A Note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices

0.13 MB·English

by  Sandrine Dallaporta

A NOTE ON THE CENTRAL LIMIT THEOREM FOR THE
EIGENVALUE COUNTING FUNCTION OF WIGNER MATRICES
1
SANDRINEDALLAPORTAANDVANVU
1
0
2

n Abstract. ThepurposeofthisnoteistoestablishaCentralLimitTheorem
a for the number of eigenvalues of a Wigner matrix in an interval. The proof
J relies on the correct aymptotics of the variance of the eigenvalue counting

3 functionofGUEmatricesduetoGustavsson,anditsextensiontolargefamilies
1 of Wigner matrices by means of the Tao and VuFour Moment Theorem and
recentlocalizationresultsbyErd¨os,YauandYin.

]
R
P
.
h
t
a 1. Introduction
m
[
This note is concerned with the asymptotic behavior of the eigenvalue counting

1 function, that is the number N of eigenvalues falling in an interval I, of families
I
v
of Wigner matrices, when the size of the matrix goes to infinity. Wigner matrices
3
are random Hermitian matrices M of size n such that, for i < j, the real and
5 n
5 imaginaryparts of (Mn)ij are iid, with mean 0 and variance 12, (Mn)ii are iid with
2 mean0andvariance1. AnimportantexampleofWignermatricesisthecasewhere
1. the entries are Gaussian, giving rise to the so-called Gaussian Unitary Ensemble
0 (GUE). GUE matrices will be denoted by Mn′. In this case, the joint law of the
1 eigenvalues is known, allowing for complete descriptions of their limiting behavior
1
both in the global and local regimes (cf. for example [1]).
:
v
i Denote by λ ,...,λ the real eigenvalues of the normalized Wigner matrix
X 1 n
W = 1 M . The classical Wigner theorem states that the empirical distribution
r n √n n
a 1 n δ on the eigenvalues of W convergesweakly almost surely as n to
n j=1 λj n →∞
thPesemicirclelawdρ (x)= 1 √4 x21 (x)dx. Consequently,foranyinterval
I R, sc 2π − [−2,2]
⊂
n
1 1
N (W )= 1 ρ (I) almost surely.
n I n nXj=1 {λj∈I} n→→∞ sc
At the fluctuation level, it is known, due to the particular determinantal structure
of the GUE, that
Theorem 1 (Costin-Lebowitz [2], Soshnikov [7] (see [1])). Let M be a GUE ma-
n′
trix. Set W = 1 M . Let I be an interval in R. If Var(N (W )) ,
n′ √n n′ n In n′ n→ ∞
→∞
V.VuissupportedbyresearchgrantsfromAFORSandNSF.
1
2 SANDRINEDALLAPORTAANDVANVU
then
N (W ) E[N (W )]
(1) In n′ − In n′ (0,1)
Var(N (W )) n→ N
In n′ →∞
p
in distribution.
In 2005, Gustavsson [5] was able to fully describe for GUE matrices the as-
ymptotic behavior of the variance of the counting function N (W ) for intervals
I n′
I = [y,+ ) with y ( 2,2) strictly in the bulk of the semicircle law. He estab-
∞ ∈ −
lished that
logn 1
(2) E[NI(Wn′)]=nρsc(I)+O n and Var(NI(Wn′))= 2π2+o(1) logn.
(cid:16) (cid:17) (cid:16) (cid:17)
In particular therefore, if I =[y,+ ) with y ( 2,2),
∞ ∈ −
N (W ) E[N (W )]
(3) I n′ − I n′ (0,1).
Var(N (W )) n→ N
I n′ →∞
p
as well as
N (W ) nρ (I)
(4) I n′ − sc (0,1)
1 logn n→→∞N
2π2
q
(which we call below the CLT with numerics).
The purpose of this note is to extend these conclusions to non-GaussianWigner
matrices. The class of Wigner matrices covered by our results is described by the
following condition. Say that M satisfies condition (C0) if the real part ξ and the
n
imaginary part ξ˜of (M ) are independent and have an exponential decay: there
n ij
are two constants C and C such that
′
P(ξ tC) e t and P(ξ˜ tC) e t,
− −
| |≥ ≤ | |≥ ≤
for all t C .
′
≥
Say that two complex random variables ξ and ξ match to order k if
′
E Re(ξ)mIm(ξ)l =E Re(ξ )mIm(ξ )l
′ ′
(cid:2) (cid:3) (cid:2) (cid:3)
for all m,l 0 such that m+l k.
≥ ≤
The following theorem is the main result of this note.
Theorem2. LetM bearandomHermitianmatrixwhoseentriessatisfycondition
n
(C0) and match the corresponding entries of GUE up to order 4. Set W = 1 M .
n √n n
Then, for any y ( 2,2) and I(y)=[y,+ ), setting Y :=N (W ), we have
n I(y) n
∈ − ∞
1
E[Y ]=nρ (I(y))+o(1) and Var(Y )= +o(1) logn,
n sc n 2π2
(cid:16) (cid:17)
and the sequence (Y ) satisfies the CLT
n
Y E[Y ]
n n
− (0,1).
Var(Y ) n→ N
n →∞
p
A NOTE ON THE CLT FOR THE EIGENVALUE COUNTING FUNCTION 3
The theorem is established in the next two sections. In a first step, relying on
Gustavsson’s results and its extension to Wigner matrices by Tao and Vu [9], we
establish that (Y ) satisfies the CLT with numerics (4). In a second step, we use
n
recent results of Erdo¨s, Yau and Yin [3] on the localization of eigenvalues in order
to prove that E[Y ] and Var(Y ) are close to those of M (GUE) and therefore
n n n′
satisfy (2).
2. CLT with numerics and eigenvalues in the bulk
On the basis of the CLT with numerics, Gustavsson [5] described the Gaussian
behavior of eigenvalues in the bulk of the semicircle law in the form of
4 t(i/n)2λ (W ) t(i/n)
(5) − i n′ − (0,1)
r 2 √logn n→ N
n →∞
in distribution, where t(x) [ 2,2] is defined for x [0,1] by
∈ − ∈
t(x) 1 t(x)
x= dρ (t)= 4 x2dx.
sc
Z 2π Z −
2 2 p
− −
Moreinformally,λ (W ) t(i/n)+ (0, 2logn ). Thisisachievedbythetight
i n′ ≈ N (4 t(i/n)2)n2
relationbetweeneigenvaluesandthecounti−ngfunctionexpressedbytheelementary
equivalence, for I(y)=[y,+ ), y R,
∞ ∈
(6) N (W ) n i if and only if λ y.
I(y) n i
≤ − ≤
The result(5)wasextendedin[9]to largefamiliesofWignermatricessatisfying
condition (C0) by means of the Four Moment Theorem (see [9] and [10]). Now
using the reverse strategy based on (6), (5) may be shown to imply back the CLT
with numerics (4) for Wigner matrices whose entries match those of the GUE up
to order 4. We provide some details in this regard relying on the following simple
consequence of the Four Moment Theorem.
Proposition3. Let M and M betworandom matrices satisfying condition (C0)
n n′′
such that their entries match up to order 4. There exists c > 0 such that, if n is
large enough, for any y ( 2,2) and any (i,j) 1,...,n 2, if I(y)=[y,+ ),
∈ − ∈{ } ∞
P(λ I(y)) P(λ I(y)) n c,
i ∈ − ′i′ ∈ ≤ −
(cid:12) (cid:12)
and (cid:12) (cid:12)
P(λ I(y) λ I(y)) P(λ I(y) λ I(y)) n c.
i ∈ ∧ j ∈ − ′i′ ∈ ∧ ′j′ ∈ ≤ −
(cid:12) (cid:12)
(cid:12) (cid:12)
Asannounced,wewouldliketoshowthatthebehaviorofeigenvaluesinthebulk
(5)extendedtoWignermatricesleadstotheCLTwithnumericsforsuchmatrices,
namely,
N (W ) nρ (I(y))
I(y) n sc
(7) − (0,1),
1 logn n→→∞N
2π2
q
4 SANDRINEDALLAPORTAANDVANVU
indistributionforWignermatricesW satisfying(C0). Toprovethis,observethat
n
for every x R.
∈
N (W ) nρ (I(y))
P I(y) n − sc x =P(N (W ) n i )
I(y) n n
(cid:16) 1 logn ≤ (cid:17) ≤ −
2π2
q
where i =nρ (( ,y]) x 1 logn. Then, by (6),
n sc −∞ − q2π2
N (W ) nρ (I(y))
P I(y) n − sc x =P(λ (W ) y)
(cid:16) 1 logn ≤ (cid:17) in n ≤
2π2
q
4 t(i /n)2λ (W ) t(i /n)
=P − n in n − n x ,
(cid:16)r 2 √logn ≤ n(cid:17)
n
where xn = q4−t(i2n/n)2y−√t(lonignn/n). Now inn → ρsc((−∞,y]) ∈ (0,1). Furthermore,
x x since
n
→
x 1
t(i /n) = t ρ (( ,y]) logn
n (cid:18) sc −∞ − nr2π2 (cid:19)
x 1 √logn
= t ρ (( ,y]) t ρ (( ,y]) logn+o
(cid:16) sc −∞ (cid:17)− ′(cid:16) sc −∞ (cid:17)nr2π2 (cid:16) n (cid:17)
2 √logn √logn
= y x +o .
− r4 y2 n (cid:16) n (cid:17)
−
Hence y−√t(loignn/n) =x 4 2y2 +o(1), from which xn →x.
n q −
Applying (5) (extended to Wigner matrices), we obtain that
4 t(i /n)2λ (W ) t(i /n)
P − n in n − n x P(X x),
(cid:16)r 2 √logn ≤ n(cid:17)n→ ≤
n →∞
where X (0,1), implying (7).
∼N
3. Estimating the mean and the variance of Y
n
ToreachtheCLTofTheorem2fromtheCLTwithnumerics(7),itisnecessaryto
suitably controlthe expectation and variance E[Y ] and Var(Y ) of the eigenvalue
n n
counting function, and to show that their behaviors are identical to the ones for
GUE matrices. The direct use of the Four Moment Theorem is unfortunately
not enough to this purpose since it only provides proximity on a small number of
eigenvalues. ButrecentresultsofErdo¨s,YauandYin[3],presentedinthefollowing
statement,describestronglocalizationoftheeigenvaluesofWignermatriceswhich
provides the additional step necessary to complete the argument.
Theorem4. LetM bearandomHermitianmatrixwhoseentriessatisfycondition
n
(C0). There is a constant C >0 such that, for any i 1,...,n ,
∈{ }
P(λ t(i/n) (logn)Cloglognmin(i,n i+1) 1/3n 2/3) n 3.
i − − −
| − |≥ − ≤
A NOTE ON THE CLT FOR THE EIGENVALUE COUNTING FUNCTION 5
Note that if nε i (1 ε)n for some small ε>0, then min(i,n i+1) nε
≤ ≤ − − ≥
so that
(8) P λ t(i/n) n 1ε 1/3(logn)Cloglogn n 3.
i − − −
(cid:16)| − |≥ (cid:17)≤
The next lemma presents the main conclusion on the expectation and variance
of the eigenvalue counting function, extending Gustavsson’s conclusion (2) for the
GUEtoWignermatricesoftheclass(C0). Oncethislemmaisestablished,Theorem
2 will follow.
Lemma 5. Set W = 1 M , I(y) = [y,+ ) where y ( 2,2), and Y =
n √n n ∞ ∈ − n
N (W ). Then
I(y) n
1
E[Y ]=nρ (I(y))+o(1) and Var(Y )= +o(1) logn.
n sc n 2π2
(cid:16) (cid:17)
Proof. By Gustavsson’s results (2) therefore, if Y denotes N (W ) in the case
n′ I(y) n′
M is GUE,
n′
logn 1
E[Y ]=nρ (I(y))+O and Var(Y )= +o(1) logn.
n′ sc n n′ 2π2
(cid:16) (cid:17) (cid:16) (cid:17)
Hence, to establish Lemma 5, it suffices to show that E[Y ] = E[Y ]+o(1) and
n n′
Var(Y ) = Var(Y )+o(1). Below, we only deal with the variance, the argument
n n′
for the expectation being similar and actually simpler.
Set A =1 , for i 1,...,n . Notice that
i {λi∈I} ∈{ }
|Var(Yn)−Var(Yn′)|≤ |(E[AiAj]−E[Ai]E[Aj])−(E[A′iA′j]−E[A′i]E[A′j])|.
1 Xi,j n
≤ ≤
Call an index i first class if E[A ] 1 n 3 or n 3 and second class otherwise.
i − −
≥ − ≤
Note that if j is first class, then, for all i 1,...,n ,
∈{ }
E[A A ] E[A ]E[A ] =O(n 3).
i j i j −
| − |
Indeed, if E[A ] n 3, then both terms between the absolute value signs are
j −
≤
O(n 3), so that E[A A ] E[A ]E[A ] =O(n 3). The other case can be brought
− i j i j −
| − |
back to this case by the identity
E[A A ] E[A ]E[A ] = E[B B ] E[B ]E[B ],
i j i j i j i j
| − | | − |
where B is the complement of A . Consequently,
i i
(9) E[A A ] E[A ]E[A ] E[A A ] E[A ]E[A ] =O(n 1).
| i j − i j − ′i ′j − ′i ′j | −
iXor j (cid:0) (cid:1) (cid:0) (cid:1)
1st class
Theorem 4 shows that there are only O((logn)Cloglogn) second class indices.
Indeed, set η = n 1ε 1/3(logn)Cloglogn and suppose first that i 1,...,n is
n − −
∈ { }
such that t(i/n)<y η :
n
−
6 SANDRINEDALLAPORTAANDVANVU
if t(i/n)>t(ε), (8) is true for W . Then
n
•
P(λ I ) P λ t(i/n) η n 3.
i n i n −
∈ ≤ (cid:16)| − |≥ (cid:17)≤
if t(i/n) < t(ε), choose j such that t(ε) < t(j/n) < y η (take ε small
n
• −
enough and n large enoughsuch that there is such a j). Then λ λ and
i j
≤
P(λ I )=P(λ y) P(λ y)=P(λ I ) n 3.
i n i j j n −
∈ ≥ ≤ ≥ ∈ ≤
Similarly one can show that if i 1,...,n is such that t(i/n)>y+η , then i is
n
∈{ }
first class.
As a consequence of this discussion, i 1,...,n can only be second class if
∈ { }
y η < t(i/n) < y +η . We need to count these possible i’s. By definition of
n n
−
t(i/n), i= n t(i/n)√4 x2dx. Thus,
2π 2 −
R−
n y−ηn n y+ηn
4 x2dx i 4 x2dx.
2π Z − ≤ ≤ 2π Z −
2 p 2 p
− −
In this case i belongs to an interval of length
n y+ηn 2
4 x2dx (logn)Cloglogn.
2π Z − ≤ πε1/3
y−ηn p
Therefore,thereareatmost 2 (logn)Cloglogn+1=O((logn)Cloglogn)second
πε1/3
class i’s.
Next, by Proposition3, it is easily seen that if both i,j are second class, then
(E[A A ] E[A ]E[A ]) (E[A A ] E[A ]E[A ]) =O(n c)
| i j − i j − ′i ′j − ′i ′j | −
forsomepositiveconstantc. Since thenumberofsuchpairsisO((logn)2Cloglogn),
we have
(10)
| E[AiAj]−E[Ai]E[Aj] − E[A′iA′j]−E[A′i]E[A′j] |=O(n−c(logn)2Cloglogn).
iXand j(cid:0) (cid:1) (cid:0) (cid:1)
2nd class
To conclude,
Var(Y ) Var(Y ) E[A A ] E[A ]E[A ] E[A A ] E[A ]E[A ]
| n − n′ |≤ | i j − i j − ′i ′j − ′i ′j |
iXor j (cid:0) (cid:1) (cid:0) (cid:1)
1st class
+ E[A A ] E[A ]E[A ] E[A A ] E[A ]E[A ] ,
| i j − i j − ′i ′j − ′i ′j |
iXand j(cid:0) (cid:1) (cid:0) (cid:1)
2nd class
so that (9) and (10) lead to
Var(Y ) Var(Y ) O(n 1)+O(n c(logn)2Cloglogn)=o(1),
| n − n′ |≤ − −
as claimed. This shows that Var(Y ) = 1 +o(1) logn. As mentioned earlier,
n 2π2
(cid:16) (cid:17)
it may be shown similarly that E[Y ]=nρ (I(y))+o(1) and the proof of Lemma
n sc
5 is thus complete. (cid:3)
A NOTE ON THE CLT FOR THE EIGENVALUE COUNTING FUNCTION 7
4. About real Wigner matrices
In this section, we briefly indicate how the preceding results for Hermitian ran-
dommatrices may be stated similarly for realWigner symmetric matrices. To this
task, we follow the same scheme of proof, relying in particular on the corollary of
Tao and Vu Four Moment Theorem (Proposition 3) which also holds in the real
case (cf. [6]).
RealWignermatricesarerandomsymmetricmatricesM ofsizensuchthat,for
n
i<j, (M ) are iid, with mean 0 and variance 1, (M ) are iid with mean 0 and
n ij n ii
variance 2. As in the complex case, an important example of real Wigner matrices
is the case where the entries are Gaussian, giving rise to the so-called Gaussian
OrthogonalEnsemble (GOE).
The main issue is actually to establish first the conclusions for the GOE. This
has been suitably developed by O’Rourke in [6] by means of interlacing formulas
(cf. [4]).
Theorem 6 (Forrester-Rains). The following relation holds betweenmatrix ensem-
bles:
GUE =even(GOE GOE ).
n n n+1
∪
This statement can be interpreted in the following way. Take two independent
matricesfromtheGOE,oneofsizenandtheotherofsizen+1. Ifwesurimperpose
the2n+1eigenvaluesonthe reallineandthentakethe nevenones,they havethe
same distribution as the eigenvalues of a n n matrix from the GUE.
×
Let I be anintervalin R. Let MR be a GOEmatrix andMC be a GUE matrix.
n n
WR andWC arethe correspondingnormalizedmatrices. Theprecedinginterlacing
n n
formula leads to
E[N (WR)]=E[N (WC)]+O(1)
• Var(IN (nWR))=2VI ar(nN (WC))+O(1), if Var(N (WC)) .
• I n I n I n n→ ∞
→∞
Relying on this result and on the GUE case, O’Rourke proved the following
theorem:
Theorem 7. Let MR be a GOE matrix. Set WR = 1 MR. Then, for any y
n n √n n ∈
( 2,2) and I(y)=[y,+ ), setting YR :=N (WR), we have
− ∞ n I(y) n
1
E[YR]=nρ (I(y))+O(1) and Var(YR)= +o(1) logn,
n sc n π2
(cid:16) (cid:17)
and the sequence (YR) satisfies the CLT
n
YR E[YR]
n − n (0,1).
Var(YR) n→ N
n →∞
p
Following exactly the same scheme as for complex Wigner matrices leads to the
sameconclusion: Theorem7 is true for Wigner symmetricmatrices,providedtheir
8 SANDRINEDALLAPORTAANDVANVU
entriesmatchthe correspondingentriesofGOEuptoorder4andsatisfycondition
(C0).
The CLT for the eigenvalue counting function has been investigated as well for
familiesofcovariancematrices. Themainconclusionofthisworkholdssimilarlyin
thiscaseconditionedhoweveronthevalidityoftheErdo¨s-Yau-Yinrigiditytheorem
for covariance matrices. There is strong indication that the current approach by
Erdo¨s, Yin and Yau for Wigner matrices will indeed yield such a result. All the
other ingredients of the proof are besides available. Indeed, Su (cf. [8]) carried
out computations for Gaussian covariance matrices and proved both the CLT and
the correct asymptotics for mean and variance. Tao and Vu in [11] extended their
Four MomentTheoremto suchmatrices. Arguing then as for Wigner matrices, we
couldreachinthesamewayaCLTwithnumericsforsuitablefamiliesofcovariance
matrices.
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InstitutdeMath´ematiquesdeToulouse,UMR5219duCNRS,Universit´edeToulouse,
F-31062 Toulouse,France
E-mail address: [email protected]
Departmentof Mathematics,Rutgers, Piscataway,NJ 08854
E-mail address: [email protected]