓ,N)
2 N 2 Γ n +ℓ − (2ℓ)!
ℓ=0 (cid:18) (cid:19) 2(cid:0) (cid:1) !
X
1 ∞ Γ n (cid:0) (cid:1) 1 ℓ
= 2 c(ℓ,N) (rt)2 . (5.4)
N Γ n +ℓ ℓ! −4
ℓ=0 2 (cid:0) (cid:1) (cid:18) (cid:19)
X
(cid:0) (cid:1)9
Forconvenience,fix N andr, and set
1 ∞ Γ n
ψ (u) := 2 c(ℓ,N)uℓ, (5.5)
N N Γ n +ℓ ℓ!
ℓ=0 2 (cid:0) (cid:1)
X
(cid:0) (cid:1)
sothatψ 1(rt)2 = φ(t,S (N,r)). Thecoefficientsofthispowerseriesareaproductofmore
N −4 2
familiarpowerseriescoefficients:
(cid:0) (cid:1)
∞ Γ n n
2 zℓ = F ,z , (5.6)
Γ n +ℓ ℓ! 0 1 2
Xℓ=0 2 (cid:0) (cid:1) (cid:16) (cid:17)
where F is aconfluenthypergeomet(cid:0)riclim(cid:1)itfunction,and by definition
0 1
∞
c(ℓ,N)zℓ = B(z) (5.7)
ℓ=0
X
where B(z) is the rational function on the right hand side of (5.2). From contour integration, we
find
1 n dz
ψ (u) = F ,uz B(z 1) . (5.8)
N 0 1 −
2πiN 2 z
I|z|=2 (cid:16) (cid:17)
Weevaluatethecontourintegralbyexpandingbothfunctionsatz = 1,whichisthelocationofthe
poleofB(z 1). TheexpansionofB is easy:
−
N N ℓ
dz 1 2 1 N 2
B(z 1) = 1+ 1 dz = dz. (5.9)
−
z 2 z 1 − 2 ℓ z 1
!
(cid:18) − (cid:19) ℓ=1 (cid:18) (cid:19)(cid:18) − (cid:19)
X
Theexpansionof F ismoreinvolvedbutelementary:
0 1
n ∞ Γ n
F ,uz = 2 (uz)ℓ
0 1 2 Γ n +ℓ ℓ!
(cid:16) (cid:17) Xℓ=0 2 (cid:0) (cid:1)
∞ (cid:0)Γ n (cid:1)
= 2 uℓ((z 1)+1)ℓ
Γ n +ℓ ℓ! −
ℓ=0 2 (cid:0) (cid:1)
X
∞ (cid:0)Γ n (cid:1) ∞ ℓ
= 2 uℓ (z 1)j
Γ n +ℓ ℓ! j −
ℓ=0 2 (cid:0) (cid:1) j=0 (cid:18) (cid:19)
X X
∞ Γ(cid:0)n (cid:1)∞ 1
= 2 uℓ (z 1)j
j! Γ n +ℓ (ℓ j)! −
j=0 (cid:0) (cid:1) ℓ=j 2 − !
X X
= ∞ Γ n2 (cid:0)∞ (cid:1)Γ n2 +j uℓ′+j (z 1)j
Γ n +j j! Γ n +j +ℓ ℓ! −
j=0 2 (cid:0) (cid:1) ℓ′=0 2 (cid:0) (cid:1)′ ′ !
X X
∞ (cid:0)Γ n (cid:1) n (cid:0) (cid:1)
= 2 F +j,u uj(z 1)j. (5.10)
Γ n +j j!0 1 2 −
Xj=0 2 (cid:0) (cid:1) (cid:16) (cid:17)
(cid:0) (cid:1) 10