Science in China Series A: MathematicsJul., 2009, Vol. 52, No. 7, 1467–1477www.scichina.com math.scichina.comwww.springerlink.com

Circular β ensembles, CMV representation, char-acteristic 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: characteristic polynomials, circular β ensembles, CMV representation, moment iden-

tity

MSC(2000): 15A52

1 Introduction and main results

Consider the following circular β ensemble with N points. It is a random process of N pointsdistributed on the unit circle T in the complex plane C with the joint probability density

pN (eiθ1 , . . . , eiθN ) =1

ZN,β

∏

j<k

|eiθj − eiθk |β, θ1, . . . , θN ∈ [0, 2π], (1.1)

where β > 0 is a model parameter and ZN,β is a normalization constant, which equals bySelberg’s integral formula

(2π)NΓ(1 + Nβ

2 )

(Γ(1 + β2 ))N

.

The more classic and well-established are spacial cases β = 1, 2, and 4. These three ensembleswere introduced by Dyson as early as in 1962 with a view to modifying Wigner’s treatment ofenergy level behavior in complex quantum systems. They correspond to random orthogonal,unitary and sympletic matrix models with the normalized Haar measure, respectively. Thereader is referred to [1] for remarkable facts and wide applications and various links to otherfields.

There has been an increasing interest in the study of circular β ensembles with generalparameter β in the past decade. For instance, Okounkov[2] studied the asymptotics of n-point correlations when β is rational, and one of crucial tools he used is orthogonality of

Received October 11, 2008; accepted April 22, 2009DOI: 10.1007/s11425-009-0099-2This 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

1468 Su Z G

Jack polynomials with respect to the probability measure pN . Johansson[3] and Forrester andKeating[4] pointed out that the Szego-type strong limit theorem still hold: suppose that f(eiθ) =∑∞

k=−∞ fkeikθ and the Fourier coefficients falling off fast enough, then

E

( N∏

j=1

ef(eiθj )

)= eNf0+

∑∞k=1 kfk f−k+o(1).

It is now natural to ask: does a matrix model exist for the circular β ensemble? By a matrixmodel we mean an ensemble of random matrices whose eigenvalues follow the desired law. Thisproblem was first posed by Dumitriu and Edelman[5]. In fact, they considered the problem ofconstructing random matrix models for general ensembles and successfully obtained tri-diagonalsparse 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 followingprobability measure on R

N :

pHN (x1, . . . , xN ) =1

ZN,H,β

∏

j<k

|xj − xk|βN∏

j=1

e−x2j/2, x1, . . . , xN ∈ (−∞,∞), (1.2)

where the normalization constant

ZN,H,β = (2π)N/2N∏

j=1

Γ(1 + β2 j)

Γ(1 + β2 )

.

Again when β = 1, 2 or 4, it arises as the eigenvalue distribution in the classical Gaussianensembles of random matrices theory. The matrix modelHβ Dumitriu and Edelman constructedfor every β > 0 is a random real symmetric, tri-diagonal matrix with independent entries whosedistribution is depicted as follows:

Hβ =1√2

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

N(0, 2) χ(N−1)β 0 0 0

χ(N−1)β N(0, 2) χ(N−1)β 0 0...

......

......

0 0 χ2β N(0, 2) χβ

0 0 0 χβ N(0, 2)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (1.3)

Similarly, the β-Laguerre ensemble is the probability measure on [0,∞)N :

pLN(x1, . . . , xN ) =1

ZN,L,β

∏

j<k

|xj − xk|βN∏

j=1

xaj e−xj , x1, . . . , xN ∈ [0,∞), (1.4)

where a > −1 and the normalization constant

ZN,L,β = 2NaN∏

j=1

Γ(1 + β2 j)Γ(a− β

2 (N − j))

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

B =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

χ2a 0 0 0 0

χ(N−1)β χ2a−β 0 0 0...

......

......

· · · · · · χ2β χ2a−β(N−2) 0

0 0 0 χβ χ2a−β(N−1)

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (1.5)

Dumitriu and Edelman proved that Lβ = BB′ is the tri-diagonal matrix whose eigenvaluesfollow 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] andincorporated the theory of polynomials orthogonal on the unit circle as well. To state theirconstruction, we need to introduce the CMV matrix, a class of unitary matrices recently inves-tigated by Cantero, Moral and Velazquez[1], and the Θν-distributed random variables. It turnsout that the CMV matrix play the same role among unitary matrices as Jacobi matrices in allHermitian matrices.

Let D be the open unit disk in C and T = ∂D the boundary of D. Given a sequence ofcoefficients 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 =

⎛

⎝αk ρk

ρk −αk

⎞

⎠

for 0 � k � N − 2, while Ξ−1 = 1 and ΞN−1 = αN−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 onthe unit circle. Doing the multiplication yields

CN =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

α0 α1ρ0 ρ1ρ0 0 0 · · ·ρ0 −α1α0 −ρ1α0 0 0 · · ·0 α2ρ1 −α2α1 α3ρ2 ρ3ρ2 · · ·0 ρ2ρ1 −ρ2α1 −α3α2 −ρ3α2 · · ·0 0 0 α4ρ3 −α4α3 · · ·0 0 0 −α4α3 −ρ4α3 · · ·· · · · · · · · · · · · · · · · · ·

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (1.6)

1470 Su Z G

Clearly, CN has a 4× 2 block structure and is “barely” five-diagonal, that is, only the elementsin those diagonals CN (k, k + j) with j = 0,±1,±2 are nonzero and half of the elements withj = ±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

〈e1, CkNe1〉 =∫

T

zkdμ (1.7)

for all integers k.Applying the Gram-Schmidt procedure to {1, z, . . . , zN−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

Szego[8], these polynomials obey a recurrence relation,

Φn+1(z) = zΦn(z) − αnΦ∗n(z), Φ∗

n+1(z) = Φ∗n(z) − αnzΦn(z) (1.8)

with Φ∗0 = Φ0 = 1. Here Φ∗

n(z) denotes the reversed polynomial:

Φn(z) =n∑

l=0

c(n)l zl =⇒ Φ∗

n(z) =n∑

l=0

c(n)n−lz

l. (1.9)

Equivalently,

Φ∗n(z) = znΦn

(1z

).

Secondly, the characteristic polynomial of CN can be expressed in terms of these monic poly-nomials. Specifically, let ΦN (z) = zΦN−1(z) − αN−1Φ∗

N−1(z), then

det(z − CN ) = ΦN (z). (1.10)

We refer the reader to [9, 10] for the proof of (1.10).As an immediate consequence, if we write ΦN (z) =

∑Nn=0 c

(N)n zn, then c

(N)n is the so-called

(N − n)-th secular coefficient of the characteristic polynomial (see [11, 12]). This suggests analternative argument for secular coefficients via orthogonal polynomials.

What is of interest to us is actually a random CMV matrix. We say that a complex randomvariable, X , with values in unit disk D, is Θν-distributed (ν > 1) if

Ef(X) =ν − 12π

∫

D

f(z)(1 − |z|2)(ν−3)/2d2z. (1.11)

X is by definition Θ1-distributed if it is uniformly distributed on T. Note that for ν > 1 aninteger, this can be realized by the following geometric way: if u = (u1, . . . , uν+1) is chosen atrandom from the unit sphere S

ν 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 argX are independent random vari-

ables with the following distributions

argX ∼ 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

EXk = 0, E|X |2k =k!Γ(ν+1

2 )Γ(k + ν+1

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 for0 � k � N − 1, let CN be a CMV matrix induced by αk’s. Then the eigenvalues of CN aredistributed according to the circular β ensemble (1.1).

The proof is based on the following two fundamental facts: (1) suppose that the spectralmeasure dμ corresponding to the (CN , e1) concentrate on N distinct points eiθ1 , . . . , eiθN withmasses μ1, . . . , μN , where

∑Ni=1 μi = 1, then

∏

j<k

|eiθj − eiθk |2N∏

j=1

μj =N−2∏

k=0

|1 − |αk|2|N−k−1;

(2) under the hypothesis of Theorem 1.1, (eiθ1 , . . . , eiθN ) is distributed as in (1.1) and (μ1, . . . , μN )

is distributed like∏Nj=1 μ

β2 −1j subjective to

∑Ni=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 aroundthe study of random matrices and put forward new challenge to mathematicians. Just recentlydid Killip[14] investigate the Gaussian fluctuations for the eigenvalue counting functions ofcircular β ensembles with the help of (1.10); Killip and Stoiciu[15] discussed the continuoustransition 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 ) invokingthe 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) > − 1

2 ,

E| det(z − CN )|2s =N−1∏

k=0

Γ(1 + β2 k)Γ(1 + 2s+ β

2k)

Γ(1 + s+ β2 k))

2; (1.14)

(2) for |s| < 1,

E

(det(z − CN)det(z − CN)

)s

= z2NsN−1∏

k=0

(Γ(1 + kβ2 ))2

Γ(1 + kβ2 + 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 interestedreader is referred to the book by Weber[17,Chapter 8].

Lemma 2.1. We have

1472 Su Z G

(1) for Re(s) > − 12 ,

∞∑

l=0

(s

l

)2

=Γ(1 + 2s)

(Γ(1 + s))2; (2.1)

(2) for s ∈ C,1

Γ(s)= seγs

∞∏

k=1

(1 +

s

k

)e−s/k, (2.2)

where γ = 0.57721 · · · , is the Euler-Mascheroni constant;(3) for |s| < 1,

∞∑

l=0

(s

l

)(−sl

)=

1Γ(1 + s)Γ(1 − s)

. (2.3)

Proof. (1) is a direct consequence of Legendre’s duplication formula and the following

∞∑

l=0

(s

l

)2

= 22sΓ(12 )Γ(1

2 + s)πΓ(1 + s)

.

(2) follows from the infinite product definition for Gamma function due to Weierstrass.(3) follows from (2.2) and

∞∑

l=0

(s

l

)(−sl

)=

∞∏

k=1

(1 − s2

k2

).

Lemma 2.2. Fix l � 0 an integer. We have for x � 1,

∞∑

m=l+1

(m− 1)!(m+ x) · · · (1 + x)

=l!

(l + x) · · ·x. (2.4)

Proof. It is sufficient to show for x � 1

∞∑

m=1

(m− 1)!(m+ x) · · · (1 + x)

=1x. (2.5)

Obviously, (2.5) is true when x = 1, that is,

∞∑

m=1

1m(m+ 1)

= 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

(m− 1)!(m+ 1 + x) · · · (2 + x)

=∞∑

m=1

m!(m+ 1 + x) · · · (1 + x)

=∞∑

m=1

(m− 1)!(m+ x) · · · (1 + x)

− 11 + x

,

Circular β ensembles, CMV representation, characteristic polynomials 1473

which in turn implies∞∑

m=1

(m− 1)!(m+ 1 + x) · · · (2 + x)

=1

1 + x. (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) − αN−1Φ∗N−1(z) and expand (ΦN (z))s. It follows

(ΦN (z))s =∞∑

l=0

(−1)l(s

l

)(zΦN−1(z))

(s−l) (αN−1Φ∗

N−1(z))l. (2.7)

Since |z| = 1, |Φ∗N (z)| = |ΦN (z)|. It is now easy to see

E|ΦN (z)|2s =∞∑

l=0

(s

l

)2

E|αN−1|2lE|ΦN−1(z)|2s.

A repeated argument leads to

E|ΦN (z)|2s =N−1∏

k=0

∞∑

l=0

(s

l

)2

E|αk|2l. (2.8)

Recall αk ∼ Θβ(N−k−1)+1, then (1.13) yields

E|αk|2l =l!

(l + β2 (N − k − 1)) · · · (1 + β

2 (N − k − 1)). (2.9)

It is sufficient to show that for any positive real number x

∞∑

l=0

(s

l

)2l!

(l + x) · · · (1 + x)=

Γ(1 + x)Γ(1 + x+ 2s)(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∑

l=0

(m

l

)2l!

(l + x) · · · (1 + x)=

Γ(1 + x)Γ(1 + x+ 2m)(Γ(1 + x+m))2

.

Equivalently,

m∑

l=0

(m

l

)2

l!(m+ x) · · · (l + 1 + x) = (2m+ x) · · · (m+ 1 + x). (2.11)

1474 Su Z G

Since both sides of (2.11) are polynomials in x, then we need only to show (2.11) holds true forany nonnegative integer q. Indeed, we have

m∑

l=0

(m

l

)2

l!(m+ q) · · · (l + 1 + q) =m!m∑

l=0

(m

l

)(m+ q

m− l

)

=m!(

2m+ q

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)(Γ(1 + x+ s))2

=∞∏

k=1

(k + x+ s)2

(k + x)(k + x+ 2s).

Thus it suffices to show that

∞∑

l=0

(s

l

)2l!

(l + x) · · · (1 + x)=

∞∏

k=1

(k + x+ s)2

(k + x)(k + x+ 2s). (2.12)

To this end, put

a(x) =∞∑

l=0

(s

l

)2l!

(l + x) · · · (1 + x).

Then we need only to show for each x � 0

a(x) − a(x+ 1) =s2

(1 + x)(1 + x+ 2s)· a(x+ 1). (2.13)

Indeed, it follows from (2.13)

a(x)a(x+ 1)

=(1 + x+ s)2

(1 + x)(1 + x+ 2s). (2.14)

Therefore, by the arbitrariness of x in (2.13),

a(x) =∞∏

k=1

a(x+ k − 1)a(x+ k)

=∞∏

k=1

(k + x+ s)2

(k + x)(k + x+ 2s).

It remains to proving (2.13). To this end, note

a(x) − a(x + 1)

=∞∑

l=1

s2

l2

l−1∏

i=1

(1 − s

i

)2[l!

(l + x) · · · (1 + x)− l!

(l + 1 + x) · · · (2 + x)

]

= s2∞∑

l=1

l−1∏

i=1

(1 − s

i

)2 (l − 1)!(l + 1 + x) · · · (1 + x)

=s2

1 + x

∞∑

l=1

l−1∏

i=1

(1 − s

i

)2 (l − 1)!(l + 1 + x) · · · (2 + x)

, (2.15)

Circular β ensembles, CMV representation, characteristic polynomials 1475

where by convention∏l−1i=1(1 − s

i )2 = 1 if l = 1.

We shall below prove for each x � 1,

∞∑

l=1

l−1∏

i=1

(1 − s

i

)2 (l − 1)!(l + x) · · · (1 + x)

=1

x+ 2s

{1 +

∞∑

l=1

s2

l2

l−1∏

i=1

(1 − s

i

)2l!

(l + x) · · · (1 + x)

}. (2.16)

Applying Lemma 2.2 yields

1x

+∞∑

l=1

(s2

l2− 2s

l

) l−1∏

i=1

(1 − s

i

)2l!

(l + x) · · · (1 + x)x

=∞∑

l=1

(l − 1)!(l + x) · · · (1 + x)

+∞∑

l=1

(s2

l2− 2s

l

) l−1∏

i=1

(1 − s

i

)2 ∞∑

m=l+1

(m− 1)!(m+ x) · · · (1 + x)

=∞∑

l=1

(l − 1)!(l + x) · · · (1 + x)

+∞∑

m=2

m−1∑

l=1

(s2

l2− 2s

l

) l−1∏

i=1

(1 − s

i

)2 (m− 1)!(m+ x) · · · (1 + x)

=1

1 + x+

∞∑

m=2

[1 +

m−1∑

l=1

(s2

l2− 2s

l

) l−1∏

i=1

(1 − s

i

)2] (m− 1)!(m+ x) · · · (1 + x)

=1

1 + x+

∞∑

l=2

l−1∏

i=1

(1 − s

i

)2 (l − 1)!(l + x) · · · (1 + x)

. (2.17)

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

E

(det(z − CN)det(z − CN)

)s= E

(ΦN(z)ΦN(z)

)s= z2s

∞∑

l=0

(s

l

)(−sl

)E|αN−1|2lE

(ΦN−1(z)ΦN−1(z)

)s.

A repeated argument leads to

E

(ΦN (z)ΦN (z)

)s= z2Ns

N−1∏

k=0

∞∑

l=0

(s

l

)(−sl

)E|αk|2l.

Note (2.12). It is sufficient to show for any nonnegative real number x

∞∑

l=0

(s

l

)(−sl

)l!

(l + x) · · · (1 + x)=

(Γ(1 + x))2

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) =∞∑

l=0

(s

l

)(−sl

)l!

(l + x) · · · (1 + x).

It suffices to show

b(x+ 1) − b(x) =s2

(x+ 1)2b(x+ 1). (2.19)

1476 Su Z G

Indeed, it directly follows from (2.19)

b(x)b(x+ 1)

= 1 − s2

(x+ 1)2.

Hence we have

b(x) =∞∏

k=1

b(x+ k − 1)b(x+ k)

=∞∏

k=1

(1 − s2

(x+ k)2

),

which together with (2.2) implies (2.18).It remains to proving (2.19). Observe

b(x+ 1) − b(x) =∞∑

l=1

s2

l2

l−1∏

i=1

(1 − s2

i2

) [l!

(l + x) · · · (1 + x)− l!

(l + 1 + x) · · · (2 + x)

]

= s2∞∑

l=1

l−1∏

i=1

(1 − s2

i2

)(l − 1)!

(l + 1 + x) · · · (1 + x)

=s2

1 + x

∞∑

l=1

l−1∏

i=1

(1 − s2

i2

)(l − 1)!

(l + 1 + x) · · · (2 + x),

where as above by convention∏l−1i=1(1 − s2

i2 ) = 1 if l = 1.Thus (2.19) holds true if we have for any x � 1

∞∑

l=1

l−1∏

i=1

(1 − s2

i2

)(l − 1)!

(l + x) · · · (1 + x)=

1x−

∞∑

l=1

s2

l2

l−1∏

i=1

(1 − s2

i2

)l!

(l + x) · · ·x. (2.20)

Use again (2.4). It follows

1x−

∞∑

l=1

s2

l2

l−1∏

i=1

(1 − s2

i2

)l!

(l + x) · · ·x

=∞∑

l=1

(l − 1)!(l + x) · · · (1 + x)

−∞∑

l=1

s2

l2

l−1∏

i=1

(1 − s2

i2

) ∞∑

m=l+1

(m− 1)!(m+ x) · · · (1 + x)

=∞∑

l=1

(l − 1)!(l + x) · · · (1 + x)

−∞∑

m=2

m−1∑

l=1

s2

l2

l−1∏

i=1

(1 − s2

i2

)(m− 1)!

(m+ x) · · · (1 + x)

=1

1 + x+

∞∑

m=2

[1 −

m−1∑

l=1

s2

l2

l−1∏

i=1

(1 − s2

i2

)](m− 1)!

(m+ x) · · · (1 + x)

=1

1 + x+

∞∑

m=2

m−1∏

i=1

(1 − s2

i2

)(m− 1)!

(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 forspecial case β = 2.

Also, let Bn(z) be the Blaschke product of degree n+ 1 defined by

Bn(z) = zΦn(z)Φ∗n(z)

, 0 � n � N (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 wechoose a branch for the logarithm. Note that ψn is an increasing function of θ, and so the numberof points lying in the arc [a, b] ⊂ (−π, π) is approximately 1

2π [ψN−1(b) − ψN−1(a)]; indeed theerror is plus or minus one. Moreover, ψN (θ) − (N + 1)θ is a sum of independent randomvariables. 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 eigenvaluecounting function. In fact, it easily follows for any θ ∈ [0, 2π],

Im log det(1 − e−iθCN) =12Im log

det(1 − e−iθCN )det(1 − e−iθCN )

=12Im log e−i(N+1)θ · eiθΦN (eiθ)

Φ∗N (eiθ)

=12Im log[e−i(N+1)θ ·BN (eiθ)]

=12(ψN (θ) − (N + 1)θ). (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 The author is grateful to Shao QiMan for useful discussions and re-marks.

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