Expected number of critical points of random holomorphic sections over complex projective space

0.17 MB·

by  Xavier Garcia

EXPECTED NUMBER OF CRITICAL POINTS OF RANDOM HOLOMORPHIC SECTIONS OVER
COMPLEX PROJECTIVE SPACE
7
XAVIERGARCIA
1
0
2

n
a
J ABSTRACT. Westudythehighdimensionalasymptoticsoftheexpectednumberofcriticalpointsofagiven
MorseindexofGaussianrandomholomorphicsectionsovercomplexprojectivespace. Weexplicitlycompute
5
theexponentialgrowth rate ofthe expectednumberof critical pointsofthelargest indexand ofdiverging
2
indicesatvariousratesaswellastheexponentialgrowthratefortheexpectednumberofcriticalpoints(re-

gardlessofindex). Wealsocomputethedistributionofthecriticalvaluesfortheexpectednumberofcritical
]
pointsofsmallestindex.
R
P
.
h
t
a 1. INTRODUCTION AND MAINRESULTS
m
Thepurposeofthispaperistodeterminethehighdimensionalasymptoticsoftheexpectednumberof
[
criticalpointsofagivenMorseindexofGaussianSU(m+1)multivariatepolynomialsofafixeddegree

2 N asthe dimension m tends to infinity. Bydefinition, these criticalpoints arethose of the holomorphic
v sectionsofthelinebundle (N) CPm equippedwiththeChernconnectioninducedfromtheFubini-
2 O →
Studymetric. ThestatisticsofcriticalpointsofGaussianrandomholomorphicsectionshavebeenstudied
5
extensivelyinDouglas,Shiffman,andZelditch[7][8],mainlyasatooltounderstandthevacuumselection
7
6 problem in string theory. The main focus of [8] (as well as most of the literature on Gaussian random
0 holomorphicsections)isonthelargedegreelimit,namelyas N tendstoinfinity. Inthispaper,weadopt
1. adifferentpointofviewandfocusinsteadonhighdimensionallimits. InBaugher[5],itwasproventhat
0 thenumberofcriticalpointswithMorseindexclosetom(i.e.saddlepoints)growsexponentially. Wenot
7 onlyrecoverthisresult,butalsoobtainestimatesfortheexpectednumberofcriticalpoints,regardlessof
1
theirindices. Wewillalsocomputetheexactdistributionofthecriticalvaluesofindexm,whichrecovers
:
v theformulainTHEOREM1.4ofBaugher[5].
i We will approach our problems by random matrix theory, in particular we will use large deviation
X
resultsconcerningtheeigenvaluesofaWishartmatrixensemble,alsoknowninthestatisticsliteratureas
r
a sample covariancematrices. This approachexpandsthe connection between criticalpoints of Gaussian
fields and random matrix theory initiated in the seminal paper of Auffinger, Ben Arous and Cˇerny´ [4],
in which they established a link between critical points of isotropic Gaussian fields on the sphere and
eigenvaluesoftheGaussianorthogonalensemble(GOE).Theunderlyingreasonforthesuccessofrandom
matrix theory in both areas is the existence of large symmetry groups, namely SO(m+1) on Sm and
SU(m+1) on CPm. The techniques in Auffinger and Ben Arous [3] help address the related problem,
namely the behavior of Gaussian random critical points of spherical harmonics on Sm of large degrees,
sincethecovariancekernelwhicharisesthereisalsoinvariantunderSO(m+1)asinthespinglasscase.
Wewillsaymoreonthislater.
We now describe the setting and our main results. We consider the line bundle (N) over CPm
O
equipped with Fubini-Study metric h and induced Chern connection . We endow the space of holo-
∇
morphicsectionsH0(CPm, (N))withtheinnerproductinducedbythemetric,namelyfortwosections
O
s ,s weset
1 2
s ,s := h (s ,s )v(dz),
1 2 z 1 2
h i CPm
Z
1
2 XAVIERGARCIA
where v is the Fubini-Study volume element. We view H0(CPm, (N)) as a finite dimensional Hilbert
O
spaceandchooseanorthonormalbasissN. Withthisbasis,wecanformtheGaussianfield
i
(1.1) s= ∑c sN,
i i
i
wherethec arecircularlysymmetriccomplexGaussianswiththenormalizedvariance
i
Vol(CPm) N!πm
E c 2 = = .
| i| dimH0(CPm, (N)) (N+m)!
O
Itisclearthatthedistributionofsisindependentofthechoiceoftheorthonormalbasis. ForanyBorelset
B ⊂ R+ = [0,∞)andintegerm ≤ k ≤ 2m,weconsiderNm,k,N(B)(s),thenumberofcriticalpointswith
Morseindexkforasectionswithh (s,s)= s(z) 2 (m+1)B;symbolically,
z || ||h ∈
(B)= ∑ ( s(z) 2),
Nm,k,N 1(m+1)B || ||h
z: s(z)=0,Ind( 2s)(z)=k
∇ ∇
where we understand Ind( 2s) as the index of the real Hessian of log s(z) 2. Thus (B) is an
∇ || ||h Nm,k,N
integer-valuedrandomnumberifsissampledfromtheGaussianfield(1.1). Therandomvariable
(B)=∑ (B)
Nm,N Nm,k,N
k
isthetotalnumberofcriticalpointsregardlessoftheirMorseindices.
We will prove two types of asymptotics for (B) as the dimension m goes to infinity: with
fixed k and with linearly growing k. MorespecNifimca,2lmly−,ki,nNthe lattercase we will consider the relationof
theformk(m)/m γ (0,1)asm ∞.
→ ∈ →
Wenowstateourmainresults. Foragivenγ (0,1)defines by
γ
∈
4
(1.2) f (x)dx =γ,
MP
Zsγ
where
(4 x)x
f (x)= −
MP
2πx
p
istheMarchenko-Pasturdensityfunctionon[0,4].
OurfirstmainresultconcernstheexponentialgrowthrateoftheexpectednumberE (x,∞)
Nm,2m−k,N
ofcriticalpoints.
Theorem1.1. Fixanintegerk.
(1)Supposethatx 0andletx = N x. Ifx 4,then
≥ N N 1 N ≥
−
lim 1 logE [x,∞)=log(N 1) xN 1 2 (k+1) xN t−4 dt.
m→∞ m Nm,2m−k,N − − 2 (cid:18) − N(cid:19)− Z4 r 4t
1 2
lim logE [0,x)=log(N 1) 2 1 .
m ∞m Nm,2m−k,N − − − N
→ (cid:18) (cid:19)
Ifx 4,then
N
≤
1
lim logE [0,x)= ∞.
m ∞m Nm,2m−k,N −
→
1 2
lim logE [x,∞)=log(N 1) 2 1 .
m ∞m Nm,2m−k,N − − − N
→ (cid:18) (cid:19)
(2)Ifk =k(m)suchthat k γ (0,1),then
m → ∈
mlim∞m1 logENm,2m−k,N(R+) =log(N−1)− 1− N2 s2γ,
→ (cid:18) (cid:19)
wheres isthenumberuniquelydefinedbytherelation(1.2).
γ
EXPECTEDNUMBEROFCRITICALPOINTSOFRANDOMHOLOMORPHICSECTIONSOVERCOMPLEXPROJECTIVESPACE 3
Theaboveresultsdonotincludethecasek(m) = m. However,inthiscasewecancomputeexplicitly
the expected value E m,m,N(R+) and recover the formula in Baugher [5]. Define the density function
N
p by
m,m,N
1
E ∑ f s(z) 2 = f(x)p (x)dx
 m+1|| ||h  R m,m,N
z: s(z)=0,Ind( 2s)(z)=k (cid:18) (cid:19) Z +
∇ ∇
for any positive continuous function f on R+. Note that theabove sum is simply the total number of
criticalpointsofMorseindexm. Oursecondmainresultisanexplicitformulafor p .
m,m,N
Theorem1.2. Foranyx 0,
≥
pm,m,N(x)=(N−1)m(m+1)2e−(2m(N+−1)1N)(2−N2+m)x.
Wecandrawtwoconsequencesfromthisexplicitdensity.
Corollary1.3. Foranyx 0,
≥
ENm,m,N[x,∞)= 2(N−2 1)m2++1(mm+1)e−(2m(N+−1)1N)(2−N2+m)x.
− N
Proof. Integratingthedensityfunctionover[x,∞). (cid:3)
Forx =0,theabovecorollaryrecoverstheformula
2(m+1)
(1.3) ENm,m,N(R+) = 2(N 1)+mN(N−1)m+1
−
provedinBaugher[5]. Forx >0,itfollowsfromthecorollarythatthereexistpositiveconstantsc andc
1 2
suchthatE m,m,N(x,∞) c1e−c2m2x,whichshowsthatitbecomesexponentiallyunlikelytofindcritical
N ≤
valuesawayfrom0whoseMorseindexism.
k >Th0.esecondconsequence isthatwecanrecoverthe exponentialrateofENm,m+k,N(R+)foranyfixed
Corollary1.4. Forafixedk >0,wehave
1
mlim∞mlogENm,m+k,N(R+) =log(N−1).
→
Proof. According to THEOREM 1.4 of Baugher [5], the total number of critical points Nm,m+k,N(R+) de-
creasesaskincreases. Thus,givenγ (0,1)andq(m)/m γ,wehaveforlargem,
∈ →
Nm,2m−q(m),N(R+) ≤Nm,m+k,N(R+) ≤Nm,m,N(R+).
Fortherighthandside,wehaveby(1.3)
1
mlim∞m logENm,m,N(R+) =log(N−1).
→
Forthelefthandside,wehavebythesecondpartofTHEOREM1.1,
mlim∞logENm,2m−q(m),N(R+) =log(N−1)− 1− N2 s2γ.
→ (cid:18) (cid:19)
Wehaves 0asγ 1,andtheabovelimitreducestothatoftherighthandside. Theresultfollows
γ
→ → (cid:3)
immediately.
Finally,ourthirdandlastmainresultconcernsthetotalnumberofcriticalpoints.
4 XAVIERGARCIA
Theorem1.5. Asbefore,weletx 0andx = N x. Then:
≥ N N 1
−
log(N 1) 1 2 xN + xN 4−tdt, x 4
mli→m∞m1 logENm,N(x,∞)= log(N−1)−(cid:18)1− N2 (cid:19) x2N, Z4 r t xNN ≥<4
− − − N 2
(cid:18) (cid:19)
Theremainingpartofthepaperisorganizedasfollows. InSECTION 2westatesomebasicfactsfrom
complex geometry essential for the understanding of the paper. In SECTION 3 we discuss the Wishart
ensemble and its large deviations needed in the proof of the main results. SECTION 4 is devoted to ex-
plainingtherelationbetweentheexpectednumberofcriticalpointsandtheWishartensemble. Themain
resultsTHEOREM1.1andTHEOREMS1.2and1.5areprovedinSECTIONS5. InSECTION6wediscussthe
analogouscaseofrandomsphericalharmonics.
Acknowledgements. This material is based upon work supported by the National Science Foundation
GraduateResearchFellowship. IwouldliketothankAntonioAuffingerandSteveZelditchforinspiring
conversationsandunwaveringpatience.IwouldalsoliketothankmyadvisorEltonHsuforhisrevision
ofthismanuscriptandforservingasaconstantsourceofencouragement.
2. COMPLEXPROJECTIVE SPACE ANDLINE BUNDLES
Inthissectionwe recallsome basicfactsfromcomplex geometrywhich areusefulforunderstanding
thesettingofthepaper.
ThecomplexprojectivespaceCPm isthequotientspaceofCm+1 0 bytheequivalencerelation
\{ }
λ(Z0,...,Zm) (Z0,...,Zm), λ C∗ =C 0 .
∼ ∈ \{ }
ThisisacompactcomplexmanifoldwithlocalchartsU = [Z ,Z ,...,Z ] Z =0 andtrivializingmaps
i 0 1 m i
Φ : U Cmdefinedby { | 6 }
i i
→ \
Φ (Z) =(Z /Z,...,Z/Z,...,Z /Z).
i 0 i i i m i
Wedenoteby (N)thelinebundlewiththetransitionfunctions
O
N
Z
σij :Ui Uj C∗, σij(Z)= i .
∩ → Zj!
The sections of this bundle correspond to homogeneous holomorphic polynomials of degree N in the
variables Z ,...,Z . To see this, given a homogeneous holomorphic polynomial p(Z ,...,Z ) we define
0 m 0 m
the functions f on U by f (Z) = p(Z/Z ). It is easy to verify that these functions glue up and yield a
j j j j
sectiononCPm. Indeed,ontheintersectionU U,wehave
i j
∩
N
Z Z Z
f (Z)σ (Z)= p i = p = f (Z).
i ij j
(cid:18)Zi(cid:19) Zj! Zj!
Conversely,asectionisjustacollectionofpolynomials f onthechartsU satisfying f (Z)σ (Z) = f (Z)
j j i ij j
onthe intersectionU U, whichdefine ahomogenous polynomial inaunique waybysetting p(Z) =
i j
∩
ZNf (Z).
j j
We equip CPm with the Fubini-Study metric h and denote the corresponding Chern connection on
(1) by . This induces canonically a connection on (N), also denoted by , by requiring that it
O ∇ O ∇
satisfyLeibniz’sruleontensorsofsections. Moreexplicitly,asections H0(CPm, (N))canbewritten
∈ O
locallyass= feN,whereeN = N eforatrivializinglocalframeefor (1)andaholomorphicfunction
f onachartofCPm. Thenthec⊗oni=n1ection canbeexpressedexplicitlyOas
∇
m
(2.1) s= ∑(∂ f + f ∂ K )dz eN,
∇ zj zj N j⊗
j=1
EXPECTEDNUMBEROFCRITICALPOINTSOFRANDOMHOLOMORPHICSECTIONSOVERCOMPLEXPROJECTIVESPACE 5
whereK isgivenby
N
(2.2) K =K (z,z¯)= Nlog(1+ z 2).
N N
| |
Since also actson 1-formscanonically, the Hessian 2 on holomorphic sections is well defined. This
∇ ∇
actioncanbe explicitlywritteninlocalcoordinatesasfollows. For simplicity we introducethe notation
f := ∂ f + f ∂ K and 2 f = ( f). In the local basis dz dz , we can view 2s as the
∇zj zj zj N ∇zi,zj ∇zi ∇zj i⊗ j ∇
2m 2msquarematrix
×
2 f fΘ
∇zi,zj N
2s(z) =
∇  
fΘ 2 f
 N ∇zi,zj 
 
whereΘ = ∂2 K . NotethatthismatrixisnotHermitian. Forthisreason,whendiscussingcritical
N N
zi,zj
points of a senction s, itois more convenient to use the real Hessian of log s(z) 2 by viewing CPm as a
|| ||h
smooth manifoldofrealdimension 2m. Byaslightabuseofnotation, weuseInd( 2s)(z)todenotethe
∇
index of this matrix. From LEMMA 7.1 of Douglas, Shiffman, and Zelditch [8], we know that in local
coordinates
Ind(∇2log||s(z)||2h) =m+Ind(∇2zi,zjf Θ∗N∇2zi,zjf −ΘN),
whereΘ istheconjugatetransposeofΘ .
∗N N
3. THE WISHART ENSEMBLEANDRELATEDLARGEDEVIATIONS
LetXbeareal(m+1) mrandommatrixwhoseentriesarei.i.d. Gaussianswithmeanzerovariance
×
1/m and W = XTX. We denote the law of W, the Wishart ensemble, by P and the corresponding
m
expectationbyE .
m
TheonlyinformationwewillneedabouttheWishartensembleistheexplicitdistributionofitseigen-
values. For a vector λ = (λ1,...,λm), we define ∆(λ) = ∏i<j(λi−λj), the Vandermonde determinant.
Wewritetheeigenvaluesλ = (λ ,...,λ )ofW indescendingorder,sothatthevector λbelongstothe
1 m
region
Rm = λ Rm : λ ... λ 0 .
≥0 { ∈ 1 ≥ ≥ m ≥ }
Theorem 3.1. The joint density function of the decreasingly ordered eigenvalues of the Wishart ensemble with
respecttotheLebesguemeasureonRm is
0
≥
1 m m
∆(λ)exp ∑λ ,
ZW(m) −2 i=1 i!
whereZ (m)isthenormalizingconstantgivenby
W
m
(3.1) ZW(m)=2mm−m(m+1)/2∏j!
j=1
Proof. SeeTHEOREM13.3.2inAnderson[2]forthedensity,andCOROLLARY2.5.9ofAnderson,Guionnet
andZeitouni[1]fortheexplicitformulaforZ (m). (cid:3)
W
WenowturntothelargedeviationsofthelargesteigenvaluesoftheWishartensemble. Wewillneed
thefollowinglargedeviationprincipleforthekthlargesteigenvalueunderP .
m
Theorem3.2. UnderP ,thekthlargesteigenvalueλ satisfiesthelargedeviationprinciple(LDP)withthespeed
m k
mandthegoodratefunctionkI ,where
MP
x t 4
I (x)= − dt
MP
Z4 r 4t
forx 4and∞otherwise.
≥
6 XAVIERGARCIA
Proof. Itisobviousthat I isagood ratefunction. Withthisinmind, thistheoremisequivalenttothe
MP
followingtwoassertions:
(1) limsup 1 logP (λ x)= ∞for0< x <4.
m ∞ m m k ≤ −
(2) limm→∞ →m1 logPm(λk ≥ x)=−kIMP(x)forx ≥4.
Fortheproof,weneedtwopreviousresults.
(a)Under the Wishart ensemble, the empiricalmeasure L = 1 ∑ δ of the eigenvaluessatisfies an
m m i λi
LDPwithspeedm2. ItsratefunctionisminimizeduniquelyattheMarchenko-Pasturdistributionµ on
MP
[0,4]
1 (4 x)x
µ (dx)= − dx.
MP
2π x
p
ThisLDPisthecontentofTHEOREM5.5.7ofHiaiandPetz[11].
(b)Thefunctional
z
φ(µ,z)= log z y µ(dy)
R | − | − 2
Z +
definedonP(R+) R+isuppersemi-continuouswhenwerestrictittoP[0,M] [0,M]foranyM>0,
and in fact it is con×tinuous on P[0,r] [x,y] for y > x > r 4, see e.g. Auf×finger, Ben Arous and
× ≥
Cˇerny´ [4]. Here P(A) is the space of probability measures on a set A R+ with a metric compatible
⊂
with the usual weak convergence of probability measures. The distribution µ and the rate function
MP
I arerelatedthroughthefunctionalby
MP
(3.2) φ(µ ,x)= I (x) 1.
MP MP
− −
SeeFeral[10],page48.
To prove assertion (1), we note that by definition, the inequality λ x for some x < 4 implies that
k
Lm[x,4] (k 1)/m. Since µMP[x,4] > 0,thereexistsaclosedsetC ≤P(R+)suchthatµMP / Cand
≤ − ⊂ ∈
λ x L C forsufficientlylargem. TheLDPfor L recalledaboveimpliesthatthereexistsa
k m m
{c >0≤suc}h⊂th{at ∈ }
Pm(λk x) Pm(Lm C) Ke−cm2,
≤ ≤ ∈ ≤
whichprovesassertion(1).
Toproveassertion(2),wefirstnotethatforthelargesteigenvalueλ ,
1
1
(3.3) lim limsup logP (λ > M)= ∞,
m 1
M ∞ m ∞ m −
→ →
whichispreciselyLEMMA2.6.7ofAnderson,GuionnetandZeitouni[1]. Nowwehave
P (λ x) P (λ > M)+P (λ x,λ < M).
m k m 1 m k 1
≥ ≤ ≥
Inviewof(3.3),itissufficienttoshowthatforsufficientlylarge M,
1
(3.4) lim logP (λ x,λ < M)= kI (x).
m ∞ m m k ≥ 1 − MP
→
Wefirstprovetheupperbound. Weintroducenewvariablesη = m λ for1 i mandwritethe
i m k i ≤ ≤
densityofP intermsoftheη. Ontheset −
m i
x η η <2M x λ λ < M
k 1 k 1
{ ≤ ≤ ··· ≤ }⊃ { ≤ ≤··· ≤ }
EXPECTEDNUMBEROFCRITICALPOINTSOFRANDOMHOLOMORPHICSECTIONSOVERCOMPLEXPROJECTIVESPACE 7
wehave η η 2M,andhence
i j
| − | ≤
1 m m
P (dλ)= ∆(λ)exp ∑λ dλ dλ
m ZW(m) "−2 i=1 i# 1··· m
m k m(m+1)/2 1 m k m
= − ∆(η)exp − ∑η dη dη
(cid:18) m (cid:19) ZW(m) "− 2 i=1 i# 1··· m
(2M)(k 1)k/2 m k m(m+1)/2 k m m k k
− − ∏ ∏ (η η ) exp − ∑η dη dη
≤ ZW(m) (cid:18) m (cid:19) i=1j=k+1 i− j · "− 2 i=1 i# 1··· k×
m k m
∏ (η η ) exp − ∑ η dη dη
k+1 i<j m i− j · "− 2 i=k+1 i# k+1··· m
≤ ≤
m k m(m+1)/2 Z (m k)
=(2M)(k−1)k/2 m− WZ (m−) ·dη1···dηk×
(cid:18) (cid:19) W
k
exp (m k)∑φ(L˜ ,η) P (dη dη ),
" − i=1 m−k i # m−k k+1··· m
whereL˜ istheempiricaldistribution
m k
−
L˜ = 1 m∑−kδ .
m−k m k ηk+i
− i=1
For ǫ > 0,let B P[0,M]betheballofradiusǫ centeredaroundµ and Bc itscomplement. Onthe
ǫ ⊂ MP ǫ
set x η η <2M ,wehaveexp (m k)∑k φ(L˜ ,η) (2M)k(m k)andthus
{ ≤ k ≤··· ≤ 1 } − i=1 m−k i ≤ −
h i
k
exp"(m−k)i∑=1φ(L˜m−k,ηi)# ≤exp"k(m−k)µ∈Bǫs,yu∈p[x,2M]φ(µ,y)#1Bǫ(L˜m−k)+(2M)k(m−k)1Bǫc(L˜m−k).
Integrationover x η η <2M yieldsanupperboundforP (λ x,λ < M):
k 1 m k 1
{ ≤ ≤··· ≤ } ≥
m(m+1)
(cid:18)mm−k(cid:19) 2 (2M)k(k2−1)ZWZ(Wm(m−)k) exp"k(m−k)µ∈Bǫs,yu∈p[x,2M]φ(µ,y)#+(2M)k(m−k)Pm−k(L˜m−k ∈/ Bǫ)!.
Twoobservationsareinorder. Thefirstobservationisthat L˜ withrespecttoP satisfiesthesame
m k m k
LDPas L withrespecttoP . Inparticular,thisimpliesthatf−ormlargeenoughthe−reexistsac > 0for
m m
which
P (L˜ / B ) exp( cm2),
m−k m−k ∈ ǫ ≤ −
hencetheprobabilityP (L˜ / B )isnegligibleinthelimit. Thesecondobservationisthatbyuseof
m k m k ǫ
− − ∈
(3.1),onecancompute
1 m k m(m+1)/2 Z (m k)
lim log − W − =k.
m→∞m "(cid:18) m (cid:19) ZW(m) #
Inlightofthesetwoobservation,wearriveattheinequality
1
limsup logP (λ x) k+klim sup φ(µ,y).
m k
m→∞ m ≥ ≤ ǫ↓0 µ∈Bǫ,y∈[x,2M]
Thesecondtermcanbecomputedexplicitly,
lim sup φ(µ,z) = sup φ(µ ,y)= I (x) k,
MP MP
ǫ↓0 µ∈Bǫ,y∈[x,2M] y∈[x,2M) − −
8 XAVIERGARCIA
wherethefirstequalityfollowsfromtheupper-semicontinuityofφandthesecondequalityfollowsfrom
(3.2)andthemonotonicityof I .
MP
To obtain the lower bound, fix y > x > r 4 and ǫ > 0. We retainthe definition of the η asin the
i
≥
proofoftheupperbound,andandontheset
m m k
y η η x = − y λ λ x λ x
≥ 1 ≥ ···≥ k ≥ m k m ≥ 1 ≥··· ≥ k ≥ ⊂{ k ≥ }
(cid:26) − (cid:27) (cid:26) (cid:27)
wecanproducetheinequality
1 m m
P (dλ)= ∆(λ)exp ∑λ dλ dλ
m ZW(m) "−2 i=1 i# 1··· m
m k m(m+1)/2 Z (m k)
− W − ∏ η η dη dη
≥ m Z (m) | i− j|· 1··· k×
(cid:18) (cid:19) W 1 i<j k
≤ ≤
1Bǫ∩P[0,r](L˜m−k)exp(cid:20)k(m−k)µ∈Bǫ∩Pi[n0f,r],z∈[x,y]φ(µ,z)(cid:21)Pm−k(dηk+1···dηm).
wherebyB P[0,r],ImeanthesetofmeasuresinB whosesupportiscontainedin[0,r].Byintegrating
ǫ ǫ
∩
over y η η m x ,weobtain
≥ 1 ≥ ··· ≥ k ≥ m k
−
n o
m
P (λ x) P y η η x
m k m 1 k
≥ ≥ ≥ ≥··· ≥ ≥ m k
(cid:18) − (cid:19)
m k m(m+1)/2 Z (m k)
− W − ∏ η η dη dη
≥ m Z (m) | i− j| 1··· k×
(cid:18) (cid:19) W Z 1 i<j k
≤ ≤
exp k(m k) inf φ(µ,z) P (L˜ B P[0,r])
m k m k ǫ
(cid:20) − µ∈Bǫ∩P[0,r],z∈[x,y] (cid:21) − − ∈ ∩
wheretheinnerintegralisovertheset
m
y η η x .
≥ 1 ≥ ··· ≥ k ≥ m k
(cid:26) − (cid:27)
Theinnerintegralisboundedawayfromzeroandfromabove,soitwillhavenoeffectinthelimit. The
factor P (L˜ B P[0,r]) convergesto one by the previously mentioned LDP, hence it too will
m−k m−k ∈ ǫ∩
notaffectthelimit. Itfollowsthatinthelimittheinequalitybecomes
1
liminf logP (λ x) k+klim inf φ(µ,z).
m→∞ m m k ≥ ≥ ǫ↓0 µ∈Bǫ∩P[0,r],z∈[x,y]
Weusethecontinuityofφand(3.2)toobtain
lim inf φ(µ,z)= I (y) 1.
MP
ǫ↓0 µ∈Bǫ∩P[0,r],z∈[x,y] − −
(cid:3)
Finally,welety xandusethecontinuityof I obtainourdesiredresult.
MP
→
4. EXPECTEDNUMBEROF CRITICAL POINTS ANDTHE WISHART ENSEMBLE
In this section we relate E (B) to the (k+1)th largest eigenvalue of an (m+1) (m+1)
Nm,2m−k,N ×
Wishartmatrix.
Theorem4.1. ForaBorelsetB R+,
⊂
(4.1) ENm,2m−k,N(B)= 2(N−N1)m+1 Em+1 e−(1−N2)m2+1λk+1; λk+1 ∈ NN 1B .
(cid:20) − (cid:21)
TheproofofthisidentityisbasedonthefollowingKac-Riceformulaadaptedtooursetting.
EXPECTEDNUMBEROFCRITICALPOINTSOFRANDOMHOLOMORPHICSECTIONSOVERCOMPLEXPROJECTIVESPACE 9
Proposition4.2. Letρ denotetheprobabilitydensityfunctionof s(z)asa(random)vectorinCm(see(2.1)).
ThenE (B)∇eqsu(za)ls ∇
Nm,2m−k,N
ρ (0)E[ det 2s(z) ( s(z) 2) s(z) =0]v(dz).
ZCPm ∇s(z) | ∇ |1(m+1)B || ||h 1Ind∇2s(z)=2m−k|∇
(cid:3)
Proof. SeeTHEOREM4.4ofDouglas,Shiffman,andZelditch[7].
Remark 4.3. In general ρ depends on our choice of sN. Nevertheless, its value ρ (0) at the origin is
s(z) i s(z)
∇ ∇
independentofthechoice.
By SU(m+1)-invariance, the integrand in the above Kac-Rice formula is independent of z, thus the
z-integrationcanbereplacedbythemultiplicationofvol(CPm)andweneedtoevaluatetheexpectation
atthepointz =0. Forthispurpose,wewrites(z) = f(z)eN inlocalcoordinatesnearthepointz =0. We
have f =∂ f :=∂ f atz =0.
∇zi zi i
Lemma4.4. Thecovarianceof f anditsfirstandsecondderivativesatz =0aregivenasfollows.
E[f(0)f(0)] =1,
E[f(0)∂ f(0)] =0,
i
E[f(0)∂∂ f(0)] =0,
i j
E[∂ f(0)∂ f(0)] = Nδ ,
i j ij
E[∂ f(0)∂ ∂ f(0)] =0,
i j k
E[∂∂ f(0)∂ ∂ f(0)] = N(N 1)(δ δ +δ δ ).
i j k l − il jk ik jl
Proof. TheGaussianfielddefinedin(1.1)isuniquelydeterminedbyitscovariancekernel
N!πm
E[s(x) s(y)] = Π (x,y).
⊗ (N+m)! N,m
HereΠ isthekerneloftheprojectionfromL2(CPm, (N))intoH0(CPm, (N)).Notethatthiskernel
N,m
O O
isindependentofourchoiceofanorthonormalbasissN in(1.1). Inlocalcoordinates,itcanbeexplicitly
i
writtenas
N!πm
Π (x,y)=(1+z w¯)NeN(z) eN(w),
(N+m)! N,m · ⊗
wherezandwarethe(inhomogeneous)coordinatesofxandy. Thecovariancesinthestatementfollows
(cid:3)
bystraightforwardcomputations.
AsimmediateconsequencesofLEMMA4.4,weseethatρ f(0)(0)=1/(Nπ)mandthatboththematrix
∇
∂2 f(0)and f(0)areindependentoftheevent∂ f(0,hencealsoindependentof s(0)=0. From(2.2)we
ij k ∇
have∂2 K (0)= Nδ ,hencefrom(2.1)wehave
zi,zj N ij
det 2s(0)=det(YY∗ N2 f(0) 2Im),
∇ − | |
where the matrix Y = ∂2 f(0) and I is the m m identity matrix. Obviously the value of the de-
ij m ×
terminant depends onlynon the oeigenvalues of YY∗. Therefore we need to study the distribution of the
eigenvaluesofYY ,whichisaSym(m,C)-valuedrandommatrix.
∗
Proposition4.5. ThelawoftheeigenvaluesofW =YY /mN(N 1)isidenticalwiththelawoftheeigenvalues
∗
−
undertheWishartensemble.
Proof. ThenaturalLebesguemeasureonSym(m,C)asarealvectorspaceis
dH = ∏RedH ImdH .
ij ij
i j
≤
10 XAVIERGARCIA
FromthelastcovarianceidentificationinLEMMA4.4thedensityfunctionofYwithrespecttotheLebesgue
measuredHis
(4.2) 1 e−2N(N1 1)Tr(HH∗).
m(m+1) −
2m(N(N 1)π) 2
−
DefinethemapΦ :U(m) Rm Sym(m,C)by
× 0 →
≥
Φ(U,λ)=Udiag(√λ)UT,
where diag(√λ) the diagonal matrix whose entries are √λ ,...,√λ and UT is the transpose of U. By-
1 m
Takagi’sfactorization(seeCOROLLARY4.4.4ofHornandJohnson[12]),almosteveryX Sym(m,C)can
∈
bewrittenuniquelyasX =Udiag( λ(XX )UT,whereUisaunitarymatrixandλ (XX )aretheeigen-
∗ i ∗
values of XX in decreasing order. A well known computation shows that the image of the Lebesgue
∗ p
measure dH under Φ becomes Φ (dH) = ∆(λ)dλdU,where dU isthe properlynormalized Haarmea-
sureon U(m). Note thattheJaco∗bianin thiscaseis ∆(λ), afunction of λ alone. Onthe other hand, the
exponentin(4.2)is
1 1 m n
N(N 1)Tr(YY∗) = N(N 1) ∑λi(YY∗) =m∑λi(W).
− − i=1 i=1
By passing from Sym(m,C) to U(m) Rm , we see from (4.2) that the density functions for the distri-
× 0
butionoftheeigenvaluesofW = YY /N(≥N 1)mustbeaconstantmultipleof∆(λ)exp m ∑m λ .
∗ − −2 i=1 i
Comparingthiswiththe densityfunctionoftheeigenvaluesundertheWishartensemblein LEMMA 3.1
(cid:2) (cid:3)(cid:3)
weobtaintheresultimmediately.
Summarizingwhatwehaveprovedsofar,fromtheKac-RiceformulainPROPOSITION4.2weconclude
thatE (B)equals
Nm,2m−k,N
Vol(CPm)mm(N 1)m N f(0) 2 m N f(0) 2
− E ( f(0) 2) | | ∏ λ | | ,
πm "1(m+1)B | | 1[λk+1,λk](cid:18)(N−1)m(cid:19)i=1(cid:12) i− (N−1)m(cid:12)#
(cid:12) (cid:12)
whereλi =λi(W)withWobeyingtheWishartensembleand f(0)is,according(cid:12)(cid:12) toLEMMA4.4,(cid:12)(cid:12)astandard
complex Gaussian random variable independent of W. It remains to identify this with (4.1). For this
purpose,wenotethat N|f(0)|2 isexponentiallydistributedwithmean N . Thustheexpectationis
(N 1)m (N 1)m
− −
(4.3) mN(ZNW−(m1)) ZmN((mN+11))BZ ∏i=m1|λi−x|∆(λ)e−m2(1−N2)xe−m2(∑mi=1λi+x)dλdx,
−
wheretheinnerintegralwithrespecttoλisovertheset
λ >...> λ > x > λ >...> λ >0 .
{ 1 k k+1 m }
Thisdomainsuggestswetreatxasifitisanotherλ. Moreprecisely,introducethenewvariablesµ = λ
i i
for 1 i k, µ = x, and µ = λ for k+2 i m. For the Vandermonde polynomial we have
∆(µ) =≤∆(≤λ)∏mk+1λ x . Inteirmsof−t1henewvar≤iable≤sµ,theintegral(4.3)becomes
i=1| i− |
mN(ZNW−(m1)) ZRm0+11(N−1m)N(m+1)B(µk+1)e−(1−N2)m2µk+1exp"−m2 mi∑=+11µi#∆(µ)dµ.
≥
ComparingthiswithLEMMA3.1,thisisexactlytheexpectationwithrespecttoPm+1uptoaconstant. We
will omit the identification of the constant statedin the theorem, it beinga straightfowardcomputation
using Selberg’sintegral formula for ZW(m). This completes the proof THEOREM 4.1, our main resultof
thissection.
An immediate consequence of THEOREM 4.1 is that E m,q+1,N(R+) is decreasing in q in the range
m q < 2m, agreeingwith THEOREM 1.4of Baugher [5].NAlso, summing over k in (4.1), we obtain the
≤
followingcorollary.