Mathematical Physics, Analysis and Geometry - Volume 4

390 Pages·2001·2.63 MB·English

by  A. Boutet de Monvel, V. A. Marchenko, H. McKean (Editors)

Mathematical Physics, Analysis and Geometry 4: 1–36, 2001.
1
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
On the Law of Multiplication of Random Matrices
VLADIMIR VASILCHUK
Université Paris 7 Denis Diderot, Mathématiques, case 7012, Paris, France,
Institute for Low Temperature Physics, 47 Lenin ave., 310164 Kharkov, Ukraine
(Received: 22 February 2001; in revised form: 2 April 2001)
Abstract. We recover Voiculescu’s results on multiplicative free convolutions of probability mea-
sures by different techniques which were already developed by Pastur and Vasilchuk for the law of
addition of random matrices. Namely, we study the normalized eigenvalue counting measure of the
product of two n × n unitary matrices and the measure of the product of three n × n Hermitian
(or real symmetric) positive matrices rotated independently by random unitary (or orthogonal) Haar
distributed matrices. We establish the convergence in probability as n → ∞ to a limiting nonrandom
measure and obtain functional equations for the Herglotz and Stieltjes transforms of that limiting
measure.
Mathematics Subject Classifications (2000): 15A52, 60B99, 60F05.
Key words: random matrices, counting measure, limit laws.
1. Introduction
This paper deals with the eigenvalue distribution of products of n×n random matri-
ces. We consider two models: (i) the product of two unitary (resp. orthogonal) ma-
trices and (ii) the product of three Hermitian (or real symmetric) positive matrices.
We study the eigenvalue distribution of these two ensembles in the limit n → ∞.
Namely, we express the limiting normalized counting measure of eigenvalues of the
product via the limits of the same counting measures of the corresponding factors.
We assume that these exist and that factors are randomly rotated one with respect
to another by a unitary (or orthogonal) random matrix uniformly distributed over
the group U(n) (resp. O(n)).
In this paper, under weaker assumptions, we obtain the analog of Voiculescu’s
results concerning the free multiplicative convolutions of probability measures on
the real axis and unit circle. They were studied within the context of free (non-
commutative) probability theory introduced by Voiculescu at the beginning of the
90’s (see [2, 8, 10] for results and references). This theory deals with free random
variables (operators in von Neumann algebras) that can be modeled by unitary (or-
thogonal) invariant random matrices [5, 6]. The notion of S-transform introduced
in this theory allows one to generalize the functional equations for transforms of
limiting counting measures of certain multiplicative unitary invariant models first
proposed and studied by Marchenko and Pastur [4].
2 VLADIMIR VASILCHUK
Motivation for studying multiplicative random matrix ensembles is given by the
fact that they appear in some physics studies (see, e.g., [3]).
We use a simple method of deriving the functional equations for limiting eigen-
value distributions. It is a natural extension of the method proposed in [5] to study
an additive analog of our ensembles. The basic idea is the same as in [4]: to
study not the counting measure itself but rather some integral transforms that are
generating functions of the moments of that measure. We derive functional rela-
tions for these transforms using the resolvent identity and differential identities for
expectations of smooth functions with respect to the Haar measure of U(n) (or
O(n)).
The paper is organized as follows. In Section 2, we state and discuss the main
results (Theorem 2.2 for unitary matrices and Theorem 2.1 for Hermitian posi-
tive matrices). In Section 3, we prove auxiliary Theorems 3.1 and 3.2 concerning
Hermitian positive definite matrices under the conditions of uniform in n bound-
edness of the fourth moment of the normalized counting measure of the factors. In
Section 4, we use these results to prove Theorem 2.1. Here the main condition
is the uniform boundedness of the second moment of the normalized counting
measure of the factors. In Section 5 we prove Theorems 5.1 and 5.2 and then prove
Theorem 2.2 giving the solution of the problem for unitary matrices. In Section 6
we prove the auxiliary facts that we need. We also describe generalizations of our
results in the cases of orthogonal and real symmetric matrices.
2. Models and Main Results
We study two ensembles of n × n random matrices Vn and Hn of the form:
Vn = V1,nV2,n, (2.1)
where
∗ ∗
V1,n = W
n SnWn, V2,n = Un TnUn
and
1/2 1/2
Hn = H
1,n H2,nH1,n , (2.2)
where
∗ ∗
H1,n = W
n AnWn, H2,n = Un BnUn.
We assume that Sn, Tn, An, Bn, Un and Wn are mutually independent. Sn and Tn
are random unitary matrices, Un and Wn are unitary (resp. orthogonal) random ma-
trices uniformly distributed over the unitary (orthogonal) group U(n) (resp. O(n))
with respect to the Haar measure. An and Bn are Hermitian positive definite random
matrices.
ON THE LAW OF MULTIPLICATION OF RANDOM MATRICES 3
We will restrict ourselves to the case of Hermitian matrices (resp. to the group
U(n)). The results for symmetric real matrices (i.e. for the group O(n)) are similar,
although their proofs are more difficult (see Section 6).
We are interested in the asymptotic behavior, as n → ∞, of the normalized
eigenvalue counting measure (NCM) νn of the ensemble (2.1), whose value on any
Borel set ⊂ [0, 2π] is given by
(n)
#{µ ∈ }
i
νn() = , (2.3)
n
(n)
where µ
i are the eigenvalues of Vn. We are also interested in the asymptotic
behavior of the NCM Nn of the ensemble (2.2), whose value on any Borel set
⊂ R is given by
(n)
#{λ ∈ }
i
Nn() = , (2.4)
n
(n)
where λ
i are the eigenvalues of Hn.
The problem was studied recently by Voiculescu [2, 8, 10] within the context of
free (noncommutative) probability. Combining Voiculescu’s results on free multi-
plicative convolution of measures having nonzero first moment [2, 10] with results
of asymptotic freeness of n × n Haar distributed unitary matrices and nonrandom
diagonal matrices [6, 9, 10], one can easily obtain the following result:
PROPOSITION. If the matrices Sn, Tn, An and Bn are nonrandom, if the norms of
An and Bn are uniformly bounded in n, i.e. their NCMs N1,n and N2,n have compact
support uniformly in n, and if these measures have weak limits as n → ∞
ν1,n → ν1, ν2,n → ν2, (2.5)
N1,n → N1, N2,n → N2, (2.6)
where ν1,n and ν2,n are the NCM’s of Sn and Tn, then the NCM’s (2.3) and (2.4)
of (2.1) and (2.2) converge weakly with probability 1 to nonrandom measures ν
and N.
Here and below the convergence with probability 1 is understood as that in the
natural probability spaces
∏ ∏
= n, ˜ = ˜ n, (2.7)
n n
where n is the probability space of matrices (2.1), that is the product of respective
˜
spaces of Sn and Tn and two copies of the group U(n) for Un and Wn, and where n
is the probability space of matrices (2.2), that is the product of respective spaces of
An and Bn and two copies of the group U(n) for Un and Wn.
Besides, according to [10], one can define the S-transforms S1, S2 and S of
˜ ˜ ˜
the measures ν1, ν2 and ν, respectively, and the S-transforms S1, S2 and S of the
4 VLADIMIR VASILCHUK
measures N1, N2 and N (see our remark after Theorem 2.2) and one can find the
˜ ˜
following simple expressions S and S via S1,2 and S1,2
˜ ˜ ˜
S = S1S2, S = S1S2.
The proof of the asymptotic freeness of n×n Haar distributed unitary matrices and
nonrandom diagonal matrices having, uniformly in n, compactly supported NCMs
in [6, 8], is based on the asymptotic analysis of the expectations of normalized
traces of mixed products of matrices Sn, Tn, Un and Wn and An, Bn, Un and Wn,
respectively. It requires a considerable amount of combinatorial analysis, the exis-
tence of all moments measures Nr , r = 1, 2, and their rather regular behavior as
n → ∞ to obtain the convergence of expectations.
In this paper we obtain analogous results under more weak assumptions by a
method that does not involve combinatorics. This is because we work with the
Stieltjes transforms of the measures (2.4) and (2.6) and with the Herglotz trans-
forms of the measures (2.3) and (2.5). We directly derive functional equations for
their limits by using simple identities for expectations of matrix-valued functions
with respect to the Haar measure (Proposition 3.2 below) and elementary facts
concerning resolvents of Hermitian and unitary matrices. This method was already
used in [5] for the additive random ensembles.
We list below the properties of the Stieltjes and Herglotz transforms that we will
need below (see, e.g., [1]).
PROPOSITION 2.1. Let
∫
m(dλ)
s(z) = , Im z ≠ 0 (2.8)
R λ − z
be the Stieltjes transform of a probability measure m on R, then
(i) s(z) is analytic in C \ R and
−1
|s(z)| ⩽ |Im z| . (2.9)
(ii) Im s(z)Im z > 0, Im z ≠ 0. (2.10)
(iii) lim y |s(iy)| = 1. (2.11)
y→∞
(iv) For any continuous function ϕ with compact support we have the Frobenius–
Perron inversion formula
∫ ∫
1
φ(λ)m(dλ) = lim φ(λ)Im s(λ + iε). (2.12)
R ε→0 π R
(v) Conversely, any function satisfying (2.9)–(2.11) is the Stieltjes transform of
a probability measure and this one-to-one correspondence between measures
and their Stieltjes transforms is continuous for the topology of weak conver-
gence for measures and for the topology of convergence on compact subsets
of C \ R for the Stieltjes transforms.
ON THE LAW OF MULTIPLICATION OF RANDOM MATRICES 5
PROPOSITION 2.2. Let
∫
2π iθ
e + z
t(z) = µ(dθ), |z| < 1 (2.13)
iθ
0 e − z
be the Herglotz transform of a probability measure µ on [0, 2π], then
(i) t(z) is analytic for |z| < 1 and
−1
|t(z) − 1| ⩽ 2|z|(1 − |z|) , |z| < 1. (2.14)
(ii) t (0) = 1, Re t(z) > 0, |z| < 1. (2.15)
(iii) For any continuous on [0, 2π] function ϕ we have the inversion formula
∫ ∫
2π 2π
1
−iθ
φ(θ)µ(dθ) = lim φ(θ)Re t(r e ) dθ. (2.16)
0 r→1− 2π 0
(iv) Conversely, any function satisfying (2.14)–(2.15) is the Herglotz transform
of a probability measure on the unit circle. This one-to-one correspondence
between measures and their Herglotz transforms is continuous for the topol-
ogy of weak convergence for measures and for the topology of convergence
on compact subsets of {z ∈ C | |z| < 1} for the Herglotz transforms.
Now we state our main results. Since the set of eigenvalues of unitary and
Hermitian matrices are unitary invariant, we can replace matrices (2.1) and (2.2)
by
∗
Vn = SnU
n TnUn (2.17)
and
1/2 ∗ 1/2
Hn = A
n Un BnUnAn , (2.18)
where Sn, Tn, An, Bn and Un are as in (2.1) and (2.2). However, it is useful to keep
in mind that the problem is symmetric in Sn and Tn and in An and Bn (as we will
see below).
THEOREM 2.1. Let Hn be a positive definite random n × n matrix of the form
(2.2). Assume that the normalized counting measures N1,n, N2,n of An, Bn converge
weakly in probability as n → ∞ to nonrandom probability measures N1, N2. We
also assume
∫
+∞
2 (2)
sup λ E{Nr,n(dλ)} ⩽ m < ∞, r = 1, 2, (2.19)
n 0
and
∫
+∞
mr = λNr(dλ) > 0, r = 1, 2, (2.20)
0
6 VLADIMIR VASILCHUK
i.e. the measures N1, N2 are not concentrated at zero. Then the normalized count-
ing measure Nn of Hn converge in probability to a nonrandom probability measure
whose Stieltjes transform
∫
+∞
N(dλ)
f (z) = , Im z ≠ 0 (2.21)
0 λ − z
is the unique solution of the system
f (z)(1 + zf (z)) = 1(z) 2(z),
( )
z 2(z)
1(z) = f2 , (2.22)
1 + zf (z)
( )
z 1(z)
2(z) = f1
1 + zf (z)
in the class of functions f (z), 1,2(z) which are analytic for Im z ≠ 0 and which
satisfy (2.9)–(2.11) and
−1
z r(z) = −mr + O(|Im z| ), r = 1, 2,
|Re z| ⩽ |Im z|, z → ∞. (2.23)
f1(z), f2(z) are the Stieltjes transforms of N1, N2 and E{ · } denotes the expectation
with respect to the probability measure generated by An, Bn, Un and Wn.
THEOREM 2.2. Let Vn be a random n × n matrix of the form (2.1). Assume that
the normalized counting measures ν1,n, ν2,n of Sn, Tn converge weakly in probabil-
ity as n → ∞ to nonrandom probability measures on the unit circle ν1, ν2. Then
the normalized counting measure νn of Vn converge in probability to a nonrandom
probability measure ν whose Herglotz transform
∫
2π iµ
e + z
h(z) = ν(dµ), |z| < 1 (2.24)
iµ
0 e − z
is the unique solution of the system
2
h (z) = 1 + 4z 1(z) 2(z),
( )
2z 2(z)
h(z) = h2 , (2.25)
1 + h(z)
( )
2z 1(z)
h(z) = h1
1 + h(z)
in the class of functions h(z), 1,2(z) which are analytic for |z| < 1 and which
satisfy (2.14)–(2.15) and
−1
| 1,2(z)| ⩽ (1 − |z|) , |z| < 1. (2.26)
h1,2(z) are the Herglotz transforms of the measures ν1,2.
ON THE LAW OF MULTIPLICATION OF RANDOM MATRICES 7
Both theorems will be proved in Sections 3 and 5. Here we interpret them in
∗
terms of S-transform introduced by Voiculescu in the context of C -algebras.
2.1. VOICULESCU’S FORMULATION
Consider a probability measure µ on the unit circle and assume that its first moment
is nonzero
∫
2π
iθ
µ1 = e µ(dθ) ≠ 0.
0
Consider the function
−1
1 + t(z )
ϕµ(z) = −
2
′
where t(z) is the Herglotz transform of µ. Since ϕ
µ(z) = µ1 + o(1), z → ∞,
then, according to the local inversion theorem, there exists a unique inverse func-
tion χµ(ϕ) of ϕµ(z), χµ(ϕµ(z)) = z defined and analytic in a neighborhood of
−1 and assuming its values in a neighborhood of infinity. On the other hand, for
any probability measure m on the real nonnegative semi-axis having nonzero first
moment
∫
∞
m1 = λm(dλ) > 0,
0
we can consider the function
−1 −1
ϕm(z) = −(1 + z s(z )),
′
where s(z) is the Stieltjes transform of the measure m. Since ϕ
m(0) = m1, then,
according to the local inversion theorem, ϕm(z) also has a unique inverse function
χm(ϕ) defined and analytic in a neighborhood of zero and assuming its values in a
neighborhood of zero. Denote
−1
Sµ(ϕ) = χµ(ϕ)ϕ (1 + ϕ),
−1
Sm(ϕ) = χm(ϕ)ϕ (1 + ϕ)
and, following Voiculescu [10], call Sµ(ϕ) the S-transform of the probability mea-
sure µ on the unit circle and Sm(ϕ) the S-transform of the probability measure m
on the real nonnegative semi-axis. By using the S-transforms S1, S2 of ν1, ν2, we
can rewrite (2.25) in the form
1 + ψ(z) 1 + ψ(z)
S(ψ(z)) = , (2.27)
z 1(z) z 2(z)
1 + ψ(z)
Sr(ψ(z)) = − , r = 1, 2, (2.28)
z r(z)
8 VLADIMIR VASILCHUK
−1
where ψ(z) = ϕ(z ) = −(1 + h(z))/2 and S(ψ) denotes the S-transform of
the limiting normalized counting measure ν of (2.1), whose Herglotz transform is
h(z). Then we derive from (2.27) Voiculescu’s very simple expression of S via S1
and S2
S(ψ) = S1(ψ)S2(ψ). (2.29)
It is easy to check that (2.22) leads to the relations (2.27) for ψ(z) = −(1+ zf (z))
and S(ψ) denotes the S-transform of the limiting normalized counting measure N
of (2.2), whose Stieltjes transform is f (z). Thus, (2.22) leads to the same expres-
sion (2.29) where S1,2 will be the S-transforms of the measures N1,2. The relation
∗
(2.29) was obtained by Voiculescu in the context of C -algebra studies (see [9, 10]
for results and references).
3. Convergence with Probability 1 for Nonrandom An, Bn
In this section, we start the proof of Theorem 2.1. As a first step we prove the
following theorem:
THEOREM 3.1. Let Hn be a positive definite nonrandom n×n matrix of the form
(2.2) in which An and Bn are nonrandom Hermitian positive matrices, Un and
Wn are random independent unitary matrices distributed according to the Haar
measure on U(n). Assume that the normalized counting measures N1,n, N2,n of An
and Bn converge weakly as n → ∞ to nonrandom probability measures N1, N2,
∫ ∫
+∞ +∞
lim λNr,n(dλ) = mr = λNr(dλ) > 0, r = 1, 2, (3.1)
n→∞
0 0
∫
+∞
4
sup λ Nr(dλ) ⩽ m4 < ∞. (3.2)
n 0
Then the normalized counting measure Nn of Hn converges with probability 1 to
a nonrandom probability measure N whose Stieltjes transform f (z) (2.21) is the
unique solution of (2.22) in the class of functions f (z), 1,2(z) which are analytic
for z ∈ C \ R+ and which satisfy (2.9)–(2.11) and (2.23).
We use the technique introduced in [5]. Let us recall its basic means. First we
collect elementary facts of linear algebra.
n
PROPOSITION 3.1. LetMn be the algebra of linear endomorphisms of C equipp-
n
ed with the norm induced by the standard Euclidean norm of C . Then
n n
(i) If {Mjk}
j,k=1 is the matrix of M ∈ Mn in any orthonormal basis of C then
|Mjk| ⩽ ||M||. (3.3)
ON THE LAW OF MULTIPLICATION OF RANDOM MATRICES 9
(ii) If Tr M denotes the trace of M ∈ Mn, then
|Tr M| ⩽ n||M||,
2 ∗ ∗
|Tr M1M2| ⩽ Tr M1M
1 Tr M2M2 , (3.4)
∗
where M is the adjoint of M. Furthermore, if P ∈ Mn is positive definite,
then
|Tr MP | ⩽ ||M||Tr P. (3.5)
(iii) Let
−1
G(z) = (M − z) (3.6)
be the resolvent of M ∈ Mn. It is defined for all nonreal z, Im z ≠ 0 if M is
Hermitian. It is defined for all z, |z| ≠ 1 if M is unitary.
(iv) If G(z1) and G(z2) are defined,
G(z1) − G(z2) = (z1 − z2)G(z1)G(z2). (3.7)
(v) If M ∈ Mn is Hermitian and Im z ≠ 0, then
−1
||G(z)|| ⩽ |Im z| . (3.8)
n
(vi) If M ∈ Mn is invertible and if {Gjk(z)}
j,k=1 is the matrix of its resolvent
G(z) in any orthonormal basis, then
−1
||M ||
|Gjk(z)| ⩽ . (3.9)
−1
1 − |z| ||M ||
(vii) If M1, M2 ∈ Mn their resolvents G1(z), G2(z) satisfy the “resolvent iden-
tity”
G2(z) = G1(z) − G1(z)(M2 − M1)G2(z). (3.10)
′ −1
(viii) The differential G (z) of the resolvent G(z) = (M − z) viewed as function
of M satisfies
′
G (z) · X = −G(z)XG(z) (3.11)
for any X ∈ Mn. In particular,
′ 2 −2
||G (z)|| ⩽ ||G(z)|| ⩽ |Im z| . (3.12)
Now we present the main technical tool.
1
PROPOSITION 3.2 ([5]). Let :Mn → C be of class C . Then, for any M ∈ Mn
and any Hermitian X ∈ Mn:
∫
′ ∗ ∗
(U MU) · [X, U MU] dU = 0, (3.13)
U(n)
where [M1, M2] denotes the commutator M1M2 − M1M2 and the integral denotes
integration over U(n) with respect to the Haar measure.
10 VLADIMIR VASILCHUK
Proof. cf. [5], Proposition 3.2. The integral
∫
iεX ∗ −iεX ∗
( e U MUe )[X, U MU] dU = 0
U(n)
does not depend on ε. The derivative at ε = 0 gives (3.13). ✷
PROPOSITION 3.3. The system (2.22) has a unique solution in the class of func-
tions f (z), 1,2(z) which are analytic for Im z ≠ 0 and which satisfy (2.9)–(2.11)
and (2.23).
′ ′ ′′ ′′
Proof. Assume that there exist two solutions (f , ) and (f , ) of (2.22).
1,2 1,2
′ ′′ ′ ′′
Denote δf = f −f , δ 1,2 =
1,2− 1,2. Then, by using (2.22) and the following
relations
∫
+∞
−1 −1 λNr(dλ)
fr(z) = −z + z , r = 1, 2
0 λ − z
we obtain a linear system for δφ = zδf , δ 1, δ 2:
(1 + a1(z))δφ − b1(z)δ 1 − c1(z)δ 2 = 0,
(1 + a2(z))δφ − b2(z)δ 2 = 0, (3.14)
(1 + a3(z))δφ − b3(z)δ 1 = 0,
where
′ ′′ ′′ ′
a1 = zf + zf , b1 = z
2, c1 = z 1, (3.15)
′ ′ ′′ ′ ′′ ′,′′
z
2J2(s2, s2 ) J2(s2, s2 ) ′,′′ z 2
a2 =
′ ′′ , b2 = z ′ , s2 = ′,′′ ,
(1 + zf )(1 + zf ) 1 + zf 1 + zf
∫
+∞
′ ′′ λN2(dλ)
J2(z , z ) = (3.16)
′ ′′
0 (λ − z )(λ − z )
′ ′′
and a3, b3 can be obtained from a2 and b2 by replacing N2 and
2 by N1 and 1
in the above formulas.
For any y0 > 0, consider the domain
E(y0) = {z ∈ C | |Im z| ⩾ y0, |Re z| ⩽ |Im z|}. (3.17)
Due to condition (2.23) and the first equation of the system (2.22), we have for
z ∈ E(y0)
′,′′ −1 −1
1 + zf (z) = −z (m1m2 + O(|Im z| )), z → ∞. (3.18)
Besides, if
∫
+∞
λm(dλ)
′ ′′
t(z , z ) =
′ ′′
0 (λ − z )(λ − z )