in PROBABILITY
THE SPECTRAL LAWS OF HERMITIAN BLOCK-MATRICES
WITH LARGE RANDOM BLOCKS
TAMER ORABY
Department of Mathematical Sciences, University of Cincinnati, 2855 Campus Way, P.O. Box 210025, Cincinnati, OH 45221-0025, USA.
email: orabyt@math.uc.ed
Submitted October 2, 2007, accepted in final form November 29, 2007 AMS 2000 Subject classification: 60F05
Keywords: Random Matrices, Matrices, Limiting Spectral Measure, Circulant Block-Matrix, Semicircular Noncommutative Random Variables.
Abstract
We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix withk−weakly dependent entries need not to be the semicircle law in the limit.
1
Preliminaries and main results
Let Mn(C) be the space of all n×nmatrices with complex-valued entries. Define the nor-malized trace of a matrixA= (Aij)ni,j=1∈ Mn(C) to be trn(A) := n1Pni=1Aii.
Definition 1. The spectral measure of a Hermitiann×nmatrixAis the probability measure µA given by
µA=
1 n
n
X
j=1
δλj
whereλ1≤λ2≤ · · · ≤λn are the eigenvalues ofAandδxis the point mass atx.
The weak limit of the spectral measures µAn of a sequence of matrices {An} is called the limiting spectral measure. We will denote the weak convergence of a probability measure µn toµby
µn−→D µas n→ ∞.
Definition 2. A finite symmetric block-structureBk(a, b, c, . . .) (or shortlyBk) over a finite alphabetK={a, b, c, . . .} is ak×ksymmetric matrix whose entries are elements in K.
IfBkis ak×ksymmetric block-structure andA,B,C, . . . aren×nHermitian matrices, then
Bk(A,B,C, . . .) is an nk×nk Hermitian matrix. One of the interesting block structures is thek×ksymmetric Circulant over{a1, a2, . . . , ak} that is defined as
Ck(a1, a2, . . . , ak) =
1 √ k
a1 a2 a3 . . . ak
ak a1 a2 . . . ak−1
ak−1 ak a1 . . . ak−2
..
. ... ... . .. ... a2 a3 a4 . . . a1
(1)
where aj =ak−j+2 forj= 2,3, . . . , k.
A random matrixAis a matrix whose entries are random variables. IfBk is a block-structure
andA,B,C, . . . are random matrices, thenBk(A,B,C, . . .) is a random block-matrix. Definition 3. We call ann×nHermitian random matrixA= √1
n(Xij) n
i,j=1a Wigner matrix
if {Xij; 1 ≤ i < j} is a family of independent and identically distributed complex random variables such that E(X12) = 0 and E(|X12|2) = σ2. In addition, {Xii;i ≥ 1} is a family
of independent and identically distributed real random variables that is independent of the upper-diagonal entries. We will denote all such Wigner matrices of ordernbyWigner(n, σ2).
If{An}is a sequence of Wigner(n, σ2) matrices, then by Wigner’s Theorem (cf. [2]),
µAn−→D γ0,σ2 asn→ ∞ a.s.
where γα,σ2 is the semicircle law centered at αand of varianceσ2 which is given as
γα,σ2(dx) = 1
2πσ2
p
4σ2−(x−α)2 1
[α−2σ,α+2σ](x)dx.
Now we are ready to state the main result of this paper.
Theorem 1 (Existence Theorem). Consider a family of independent Wigner(n,1) ma-trices ³{A(ni)}:i= 1, . . . , h
´
for whichE(|A(12i)|4)<∞ andE(A(i)
11)2 <∞ for every i. For a
fixed k×k symmetric block-structureBk, define
Xn,k:=Bk(A(1)n ,A(2)n , . . . ,A(nh)).
Then there exists a unique non-random symmetric probability measure µBk with a compact support in Rwhich depends only on the block-structureBk such that
µXn,k −→D µBk as n→ ∞ a.s.
Consider the symmetric Circulant block-matrixCk defined in (1). IfA(1)n ,A(2)n , . . . , A(⌊
k
2⌋+1)
n are independentWigner(n,1) for everyn, then Theorem 1 insures the existence of a non-random probability measureνk such that
µ
However, Theorem 1 doesn’t specifyνk but we will identify it in the following proposition. Proposition 1. IfA(1)n ,A(2)n , . . . ,A(⌊
k
2⌋+1)
n are independentWigner(n,1) for everyn, then µ the proof of the odd case.
In [2, p.626], Bai raised the question of whether Wigner’s theorem is still holding true when the independence condition in the Wigner matrix is weakened. Schenker and Schulz-Baldes [11] provided an affirmative answer under some dependency assumptions in which the number of correlated entries doesn’t grow too fast and the number of dependent rows is finite. After the first draft of the underlying paper was completed, we have learnt that Anderson and Zeitouni [1] showed that it doesn’t hold in general and they gave an example in which the limiting spectral distribution is the free multiplicative convolution of the semicircle law and shifted arcsine law. In the rest of this section, we are going to use the following corollary of Proposition 1 to give another example.
LetW(a11, a12, . . . , ann) be the Wigner symmetric block-structure,i.e.,
W(a11, a12, . . . , ann) =
a11 a12 . . . a1n a12 a22 . . . a2n
..
. ... . .. ... a1n a2n . . . ann
.
Consider the family of k×k random matrices {Aij : i, j ≥ 1} such that Aij = Aji and Aij = Ck(aij, bij, cij, . . .) where {aij, bij, cij, . . . : i, j ≥ 1} are independent and identically distributed random variables with variance one. Then Kn,k := W(A11,A12, . . . ,Ann) is an
kn×knsymmetric matrix.
Corollary 1. Fixk∈N. The limiting spectral measure ofKn,k is given by µKn,k
D
−→νk asn→ ∞ a.s.
In order to prove this corollary we need the following definitions. LetAandBben×mand k×ℓmatrices, respectively. By⊗ we mean here the Kronecker product for whichA⊗B= (AijB)i=1,...,n;j=1,...,mis annk×mℓmatrix. The (p, q)-commutation matrixPp,q is apq×pq matrix defined as
Pp,q= p
X
i=1
q
X
j=1
Eij⊗ETij
whereEij is thep×qmatrix whose entries are zero’s except the (i, j)−entry is 1. It is known that P−p,q1=PTp,q =Pq,p andPn,k(A⊗B)Pℓ,m=B⊗A(cf. [7]).
Proof. SinceKn,k =Pni,j=1Eeij⊗Aij whereEeij is then×nmatrix whose entries are zero’s except the (i, j)−entry is 1. Hence
Pk,nKn,kPn,k = Pni,j=1Aij⊗Eeij
= Pni,j=1Ck(aij, bij, cij, . . .)⊗Eeij = Pni,j=1Ck(aijEeij, bijEeij, cijEeij, . . .)
= Ck(Pni,j=1aijEeij,Pni,j=1bijEeij,Pni,j=1cijEeij, . . .) = Ck(An,Bn,Cn, . . .)
where An = (aij)ni,j=1, Bn = (bij)ni,j=1, Cn = (cij)ni,j=1, . . . are independent Wigner(n,1)
Now, we define the distance onN2byd((i, j),(i′, j′)) =max{|i−i′|,|j−j′|}and forS, T⊂N2;
d(S, T) =min{d((i, j),(i′, j′)) : (i, j)∈S,(i′, j′)∈T}. We say the random field{Xij : (i, j)∈
N2≤}is (k−1)-dependent if theσ-fieldsFS =σ({Xij : (i, j)∈S}) andFT =σ({Xij : (i, j)∈ T}) are independent for allS, T ⊂N2
≤ such thatd(S, T)> k−1.
The matrix Kn,k = W(A11,A12, . . . ,Ann), defined in Corollary 1, is an kn×kn matrix
with (k−1)-dependent entries, up to symmetry. That is, if we writeKn,k = (Xij)nk
i,j=1, then
{Xij : (i, j)∈N2
≤}is a (k−1)-dependent random field. However, the limiting spectral measure
ofKn,kis not the semicircle law but rather a mixture of two semicircle laws due to Corollary 1.
Our example violates the conditions imposed on the Wigner matrix by Schenker and Schulz-Baldes in [11] in both the number of correlated entries and the number of dependent rows grow asO(n2) and noto(n2).
Unfortunately,{Xij : (i, j)∈N2
≤}, in our example, is not strictly stationary as the distributions
remain the same only when shifts are made by multiple ofk.
2
Proofs
In order to prove Theorem 1 we need to introduce some definitions from free probability theory. A noncommutative probability space (A, τ) is a pair of a unital algebraAwith a unit element
Iand a linear functional τ, called the state, for whichτ(I) = 1. We call an elementa∈ Aa noncommutative random variable and callτ(an) itsnthmoment. We say thatAis a *-algebra if the involution * is defined onA. In addition, we assume thatτ(a∗) =τ(a) andτ(a∗a)≥0.
Henceforth, we will consider only *-algebras. We say thata∈ Ais selfadjoint ifa∗=a.
Fix a noncommutative probability space (A, τ). For each selfadjoint a ∈ A there exists a probability measureµaonRsuch that
τ(an) =
Z
R xnµ
a(dx)
for all n ≥ 1, see [9, p.2]. The probability measureµa is unique if|τ(an)| ≤ Mn for some
M >0 and for alln≥1.
Definition 4 ([8]). A family of subalgebras (Aj;j ∈J) of A, which contain I, is said to be free with respect toτ if for everyk≥1 andj16=j26=. . .6=jk∈J⊂N
τ(a1a2· · ·ak) = 0
for allai∈ Aji wheneverτ(ai) = 0 for every 1≤i≤k.
Random variables in a noncommutative probability space (A, τ) are said to be free if the subalgebras generated by them andIare free.
Definition 5. We say that a family of sequences of random matrices ({A(nl)};l = 1, . . . , m) is almost surely asymptotically free (cf. [8]) if for every noncommutative polynomial pin m variables
trn³p(A(1)
n , . . . ,A(nm))
´ n→∞
−−−−→τ(p(a1, . . . ,am)) a.s.
where (a1, . . . ,am) is a family of free noncommutative random variables in some
In [3], Capitaine and Donati-Martin showed the asymptotic freeness for independent Wigner matrices when the distribution of the entries is symmetric and satisfies Poincar´e inequality. Recently, Guionnet [6] gave a proof where she assumes that all the moments of the entries exist. Szarek [12] showed us a proof for symmetric and non-symmetric matrices with uniformly bounded entries. Szarek’s proof, in brief, is based on concentration inequalities and some tools of operator theory. In this paper, we give a combinatorial proof of the almost sure asymptotic freeness for Wigner matrices under the assumption of finite variance and fourth moment of the entries.
Theorem 2. If ({An(l)};l= 1, . . . , m) is a family of independentWigner(n,1) matrices for which E(|A(12l)|4)<∞ andE(A
(l)
11)2 <∞, then ({A (l)
n };l= 1, . . . , m) is almost surely asymp-totically free.
2.1
Proof of Theorem 2
The Schatten p-norm of a matrixAis defined as kAkp := (trn|A|p)1p whenever 1≤p <∞, where |A| = (ATA)1
2. The operator norm is defined as kAk := max1
≤i≤n|λi| where λi; i= 1,2, . . . , nare the eigenvalues ofA. The following three inequalities hold true;
1. Domination inequality [8, p.154]
|trn(A)| ≤ kAk1≤ kAkp≤ kAk (5) 2. H¨older’s inequality [8, p.154]
kABkr≤ kAkpkBkq (6) whenever 1
r =
1
p+
1
q forp, q >1 andr≥1. 3. Generalized H¨older’s inequality
kA(1)A(2)· · ·A(m)k1≤ kA(1)kp1kA
(2)
kp2· · · kA
(m)
kpm (7) whereA(1),A(2), . . . ,A(m)aren×nmatrices andPmi=1 1
pi = 1. This inequality follows from (6) by induction.
Let A = √1
n(Xij) n
i,j=1 be a Wigner(n,1) matrix. We define Ae(c) = √1n(Xije (c)) n
i,j=1 to be
the matrix whose off-diagonal entries are those ofAtruncated byc/√nand standardized. We will also assume that the diagonal entries of A(c) are equal to zero. In other words,e
e
Xij(c) =
½ 1
σ(c)
£
Xij1(|Xij|≤c)−E(Xij1(|Xij|≤c))¤, fori < j;
0, fori=j
where 1(|Xij|≤c)is equal to one if|Xij| ≤c and zero otherwise; and
σ2(c) =E|Xij1(|Xij|≤c)−E(Xij1(|Xij|≤c))|2≤1.
We would choose sufficiently large c so thatσ(c)>0 andXije (c) would be well defined. Note that σ2(c)→1 asc→ ∞and Var(X
121(|X12|>c))≤1−σ
2(c).
Lemma 1. Let³{A(nl)}:l= 1, . . . , m
´
be a family of independent sequences ofWigner(n,1) matrices for which E(|A(12l)|4)<∞and E(A
Proof. First, we can write the difference between products of matrices as a telescopic sum, i.e., two inequalities are due to the generalized H¨older’s inequality (7).
It is known that ifE(|X12(l)|4)<∞andE(X
almost surely (cf. [2, Theorem 2.12]). Similarly, we can see that lim
n→∞kAe
(l)
n (c)k= 2 almost surely for eachl. By the domination inequality (5)
By definition, of Large Numbers (SLLN). Once more theSLLN implies that
lim almost surely. Consequently, for allc >0
lim sup
is a family of independent sequences ofWigner(n,1) matrices whose entries are bounded, then
lim n→∞E
³
trn³Ae(1)n Ae(2)n · · ·Aen(m)´´=τ(a1a2· · ·am) (8)
where ai’s are some free noncommutative random variables in (A, τ) such that ai has the semicircle law γ0,1 for alli.
We say that a partition π={B1, . . . , Bp} of a set of integers is non-crossing ifa < b < c < d
partitions of {1, . . . , k} by NC(k). Also let NC2(k) be the family of all non-crossing pair
partitions which is empty unlesskis even. The Catalan number
Ck= 1
If (al;l= 1, . . . m) is a family of free semicircular random variables which have mean zero and variance one, then (cf. [10, Equation (8)])
τ(ai1ai2· · ·aik) =
is a family of independent sequences of Wigner(n,1) matrices whose entries are bounded, then
∞
Proof. It is enough to show that
Var
Var
Ã
trn
Ãm
Y
i=1
e
A(nli)
!!
≤ nC2.
Concusion of the proof of Theorem 2. Any noncommutative polynomialpcan be written as a linear combination of noncommutative monomials,i.e.,
p³A(1)n , . . . ,An(m)´=Xui1,...,ikA
(i1)
n A(ni2)· · ·A(nik)
where the sum runs overi1, . . . , ik∈ {1, . . . , m}fork≥1 andui1,...,ik ∈Care constants. Note that phas a finite number of terms. Therefore,
|trnp³A(1)n ,An(2), . . . ,A(nm)´−trnp(Ae(1)n (c),Ae(2)n (c), . . . ,Ae(nm)(c))| ≤X|ui1,...,ik| |trn
³
A(i1)
n A(ni2)· · ·A(nik)
´
−trn(Ae(i1)
n (c)Ae(ni2)(c)· · ·Ae(nik)(c))| Hence, givenǫ >0
lim sup n→∞ |trnp
³
A(1)n ,A(2)n , . . . ,A(nm)´−trnp(Ae(1)n (c),Ae(2)n (c), . . . ,Ae(nm)(c))| ≤ǫX|ui1,...,ik|
(10)
almost surely, for sufficiently largec. Due to Lemma 2
lim n→∞E
³
trn(Ae(i1)
n (c)Ae(ni2)(c)· · ·Ae(nik)(c))
´
=τ(ai1ai2· · ·aik) (11)
for all i1, . . . , ik ∈ {1, . . . , m} and k ≥ 1 where ai’s are some free noncommutative random
variables in some noncommutative probability space (A, τ) such thataihas the semicircle law γ0,1 for alli. Lemma 3 implies that the limit in (11) is holding true in the almost sure sense
due to Borel-Cantelli lemma. Consequently, for everyc >0 lim
n→∞trnp(Ae
(1)
n (c),Ae(2)n (c), . . . ,Aen(m)(c)) =τ(p(a1,a2, . . . ,am)) (12)
almost surely, since τ is a linear functional. Equation(10) implies that
lim sup n→∞ |trnp
³
A(1)n ,A(2)n , . . . ,An(m)´−τ(p(a1,a2, . . . ,am))|
≤lim sup n→∞ |
trnp³A(1)n ,A(2)n , . . . ,A(nm)´−trnp(Ae(1)n (c),Ae(2)n (c), . . . ,Ae(nm)(c))| + lim sup
n→∞ |
trnp(Ae(1)n (c),Ae(2)n (c), . . . ,Aen(m)(c))−τ(p(a1,a2, . . . ,am))|
≤ǫ1
(13)
2.2
Proof of Theorem 1
Fixk≥1 and a symmetric block-structureBk. Let us introduce the noncommutative proba-bility space (ANMk(C), τNtrk), whereNstands for the tensor product. A typical element in ANMk(C) is ak×k matrix whose entries are noncommutative random variables in A. For example, Bk(a1, . . . ,ah)∈ ANMk(C) for any a1, . . . ,ah ∈ A. The state τNtrk is de-fined by τNtrk(A) = 1
k
Pk
i=1τ(Aii) for any A ∈ A
N
Mk(C). The involution is given by the *-transpose,i.e., for anyA= (aij)i,j=1,...,k∈ ANMk(C) the involution ofAis given by (a∗
ij)Ti,j=1,...,k.
The proof of Theorem 1 is based on the method of moments. First, we are going to show that for everys∈N, the limit of trnk³Bk³A(1)n , . . . ,A(nh)
´s´
exists asn→ ∞, almost surely. Fix s ≥1. We can see that the trace for the s-power of Xn,k =Bk
³
A(1)n , . . . ,A(nh)
´
is the trace of some noncommutative polynomials in the matricesA(1)n , . . . ,A(nh). In other words,
trnk¡Xsn,k¢= 1 k
k
X
i=1
trn³pi³A(1)n , . . . ,A(nh)´´
for some noncommutative polynomialpi and 1≤i≤k. Theorem 2 implies that for eachi trn³pi³A(1)n , . . . ,A(nh)´´→τ(pi(a1, . . . ,ah)) as n→ ∞ a.s.
where {al :l= 1, . . . , m} is a family of free noncommutative random variables that have the semicircle law with variance equals to one. Therefore
trnk³Bk³A(1)n , . . . ,A(nh)´ s´
→ 1 kτ
à k
X
i=1
pi(a1, . . . ,ah)
!
as n→ ∞ a.s.
Thus,
trnk¡Xsn,k¢→τOtrk(Bk(a1, . . . ,ah)s) asn→ ∞ a.s.
SinceBk(a1, . . . ,ah) is self-adjoint, then there exists a probability measureµBk such that τOtrk(Bk(a1, . . . ,ah)s) =
Z
R xsµB
k(dx).
Note that ifs is an odd integer thenτNtrk(Bk(a1, . . . ,ah)s) is zero by Equation (9). This
implies thatµBk is a symmetric probability measure.
To complete the proof, we need to prove thatµBk is unique and has a compact support in R. Both follows by showing that there existM >0 andC >0 such thatτNtrk³Bk(a1, . . . ,ah)2s´≤
C M2s for alls≥1. But for a fixeds≥1 τOtrk³Bk(a1, . . . ,ah)2s
´
= X
J(2s,k)
τ(Bj1j2Bj2j3· · ·Bj2sj1)
where Bij ∈ {a1, . . . ,ah} and J(m, k) := {(j1, . . . , jm) : 1 ≤j1, . . . , jm ≤k}. But again by
Equation (9), X
J(2s,k)
τ(Bj1j2Bj2j3· · ·Bj2sj1)≤k
where the Catalan numberCsis at most 4s.
Therefore, there exists a unique non-random symmetric probability measureµBk with a com-pact support in R that has the moments τNtrk(Bk(a1, . . . ,ah)s), for every s ≥ 1, such
that
µXn,k−→D µBk asn→ ∞ a.s.
Acknowledgements
I would like to thank my advisor professor Bryc for his suggestions and his continuous and sincere help and support. I am also grateful to professor Szarek for references pertinent to Theorem 2 and for [12].
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