Science in China Series A: Mathematics
Jul., 2009, Vol. 52, No. 7, 1467–1477
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Circular β ensembles, CMV representation, characteristic polynomials
SU ZhongGen
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
(email: suzhonggen@zju.edu.cn)
Abstract In this note we first briefly review some recent progress in the study of the circular β
ensemble on the unit circle, where β > 0 is a model parameter. In the special cases β = 1, 2 and 4,
this ensemble describes the joint probability density of eigenvalues of random orthogonal, unitary and
sympletic matrices, respectively. For general β, Killip and Nenciu discovered a five-diagonal sparse
matrix model, the CMV representation. This representation is new even in the case β = 2; and it has
become a powerful tool for studying the circular β ensemble. We then give an elegant derivation for
the moment identities of characteristic polynomials via the link with orthogonal polynomials on the
unit circle.
Keywords:
tity
MSC(2000):
1
characteristic polynomials, circular β ensembles, CMV representation, moment iden15A52
Introduction and main results
Consider the following circular β ensemble with N points. It is a random process of N points
distributed on the unit circle T in the complex plane C with the joint probability density
1 iθj
pN (eiθ1 , . . . , eiθN ) =
|e − eiθk |β , θ1 , . . . , θN ∈ [0, 2π],
(1.1)
ZN,β
j<k
where β > 0 is a model parameter and ZN,β is a normalization constant, which equals by
Selberg’s integral formula
Γ(1 + N2β )
(2π)N
.
(Γ(1 + β2 ))N
The more classic and well-established are spacial cases β = 1, 2, and 4. These three ensembles
were introduced by Dyson as early as in 1962 with a view to modifying Wigner’s treatment of
energy level behavior in complex quantum systems. They correspond to random orthogonal,
unitary and sympletic matrix models with the normalized Haar measure, respectively. The
reader is referred to [1] for remarkable facts and wide applications and various links to other
fields.
There has been an increasing interest in the study of circular β ensembles with general
parameter β in the past decade. For instance, Okounkov[2] studied the asymptotics of npoint correlations when β is rational, and one of crucial tools he used is orthogonality of
Received October 11, 2008; accepted April 22, 2009
DOI: 10.1007/s11425-009-0099-2
This work was supported by National Natural Science Foundation of China (Grant No. 10671176)
Citation: Su Z G. Circular β ensembles, CMV representation, characteristic polynomials. Sci China Ser A, 2009,
52(7): 1467–1477, DOI: 10.1007/s11425-009-0099-2
Su Z G
1468
Jack polynomials with respect to the probability measure pN . Johansson[3] and Forrester and
Keating[4] pointed out that the Szegö-type strong limit theorem still hold: suppose that f (eiθ ) =
∞
ˆ ikθ and the Fourier coefficients falling off fast enough, then
k=−∞ fk e
E
N
e
f (eiθj )
ˆ
= eN f0 +
∞
k=1
kfˆk fˆ−k +o(1)
.
j=1
It is now natural to ask: does a matrix model exist for the circular β ensemble? By a matrix
model we mean an ensemble of random matrices whose eigenvalues follow the desired law. This
problem was first posed by Dumitriu and Edelman[5] . In fact, they considered the problem of
constructing random matrix models for general ensembles and successfully obtained tri-diagonal
sparse matrix models for the β-Hermite and β-Laguerre ensembles. For reader’s convenience,
we simply describe their matrix models. Recall that the β-Hermite ensemble is the following
probability measure on RN :
pH
N (x1 , . . . , xN )
=
1
ZN,H,β
β
|xj − xk |
N
2
e−xj /2 ,
x1 , . . . , xN ∈ (−∞, ∞),
(1.2)
j=1
j<k
where the normalization constant
ZN,H,β = (2π)N/2
N
Γ(1 + β2 j)
j=1
Γ(1 + β2 )
.
Again when β = 1, 2 or 4, it arises as the eigenvalue distribution in the classical Gaussian
ensembles of random matrices theory. The matrix model Hβ Dumitriu and Edelman constructed
for every β > 0 is a random real symmetric, tri-diagonal matrix with independent entries whose
distribution is depicted as follows:
⎛
⎞
N (0, 2) χ(N −1)β
0
0
0
⎜
⎟
⎜
⎟
0
0 ⎟
⎜ χ(N −1)β N (0, 2) χ(N −1)β
⎜
⎟
1 ⎜
..
..
..
..
..
⎟
(1.3)
Hβ = √ ⎜
⎟.
.
.
.
.
.
⎜
⎟
2⎜
⎟
⎜
0
0
χ2β N (0, 2) χβ ⎟
⎝
⎠
0
0
0
χβ N (0, 2)
Similarly, the β-Laguerre ensemble is the probability measure on [0, ∞)N :
pL
N (x1 , . . . , xN ) =
1
ZN,L,β
|xj − xk |β
j<k
N
xaj e−xj ,
x1 , . . . , xN ∈ [0, ∞),
(1.4)
j=1
where a > −1 and the normalization constant
ZN,L,β = 2
Na
N
Γ(1 + β2 j)Γ(a − β2 (N − j))
j=1
Γ(1 + β2 )
.
Let B be an N ×N matrix with independent χ-distributed entries on the main and sub-diagonal
Circular β ensembles, CMV representation, characteristic polynomials
1469
and zero everywhere else as follows
⎞
⎛
χ2a
0
0
0
0
⎜
⎜
0
⎜ χ(N −1)β χ2a−β 0
⎜
..
..
..
..
⎜
B=⎜
.
.
.
.
⎜
⎜
⎜ ···
· · · χ2β χ2a−β(N −2)
⎝
0
0
0
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎠
0
..
.
0
(1.5)
χ2a−β(N −1)
χβ
Dumitriu and Edelman proved that Lβ = BB is the tri-diagonal matrix whose eigenvalues
follow the desired distribution (1.4).
However, Dumitriu and Edelman[5] did not give the matrix model for circular β ensembles.
It was later solved by Killip and Nenciu[6] . They followed rather closely the ideas of [5] and
incorporated the theory of polynomials orthogonal on the unit circle as well. To state their
construction, we need to introduce the CMV matrix, a class of unitary matrices recently investigated by Cantero, Moral and Velázquez[1] , and the Θν -distributed random variables. It turns
out that the CMV matrix play the same role among unitary matrices as Jacobi matrices in all
Hermitian matrices.
Let D be the open unit disk in C and T = ∂D the boundary of D. Given a sequence of
coefficients α0 , α1 , . . . , αN −2 in D and αN −1 ∈ T, let ρk = 1 − |αk |2 for 0 k N − 1. We
define 2 × 2 matrices
⎛
⎞
Ξk = ⎝
ᾱk ρk
ρk −αk
⎠
for 0 k N − 2, while Ξ−1 = 1 and ΞN −1 = ᾱN −1 . From these, we form an N × N
block-diagonal matrices
L = diag(Ξ0 , Ξ2 , Ξ4 , . . .),
M = diag(Ξ−1 , Ξ1 , Ξ3 , . . .).
The finite CMV matrix associated to the coefficients α0 , α1 , . . . , αN −1 is defined by
CN = LM.
As each of Ξk is unitary, so are L and M. As a result, CN is unitary and all eigenvalues lie on
the unit circle. Doing the multiplication yields
⎛
CN
ᾱ0 ᾱ1 ρ0
ρ1 ρ0
0
0
···
⎞
⎜
⎟
⎜ ρ −ᾱ α −ρ α
0
0 · · ·⎟
1 0
1 0
⎜ 0
⎟
⎜
⎟
⎜ 0 ᾱ ρ −ᾱ α ᾱ ρ
ρ3 ρ2 · · · ⎟
2 1
2 1
3 2
⎜
⎟
⎜
⎟
⎟
=⎜
⎜ 0 ρ2 ρ1 −ρ2 α1 −ᾱ3 α2 −ρ3 α2 · · · ⎟ .
⎜
⎟
⎜ 0
0
0
ᾱ4 ρ3 −ᾱ4 α3 · · · ⎟
⎜
⎟
⎜
⎟
⎜ 0
⎟
α
−ρ
α
·
·
·
0
0
−
ᾱ
4 3
4 3
⎝
⎠
··· ···
···
···
··· ···
(1.6)
Su Z G
1470
Clearly, CN has a 4 × 2 block structure and is “barely” five-diagonal, that is, only the elements
in those diagonals CN (k, k + j) with j = 0, ±1, ±2 are nonzero and half of the elements with
j = ±2 are zero.
Let e1 = (1, 0, . . . , 0) . Consider the spectral measure associated to the pair (CN , e1 ), that is,
the unique measure dμ on T that obeys
k
e1 =
z k dμ
(1.7)
e1 , CN
T
for all integers k.
Applying the Gram-Schmidt procedure to {1, z, . . . , z N −1 } leads to an orthogonal basis for
L2 (T, dμ) built of monic polynomials. We write 1 ≡ Φ0 (z), . . . , ΦN −1 (z) for these polynomials.
Two important properties carry over from the Jacobi matrix case. Firstly, as discovered by
Szegö[8] , these polynomials obey a recurrence relation,
Φn+1 (z) = zΦn (z) − ᾱn Φ∗n (z),
Φ∗n+1 (z) = Φ∗n (z) − αn zΦn (z)
(1.8)
with Φ∗0 = Φ0 = 1. Here Φ∗n (z) denotes the reversed polynomial:
Φn (z) =
n
l=0
(n)
cl z l =⇒ Φ∗n (z) =
Equivalently,
Φ∗n (z) = z n Φn
n
l=0
(n)
cn−l z l .
(1.9)
1
.
z̄
Secondly, the characteristic polynomial of CN can be expressed in terms of these monic polynomials. Specifically, let ΦN (z) = zΦN −1 (z) − ᾱN −1 Φ∗N −1 (z), then
det(z − CN ) = ΦN (z).
(1.10)
We refer the reader to [9, 10] for the proof of (1.10).
(N ) n
(N )
As an immediate consequence, if we write ΦN (z) = N
is the so-called
n=0 cn z , then cn
(N − n)-th secular coefficient of the characteristic polynomial (see [11, 12]). This suggests an
alternative argument for secular coefficients via orthogonal polynomials.
What is of interest to us is actually a random CMV matrix. We say that a complex random
variable, X, with values in unit disk D, is Θν -distributed (ν > 1) if
ν−1
Ef (X) =
f (z)(1 − |z|2 )(ν−3)/2 d2 z.
(1.11)
2π D
X is by definition Θ1 -distributed if it is uniformly distributed on T. Note that for ν > 1 an
integer, this can be realized by the following geometric way: if u = (u1 , . . . , uν+1 ) is chosen at
random from the unit sphere Sν in Rν+1 according to the usual surface measure, then u1 + iu2
is a Θν -distributed random variable.
It follows immediately from the definition that |X| and arg X are independent random variables with the following distributions
arg X ∼ U [0, 2π],
|X| ∼ p(r) ≡ (ν − 1)r(1 − r2 )(ν−3)/2 ,
0 r 1.
(1.12)
Circular β ensembles, CMV representation, characteristic polynomials
1471
In particular, we have for positive integer k
EX k = 0,
E|X|2k =
k!Γ( ν+1
2 )
ν+1 .
Γ(k + 2 )
(1.13)
The main result of [6] now reads as follows.
Theorem 1.1. Given β > 0, let αk ∼ Θβ(N −k−1)+1 be independent random variables for
0 k N − 1, let CN be a CMV matrix induced by αk ’s. Then the eigenvalues of CN are
distributed according to the circular β ensemble (1.1).
The proof is based on the following two fundamental facts: (1) suppose that the spectral
measure dμ corresponding to the (CN , e1 ) concentrate on N distinct points eiθ1 , . . . , eiθN with
N
masses μ1 , . . . , μN , where i=1 μi = 1, then
|eiθj − eiθk |2
N
μj =
j=1
j<k
N
−2
|1 − |αk |2 |N −k−1 ;
k=0
(2) under the hypothesis of Theorem 1.1, (eiθ1 , . . . , eiθN ) is distributed as in (1.1) and (μ1 , . . . , μN )
β
N
N
−1
is distributed like j=1 μj2
subjective to i=1 μi = 1, and moreover these two vectors are
independent of each other. The interested reader is referred to [10, 13] for more information.
We remark that Theorem 1.1 provides us a totally new matrix representation even for the
classical circular unitary ensemble (β = 2). This discovery has stimulated much activity around
the study of random matrices and put forward new challenge to mathematicians. Just recently
did Killip[14] investigate the Gaussian fluctuations for the eigenvalue counting functions of
circular β ensembles with the help of (1.10); Killip and Stoiciu[15] discussed the continuous
transition from Poisson to Clock statistics as the model parameter β varies from 0 to ∞.
In this note we derive moment identities of characteristic polynomials det(z − CN ) invoking
the equation (1.10) and recurrence relations (1.9). Our main result is as follows.
Theorem 1.2. Fix N 1, β > 0, define CN as in Theorem 1.1. Assume |z| = 1, then
(1) for Re(s) > − 12 ,
E| det(z − CN )|2s =
N
−1
Γ(1 + β2 k)Γ(1 + 2s + β2 k)
k=0
Γ(1 + s + β2 k))2
;
(1.14)
(2) for |s| < 1,
det(z − CN )
E
det(z − CN )
s
= z 2N s
N
−1
k=0
(Γ(1 +
Γ(1 +
kβ
2
kβ 2
2 ))
+ s)Γ(1 +
kβ
2
− s)
.
(1.15)
We remark that the special cases β = 1, 2 and 4 have been obtained by Keating and Snaith[16]
using Selberg’s integral.
2
Proof of Theorem 1.2
We begin with the following classical identities about the Gamma function. The interested
reader is referred to the book by Weber[17, Chapter 8] .
Lemma 2.1.
We have
Su Z G
1472
(1) for Re(s) > − 12 ,
∞ 2
s
l
l=0
(2) for s ∈ C,
=
Γ(1 + 2s)
;
(Γ(1 + s))2
∞ 1
s −s/k
= seγs
,
1+
e
Γ(s)
k
(2.1)
(2.2)
k=1
where γ = 0.57721 · · · , is the Euler-Mascheroni constant;
(3) for |s| < 1,
∞ s
−s
1
.
=
Γ(1
+
s)Γ(1
− s)
l
l
l=0
Proof.
(2.3)
(1) is a direct consequence of Legendre’s duplication formula and the following
∞ 2
s
= 22s
l
l=0
Γ( 12 )Γ( 12 + s)
.
πΓ(1 + s)
(2) follows from the infinite product definition for Gamma function due to Weierstrass.
(3) follows from (2.2) and
∞ s
−s
l=0
Lemma 2.2.
l
=
k=1
s2
1− 2 .
k
Fix l 0 an integer. We have for x 1,
∞
m=l+1
Proof.
l
∞ l!
(m − 1)!
=
.
(m + x) · · · (1 + x)
(l + x) · · · x
(2.4)
It is sufficient to show for x 1
∞
1
(m − 1)!
= .
(m + x) · · · (1 + x)
x
m=1
Obviously, (2.5) is true when x = 1, that is,
∞
1
= 1.
m(m + 1)
m=1
Note that
∞
(m−1)!
m=1 (m+x)···(1+x)
is convergent for an arbitrary x 1. We have
∞
∞
(m − 1)!
(m − 1)!
−
(m
+
x)
·
·
·
(1
+
x)
(m
+
1
+ x) · · · (2 + x)
m=1
m=1
=
=
∞
m!
(m
+
1
+
x)
· · · (1 + x)
m=1
∞
1
(m − 1)!
−
,
(m + x) · · · (1 + x) 1 + x
m=1
(2.5)
Circular β ensembles, CMV representation, characteristic polynomials
which in turn implies
1473
∞
1
(m − 1)!
=
.
(m
+
1
+
x)
·
·
·
(2
+
x)
1
+
x
m=1
(2.6)
This shows (2.5) for x 2, and so (2.4) follows by uniqueness property of analytic continuation.
Proof of Theorem 1.2.
(1) Using (1.10) we have
E| det(z − CN )|2s = E|ΦN (z)|2s .
Note that ΦN (z) = zΦN −1 (z) − ᾱN −1 Φ∗N −1 (z) and expand (ΦN (z))s . It follows
(ΦN (z))s =
∞
(−1)l
s
l
l=0
(s−l)
(zΦN −1 (z))
l
ᾱN −1 Φ∗N −1 (z) .
(2.7)
Since |z| = 1, |Φ∗N (z)| = |ΦN (z)|. It is now easy to see
E|ΦN (z)|2s =
∞ 2
s
l
l=0
E|αN −1 |2l E|ΦN −1 (z)|2s .
A repeated argument leads to
E|ΦN (z)|2s =
N
−1 ∞ s
k=0 l=0
l
2
E|αk |2l .
(2.8)
Recall αk ∼ Θβ(N −k−1)+1 , then (1.13) yields
E|αk |2l =
(l +
β
2 (N
l!
.
− k − 1)) · · · (1 + β2 (N − k − 1))
(2.9)
It is sufficient to show that for any positive real number x
∞ 2
s
l=0
l
Γ(1 + x)Γ(1 + x + 2s)
l!
=
.
(l + x) · · · (1 + x)
(Γ(1 + x + s))2
(2.10)
As a warm-up, let us first consider the case that s = m is a nonnegative integer. In this case
m
l
= 0 when l > m,
and so (2.10) becomes
m m 2
l=0
l
Γ(1 + x)Γ(1 + x + 2m)
l!
=
.
(l + x) · · · (1 + x)
(Γ(1 + x + m))2
Equivalently,
m m 2
l=0
l
l!(m + x) · · · (l + 1 + x) = (2m + x) · · · (m + 1 + x).
(2.11)
Su Z G
1474
Since both sides of (2.11) are polynomials in x, then we need only to show (2.11) holds true for
any nonnegative integer q. Indeed, we have
m m 2
l
l=0
l!(m + q) · · · (l + 1 + q) = m!
m m
m+q
l=0
l
m−l
2m + q
= m!
m
= (2m + q) · · · (m + 1 + q).
Next let us turn to the general case Re(s) > − 12 . Obviously, (2.1) implies that (2.10) is valid
for x = 0.
For x > 0, we shall use (2.2), from which it follows
∞
Γ(1 + x)Γ(1 + x + 2s)
(k + x + s)2
.
=
(Γ(1 + x + s))2
(k + x)(k + x + 2s)
k=1
Thus it suffices to show that
∞ 2
s
l=0
l
∞
l!
(k + x + s)2
=
.
(l + x) · · · (1 + x)
(k + x)(k + x + 2s)
(2.12)
k=1
To this end, put
a(x) =
∞ 2
s
l
l=0
l!
.
(l + x) · · · (1 + x)
Then we need only to show for each x 0
a(x) − a(x + 1) =
s2
· a(x + 1).
(1 + x)(1 + x + 2s)
(2.13)
Indeed, it follows from (2.13)
a(x)
(1 + x + s)2
=
.
a(x + 1)
(1 + x)(1 + x + 2s)
(2.14)
Therefore, by the arbitrariness of x in (2.13),
a(x) =
∞
∞
a(x + k − 1)
(k + x + s)2
=
.
a(x + k)
(k + x)(k + x + 2s)
k=1
k=1
It remains to proving (2.13). To this end, note
a(x) − a(x + 1)
2 l−1 ∞
l!
l!
s2 s
−
1
−
l2 i=1
i
(l + x) · · · (1 + x) (l + 1 + x) · · · (2 + x)
l=1
∞ l−1
2
(l − 1)!
s
= s2
1−
i (l + 1 + x) · · · (1 + x)
l=1 i=1
2
∞ l−1
(l − 1)!
s2 s
,
=
1−
1+x
i (l + 1 + x) · · · (2 + x)
i=1
=
l=1
(2.15)
Circular β ensembles, CMV representation, characteristic polynomials
1475
l−1
where by convention i=1 (1 − si )2 = 1 if l = 1.
We shall below prove for each x 1,
2
l−1 ∞ (l − 1)!
s
1−
i (l + x) · · · (1 + x)
l=1 i=1
2
∞
l−1 s2 l!
1
s
=
1−
1+
.
x + 2s
l2 i=1
i (l + x) · · · (1 + x)
(2.16)
l=1
Applying Lemma 2.2 yields
∞
1 +
x
l=1
l−1
2
l!
s
1−
i (l + x) · · · (1 + x)x
i=1
l−1 2 ∞
∞ 2
∞
s
(l − 1)!
(m − 1)!
2s s
+
=
−
1
−
2
(l + x) · · · (1 + x)
l
l i=1
i
(m + x) · · · (1 + x)
s2
2s
−
2
l
l
l=1
l=1
m=l+1
∞ m−1
s2
l−1 2
(l − 1)!
(m − 1)!
2s s
+
−
=
1−
2
(l + x) · · · (1 + x) m=2
l
l i=1
i (m + x) · · · (1 + x)
l=1
l=1
∞
m−1
l−1
2
s2
(m − 1)!
2s 1
s
+
−
=
1+
1
−
1 + x m=2
l2
l i=1
i
(m + x) · · · (1 + x)
l=1
2
∞ l−1
(l − 1)!
1
s
+
.
(2.17)
=
1−
1+x
i
(l
+
x)
· · · (1 + x)
i=1
∞
l=2
Some simple algebra yields (2.16). The proof of (1.14) is now completed.
(2) The proof is very similar to that of (1.14). Use again (1.10) and (2.7). Then we have
det(z − CN )
E ¯
det(z − CN )
s
ΦN (z)
=E
Φ̄N (z)
s
=z
2s
s
∞ s
−s
ΦN −1 (z)
2l
.
E|αN −1 | E
Φ̄N −1 (z)
l
l
l=0
A repeated argument leads to
E
ΦN (z)
Φ̄N (z)
s
= z 2N s
∞ N
−1 s
k=0 l=0
l
−s
l
E|αk |2l .
Note (2.12). It is sufficient to show for any nonnegative real number x
∞ s
−s
l=0
l
l
l!
(Γ(1 + x))2
=
.
(l + x) · · · (1 + x)
Γ(1 + x + s)Γ(1 + x − s)
(2.18)
Obviously, in virtue of (2.2), (2.18) is valid for x = 0. Define for any x 0
b(x) =
∞ s
−s
l=0
l
l
It suffices to show
b(x + 1) − b(x) =
l!
.
(l + x) · · · (1 + x)
s2
b(x + 1).
(x + 1)2
(2.19)
Su Z G
1476
Indeed, it directly follows from (2.19)
s2
b(x)
=1−
.
b(x + 1)
(x + 1)2
Hence we have
b(x) =
∞
∞ b(x + k − 1)
s2
=
1−
,
b(x + k)
(x + k)2
k=1
k=1
which together with (2.2) implies (2.18).
It remains to proving (2.19). Observe
l−1 ∞
l!
l!
s2 s2
b(x + 1) − b(x) =
−
1− 2
l2 i=1
i
(l + x) · · · (1 + x) (l + 1 + x) · · · (2 + x)
l=1
∞ l−1
(l − 1)!
s2
= s2
1− 2
i
(l
+
1
+
x) · · · (1 + x)
l=1 i=1
∞ l−1 (l − 1)!
s2 s2
,
=
1− 2
1+x
i
(l
+
1
+
x) · · · (2 + x)
i=1
l=1
l−1
2
where as above by convention i=1 (1 − si2 ) = 1 if l = 1.
Thus (2.19) holds true if we have for any x 1
l−1 l−1 ∞ ∞
(l − 1)!
l!
1 s2 s2
s2
= −
.
1− 2
1− 2
2
i (l + x) · · · (1 + x)
x
l i=1
i (l + x) · · · x
i=1
l=1
(2.20)
l=1
Use again (2.4). It follows
l−1 ∞
l!
1 s2 s2
−
1
−
x
l2 i=1
i2 (l + x) · · · x
l=1
=
∞
l=1
l−1 ∞
∞
s2
(l − 1)!
s2 (m − 1)!
−
1
−
(l + x) · · · (1 + x)
l2 i=1
i2
(m + x) · · · (1 + x)
l=1
∞ m−1
l−1 2 m=l+1
(m − 1)!
(l − 1)!
s
s2
−
1
−
(l + x) · · · (1 + x) m=2
l2 i=1
i2 (m + x) · · · (1 + x)
l=1
l=1
l−1
∞
m−1
2
s (m − 1)!
1
s2
+
=
1−
1
−
2
2
1 + x m=2
l i=1
i
(m + x) · · · (1 + x)
l=1
∞ m−1
(m − 1)!
1
s2
=
+
.
1− 2
1 + x m=2 i=1
i (m + x) · · · (1 + x)
=
∞
(2.21)
Now some simple algebra yields (2.20). So we complete the proof of (1.15).
To conclude the note, let us point out that one can easily derive from Theorem 1.2 the central
limit theorem for Re log det(z − CN ) and Imlog det(z − CN ) as Keating and Snaith[16] did for
special case β = 2.
Also, let Bn (z) be the Blaschke product of degree n + 1 defined by
Bn (z) = z
Φn (z)
,
Φ∗n (z)
0nN
(2.22)
Circular β ensembles, CMV representation, characteristic polynomials
1477
and introduce random continuous functions ψn : (−π, π) → R via Bn (eiθ ) = eiψn (θ) . Here we
choose a branch for the logarithm. Note that ψn is an increasing function of θ, and so the number
1
of points lying in the arc [a, b] ⊂ (−π, π) is approximately 2π
[ψN −1 (b) − ψN −1 (a)]; indeed the
error is plus or minus one. Moreover, ψN (θ) − (N + 1)θ is a sum of independent random
variables. Based on this argument, Killip[14] demonstrated that it has Gaussian behavior.
We remark that there is a nice link between the characteristic polynomial and the eigenvalue
counting function. In fact, it easily follows for any θ ∈ [0, 2π],
Im log det(1 − e−iθ CN ) =
det(1 − e−iθ CN )
1
Im log
2
det(1 − e−iθ CN )
ΦN (eiθ )
1
Im log e−i(N +1)θ · eiθ ∗ iθ
2
ΦN (e )
1
= Im log[e−i(N +1)θ · BN (eiθ )]
2
1
= (ψN (θ) − (N + 1)θ).
2
=
(2.23)
The above argument shows that one can establish the central limit theorem for Im log det(z−CN )
from the CMV representation.
Acknowledgements
marks.
The author is grateful to Shao QiMan for useful discussions and re-
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