Universality at the Edge for Unitary Matrix Models
Using the results on the 1/n-expansion of the Verblunsky coe±cients for a class of polynomials orthogonal on the unit circle with n varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our pr...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1067292016-10-04T03:02:31Z Universality at the Edge for Unitary Matrix Models Poplavskyi, M. Using the results on the 1/n-expansion of the Verblunsky coe±cients for a class of polynomials orthogonal on the unit circle with n varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times di®erentiable potentials and of supports, consisting of one interval. Используя результат о разложении коэффициентов Верблански для полиномов, ортогональных на единичном круге, с переменным весом по степеням 1-n, доказано, что локальная статистика собственных значений унитарного матричного ансамбля не зависит от вида потенциала, определяющего матричную модель. Доказательство применимо для любого четыре раза дифференцируемого потенциала и носителя, состоящего из одного интервала. 2012 Article Universality at the Edge for Unitary Matrix Models / M. Poplavskyi // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 4. — С. 367-392. — Бібліогр.: 17 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106729 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Using the results on the 1/n-expansion of the Verblunsky coe±cients for a class of polynomials orthogonal on the unit circle with n varying weight, we prove that the local eigenvalue statistic for unitary matrix models is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of four times di®erentiable potentials and of supports, consisting of one interval. |
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Poplavskyi, M. |
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Poplavskyi, M. Universality at the Edge for Unitary Matrix Models Журнал математической физики, анализа, геометрии |
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Poplavskyi, M. |
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Poplavskyi, M. |
| title |
Universality at the Edge for Unitary Matrix Models |
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Universality at the Edge for Unitary Matrix Models |
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Universality at the Edge for Unitary Matrix Models |
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Universality at the Edge for Unitary Matrix Models |
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Universality at the Edge for Unitary Matrix Models |
| title_sort |
universality at the edge for unitary matrix models |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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http://dspace.nbuv.gov.ua/handle/123456789/106729 |
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Universality at the Edge for Unitary Matrix Models / M. Poplavskyi // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 4. — С. 367-392. — Бібліогр.: 17 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT poplavskyim universalityattheedgeforunitarymatrixmodels |
| first_indexed |
2025-07-07T18:54:51Z |
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2025-07-07T18:54:51Z |
| _version_ |
1837015490196668416 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2012, vol. 8, No. 4, pp. 367–392
Universality at the Edge for Unitary Matrix Models
M. Poplavskyi
Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv 61103, Ukraine
E-mail: poplavskiymihail@rambler.ru
Received August 5, 2012
Using the results on the 1/n-expansion of the Verblunsky coefficients for
a class of polynomials orthogonal on the unit circle with n varying weight,
we prove that the local eigenvalue statistic for unitary matrix models is
independent of the form of the potential, determining the matrix model.
Our proof is applicable to the case of four times differentiable potentials and
of supports, consisting of one interval.
Key words: unitary matrix models, local eigenvalue statistics, universa-
lity, polynomials orthogonal on the unit circle.
Mathematics Subject Classification 2010: 15B52, 42C05.
1. Introduction
We study a class of random matrix ensembles known as unitary matrix models.
These models are defined by the probability law
pn (U) dµn (U) = Z−1
n,2 exp
{
−nTrV
(
U + U∗
2
)}
dµn (U) , (1.1)
where U = {Ujk}n
j,k=1 is an n × n unitary matrix, µn (U) is the Haar measure
on the group U(n), Zn,2 is the normalization constant, and V : [−1, 1] → R is a
continuous function called the potential of the model. Denote eiλj the eigenvalues
of the unitary matrix U . The joint probability density of λj , corresponding to
(1.1), is given by (see [1])
pn (λ1, . . . , λn) =
1
Zn
∏
1≤j<k≤n
∣∣∣eiλj − eiλk
∣∣∣
2
exp
−n
n∑
j=1
V (cosλj)
. (1.2)
Normalized Counting Measure of eigenvalues (NCM) is given by
c© M. Poplavskyi, 2012
M. Poplavskyi
Nn (∆) = n−1]
{
λ
(n)
l ∈ ∆, l = 1, . . . , n
}
, ∆ ⊂ [−π, π].
The random matrix theory deals with several asymptotic regimes of the eigen-
value distribution. The global regime is centred around the weak convergence of
NCM. It is well known (see e.g. [2]) that for some smooth conditions for the
potential V there exists a measure N ∈ M1 ([−π, π]) with a compact support σ
such that Nn converges to N in probability .
Let
p
(n)
l (λ1, . . . , λl) =
∫
pn (λ1, . . . , λl, λl+1, . . . , λn) dλl+1 . . . dλn
be the l -th marginal density of pn. The local regime of eigenvalue distribution
describes the asymptotic behaviour of marginal densities when their arguments
are on the distances of order of the typical distance between eigenvalues. The
universality conjecture of marginal densities was suggested by Dyson (see [3]) in
the early 60s. He supposed that their asymptotic behaviour depends only on the
ensemble symmetry group and does not depend on other ensemble parameters.
First rigorous proofs for the hermitian matrix models with non-quadratic V ap-
peared only in the 90s. The case of general V which is locally C3 function was
studied in [4]. The case of real analytic V was studied in [5], where the asymp-
totic behaviour of orthogonal polynomials was obtained. For the unitary matrix
models the bulk universality was proved for V = 0 (see [3]), and for the locally
C3 functions (see [6]). The edge universality was proved only in the case of the
linear V (see [7]). In the present paper we prove the universality conjecture for
UMM with a smooth potential V in the case of one-interval support σ of the
limiting NCM.
It was proved in [2] that the limiting measure can be obtained as a unique
minimizer of the functional
E [m] =
π∫
−π
V (cosλ)m(dλ)−
π∫
−π
log
∣∣∣eiλ − eiµ
∣∣∣m(dλ)m(dµ)
in the class of unit measures on the interval [−π, π] (see [8] for the existence and
properties of the solution). It is well known, in particular, that for smooth V ′ the
equilibrium measure has a density ρ which is uniquely defined by the condition
that the function
u (λ) = V (cosλ)− 2
∫
σ
log
∣∣∣eiλ − eiµ
∣∣∣ ρ (µ) dµ (1.3)
368 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
takes its minimum value if λ ∈ σ = supp ρ. From this condition in the case of
differentiable V one can obtain the following integral equation for the equilibrium
density ρ:
(V (cosλ))′ = v.p.
∫
σ
cot
λ− µ
2
ρ (µ) dµ, forλ ∈ σ. (1.4)
We also use the weak convergence of the first marginal density ρn (λ) = p
(n)
1
proved in [2].
Proposition 1.1. For any φ ∈ H1 (−π, π) ,
∣∣∣∣
∫
φ (λ) ρn (λ) dλ−
∫
φ (λ) ρ (λ) dλ
∣∣∣∣ ≤ C ‖φ‖1/2
2
∥∥φ′
∥∥1/2
2
n−1/2 ln1/2 n, (1.5)
where ‖·‖2 denotes L2 norm on [−π, π].
We consider here the case of one interval σ. Our main conditions on the
potential V are
Condition C1. The support σ of the equilibrium measure is a single sym-
metric subinterval of the interval [−π, π], i.e.,
σ = [−θ, θ] , with θ < π.
R e m a r k 1.2. In fact, there is one more possibility to have one-interval σ.
Another case is some left symmetric arc of the circle, i.e., [π − θ, π + θ]. In this
case we replace V (cosx) in (1.2) by V (cos (π − x)). This replacement will rotate
all eigenvalues on the angle π and we will have the support from condition C1.
Condition C2. The equilibrium density ρ has no zeros in (−θ, θ) and
ρ (λ) ∼ C |λ∓ θ|1/2 , for λ → ±θ,
and the function u (λ) of (1.3) attains its minimum if and only if λ belongs to σ.
R e m a r k 1.3. From this condition we obtain the necessary scaling for
marginal densities at the edge of σ
∫
∆
ρ (λ) dλ ∼ n−1 ⇒| ∆ |∼ n−2/3, (1.6)
hence the typical distance between eigenvalues is of order n−2/3.
Condition C3. V (cosλ) possesses four bounded derivatives on σε =
[−θ − ε, θ + ε].
The following simple representation of ρ plays an important role in our asymp-
totic analysis (see [9])
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 369
M. Poplavskyi
Proposition 1.4. Under conditions C1-C3 the density ρ has the form
ρ (λ) =
1
4π2
χ (λ) P (λ)1σ,
where
χ (λ) =
√
|cosλ− cos θ|, P (λ) =
θ∫
−θ
(V (cosµ))′ − (V (cosλ))′
sin (µ− λ) /2
dµ
χ (µ)
. (1.7)
The main result of the paper is the following theorem
Theorem 1.5. Consider the unitary matrix ensemble of the form (1.1), sa-
tisfying conditions C1–C3 above. Then
• for the endpoints θ± = ±θ and any positive integer l the rescaled marginal
density
(
γn2/3
)−l n!
(n− l)!
p
(n)
l
(
θ± ± t1/γn2/3, . . . , θ± ± tl/γn2/3
)
(1.8)
with the sign ± corresponding to θ± and
γ = tan1/3 θ/2
(
P (θ)
4π
)2/3
converges weakly, as n → ∞, to det {QAi (tj , tk)}l
j,k=1 , where QAi (x, y) is
the Airy kernel
QAi (x, y) =
Ai (x) Ai′ (y)−Ai′ (x) Ai (y)
x− y
; (1.9)
• if ∆ ⊂ R is a finite union of disjoint bounded intervals and
En (∆n) = P (∆ndoes not contain eigenvalues of U)
is the hole probability for ∆n = θ± ±∆/γn2/3, then
lim
n→∞En (∆n) = 1 +
∞∑
l=1
(−1)l
l!
∫
∆
dt1 . . . dtl det {K (tj , tk)}l
j,k=1 , (1.10)
i.e., the limit is the Fredholm determinant of the integral operator K∆ de-
fined by the kernel K on the set ∆.
The paper is organized as follows. In Section 2 we give a brief outline of the
orthogonal polynomials method. In Section 3 we prove the main Theorem 1.5
using some technical results. These results are proved in Section 4.
370 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
2. Orthogonal Polynomials
We prove Theorem 1.5, using the orthogonal polynomials technique. This
method is based on a simple observation. Joint eigenvalue distribution (1.2) is
expressed in terms of the Vandermonde determinant of powers of eiλk , and there-
fore by the properties of determinants, can be written in terms of the determinant
of any system of linearly independent trigonometric polynomials. We consider
a system of polynomials orthogonal on the unit circle(OPUC) with a varying
weight. Let
wn (λ) = e−nV (cos λ)
be the weight function for the system of polynomials. Then the system can
be obtained from
{
eikλ
}∞
k=0
if we use the Gram-Schmidt procedure in L(n) :=
L2 ([−π, π] , wn (λ)) with the inner product
〈f, g〉n =
π∫
−π
f (x) g (x)wn (x) dx.
Hence, for any n we get the system of trigonometric polynomials
{
P
(n)
k (λ)
}∞
k=0
which are orthonormal in L(n). One can see from the Szegö’s condition that the
system
{
P
(n)
k (λ)
}∞
k=0
is not complete in L(n). To construct the complete system
one should also include polynomials with respect to e−iλ. Thus, following [10],
we introduce the Laurent polynomials
χ
(n)
2k (λ) = eikλP
(n)
2k (−λ) ,
χ
(n)
2k+1 (λ) = e−ikλP
(n)
2k+1 (λ) .
(2.1)
It is easy to check (see, e.g., [10, 11]) that the system
{
χ
(n)
k (λ)
}∞
k=0
is an
orthonormal basis in L(n). Moreover, it was proved in [10] that the functions χ
(n)
k
satisfy some five term recurrent relations. Let α
(n)
k and ρ
(n)
k be the Verblunsky
coefficients of the system
{
χ
(n)
k (λ)
}∞
k=0
(for the definition and properties see [9]).
Denote by
Θ(n)
j =
(
−α
(n)
j ρ
(n)
j
ρ
(n)
j α
(n)
j
)
,
M (n) = E1 ⊕Θ(n)
2 ⊕Θ(n)
4 ⊕ ..., L(n) = Θ(n)
1 ⊕Θ(n)
3 ⊕Θ(n)
5 ⊕ ...,
C(n) = M (n)L(n). (2.2)
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 371
M. Poplavskyi
From the properties of the Verblunsky coefficients one can see that the semi-
infinite matrices M (n) and L(n) are symmetric, three diagonal and unitary. C(n)
is also a unitary five diagonal matrix. Finally, using the above notations, we can
write the recurrence relations as
eiλ
−−→
χ(n) = C(n)
−−→
χ(n).
Hence, C(n) is a matrix presentation of the multiplication operator by eiλ in the
basis
{
χ
(n)
k (λ)
}∞
k=0
.
The main advantage of the orthogonal polynomials technique is the determi-
nant formulas which can be obtained in the same way as in [1],
n!
(n− l)!
p
(n)
l (λ1, . . . , λl) = det
{
K(n)
n (λj , λk)
}l
j,k=1
, (2.3)
where
K(n)
m (λ, µ) =
m−1∑
k=0
χ
(n)
k (λ) χ
(n)
k (µ)w1/2
n (λ) w1/2
n (µ) (2.4)
is the reproducing kernel of the system
{
χ
(n)
k (λ)
}∞
k=0
. Similarly to [12], the weak
convergence of the kernel K
(n)
n to K as n →∞ will prove Theorem 1.5.
3. Proof of Theorem 1.5
To prove the weak convergence of the reproducing kernel (2.4), we use the
lemma (see [12])
Lemma 3.1. Consider the sequence of functions Kn : R× R→ C and define
for =ζ, ξ 6= 0,
Fn (ζ, ξ) =
∫∫
= 1
x− ζ
= 1
y − ξ
|Kn (x, y)|2 dxdy. (3.1)
Assume that there exists F (ζ, ξ) of the form
F (ζ, ξ) =
∫∫
= 1
x− ζ
= 1
y − ξ
|K (x, y)|2 dxdy, (3.2)
with K bounded uniformly in each compact in R2 and such that for any fixed
A > 0 uniformly on the set
ΩA = {ζ, ξ : 1 ≤ =ζ,=ξ ≤ A, |<ζ,<ξ| ≤ A} (3.3)
we have
|Fn (ζ, ξ)−F (ζ, ξ)| ≤ εn, εn → 0, as n →∞. (3.4)
372 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
Then for any intervals I1, I2 ⊂ R
lim
n→∞
∫
I1
dx
∫
I2
dy |Kn (x, y)|2 =
∫
I1
dx
∫
I2
dy |K (x, y)|2 .
The lemma helps to prove the convergence of |Kn|2 to |K|2. Similarly, we
can check the convergence of Kn (t1, t2)Kn (t2, t3) . . .Kn (tl, t1) for any l ∈ N. To
prove the second part of Theorem 1.5, we use another proposition from [12].
Proposition 3.2. Let ∆ ⊂ R be a system of disjoint intervals as in Theo-
rem 1.5 and let Kn : L2 (∆) → L2 (∆) be a sequence of positive definite integral
operators with kernels Kn (x, y) and K : L2 (∆) → L2 (∆) a positive definite inte-
gral operator with kernel K (x, y), such that for any l ∈ N, det {Kn (tj , tk)}l
j,k=1 →
det {K (tj , tk)}l
j,k=1 weakly as n → ∞. Assume also that for any ∆ there exists
C∆ such that ∫
∆
Kn (s, s) ds ≤ C∆. (3.5)
Then, for the Fredholm determinants of Kn and K we have
lim
n→∞det (1−Kn) = det (1−K) .
We are going to use Lemma 3.1 for the scaled reproducing kernel of the system
of OPUC. Let
Kn (x, y) = n−2/3K(n)
n
(
θ + xn−2/3, θ + yn−2/3
)
1|x,y|≤cθn2/3 (3.6)
for some small enough θ-dependent constant cθ. This will be sufficient in view of
the following lemma (the analogue of Theorem 11.1.4, [13])
Lemma 3.3. Let the model (1.1) satisfy conditions C1-C3. Then, for any
n-independent ε > 0, there exists a constant dε > 0 such that
∫
σc
ε
K(n)
n (λ, λ) dλ ≤ Ce−ndε .
Since the polynomials χ
(n)
k are functions of eiλ, it is more convenient to define
a little bit different from (3.1) transformation and estimate the difference between
it and (3.1). Hence, we consider the following transformation:
Fn (z, w) = n−4/3
∫∫
[−π,π]
G (z − λ)G (w − µ)
∣∣∣K(n)
n (λ, µ)
∣∣∣
2
dλdµ, (3.7)
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 373
M. Poplavskyi
with
G (z) = <g (z) , and g (z) =
1 + eiz
1− eiz
(3.8)
being the analogues of the Poisson and the Herglotz transformations.
Proposition 3.4. It follows from the definition of g (z) that
g (z) = i cot
z
2
, g (z − λ) =
eiλ + eiz
eiλ − eiz
.
For z = x + iy we have g (x + iy) =
i sinx + sinh y
cosh y − cosx
, hence g (z) = −g (z). And
for G (z) we get
G (x + iy) =
sinh y
cosh y − cosx
, G (z − λ) = = cot
λ− z
2
.
Moreover, G (z) is a Nevanlinna function and
|g (z)|2 = −1 + 2 coth=z ·G (z) . (3.9)
The difference between the new transformation and the old one can be esti-
mated in the following way:
Proposition 3.5. Let z = θ + ζn−2/3 and w = θ + ξn−2/3 with |ζ| , |ξ| ≤
cθn
−2/3 and =ζ,=ξ ≥ 1. Then,
|Fn (z, w)− 4Fn (ζ, ξ)| ≤ Cn−1/6 (|Fn (z, w)|+ 1) . (3.10)
The next step is to prove the convergence of Fn (z, w) to the transformation
F (3.2) of the Airy kernel QAi (1.9). F can be calculated in terms of the Airy
functions, thus we are concentrated on the calculations of Fn. First, using the
properties of CMV matrices, we present Fn (z, w) in terms of the ”resolvent” of
C(n). After that we use the asymptotic behaviour of the Verblunsky coefficients,
obtained in [9], to get an approximation of the ”resolvent”. The approximation
will be given in terms of the Airy functions. Then we will estimate the error of
the ”resolvent” approximation and prove the uniform bound (3.4).
We start with a simple corollary from the spectral theorem and Proposi-
tion 3.4.
Proposition 3.6. Let
g(n) (z) =
(
C(n) + eiz
)(
C(n) − eiz
)−1
,
374 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
be the ”resolvent” of the CMV matrix C(n). Then,
(
g(n) (z)
)†
= −g(n) (z) , G(n) (z) :=
1
2
(
g(n) (z)− g(n) (z)
)
,
g(n) (z)
(
g(n) (z)
)†
= −I + 2 cot=z ·G(n) (z)
and
Fn (z, w) = n−4/3
n−1∑
j,k=0
G
(n)
j,k (z) G
(n)
k,j (w) . (3.11)
First of all, we would like to restrict the summation above by j, k ≤ M =[
Cn1/2 log n
]
with some constant C.
Lemma 3.7. There exists V -depended constants C such that under the con-
ditions of Theorem 1.5 uniformly in ΩA of (3.3) we have
n−2/3
n∑
j=M+1
G
(n)
n−j,n−j (z) ≤ Cn−1/12 log n.
Now we present the approximation for the matrix elements G
(n)
n−j,n−k. Using
the three-diagonal matrices expansion (2.2) of the C(n), we can write the matrix
g(n) as
g(n) (z) =
(
M (n)e−iz/2 + L(n)eiz/2
) (
M (n)e−iz/2 − L(n)eiz/2
)−1
.
From the definitions of M (n) and L(n) one can find their matrix elements
M
(n)
n+k,n+k−1 = dn+kρ
(n)
n+k, M
(n)
n+k,n+k = dn+kα
(n)
n+k − dn+k+1α
(n)
n+k+1,
L
(n)
n+k,n+k−1 = dn+k+1ρ
(n)
n+k, L
(n)
n+k,n+k = dn+k+1α
(n)
n+k − dn+kα
(n)
n+k+1,
where dk = (1 + sk) /2 and sk = (−1)k. Denote
C
(n)
± (z) = M (n)e−iz/2 ± L(n)eiz/2.
At the first step we derive the representation for the matrix elements of the
inverse matrix of C
(n)
− (z). Note that C
(n)
r− is three-diagonal and symmetric, and
its entries are
C
(n)
−n+k,n+k−1
(z) = sn+kρ
(n)
n+ken+k (z) ,
C
(n)
−n+k,n+k
(z) = sn+kα
(n)
n+ken+k (z) + sn+kα
(n)
n+k+1en+k+1 (z)
with
ek (z) = cos
z
2
− isk sin
z
2
.
For the Verblunsky coefficients we use the result of [9].
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 375
M. Poplavskyi
Lemma 3.8. Consider the system of orthogonal polynomials and the Verblun-
sky coefficients defined above. Let the potential V satisfy conditions C1–C3 above.
Then, for any k,
α
(n)
n+k = (−1)k s(n)
(
cos
θ
2
− pθx
(n)
k n−2/3
)
+ O (εn,k) ,
ρ
(n)
n+k = sin
θ
2
+ cot
θ
2
pθx
(n)
k n−2/3 + O (εn,k) ,
where s(n) = 1 or s(n) = −1 and
x
(n)
k = kn−1/3, εn,k = n−4/3 log11 n
(
1 +
(
x
(n)
k
)2
)
1|k|<n + 1|k|≥n,
with pθ =
π
√
2
P (θ)
and P defined in (1.7).
To introduce the approximation for the resolvent, we define two ”rotation”
matrices which help to present the matrix C
(n)
r− in the form, similar to the discrete
Laplacian matrix. Let U (n) and V (n) be two semi-infinite matrices with the entries
U
(n)
n+j,n+k =
(
is(n)
)2nk−k−1
δjk, V
(n)
n+j,n+k =
(
is(n)
)2nk−k
δjk
and
C(n)
r± (z) = U (n)C
(n)
± V (n), R(n) (ζ) =
(
C(n)
r− (z)
)−1
, where z = θ + ζn−2/3.
Then the entries of the new matrix are
(
C(n)
r−
)
n+k,n+k−1
(z) = ρ
(n)
n+ken+k (z) ,
(
C(n)
r−
)
n+k,n+k
(z) = −is(n)sn
(
α
(n)
n+ken+k (z) + α
(n)
n+k+1en+k+1 (z)
)
.
Using the above definitions, we write
g(n) (z) = I + 2L(n)V (n)R(n) (ζ) U (n)eiz/2. (3.12)
Now we prove that the matrix elements of R(n) (ζ) can be expressed in terms
of the Airy functions. For this aim we present an approximation matrix R? and
find the difference between R? and R(n). Note that
eiz/2 = eiθ/2 + ieiθ/2ζn−2/3 + O
(
|ζ|2 n−4/3
)
,
en+k (z) = en+k (θ)− isn+ken+k (θ) ζn−2/3 + O
(
|ζ|2 n−4/3
)
,
376 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
Let y
(n)
k = x
(n)
k − n−1/3/2 and r
(n)
k,ζ = n−4/3εn,k + |ζ|2. Then
(
C(n)
r−
)
n−k,n−k−1
(ζ) = sin
θ
2
en+k (θ)− cot
θ
2
en+k (θ) pθy
(n)
k n−2/3
−isn+k sin
θ
2
en+k (θ) ζn−2/3 − 1
2
cot
θ
2
en+k (θ) pθn
−1
+n−4/3O
(
r
(n)
k,ζ
)
, (3.13)
(
C(n)
r−
)
n−k,n−k
(ζ) = − sin θ − 2 sin
θ
2
pθy
(n)
k n−2/3
− 2 cos2
θ
2
ζn−2/3 − isn+kpθ cos
θ
2
n−1 + n−4/3O
(
r
(n)
k,ζ
)
. (3.14)
The matrix elements of C
(n)
r− are similar to the matrix elements of the discrete
Laplace operator with some potential in the n−1/3 scale, but off-diagonal ele-
ments contain alternating terms isn+k sin2 θ
2
. Hence, we define the approximate
resolvent in terms of the Airy function with some shift. Set
δ
(n)
k = isn+k+1δ, δ =
1
2
tan
θ
2
, h = n−1/3
and
R?
n−k,n−j (ζ) = h−1Rζ
(
y
(n)
k + δ
(n)
k h, y
(n)
j + δ
(n)
j h
)
, (3.15)
where Rζ (z, w) , defined by
Rζ (z, w) = ab−1π
{
ψ− (z, ζ) ψ+ (w, ζ) , <z ≤ <w,
ψ+ (z, ζ) ψ− (w, ζ) , <z ≥ <w
(3.16)
with ψ± defined in the Appendix, is the extension of the resolvent of the operator L
L [f ] (x) = a3f ′′ (x)− b3xf (x) (3.17)
to the complex plane, where a3 = sin θ and b3 = 2pθ sin−1 (θ/2). For the proper-
ties, asymptotic behaviour, and the integral representation of Rζ see Appendix.
Denote by D(n) the error of the approximation
D(n) (ζ) = C(n)
r− (ζ) R? (ζ)− I. (3.18)
To present the bounds for D
(n)
n−k,n−j , we introduce the notations
d
(p)
n−k,n−j = sup
|s|≤δ+1
∣∣∣∣
∂p
∂zp
Rζ
(
y
(n)
k + sh, y
(n)
j + δ
(n)
j h
)∣∣∣∣ .
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 377
M. Poplavskyi
One can see from the definition of Rζ that
∂
∂z
Rζ is not defined for z = w. In
this case, by
∂
∂z
we denote the half of the sum of the left and the right derivatives
1
2
(
∂+
∂z
+
∂−
∂z
)
. Then D(n) satisfies the following bound.
Lemma 3.9. There exists constants C1, C2 such that uniformly in k, j and
ζ ∈ ΩA
D
(n)
n−k,n−j (ζ) ≤ C1h
2 logC2 n
((
1 + h2
∣∣∣y(n)
k
∣∣∣
2
)
d
(0)
n−k,n−j +
(∣∣∣y(n)
k
∣∣∣ + |ζ|
)
d
(1)
n−k,n−j
)
. (3.19)
Now we are ready to analyse the r.h.s of (3.11). From (3.15), (3.12), and
Lemma 3.9 one can see that G
(n)
n−k,n−j ≈ h−1=Rζ
(
y
(n)
k , y
(n)
j
)
, and if we could
neglect the remainder, then
Fn (ζ, ξ) ≈ h2
∑
=Rζ
(
y
(n)
k , y
(n)
j
)
=Rξ
(
y
(n)
j , y
(n)
k
)
.
On the other hand, changing a double sum by the double integral and using
(5.4), we obtain F [QAi]. Hence, our main goal now is to estimate the remainder
that appears after replacement of the ”resolvent” of C
(n)
r− by the resolvent of the
differential operator. We will do these calculations in several steps.
We start from the proof of the bound for
ΣM = n−2/3
M∑
j=0
G
(n)
n−j,n−j (z) (3.20)
with M =
[
C0n
1/2 log n
]
. It follows from (3.12) and the definition of G(n) that
G(n) (z) = L(n)V (n)
(
R(n) (ζ) eiz/2 −R(n)
(
ζ
)
eiz/2
)
U (n).
Using the definition of D(n), we can write R(n) as
R(n) (ζ) = R? −R(n) (ζ) D(n) (ζ) .
Then,
ΣM = n−2/3
M∑
j=0
(
L(n)V (n)
(
R?
e (ζ)−R(n)
e D(n) (ζ)
)
U (n)
)
n−j,n−j
= Σ∗M − ΣD(n)
M ,
378 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
where R?
e (ζ) = R? (ζ) eiz/2−R?
(
ζ
)
eiz/2 and the same with R(n) and R
(n)
e . Here
Σ?
M can be estimated immediately by using Proposition 5.5, and ΣD(n)
M can be
estimated by multiplying Σ1/2
M by some small factor which we get using the Cauchy
inequality and the bounds (3.19) for D
(n)
n−k,n−j . Thus we obtain the quadratic
inequality (3.23). Solving this inequality, we will obtain (3.20). Indeed,
|Σ∗M | ≤ C
M∑
j=0
∑
|k−j|≤1
h
∣∣∣=Rζ
(
y
(n)
k + δ
(n)
k h, y
(n)
j + δ
(n)
j h
)∣∣∣ (3.21)
+h3
∣∣∣Rζ
(
y
(n)
k + δ
(n)
k h, y
(n)
j + δ
(n)
j h
)∣∣∣ . (3.22)
Using Proposition 5.5, we can estimate Σ?
M as follows:
|Σ∗M | ≤ C.
To estimate ΣD(n)
M , we start with the relation
L(n)V (n)R(n)
e D(n)U (n) = L(n)V (n)R(n)
e U (n)
(
U (n)
)−1
D(n)U (n)
=
(
g(n) (z)− g(n) (z)
)
D̂(n),
where D̂(n) entries have the same bounds as D(n), and we will write below D(n)
to simplify notations. Note that
(
g(n)D(n)
)
n−j,n−j
=
〈
g(n)D(n)en−j , en−j
〉
=
〈
D(n)en−j ,
(
g(n)
)†
en−j
〉
≤
∥∥∥D(n)en−j
∥∥∥
∥∥∥∥
(
g(n)
)†
en−j
∥∥∥∥ =
((
D(n)
)†
D(n)
)1/2
n−j,n−j
((
g(n)
)†
g(n)
)1/2
n−j,n−j
,
and by the Cauchy inequality and (3.9),
∣∣∣ΣD(n)
M
∣∣∣ ≤ Cn−2/3
M∑
j=0
((
D(n)
)†
D(n)
)
n−j,n−j
1/2
×
M + 2 coth=z
M2∑
j=M1+1
G
(n)
n−j,n−j
1/2
= S
1/2
D(n)
(
O
(
n−5/6 log n
)
+ 2n−2/3 coth
(
=ζn−2/3
)
ΣM
)
.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 379
M. Poplavskyi
Using Lemma 3.9, the Cauchy inequality, and Proposition 5.4, we estimate
SD(n) as follows:
SD(n) =
M∑
j=0
((
D(n)
)†
D(n)
)
n−j,n−j
≤ C1n
−4/3 logC2 n
M∑
j=0
∞∑
k=0
(∣∣∣y(n)
k
∣∣∣
2
+ |ζ|2
) ∣∣∣d(1)
n−k,n−j
∣∣∣
2
+
∣∣∣d(0)
n−k,n−j
∣∣∣
2
+h4
(∣∣∣y(n)
k
∣∣∣
4
+ |ζ|4
) ∣∣∣d(0)
n−k,n−j
∣∣∣
2
≤ C1n
−1 logC2 n
M∑
j=0
(
1 +
∣∣∣y(n)
j
∣∣∣
)3/2
+ h4
(
1 +
∣∣∣y(n)
j
∣∣∣
)5/2
≤ C1n
−2/3 logC2 n
(
Mn−1/3
)5/2
≤ C1n
−1/4 logC2 n.
Combining this inequality with the above estimate of ΣD(n)
M , we obtain the
inequality for ΣM
|ΣM | ≤ C1 + C2n
−1/8 logC3 n
(
O
(
n−5/6 log n
)
+ |ΣM |
)1/2
(3.23)
which gives (3.20).
Now we are ready to find the limit of the r.h.s. of (3.11). Combining
Lemma 3.7 with (3.21), we get
n−2/3
n∑
j=0
G
(n)
n−j,n−j (z) ≤ C. (3.24)
Using the definition of G(n), the sum in (3.11) can be splitted into four parts
with different products of g(n) and g(n). For each sum, the Cauchy inequality
yields
n−4/3
∣∣∣∣∣∣
∑
j,k
g
(n)
n−j,n−k (z) g
(n)
n−k,n−j (w)
∣∣∣∣∣∣
≤
n−4/3
∑
j
(
g(n)
(
g(n)
)†)
n−j,n−j
(z)
1/2
×
n−4/3
∑
j
(
g(n)
(
g(n)
)†)
n−j,n−j
(w)
1/2
,
380 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
where each of the brackets is bounded because of (3.9) and (3.24). Changing the
summation limits in the previous bound to j ∈ [M,n] and using Lemma 3.7, we
obtain that under the conditions of Lemma 3.1
Fn (z, w) = n−4/3
M∑
j,k=0
G
(n)
n−k,n−j (z) G
(n)
n−j,n−k (w) + O
(
n−1/24 log n
)
.
Now we use once more the identity
G(n) = G? −G(n)D̂(n).
Repeating the above arguments, we obtain
Fn (z, w) = F ?
n (z, w) + FD(n) (z, w) ,
and
FD(n) (z, w) ≤ C1n
−1/8 logC2 n.
Since G? = L(n)V (n)R?
eU
(n) with R?
e defined above, we have
G?
n−k,n−j = n1/3=Rζ
(
y
(n)
k , y
(n)
j
)
+ rG?
k,j ,
where rG?
k,j contains terms with some derivatives of the Rζ multiplied by h in some
non-negative power. Thus, from the boundness of the corresponded integrals (see
proof of Proposition 5.4 for the arguments)
hp+q
Mn−1/3∫
0
Mn−1/3∫
0
∣∣∣∣
∂p+q
∂xp∂yq
Rζ (x, y)
∣∣∣∣
2
dxdy ≤ Cp,q,r,s,
we obtain that we can neglect terms from rG∗
k,j and
F ?
n (z, w) =
Mn−1/3∫
0
Mn−1/3∫
0
=Rζ (x, y)=Rξ (y, x) dxdy + O
(
h1/2
)
.
Finally we note that by (5.7) and (5.8),
∞∫
Mn−1/3
dx
∫
dy |Rζ (x, y)|2 ≤
∞∫
Mn−1/3
=Rζ (x, x) dx ≤ Cn−1/12 log n,
and ∞∫
0
∞∫
0
=Rζ (x, y)=Rξ (y, x) dxdy ≤ C.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 381
M. Poplavskyi
Hence,
Fn (z, w) =
∞∫
0
∞∫
0
=Rζ (x, y)=Rξ (y, x) dxdy + O
(
Cn−1/24 logC n
)
. (3.25)
Estimate (3.25), integral representation (5.4), and the following relation (see [14])
QAi (x, y) =
∞∫
0
Ai (x + t) Ai (y + t) dt
imply (3.4) with
K (x, y) = a−2b−4QAi
(
a−1b−2x, a−1b−2y
)
.
Proposition 3.2 implies that it is sufficient to check (3.5) to finish the proof
of Theorem 1.5. We use an evident relation
G (t + iε− s) =
d
dt
2 arctan
(
tan
(
t− s
2
)
cot
ε
2
)
that implies the inequality valid for any s ∈ [a, b] ⊂ R
b+1∫
a−1
G
(
(t + i− s) n−2/3
)
dt ≥ Cn2/3,
with some absolute constant C. The last inequality, the positiveness of Kn and
G, and definition of G(n) imply
b∫
a
Kn (s, s) ds ≤ Cn−2/3
b∫
a
ds
b+1∫
a−1
dtKn (s, s) G
(
(t + i− s) n−2/3
)
≤ C
b+1∫
a−1
n∑
j=1
G
(n)
n−j,n−j
(
θ + (t + i) n−2/3
)
dt.
Hence, by (3.24) for any finite ∆ ⊂ [−A + 1, A− 1] we obtain (3.5).
4. Auxiliary Results
P r o o f of Proposition 3.5. Using Lemma 3.3 with ε = 2cθ and inequality
∣∣∣K(n)
n (λ, µ)
∣∣∣
2
≤ K(n)
n (λ, λ) K(n)
n (µ, µ) , (4.1)
382 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
we obtain
∫
λ∈σc
ε
G (z − λ)
∣∣∣K(n)
n (λ, µ)
∣∣∣
2
dλ ≤ Ce−nd(ε) sup
λ∈σc
ε
G (z − λ) K(n)
n (µ, µ) .
Due to the restrictions on λ and z we get G (z − λ) ≤ C ′ when λ ∈ σc
ε. Thus,
∫∫
σc
ε
G (z − λ) G (w − µ)
∣∣∣K(n)
n (λ, µ)
∣∣∣
2
dλdµ = e−cnO
(=−1z + =−1w
)
.
Changing the variables by the scaled ones in (3.7), we get
Fn (z, w) = n−4/3
∫∫
= cot
ζ − x
2n2/3
= cot
ξ − y
2n2/3
|Kn (x, y)|2 dxdy + O
(
e−cn
)
.
Finally we estimate the difference between Fn and 4Fn
4Fn (ζ, ξ)− Fn (z, w) = n−4/3 (I1 (ζ, ξ) + I2 (ζ, ξ) + I2 (ξ, ζ)) + O
(
e−cn
)
with I1 and I2 of (4.2) and (4.3). It is easy to see that
|I1 (ζ, ξ)|=
∣∣∣∣∣
∫∫
=
(
2n2/3
ζ − x
− cot
ζ − x
2n2/3
)
=
(
2n2/3
ξ − y
− cot
ξ − y
2n2/3
)
|Kn (x, y)|2 dxdy
∣∣∣∣∣
≤ C
∫∫
|Kn (x, y)|2 dxdy ≤ Cn, (4.2)
where we have used that for 0 < |z| ≤ 2cθ
∣∣∣∣cot z − 1
z
∣∣∣∣ ≤ C.
In addition, since the kernel
∣∣∣K(n)
n (λ, µ)
∣∣∣
2
is positive definite, we can use the
Cauchy inequality to get
|I2 (ζ, ξ)| =
∣∣∣∣∣
∫∫
=
(
2n2/3
ζ − x
− cot
ζ − x
2n2/3
)
= cot
ξ − y
2n2/3
|Kn (x, y)|2 dxdy
∣∣∣∣∣
≤ |I1 (ζ, ξ)|1/2
∣∣∣n4/3Fn (z, w)
∣∣∣
1/2
≤ Cn7/6 |Fn (z, w)|1/2 . (4.3)
Finally, collecting the above bounds, we obtain
|Fn (z, w)−Fn (ζ, ξ)| ≤ Cn−1/6 |Fn (z, w)|1/2 + C ′n−1/3,
and using the Cauchy inequality, we get (3.10).
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 383
M. Poplavskyi
P r o o f of Lemma 3.9. The proof is based on the direct calculations of the
matrix elements D
(n)
n−j,n−k. We start with the case j 6= k. Then all derivatives
of Rζ are well defined and the points y
(n)
j−1, y
(n)
j , y
(n)
j+1 are laying on the same side
of y
(n)
k . Now we are going to calculate D
(n)
n−j,n−k using the Taylor expansion and
definition of the C
(n)
r− . These calculations are a little bit involved, so we present
them in several steps. First, we calculate R?
n−k∓1,n−j ,
R?
n−k∓1,n−j = h−1Rζ
(
y
(n)
k ± h− δ
(n)
k h, y
(n)
j + δ
(n)
j h
)
= h−1Rζ
(
y
(n)
k , y
(n)
j + δ
(n)
j h
)
+
(
±1− δ
(n)
k
) ∂
∂z
Rζ
(
y
(n)
k , y
(n)
j + δ
(n)
j h
)
+
(
±1− δ
(n)
k
)2
h
∂2
∂z2
Rζ
(
y
(n)
k , y
(n)
j + δ
(n)
j h
)
+ h2O
(
r?
n−k,n−j (δ + 1)
)
with the remainder
r?
n−k,n−j (d) = sup
|s|<d
∣∣∣∣
∂3
∂z3
Rζ
(
y
(n)
k + s, y
(n)
j + δ
(n)
j h
)∣∣∣∣ ,
where the last bound follows from differential equation (5.1) valid for the functions
ψ±. To simplify calculations for C
(n)
r− , we use the following notations:
Sk :=
(
C(n)
r−
)
n−k,n−k−1
+
(
C(n)
r−
)
n−k,n−k+1
,
Dk :=
(
C(n)
r−
)
n−k,n−k−1
−
(
C(n)
r−
)
n−k,n−k+1
.
Then, combining the above expansion with (3.13)–(3.14), we obtain
D
(n)
n−k,n−j = h−1Rζ
(
y
(n)
k , y
(n)
j + δ
(n)
j h
)(
Sk +
(
C(n)
r−
)
n−k,n−k
)
+
∂
∂z
Rζ
(
y
(n)
k , y
(n)
j + δ
(n)
j h
) (
Dk − δ
(n)
k Sk + δ
(n)
k
(
C(n)
r−
)
n−k,n−k
)
+ h
∂2
∂z2
Rζ
(
y
(n)
k , y
(n)
j + δ
(n)
j h
)(
1
2
Sk − δ
(n)
k Dk − δ2
2
(
Sk +
(
C(n)
r−
)
n−k,n−k
))
+ O
(
r?
n−k,n−j (δ + 1)
)
, (4.4)
where for the last term we have used the uniform bound for elements
(
C
(n)
r−
)
n−j,n−k
.
Now it is sufficient to calculate every expression in the brackets. We start
with Sk and Dk,
Sk = sin θ − 2 cos
θ
2
cot
θ
2
pθy
(n)
k h2 − 2 sin2 θ
2
ζh2 + isn+kpθ cos
θ
2
h3 + h4O
(
r
(n)
k,ζ
)
,
384 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
Dk = −2isn+k sin2 θ
2
+ 2isn+k cos
θ
2
pθy
(n)
k h2 − isn+k sin θζh2
− cos
θ
2
cot
θ
2
pθh
3 + h4O
(
r
(n)
k,ζ
)
.
Therefore, with an error of order h4O
(
r
(n)
k,ζ
)
we can write
Sk +
(
C(n)
r−
)
n−k,n−k
≈ −2h2
(
pθ sin−1 (θ/2) y
(n)
k + ζ
)
,
Dk − δ
(n)
k Sk + δ
(n)
k
(
C(n)
r−
)
n−k,n−k
≈ −2δ
(n)
k h2
(
pθ sin−1 (θ/2) y
(n)
k − ζ
+ isn+kpθ cos (θ/2) sin−2 (θ/2)h
)
.
Finally, combining the above relations and the equation for Rζ in the form
sin θ
∂2
∂z2
Rζ
(
y
(n)
k , y
(n)
j + δ
(n)
j h
)
−
(
2pθ sin−1 θ/2y
(n)
k + ζ
)
Rζ
(
y
(n)
k , y
(n)
j + δ
(n)
j h
)
= 0,
we obtain the remainder in (4.4) with all terms of order less than h2. Gathering
all these remainders and the remainder h4O
(
r
(n)
k,ζ
)
, we get (3.19). For j = k, the
calculations can be performed similarly if we take into account jump condition
(5.2).
P r o o f of Lemma 3.7. We start with estimate of
Xn (ζ) = n−2/3
∫
Kn (x, x) G
(
(ζ − x) n−2/3
)
dx,
where Kn is defined as in (3.6) but without any restriction. Let ζ = s + iε.
Changing variables to z = θ + ζn−2/3 and using (3.6) with (3.8), we obtain
Xn (ζ) = n1/3<hn (z) ,
where
hn (z) =
π∫
−π
g (z − λ) ρn (λ) dλ.
For further estimates we use the ”quadratic” equation obtained in [6],
h2
n (z)− 2iV ′ (<z) hn (z)− 2iQn (z)− 1 = − 2
n2
δn (z) ,
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 385
M. Poplavskyi
with
Qn (z) =
π∫
−π
g (z − λ)
(
V ′ (λ)− V ′ (<z)
)
ρn (λ) dλ,
δn (z) =
π∫∫
−π
∣∣∣K(n)
n (λ, µ)
∣∣∣
2
(g (z − λ)− g (z − µ))2 dλdµ.
Solving the ”quadratic” equation, we get
Xn (ζ) = n1/3<
√
fn (s, ε)− 2n−2δn (z),
where the function
fn (s, ε) = −V ′2
(
θ + sn−2/3
)
+ 2iQn
(
θ + (s + iε) n−2/3
)
+ 1
is twice differentiable in both variables. Using the symmetry of the kernel K
(n)
n
and (4.1), we can estimate δn (z) as
∣∣n−2δn (z)
∣∣ ≤ 4n−2
π∫
−π
K(n)
n (λ, λ) |g (z − λ)|2 dλ.
Then the identity (3.9) yields
∣∣n−2δn (z)
∣∣ ≤ 4n−1+2n−4/3 coth
(
εn−2/3
)
·Xn (ζ) ≤ Cn−2/3
(
n−1/3 + ε−1Xn (ζ)
)
,
as ε = O (1). Now we continue the estimation of Qn (z). For the density ρn, we
use the bound (see [6])
∣∣ρ′n (λ)
∣∣ ≤ C
(∣∣∣ψ(n)
n−1
∣∣∣
2
+
∣∣∣ψ(n)
n
∣∣∣
2
+ 1
)
,
where ψ
(n)
k = P
(n)
k w
1/2
n are orthonormal functions. Hence, the density ρn is
uniformly bounded and therefore, similarly to (2.17) of [6], we have
|Qn (z)−Qn (<z)| ≤ C=z |log=z| .
The weak convergence (1.5) with
φ (λ) =
(
V ′ (λ)− V ′
(
θ + s/γn2/3
))
cot
λ− θ − s/γn2/3
2
implies ∣∣∣Qn
(
θ + s/γn2/3
)
−Q
(
θ + s/γn2/3
)∣∣∣ ≤ Cn−1/2 log1/2 n
386 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
if |s| ≤ cθn
2/3. Hence, combining the above relations, we obtain
|fn (s, ε)− f (s)| ≤ Cn−2/3 log n
(
|log ε|+ n1/6
)
,
with f (s) := f (s, 0). The properties of the Herglotz transformation yield (see
[6])
ρ (λ) =
1
2π
lim
ε→+0
<h (λ + iε) .
Therefore, at the edge point θ we obtain f (0) = 0 and f ′ (0) < 0. Hence, by
the differentiability of f (s) , we obtain
X (ζ) = <
√
O
(
s + ε−1X (ζ) + n1/6 log n
)
. (4.5)
Solving the quadratic inequality, we estimate X (ζ) as follows:
X (ζ) ≤ C
(
ε−1 + s1/2 + n1/12 log1/2 n
)
.
Now we write (4.5) more precisely
X (ζ) = <
√
−Cs + ε−2O
(
1 + εs1/2 + εn1/12 log1/2 n
)
.
Below we need the estimate of X (ζ) for s > Cn1/6 log n and ε = O (1). Hence
we obtain
X (ζ) ≤ C1
∣∣∣s− C2n
1/6 log n
∣∣∣
−1/2
. (4.6)
Note that all constants in the above estimates depend only on V and can be
bounded by some combination of sup |V |, sup |V ′′| and sup |V ′′′|. Now we return
to the estimate of the sum in Lemma 3.7. By the spectral theorem,
I (M) = n−2/3
n∑
j=M+1
G
(n)
n−j,n−j (z) = n−2/3
n−M−1∑
j=0
∫
G (λ− z)
∣∣∣χ(n)
j (λ)
∣∣∣
2
wn (λ) dλ.
Let us consider the analogue of the joint eigenvalue distribution of model (1.1)
in the form
p
(n−M)
n−M (λ1, . . . , λn−M )=
1
Z
(n−M)
n
∏
1≤j<k≤n−M
∣∣∣eiλj − eiλk
∣∣∣
2
exp
−n
n−M∑
j=1
V (cosλj)
.
Then, by the same argument as above for model (1.1), we define the first
marginal density
ρ
(n−M)
n−M (λ) =
1
n−M
n−M−1∑
j=0
∣∣∣χ(n)
j (λ)
∣∣∣
2
wn (λ) .
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 387
M. Poplavskyi
On the other hand, this density can be considered as the first marginal density
for model (1.1) with the potential Ṽ =
n
n−M
V . Hence,
I (M) = n−2/3
∫
G (λ− z) K
(n−M,Ṽ )
n−M (λ, λ) dλ = X Ṽ
n−M (ζ) .
But it follows from the result of [15] that the support of the equilibrium density
for Ṽ is [θM , θM ] with θM = θ − cV
(
Mn−1
)
+ o
(
Mn−1
)
with some cV > 0.
Hence, by (4.6),
X Ṽ
n−M ≤ Cn−1/12,
and Lemma 3.7 is proved.
5. Appendix
In this section we present the properties and the asymptotic analysis of the
resolvent of the Airy operator. Denote by L the second order differential operator
on the set of twice continuously differentiable functions on R,
L [f ] (x) = a3f ′′ (x)− b3xf (x) .
Let Rζ (x, y) be the kernel of the resolvent (L − ζI)−1 for =ζ 6= 0. By the
general principles (for example see [16], Section 72)
Proposition 5.1. Let Ai (z) and Bi (z) be the standard Airy functions. De-
note by ψ± the following functions:
ψ− (x, ζ) = Ci (Xx,ζ) , ψ+ (x, ζ) = Ai (Xx,ζ) ,
with
Ci (X) = iAi (X)−Bi (X) and Xx,ζ = a−1bx + a−1b−2ζ.
Then these functions are the unique solutions of the differential equation
a3 ∂2
∂x2
ψ± (x, ζ)− (
b3x + ζ
)
ψ± (x, ζ) = 0, (5.1)
that are square integrable on the right (left) half axis and fixed by jump condition
ψ− (x, ζ)
d
dx
ψ+ (x, ζ)− ψ+ (x, ζ)
d
dx
ψ− (x, ζ) = a−1bπ−1. (5.2)
And the resolvent Rζ has two representations
Rζ (x, y) = ab−1π
{
ψ− (x, ζ) ψ+ (y, ζ) , x ≤ y,
ψ+ (x, ζ) ψ− (y, ζ) , x ≥ y,
(5.3)
Rζ (x, y) = a−2b−1
∫
1
t− ζ
Ai
(
a−1bx + a−1b−2t
)
Ai
(
a−1by + a−1b−2t
)
dt. (5.4)
388 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
The following asymptotic behaviour of the Airy functions can be found in [17].
Proposition 5.2. For any δ > 0, the following asymptotics are uniform in
the corresponding domains:
Ai (z) = π−1/2z−1/4e−
2
3
z3/2 (
1 + O
(
z−3/2
))
, |argz| < π − δ,
Ai (−z) = π−1/2z−1/4 sin
(
2
3
z3/2 +
π
4
) (
1 + O
(
z−3/2
))
, |argz| < 2
3
π − δ,
Ci (z) = π−1/2z−1/4e
2
3
z3/2 (
1 + O
(
z−3/2
))
, |argz| < 1
3
π − δ,
Ci (−z) = π−1/2z−1/4e
i 2
3
z3/2+i
π
4
(
1 + O
(
z−3/2
))
, |argz| < 2
3
π − δ.
The main term for the derivatives can be obtained by direct differentiation
of the asymptotics. The last proposition and the definition of the functions ψ±
yield the asymptotic behaviour of them
Proposition 5.3. The functions ψ± are entire in x and ζ and have the fol-
lowing asymptotic behaviour in x for =ζ = ε > 0:
|ψ+ (x, ζ)| =
π−1/2 |Xx,ζ |−1/4
(
1 + O
(
|Xx,ζ |−3/2
))
exp
{
−2
3
|<Xx,ζ |3/2
}
, x →∞
exp
{
a−1b−2ε |<Xx,ζ |1/2
}
, x → −∞
|ψ− (x, ζ)| =
(4π)−1/2 |Xx,ζ |−1/4
(
1 + O
(
|Xx,ζ |−3/2
))
exp
{
2
3
|<Xx,ζ |3/2
}
, x →∞
exp
{
−a−1b−2ε |<Xx,ζ |1/2
}
, x → −∞
Proposition 5.4. For any non-negative integers s, q and any A ∈ R+ there
exists a constant CA,s,q such that for any x ≥ −A and ζ ∈ ΩA
I (s; q) =
∞∫
−∞
|y|s
∣∣∣∣
∂q
∂yq
Rζ (x, y)
∣∣∣∣
2
dy ≤ CA,s,q (1 + |x|)s+q−3/2 . (5.5)
P r o o f of Proposition 5.4. In view of equation (5.1), two extra derivatives
in (5.4) give the extra factor of order |y|2 + |ζ|2 to the integrand. Therefore, we
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4 389
M. Poplavskyi
start with I (s; 0). Since |Rζ (x, y)| ≤ CAe−cA|x−y|1/2
for x ≥ −A and ζ ∈ ΩA, we
split the integral from (5.5) into two parts
I (s; 0) =
∫
|y−x|<2|x|
+
∫
|y−x|>2|x|
≤ Cs (xs + |ζ|s)
∫
|Rζ (x, y)|2 dy
+CA
∫
t>2|x|
(t + x)s e−cAt1/2
dt. (5.6)
For the first integral we note that by the spectral theorem and the resolvent
identity,
∞∫
−∞
|Rζ (x, y)|2 dy =
=Rζ (x, x)
=ζ
. (5.7)
The asymptotic behaviour of ψ± from Proposition 5.3 implies
|Rζ (x, x)| ≤ CA (1 + |x|)−1/2 , and |=Rζ (x, x)| ≤ CA (1 + |x|)−3/2 . (5.8)
Combining (5.6) with (5.7) and (5.8), we obtain (5.5) with q = 0. In view of
equation (5.1), it is sufficient to prove (5.4) only for q = 0, 1. If q = 1, similarly
to the above argument, we split the integral into two parts. In the first term,
integrating by parts, we have
∞∫
−∞
∣∣∣∣
∂
∂y
Rζ (x, y)
∣∣∣∣
2
dy =
∞∫
−∞
(c1y + c2ζ) |Rζ (x, y)|2 dy.
The r.h.s satisfies the necessary bound for q = 1, hence the proposition is
proved.
Proposition 5.5. Let h = n−1/3, M =
[
C0n
1/2 log n
]
. Also, denote by xj =
jh the equidistant set and z
(1,2)
j = xj +δ
(1,2)
j h two shifted sets, with complex shifts∣∣∣δ(1,2)
j
∣∣∣ ≤ C for some absolute constant C. Then,
h
M∑
j=0
∣∣∣=Rζ
(
z
(1)
j , z
(2)
j
)∣∣∣ ≤ C, (5.9)
h
M∑
j=0
∣∣∣Rζ
(
z
(1)
j , z
(2)
j
)∣∣∣ ≤ C (Mh)1/2 , (5.10)
and for any non-negative integer p, d = 0 or 1 and k ≤ M
h
∞∑
j=0
|xj |p
∣∣∣∣
∂d
∂zd
Rζ
(
z
(1)
j , z
(2)
k
)∣∣∣∣
2
≤ C (1 + |xk|)p+d−3/2 . (5.11)
390 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 4
Universality at the Edge for Unitary Matrix Models
P r o o f of Proposition 5.5. Since
∣∣∣z(1,2)
j − xj
∣∣∣ = O (h), |=Rζ (x, x)| ≤
C (1 + |x|)−3/2 and derivatives of Rζ are bounded near the real line, we obtain
that ∣∣∣=Rζ
(
z
(1)
j , z
(2)
j
)∣∣∣ ≤ 2C (1 + |xj |)−3/2
for n > n0 with some integer n0. Hence,
h
M∑
j=0
∣∣∣=Rζ
(
z
(1)
j , z
(2)
j
)∣∣∣ ≤ Ch
M∑
j=0
(1 + |xj |)−3/2 ≤ C.
The second statement can be checked in a similar way. The proof of the third
statement consists of several steps. First, we change zj by xj in (5.11). The error
of this change is a combination of sums of higher derivatives with extra factors
h. These sums are small, because for zj far from zk these derivatives admit the
exponential bound, and for zj ∼ zk, in view of equation (5.1) and restriction
|zk| ≤ Cn1/6 log n, every two extra derivatives will give us the sum as in (5.11)
with the factor of order n−1/2 log n. After the change of zj by xj , we obtain the
sum which can be estimated by the integral
C
∞∫
0
xp
∣∣∣∣
∂d
∂zd
Rζ
(
x, z
(2)
k
)∣∣∣∣
2
dx,
because of the smoothness and exponential decreasing of Rζ . And finally, the
identity (5.7) and Proposition 5.4 yield (5.11). We used the identity (5.7) which
is valid for real x, but it remains valid for complex x because the l.h.s and r.h.s
of the (5.7) are entire functions equal at the real line.
Acknowledgement. The author is grateful to Prof. M.V. Shcherbina for
the problem statement and fruitful discussions.
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