On a Limiting Distribution of Singular Values of Random Band Matrices
An equation is obtained for the Stieltjes transform of the normalized distribution of singular values of non-symmetric band random matrices in the limit when the band width and rank of the matrix simultaneously tend to infinity. Conditions under which this limit agrees with the quarter-circle law ar...
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irk-123456789-1198832017-06-11T03:03:48Z On a Limiting Distribution of Singular Values of Random Band Matrices Lytova, A. Pastur, L. An equation is obtained for the Stieltjes transform of the normalized distribution of singular values of non-symmetric band random matrices in the limit when the band width and rank of the matrix simultaneously tend to infinity. Conditions under which this limit agrees with the quarter-circle law are found. An interesting particular case of lower triangular random matrices is also considered and certain properties of the corresponding limiting singular value distribution are given. Получено уравнение для преобразования Стилтьеса нормированного распределения сингулярных чисел несимметричных случайных бэнд-матриц в пределе, когда ширина полосы и размерность матриц одновременно стремятся к бесконечности. Найдены условия, при которых этот предел согласуется с четверть-круговым законом. Также рассмотрен интересный частный случай нижне-треугольных случайных матриц и приведены определенные свойства соответствующего предельного распределения сингулярных чисел. 2015 Article On a Limiting Distribution of Singular Values of Random Band Matrices / A. Lytova, L. Pastur // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 4. — С. 311-332. — Бібліогр.: 20 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag11.04.311 MSC2000: 60B10, 60B20 (primary); 15B52, 15A18 (secondary). http://dspace.nbuv.gov.ua/handle/123456789/119883 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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An equation is obtained for the Stieltjes transform of the normalized distribution of singular values of non-symmetric band random matrices in the limit when the band width and rank of the matrix simultaneously tend to infinity. Conditions under which this limit agrees with the quarter-circle law are found. An interesting particular case of lower triangular random matrices is also considered and certain properties of the corresponding limiting singular value distribution are given. |
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Lytova, A. Pastur, L. |
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Lytova, A. Pastur, L. On a Limiting Distribution of Singular Values of Random Band Matrices Журнал математической физики, анализа, геометрии |
| author_facet |
Lytova, A. Pastur, L. |
| author_sort |
Lytova, A. |
| title |
On a Limiting Distribution of Singular Values of Random Band Matrices |
| title_short |
On a Limiting Distribution of Singular Values of Random Band Matrices |
| title_full |
On a Limiting Distribution of Singular Values of Random Band Matrices |
| title_fullStr |
On a Limiting Distribution of Singular Values of Random Band Matrices |
| title_full_unstemmed |
On a Limiting Distribution of Singular Values of Random Band Matrices |
| title_sort |
on a limiting distribution of singular values of random band matrices |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/119883 |
| citation_txt |
On a Limiting Distribution of Singular Values of Random Band Matrices / A. Lytova, L. Pastur // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 4. — С. 311-332. — Бібліогр.: 20 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT lytovaa onalimitingdistributionofsingularvaluesofrandombandmatrices AT pasturl onalimitingdistributionofsingularvaluesofrandombandmatrices |
| first_indexed |
2025-07-08T16:50:29Z |
| last_indexed |
2025-07-08T16:50:29Z |
| _version_ |
1837098284849561600 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2015, vol. 11, No. 4, pp. 311–332
On a Limiting Distribution of Singular Values
of Random Band Matrices
A. Lytova
Department of Mathematical and Statistical Sciences, University of Alberta
Edmonton, Alberta, Canada, T6G 2G1
E-mail: lytova@ualberta.ca
L. Pastur
B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv 61103, Ukraine
E-mail: pastur@ilt.kharkov.ua
Received April 23, 2015
An equation is obtained for the Stieltjes transform of the normalized dis-
tribution of singular values of non-symmetric band random matrices in the
limit when the band width and rank of the matrix simultaneously tend to
infinity. Conditions under which this limit agrees with the quarter-circle law
are found. An interesting particular case of lower triangular random matri-
ces is also considered and certain properties of the corresponding limiting
singular value distribution are given.
Key words: random band matrices, triangular matrices, limiting distri-
bution of singular values.
Mathematics Subject Classification 2010: 60B10, 60B20 (primary); 15B52,
15A18 (secondary).
1. Introduction: Problem and Main Results
Given a positive integer n = 2m+1, m ∈ N, consider an n×n random matrix
An = {A(n)
jk }|j|,|k|≤m, A
(n)
jk = b−1/2
n v
(
(j − k)/bn
)
a
(n)
jk , (1.1)
where {a(n)
jk }|j|,|k|≤m are real random variables, {bn} is a sequence of positive
integers such that
lim
n→∞ bn = ∞, ν := lim
n→∞ νn ∈ [1,∞], νn = n/2bn, (1.2)
c© A. Lytova and L. Pastur, 2015
A. Lytova and L. Pastur
v : R→ R is a piecewise continuous function of compact support, and we denote
ν∫
−ν
v2(t)dt = w2 < ∞, (1.3)
max
t∈R
v2(t) = K < ∞. (1.4)
In particular, if v = χ[0,1], the indicator of the interval [0, 1], then the matrix
elements {A(n)
jk }|j|,|k|≤m are non-vanishing only in the ”band” of the width bn
under the principal diagonal. If in addition 2ν = 1, then An is a lower triangular
matrix asymptotically.
We are interested in the limiting distribution of the squares of singular values
of An, i.e., the eigenvalues
0 ≤ λ
(n)
1 ≤ ... ≤ λ(n)
n < ∞ (1.5)
of the positive definite random matrix
Mn = AnAT
n . (1.6)
To this end, we introduce the Normalized Counting Measure Nn of (1.5), setting
for any interval ∆ ⊂ R
Nn(∆) = Card{l ∈ [1, n] : λ
(n)
l ∈ ∆}/n. (1.7)
It is convenient to write the matrix Mn in the form
Mn =
∑
|k|≤m
yk ⊗ yk, (1.8)
where
yk = (A(n)
−mk, ..., A
(n)
mk)
T (1.9)
are the columns of An (see (1.1)). According to (1.1), each yk corresponds to the
vector
ak = (a(n)
−mk, ..., a
(n)
mk)
T . (1.10)
We will assume that {ak}|k|≤m are jointly independent random vectors, however
the components of each vector ak can be dependent. Here are the corresponding
definitions [11, 18].
Definition 1.1. (i). [Isotropic vectors] A random vector a = (a−m, . . . , am)∈
Rn is called isotropic if
E{aj} = 0, E{ajak} = δjk, |j|, |k| ≤ m. (1.11)
312 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
(ii). [Unconditional distribution] The distribution of random vector a ∈
Rn is called unconditional if its components (a−m, . . . , am) have the same joint
distribution as (±a−m, . . . ,±am) for any choice of signs.
(iii). [Log-concave measure] A measure µ on Cn is log-concave if for any
measurable subsets A,B of Cn and any θ ∈ [0, 1],
µ(θA + (1− θ)B) ≥ µ(A)θµ(B)(1−θ)
whenever θA + (1− θ)B = {θX1 + (1− θ)X2 : X1 ∈ A, X2 ∈ B} is measurable.
Definition 1.2. [Good vectors] We say that a random vector a = (a(n)
−m, . . . ,
a
(n)
m ) ∈ Rn is good if it is an isotropic vector with an unconditional distribution
satisfying the moment conditions
m
(n)
2,2 := E{(a(n)
j )2(a(n)
k )2} = 1 + o(1), j 6= k, m
(n)
4 := E{(a(n)
j )4} = O(1)
(1.12)
as n = 2m + 1 →∞. Note that m
(n)
2,2 and m
(n)
4 do not depend on j and k.
A simple example of good vectors are the vectors with i.i.d. n-independent
components of zero mean and unit variance. Another important case is given
by the isotropic random vectors with symmetric unconditional and log-concave
distributions (see Lemma 2.1 of [11]), the simplest among them are the vectors
uniformly distributed on the unit sphere in Rn. Now we are ready to formulate
our main results.
Theorem 1.3. Let Mn, n = 2m + 1, m ∈ N, be the random matrix (1.8)–
(1.10), where for every m {ak}|k|≤m are jointly independent good vectors (see De-
finition 1.2) and corresponding vectors {yk}|k|≤m are defined in (1.9) and (1.1)–
(1.4). Let Nn be the Normalized Counting Measure (1.7) of eigenvalues of Mn.
Then there exists a non-random and non-negative measure N , N(R) = 1, such
that for any interval ∆ ⊂ R we have in probability
lim
n→∞Nn(∆) = N(∆). (1.13)
The limiting measure N is uniquely defined via its Stieltjes transform f (see
[1, 19])
f(z) =
∞∫
0
N(dλ)
λ− z
, =z 6= 0, (1.14)
by the formula
∫
R
ϕ(λ)N(dλ) = lim
ε→0+
1
π
∫
R
ϕ(λ)=f(λ + iε)dλ, ∀ϕ ∈ C0(R), (1.15)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 313
A. Lytova and L. Pastur
and we have:
(i) if ν < ∞ in (1.2), then
f(z) =
1
2ν
ν∫
−ν
f(t, z)dt, (1.16)
where f : [−ν, ν]× C \ [0,∞) → C on the right of the formula is continuous in t
for every z ∈ C \ [0,∞), is analytic in z for every |t| ≤ ν, has the property
=f(t, z)=z ≥ 0, |f(t, z)| ≤ |=z|−1, =z 6= 0, (1.17)
and is the unique solution of the equation
f(t, z) = −
(
z −
ν∫
−ν
v2(t− τ)dτ
1 +
∫ ν
−ν v2(θ − τ)f(θ, z)dθ
)−1
; (1.18)
(ii) if ν = ∞ in (1.2), then f of (1.14) is the unique solution of the quadratic
equation
zw2f2 + zf + 1 = 0 (1.19)
in the class of analytic in C \ R functions satisfying
=f(z)=z ≥ 0, |f(z)| ≤ |=z|−1, =z 6= 0, (1.20)
and we have the following formula for the density ρqc of the limiting measure N :
N(dλ) = ρqc(λ)dλ, ρqc(λ) = (2πw2)−1
√
(4w2 − λ)/λ1[0,4w2], (1.21)
known as the quarter-circle law.
Theorem 1.4. The results of Theorem 1.3 remain valid if {ak}|k|≤m are in-
dependent isotropic random vectors with independent components having finite
absolute moment of the order 2 + ε, ε > 0,
sup
n
max
|j|,|k|≤m
E{|a(n)
kj |2+ε} < ∞. (1.22)
Corollary 1.5. Under conditions of Theorem 1.3 (or Theorem 1.4) with
ν < ∞ in (1.2), N is the quarter-circle law if and only if the function v2 :
[−2ν, 2ν] → R+ is the restriction on the interval [−2ν, 2ν] of a 2ν-periodic func-
tion.
314 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
In particular, if all entries {A(n)
jk }|j|,|k|≤m are non-vanishing, then we get the
quarter-circle law, and this fact was proved long time ago [16] (see also [18, 19]).
To prove the corollary, we note first that if ν < ∞ and v2 is 2ν-periodic, then
(1.18) has a t-independent solution f satisfying (1.19). If v2 does not possess this
property, then the function
u(t) ≡
ν∫
−ν
v2(t− τ)dτ
cannot be a constant on the interval (−ν, ν). Hence, expanding the solution f of
(1.18) in the inverse powers of z,
f(z) = −1
z
− a1
z2
− a2
z3
+ O(z−4), a1 =
1
2ν
ν∫
−ν
u(t)dt, a2 =
1
2ν
ν∫
−ν
u2(t)dt,
and then applying the Schwarz inequality, we get the strict inequality
a2
1 < a2/2
if u is not identically constant. On the other hand, we have for fqc of (1.21)
fqc(z) =
1
2w2
(−1 +
√
(1− 4w2/z)) = −1
z
− w2
z2
− 2w4
z3
+ O(z−4), z →∞,
so that a2
1 = a2/2. Therefore, in the considered case of a non-periodic v2, the
limiting Normalized Counting Measure N of (1.13) cannot be the quarter-circle
law.
Note that the results of Theorem 1.3 and Corollary 1.5 agree with those
obtained in [4] and [17] for the Wigner band matrices. Indeed, according to
[4] and [17], the Stieltjes transform of the Wigner matrices (i.e., matrices with
independent, modulo symmetry conditions, entries satisfying an analog of (1.22))
is given by the same formula (1.16), where now f(t, z) solves uniquely the equation
(cf. (1.18))
f(t, z) = −
(
z +
ν∫
−ν
v2(t− τ)f(z, τ)dτ
)−1
in the same class of functions defined by (1.17).
Another corollary of Theorem 1.3 yields the limiting distribution of singular
values of lower triangular random matrices.
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 315
A. Lytova and L. Pastur
Theorem 1.6. Consider the case of Theorem 1.3, where v = χ[0,1] is the
indicator of the interval [0, 1] and 2ν = 1, so that the matrices An are lower
triangular. Then:
(i) the Stieltjes transform f (1.14) of the limiting Normalized Counting Mea-
sure (1.13) of eigenvalues of Mn (1.6) solves uniquely the equation
(1 + f(z)) ln(1 + f(z)) = −z−1 (1.23)
in the class of functions analytic in C \ [0,∞) and satisfying (1.20);
(ii)
supp N = [0, e]; (1.24)
(iii) the measure N of (1.13) is absolutely continuous and its density ρ has
the following asymptotics at the endpoints of its support [0, e]:
ρ(λ) =
1
λ(lnλ)2
(1 + o(1)), λ ↓ 0, (1.25)
ρ(λ) = const · (e− λ)1/2(1 + o(1)), λ ↑ e;
(iv) the moments of N , i.e., µk := limn→∞ n−1TrMk
n , are
µk =
kk
(k + 1)!
, k ≥ 1. (1.26)
R e m a r k s. (1). The lower edge λ = 0 of the support is a hard edge in the
random matrix terminology. The typical (or standard) soft edge asymptotic of
the density of the limiting Normalized Counting Measure near the hard edge is
ρ(λ) = const ·λ−1/2, λ ↓ 0+ [19]. The asymptotics (1.25) for the lower triangular
matrices seems the most singular among the known so far. It follows from the
results of [6] that for the matrices M
(q)
n = (An)q((An)q)T the soft edge asymptotic
of the corresponding density is ρ(λ) = Cq(λ(ln 1/λ)q+1)−1(1 + o(1)), λ ↓ 0. It is
interesting that for q = 1, 2, this asymptotic formula coincides (up to a constant
factor) with the asymptotic formula near the center of the spectrum for certain
random Jacobi matrices studied in [7, 10, 13]. This seems somewhat surprising,
since the respective matrices are quite different: the i.i.d. entries of matrices An,
hence the entries of (An)2((An)2)T , are of the same order of magnitude roughly
while the Jacobi matrices in question have non-zero i.i.d. entries only on the
diagonals adjacent to the principal one.
(2). It is of interest that if we replace the lower triangular matrix An by
An + yIn, where y > 0 and In is n × n unit matrix, then it can be shown that
the support of the corresponding limiting distribution is [a (y), a+(y)] for any
316 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
y > 0, a (y) > 0, a+(y) < ∞, but a−(y) → 0+, a+(y) → e as y → 0 and the
both edges of support are soft.
(3). Formula (1.26) was found in [6] by combining the operator and the free
probability methods. Our proof is based on the techniques of random matrix
theory.
To conclude the section, we note that the results of Theorem 1.3 for ν < ∞,
in particular those for the triangular random matrices, generalize in part various
results of works [5, 6, 9, 12] obtained for matrices with independent entries by
various methods. In Section 3, we outline the proof of Theorem 1.4 treating the
case of independent entries under condition (1.22), applicable for both finite and
infinite ν and based on the scheme developed in [19] (and recently applied in
particular in [14]), to find the limiting eigenvalue distribution of a wide variety
of random matrices.
It is also worth mentioning that in this paper we confine ourselves to the
asymptotic behavior of the Normalized Counting Measure (1.7) of eigenvalues of
matrices (1.1)–(1.4), i.e., with an analog of the Law of Large Numbers for the
Counting Measure
Nn =
n∑
l=1
δ
λ
(n)
l
of eigenvalues of (1.1)–(1.4). A natural question then is on the asymptotics of
fluctuations of Nn and, more generally, of linear eigenvalue statistics
Nn[ϕ] =
n∑
l=1
ϕ(λ(n)
l )
for a sufficiently wide class of ϕ : R → C, i.e., on an analog of the Central
Limit Theorem for the appropriately normalized Nn[ϕ] − E{Nn[ϕ]}. We leave
this interesting question for a future work basing on the combinations of the
techniques developed in [11, 15, 20] and below.
2. Proof of Theorem 1.3
Recall that if m is a non-negative measure of unit mass and
s(z) =
∞∫
−∞
m(dλ)
λ− z
, =z 6= 0, (2.1)
is its Stieltjes transform, then this correspondence is one-to-one, provided that
=s(z)=z > 0, lim
n→∞ η|s(iη)| = 1. (2.2)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 317
A. Lytova and L. Pastur
Moreover, the correspondence is continuous if we use the uniform convergence of
analytic functions on a compact set of C\R for Stieltjes transforms and the weak
convergence of probability measures (see, e.g., [1, 19]).
Let
gn(z) =
∞∫
−∞
Nn(dλ)
λ− z
(2.3)
be the Stieltjes transform of Nn of (1.7).
By using the representation (1.8) and repeating almost literally the proof of
Theorem 19.1.6 of [19], we obtain the bound
P{|Nn(∆)−E{Nn(∆)}| > ε} ≤ C(ε)/n (2.4)
for any ∆ ⊂ R, where C(ε) is independent of n and is finite if ε > 0. The bound,
the above one-to-one correspondence between the measures and their Stieltjes
transforms and the analyticity of gn of (2.3) in C \ R reduce the proof of the
theorem to that of the limiting relation
lim
n→∞E{gn(z)} = f(z) (2.5)
uniformly on the set
CK0 = {z : <z = 0, |=z| ≥ K0 > 0}, (2.6)
where K0 is large enough (see (2.35) and (2.51)).
It follows from (1.7) and the spectral theorem for real symmetric matrices
that
gn(z) =
1
n
TrG(z) =
1
n
∑
|j|≤m
Gjj(z), (2.7)
where
G(z) = (Mn − z)−1
is the resolvent of Mn. We have
|Gjj | ≤ |=z|−1,
∑
k
|Gjk|2 ≤ |=z|−2. (2.8)
Here and in what follows we use the notation
∑
j
=
∑
|j|≤m
.
We have by the resolvent identity
Gjj = −1
z
+
1
z
∑
k
(yk⊗ykG)jj . (2.9)
318 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
Let us introduce the matrix
Mk
n =
∑
l 6=k
yl⊗yl (2.10)
and its resolvent
Gk(z) = (Mk
n − zIn)−1, =z 6= 0. (2.11)
It follows from the rank-one perturbation formula
G−Gk = − Gkyk⊗ykG
k
1 + (Gkyk,yk)
(2.12)
that
(yk⊗ykG)jj =
(Gkyk)jykj
1 + (Gkyk,yk)
≡ Bkn
Akn
, (2.13)
and we obtain from (2.9)
E{Gjj} = −1
z
+
1
z
∑
k
E
{Bkn
Akn
}
. (2.14)
The moment conditions (1.11) and the fact that Gk does not depend on yk allow
us to write
Ek{Akn} = 1 + b−1
n
∑
p
v2
pkG
k
pp, Ek{Bkn} = b−1
n v2
jkG
k
jj , (2.15)
where
vjk = v((j − k)/bn)
and Ek denotes the expectation with respect to yk. It follows from (2.15) and
the identity
1
A
=
1
E{A} −
1
E{A}
A◦
A
, A◦ = A−E{A}, (2.16)
that
E{Gjj} = −1
z
+
1
zbn
∑
k
v2
jkE{Gk
jj}
1 + b−1
n
∑
p v2
pkE{Gk
pp}
+ rn, (2.17)
where
rn = −1
z
∑
k
1
E{Akn}E
{(Akn)◦Bkn
Akn
}
. (2.18)
Let us show that
rn = o(1), n →∞. (2.19)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 319
A. Lytova and L. Pastur
By the spectral theorem for the real symmetric matrices, there exists a non-
negative measure mk such that
(Gkyk,yk) =
∞∫
0
mk(dλ)
λ− z
,
and we can write
=(z(Gkyk,yk)) = =
∞∫
0
λmk(dλ)
λ− z
= =z
∞∫
0
λmk(dλ)
|λ− z|2 .
Thus =z =(z(Gkyk,yk)) ≥ 0 and
|Akn|−1 ≤
∣∣∣ z
=z + =(z(Gkyk,yk))
∣∣∣ ≤ |z||=z|−1 = 1, z ∈ CK0 , (2.20)
implying the bounds
|Ek{Akn}|−1 ≤ 1, |E{Akn}|−1 ≤ 1, z ∈ CK0 . (2.21)
This and the Schwarz inequality allow us to write for rn of (2.18):
|rn| ≤ 1
|=z|
∑
k
E{|(Akn)◦|2}1/2E{|Bkn|2}1/2. (2.22)
It follows then from (1.12), (2.8) and the bounds (see (1.4))
max
s,k
v2
sk ≤ K, max
k
1
bn
∑
s
v2
sk ≤ w2, (2.23)
valid for sufficiently large n, that
E{|Bk|2} =
v2
jk
b2
n
∑
s,t
E{Gk
jsG
k
jtvskvtkEk{a(n)
sk a
(n)
tk a
(n)2
jk }} ≤ Cv2
jk
|=z|2b2
n
, (2.24)
where C is an absolute constant. Now (2.19) follows from (2.22) – (2.24) and
(2.44).
Let us show that we can replace Gk by G in (2.17) with the error of the order
O(b−1
n ). Indeed, we have from (2.12)
Gps −Gk
ps = −(Gkyk)p(Gkyk)s
Akn
. (2.25)
320 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
Applying (2.25), (2.21) and then (1.11), (2.8) and (2.23), we get
Ek{|Gpp −Gk
pp|} ≤ Ek{|(Gkyk)p|2} =
1
bn
∑
s,t
Gk
psG
k
ptvskvtkEk{a(n)
sk a
(n)
tk }
=
1
bn
∑
s
|Gk
ps|2v2
sk ≤
K
bn|=z|2 . (2.26)
Now it follows from (2.17), in which Gk
jj is replaced by Gjj , (2.19) and (2.26)
that
E{Gjj} = −1
z
+
1
zbn
∑
k
v2
jkE{Gjj}
1 + b−1
n
∑
p v2
pkE{Gpp}
+ rnj , n →∞,
where we denote by rnj any reminder satisfying
sup
z∈CK0
max
j
|rnj | → 0, n →∞.
Hence, we have
E{Gjj} =
( 1
bn
∑
k
v2
jk
1 + b−1
n
∑
p v2
pkE{Gpp}
− z
)−1
(1 + rnj). (2.27)
Using (2.20) ant the fact that =Gpp(z)=z ≥ 0, =z 6= 0, it is easy to show that
the denominators in (2.27) do not vanish.
Fix z and n and introduce the piece-wise constant function
fn(t, z) =
{
0, t /∈ [(−m− 1)/bn; m/bn]
E{Gjj(z)}, t ∈ ((j − 1)/bn; j/bn], |j| ≤ m.
(2.28)
We have from (2.7)
E{gn(z)} =
bn
n
1
bn
∑
j
fn(j/bn, z) =
bn
n
m/bn∫
(−m−1)/bn
fn(t, z)dt
=
1
2νn
νn∫
−νn
fn(t, z)dt + o(1), n →∞, (2.29)
where νn = bn/m → ν, n →∞ (see (1.2)). Besides, (2.27) implies
fn(j/bn, z) =
( 1
bn
∑
k
v2((j − k)/bn)
1 + b−1
n
∑
p v2((p− k)/bn)fn(p/bn, z)
− z
)−1
+ rnj ,
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 321
A. Lytova and L. Pastur
and taking into account (1.3), we get for any |t| ≤ νn
fn(t, z) =
( ∫
|τ |≤νn
v2(t− τ)dτ
1 +
∫
|θ|≤νn
v2(θ − τ)fn(θ, z)dθ
− z
)−1
+ rn(t, z), (2.30)
where
lim
n→∞ sup
z∈≤CK0
sup
|t|≤νn
|rn(t, z)| = 0. (2.31)
Note now that (1.18) can be written as f = Tf where T is a contracting map
for any z ∈ CK0 . Indeed, we have for any pair f1, f2 satisfying (1.17) and any
z ∈ CK0
∣∣∣
∫
|τ |≤ν
v2(t− τ)
(
1 +
∫
|θ|≤ν
v2(θ − τ)f1,2(θ, z)dθ
)−1
dτ − z
∣∣∣
−1
≤ K−1
0 , (2.32)
∣∣∣1 +
∫
|θ|≤ν
v2(θ − τ)f1,2(θ, z)dθ
∣∣∣
−1
≤ (1− w2/|=z|)−1 ≤ (1− w2/K0)−1, (2.33)
so that
sup
|t|≤ν
|[Tf1](t, z)− [Tf2](t, z)| ≤ q sup
|t|≤ν
|f1(t, z)− f2(t, z)|, (2.34)
where
q ≤ w4
(K0 − w2)2
< 1 if |=z| ≥ K0 > 2w2. (2.35)
Hence, for all z ∈ CK0 there exists a unique solution of (1.18) satisfying (1.17).
Consider first the case ν < ∞. Then it follows from (1.4) that for any uni-
formly bounded in t, z, n functions {Fn} we have
lim
n→∞ sup
z∈CK0
sup
|t|≤νn
∣∣∣
{ ∫
|τ |≤νn
−
∫
|τ |≤ν
}
v2(t− τ)Fn(τ, z)dτ
∣∣∣ = 0. (2.36)
This, (1.18), (2.30)–(2.31), and (2.34) lead to
lim
n→∞ sup
z∈CK0
sup
|t|≤νn
|fn(t, z)− f(t, z)| = 0. (2.37)
Hence,
f(z) = lim
n→∞E{gn(z)} = lim
n→∞
1
2νn
∫
|t|≤νnn
f(t, z)dt =
1
2ν
∫
|t|≤ν
f(t, z)dt.
322 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
Consider now the case ν = ∞. In this case the unique solution of (1.19) is a
t-independent function f satisfying (1.21). In addition, by (2.32), we have
1
2νn
∫
|t|≤νn
|fn(t, z)− f(z)|dt ≤ K−2
0
1
2νn
∫
|t|≤νn
(
T
(n)
1 + T
(n)
2 + T
(n)
3
)
dt, (2.38)
where
T
(n)
1 =
∫
|τ |≤νn
v2(t− τ)
∣∣∣∣∣
(
1 +
∫
|θ|≤νn
v2(θ − τ)fn(θ, z)dθ
)−1
−
(
1 +
∫
|θ|≤νn
v2(θ − τ)f(z)dθ
)−1
∣∣∣∣∣dτ,
T
(n)
2 =
∫
|τ |≤νn
v2(t− τ)
∣∣∣∣∣
(
1 +
∫
|θ|≤νn
v2(θ − τ)f(z)dθ
)−1
−
(
1 +
∫
|θ|≤∞
v2(θ − τ)f(z)dθ
)−1
∣∣∣∣∣dτ,
T
(n)
3 = |1 + w2h(z)|−1
∫
|τ |≥νn
v2(t− τ)dτ.
It follows from (1.19) that
|1 + w2f(z)|−1 = |zf(z)| ≤ 1, ∀z ∈ CK0 ,
hence,
1
2νn
∫
|t|≤νn
T
(n)
3 dt ≤ 1
2νn
∫
|t|≤νn
dt
∫
|τ |≥νn
v2(t− τ)dτ
=
∫
|y|≥νn
v2(y)dy − 1
2νn
∫
|y|≤νn
|y|v2(y)dy,
and then (1.2) and (1.3) imply
lim
n→∞
1
2νn
∫
|t|≤νn
T
(n)
3 dt = 0. (2.39)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 323
A. Lytova and L. Pastur
Furthermore, it follows from (2.33) and (2.35) that
1
2νn
∫
|t|≤νn
T
(n)
2 dt ≤ 2
νn
∫
|t|≤νn
dt
∫
|τ |≤νn
v2(t− τ)
∫
|θ|≥νn
v2(θ − τ)dθ
=
2
νn
∫
|t|≤νn
dτ
νn−τ∫
−νn−τ
v2(y)dy
∫
|θ|≥νn
v2(θ − τ)dθ
≤ 2w2
νn
∫
|τ |≤νn
dτ
∫
|θ|≥νn
v2(θ − τ)dθ.
This and (2.39) yield
lim
n→∞
1
2νn
∫
|t|≤νn
T
(n)
2 dt = 0. (2.40)
We also have
1
2νn
∫
|t|≤νn
T
(n)
1 dt ≤ 2
νn
∫
|t|≤νn
dt
∫
|τ |≤νn
v2(t− τ)dτ
×
∫
|θ|≤νn
v2(θ − τ)|fn(θ, z)− h(z)|dθ
=
2
νn
∫
|θ|≤νn
|fn(θ, z)− f(z)|dθ
∫
|τ |≤νn
v2(θ − τ)dτ
∫
|t|≤νn
v2(t− τ)dτ
≤ 2w4
νn
∫
|θ|≤νn
|fn(θ, z)− f(z)|dθ. (2.41)
It follows from (2.38) and (2.39)–(2.41) that
(1− 4w4K−2
0 )
1
2νn
∫
|t|≤νn
|fn(t, z)− f(z)|dt = o(1), n →∞.
We conclude that if ν = ∞, then f of (2.5) coincides with the solution of (1.19)
satisfying (1.17). Thus, to finish the proof of Theorem 1.3, it remains to prove
Lemma 2.1. Denote by Ek the expectation with respect to yk and let for any
random variable ξ
ξ◦k = ξ −Ek{ξ}
be its centered version. Then we have under conditions of Theorem 1.3 :
324 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
(i)
E{|(Gkyk,yk)◦k|2} ≤ C(z)δn, (2.42)
(ii) there exists K0 > 0 such that ∀z ∈ CK0 = {z : <z = 0, |=z| ≥ K0}
Var{Gpp} ≤ C(z)δn, (2.43)
Var{(Gkyk,yk)} ≤ C(z)δn. (2.44)
where
δn = o(1), n →∞ (2.45)
does not depend on p, k, z, and we denote by C(z) any positive quantity which
depends only on z and is finite for z ∈ CK0 (see (2.6)).
P r o o f. It follows from (1.12) and from unconditionality of the distribution
of ak that
E{a(n)
pk a
(n)
qk a
(n)
sk a
(n)
tk } = m
(n)
2,2 (δpqδst + δpsδqt + δptδqs) + κ
(n)
4 δpqδpsδpt, (2.46)
where κ
(n)
4 = m
(n)
4 − 3(m(n)
2,2 )2 = O(1), n →∞. This and (2.15) yield
Ek{|(Gkyk,yk)◦k|2} (2.47)
=
1
b2
n
∑
p,q,s,t
GpqGstvpkvqkvskvtkEk{a(n)
pk a
(n)
qk a
(n)
sk a
(n)
tk } −
∣∣∣ 1
bn
∑
p
v2
pkG
k
pp
∣∣∣
2
= (m(n)
2,2 − 1)
∣∣∣ 1
bn
∑
p
v2
pkG
k
pp
∣∣∣
2
+
2m
(n)
2,2
b2
n
∑
p,s
v2
skv
2
pk|Gk
ps|2 +
κ
(n)
4
b2
n
∑
p
v4
pk|Gk
pp|4.
By (2.8) and (2.23), we have
∣∣∣ 1
bn
∑
p
v2
pkG
k
pp
∣∣∣ ≤ K
|=z| ,
1
b2
n
∑
p,s
v2
skv
2
pk|Gk
ps|2 ≤
K
b2
n
∑
p
v2
pk
∑
s
|Gk
ps|2 ≤
K2
bn|=z|2 .
This, (1.12) and (2.47) lead to (2.42).
Let us prove (2.43). We have (cf. (2.14))
E{GjjG◦
jj} =
1
z
∑
k
E
{Bkn
Akn
G◦
jj
}
=
1
z
∑
k
E
{
Ek
{Bkn
Akn
}
Gk◦
jj
}
(2.48)
+
1
z
∑
k
E
{Bkn
Akn
(Gjj −Gk
jj)◦
}
=: T1 + T2.
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 325
A. Lytova and L. Pastur
It follows from the Schwarz inequality, (2.21), (2.24) and (2.26):
|T2| ≤
∑
k
E{|Bk|2}1/2 E{|(Gpp −Gk
pp)
◦|2}1/2 ≤ C(z)b−1/2
n . (2.49)
Consider now T1 of (2.49). By (2.16), we have
Ek
{Bkn
Akn
}
=
Ek{Bkn}
Ek{Akn} −
1
Ek{Akn}Ek
{(Akn)◦kBkn
Akn
}
,
and by the Schwarz inequality, (2.21), (2.24), and (2.42),
Ek
{
Akn
−1(Akn)◦kBkn
}
≤ C(z)b−1
n |vjk|δ1/2
n .
This and (2.15) yield
T1 =
1
zbn
∑
k
v2
jkE
{ 1
Ek{Akn}Gk
jjG
k◦
jj
}
+ o(1), n →∞. (2.50)
Applying again (2.16) and then (2.15), we get
E
{
Gk
jjG
k◦
jj
Ek{Akn}
}
=
1
E{Akn}E
{
|Gk◦
jj |2
}
− 1
E{Akn}E
{(Ek{Akn})◦
Ek{Akn} Gk
jjG
k◦
jj
}
=
1
E{Akn}E
{
|Gk◦
jj |2
}
− 1
E{Akn}
1
bn
∑
p
v2
pkE
{ Gk◦
pp
Ek{Akn}Gk
jjG
k◦
jj
}
.
Note also that in view of (2.26) we can replace Gk with G with the error term of
the order O(b−1
n ), hence
T1 =
1
zbn
∑
k
v2
jk
1
E{Akn} ·E
{
|G◦
jj |2
}
− 1
zb2
n
∑
k
v2
jk
1
E{Akn}
∑
p
v2
pkE
{ Gk
jj
Ek{Akn}G◦
ppG
◦
jj
}
+ o(1), n →∞,
and by the Schwarz inequality, (2.21), and (2.23),
|T1| ≤ K
|=z|Var{Gjj}+
K2
|=z|2Var{Gjj}1/2 max
|p|≤m
Var{Gpp}1/2 + o(1), n →∞.
This and (2.48) – (2.49) yield for Vj := Var{Gjj}, |j| ≤ m, z ∈ CK0 :
Vj ≤ K
K0
Vj +
K2
K2
0
V
1/2
j max
|p|≤m
V 1/2
p + o(1), n →∞.
326 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
Choosing here K0 such that
K/K0 + K2/K2
0 < 1, (2.51)
we obtain that
max
|p|≤m
Vp = o(1), n →∞,
i.e., (2.43).
It remains to note that we have by (2.15)
(Gkyk,yk)◦ = (Gkyk,yk)◦k + b−1
n
∑
p
v2
pkG
k◦
pp.
This, together with (2.42)–(2.43), leads to (2.44) and completes the proof of the
lemma.
3. Proof of Theorem 1.4
The proof of Theorem 1.4 can be obtained by following the scheme worked
out in [15] (see also [19] and references therein) and applicable to a wide va-
riety of random matrices with independent entries. Namely, one uses first the
martingale-type argument to prove the bound (2.4) and then the so-called inter-
polation trick to reduce the initial problem to the one of finding the limit (2.5)
for the random matrices with Gaussian entries with the same first and second
moments. Since these two steps are rather standard, we will explain below just
the derivation of the limiting equations (1.16)–(1.18) for i.i.d. Gaussian entries
{ajk}|j|,|k|≤m satisfying (1.11). Note also that by using a standard truncation
technique, condition (1.22) can be replaced with the Lindeberg type condition
for the second moments (see [15, 19]).
Accordingly, consider a random matrix An (1.1)–(1.4), where {ajk}|j|,|k|≤m
are jointly independent standard Gaussian random variables of zero mean and
unit variance. We will use
Proposition 3.1. Let ξ = {ξl}p
l=1 be independent Gaussian random variables
of zero mean, and Φ : Rp → C be a differentiable function with polynomially
bounded partial derivatives Φ′l, l = 1, . . . , p. Then we have
E{ξlΦ(ξ)} = E{ξ2
l }E{Φ′l(ξ)}, l = 1, ..., p, (3.1)
and
Var{Φ(ξ)} ≤
p∑
l=1
E{ξ2
l }E
{|Φ′l(ξ)|2
}
. (3.2)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 327
A. Lytova and L. Pastur
The first formula is a version of the integration by parts. The second is a
version of the Poincaré inequality (see, e.g., [19]).
By the resolvent identity and (3.1), we have
E{Gjj(z)} = −1
z
+
1
zbn
∑
k
v2
jkE{Djk(G(z)An)jk},
where Djk = ∂/∂A
(n)
jk . It can be shown that
Djk(GAn)jk = −zG̃kkGjj − (GAn)2jk (3.3)
(see, e.g., [15]), where G̃ = (M̃ − zIn)−1, M̃ = AT
nAn. Hence,
E{Gjj(z)} = −1
z
− 1
bn
∑
k
v2
jkE{G̃αα(z)Gjj(z)}+ rn(z), (3.4)
rn(z) = − 1
bn
∑
k
v2
jkE{(GAn)2jk}.
Since ∑
k
∣∣(GAn)2jk
∣∣ = (GMnG)jj = ((In + zG)G)jj ,
then
rn = O(b−1
n ), n →∞. (3.5)
We also have by (3.2) – (3.3)
Var{Gjj(z)} ≤
∑
l,k
E{|∂Gjj/∂alk|2} (3.6)
≤ 4
bn
∑
l,k
v2
lkE{|(GAn)jlGjk|2} ≤ C(z)b−1
n .
Now it follows from (3.4) – (3.6) that
E{Gjj(z)} = −1
z
− 1
bn
∑
k
v2
jkE{G̃kk(z)}E{Gjj(z)}+ O(b−1/2
n ) (3.7)
as n →∞. Similarly,
E{G̃kk(z)} = −1
z
− 1
bn
∑
p
v2
pkE{Gpp(z)}E{G̃kk(z)}+ O(b−1/2
n ), (3.8)
as n → ∞. Solving system (3.7) – (3.8), we get (2.27) and then it suffices to
use the same argument as that in the proof of Theorem 1.3 to finish the proof of
Theorem 1.4.
328 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
4. Triangular Matrices
In this section we prove Theorem 1.6. It follows from (1.18) that if v(t) =
χ[0;1](t) and ν = 1/2, then
f(t, z) = −
(
z −
t∫
−1/2
(
1 +
1/2∫
τ
f(θ, z)dθ
)−1
dτ
)−1
. (4.1)
Denote
ϕ(t, z) =
1/2∫
t
f(θ, z)dθ. (4.2)
It follows from (4.1)–(4.2) that
ϕ′′ − ϕ′2(1 + ϕ)−1 = 0, ϕ(1/2, z) = 0, ϕ′(1/2, z) = z−1,
where ϕ′ = ∂ϕ/∂t. Solving this system, we get for f(z) = ϕ(−1/2, z):
f(z) = ec(z) − 1, c(z)ec(z) = −z−1, z ∈ C\[0;∞). (4.3)
These equations are equivalent to (1.23). Evidently, there is only one solution c
analytical in R \ [0,∞).
Let us prove (1.24). As it was firstly shown in [16] (see also [3, 19]), to find the
support of the measure N , it suffices to consider the function x = x(f), f ∈ R,
which is the functional inverse of the Stieltjes transform of N , and to find the
set L ⊂ R on which x increases monotonically. Then supp N = R \ x(L), where
x(L) = {x(f) : f ∈ L}.
It follows from (4.3) that in our case
x(f) = − 1
(1 + f) ln(1 + f)
, f > −1, f 6= 0. (4.4)
It is easy to find that x(f) increases on L = [e−1, 0) ∪ (0,∞). Thus, x(L) =
(−∞, 0) ∪ [e,∞) and supp N = R \ x(L) = [0, e].
To prove asymptotic relations (1.25), we first consider
F (x) := f(−x) =
∞∫
0
N(dλ)
λ + x
, x > 0.
It is easy to find from (1.23) that
F (x) =
1
x ln 1/x
(1 + o(1)), x ↓ 0. (4.5)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4 329
A. Lytova and L. Pastur
This and the Tauberain theorem (see [8], Chapter XIII.5) imply
N(λ) := N([0, λ]) =
1
ln 1/λ
(1 + o(1)), λ ↓ 0.
Differentiating formally this asymptotic formula, we obtain the first formula of
(1.25).
To prove this formula rigorously we use (4.3). Denoting c(λ + i0) = ξ(λ) +
iη(λ), we obtain from (4.3) and (1.15)
ρ(λ) =
sin2 η
πλη
, λ =
eη cot η sin η
η
, η ∈ [0, π]. (4.6)
Since the limit λ ↓ 0 corresponds to η = π − σ, σ ↓ 0, we have from (4.6)
lnλ = −π/σ +O(lnσ), σ ↓ 0, and eventually the first asymptotics of (1.25). The
second asymptotics of (1.25) can be obtained similarly taking into account that
the limit λ ↑ e corresponds to the limit η ↓ 0.
Let us prove (1.26). To this end, we will use the identity
µk :=
kk
(k + 1)!
=
1
2πik
∫
|ζ|=ε
ekζ
ζk+2
dζ, ε > 0, k ≥ 1.
Consider the generating function
h(z) = −
∞∑
l=0
µl
zl+1
,
which is well defined if z is sufficiently large. We have then the integral represen-
tation
zh(z) = −1 +
1
2πi
∫
|ζ|=ε
log
(
1− eζ
ζz
)
dζ
ζ2
or
(zh(z))
′
=
1
2πiz
∫
|ζ|=ε
dζ
ζ2(e−ζζz − 1)
. (4.7)
The both formulas are valid for |z| > eεε−1, where the integrands are analytic in
ζ just because the series for the integrands are convergent. It is easy to find that
the function uz(ζ) = e−ζζz−1 has a simple zero in ζ(z) = z−1(1+o(1)), z →∞,
i.e., inside the contour |ζ| = ε, if ε does not depend on z. Thus the integral on the
left of (4.7) is equal to z−1 times the residue of uz at ζ(z) (i.e., (ζ(z)(1−ζ(z)))−1)
plus the integral over a sufficiently ”small” contour, say, |ζ| = |2z|−1. Since uz
330 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 4
On a Limiting Distribution of Singular Values of Random Band Matrices
has no zeros inside this contour, the corresponding integral is just u′z(0) = −z.
Putting everything together, we obtain
(zh(z))′ = (zζ(z)(1− ζ(z)))−1 − 1, zζ(z)e−ζ(z) = 1.
On the other hand, it follows from (4.3) that (zf(z))′ = −(zc(z)(1+ c(z)))−1−1.
Thus, setting ζ = −c we obtain that h coincides with f , i.e., assertion (iv) of the
theorem.
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