# Moments of traces of circular beta-ensembles

###### Abstract

Let be random variables from Dyson’s circular -ensemble with probability density function . For each and , we obtain some inequalities on , where and is the power-sum symmetric function for partition . When , our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: for any and partitions ; for any and , where is the length of and is explicit on . These results apply to the three important ensembles: COE (), CUE () and CSE (). We further examine the nonasymptotic behavior of for . The central limit theorems of are obtained when (i) is a polynomial and is arbitrary, or (ii) has a Fourier expansion and . The main tool is the Jack function.

10.1214/14-AOP960 \volume43 \issue6 2015 \firstpage3279 \lastpage3336 \docsubtyFLA \newproclaimexampleExample

Moments of traces of circular beta-ensembles

A]\fnmsTiefeng \snmJiang\thanksrefT1label=e1]jiang040@umn.edu and B]\fnmsSho \snmMatsumoto\corref\thanksrefT2label=e2]sho-matsumoto@math.nagoya-u.ac.jplabel=e3]shom@sci.kagoshima-u.ac.jp \thankstextT1Supported in part by NSF Grants DMS-04-49365, DMS-12-08982 and DMS-14-06279. \thankstextT2Supported in part by JSPS Grant-in-Aid for Young Scientists (B) 25800062.

class=AMS] \kwd[Primary ]60B20 \kwd[; secondary ]15B52 \kwd05E05

Random matrix \kwdcircular beta-ensemble \kwdmoment \kwdJack function \kwdpartition \kwdHaar-invariance \kwdcentral limit theorem

## 1 Introduction

Let be an Haar-invariant unitary matrix, that is, the entries of unitary matrix are random variables satisfying that the probability distribution of the entries of is the same as that of and that of for any unitary matrix . Diaconis and Evans (Theorem 2.1 from [4]) proved that

(a) Consider and with . Then for ,

(1) |

where is Kronecker’s delta.

(b) For any positive integers and ,

(2) |

The idea of the proof is based on the group representation theory of unitary group . Some other derivations for (1) and (2) are given in [5, 23, 24, 25]. The right-hand side of (1) is evidently equal to where ’s are independent complex-normal random variables with for each .

Notice an Haar-invariant unitary matrix is also called a CUE, which belongs to the Circular Ensembles of three members: the Circular Orthogonal Ensemble (COE), the Circular Unitary Ensemble (CUE) and the Circular Symplectic Ensemble (CSE); see Figure 1 for the relationship, where the left circle consists of matrices which induce the Haar probability measure on the orthogonal group , Haar probability measure on the unitary group and Haar probability measure on the real symplectic group , respectively.

Let be the eigenvalues of an Haar-invariant unitary matrix, or equivalently, an CUE, it is known (see, e.g., [12, 22]) that the density function of is with , where

(3) |

with and for . The density function of for the COE is with , and that for the CSE is with .

The purpose of this paper is to study the analogues of (1) and (2) for the circular -ensembles with density function in (3) for any . Further, we develop the central limit theorems for functions of . Before stating the main results, we next introduce some background about the circular -ensembles.

The circular ensembles were first introduced by physicist Dyson [8, 9, 10] for the study of nuclear scattering data. In fact, as studied in [8], Dyson shows that the consideration of time reversal symmetry leading to the three Gaussian ensembles behaves equally well to unitary matrices. A time reversal symmetry requires that , no time reversal symmetry has no constraint, and a time reversal symmetry for a system with an odd number of spin particles requires , where denotes the quaternion dual. Choosing such matrices with a uniform probability then gives COE, CUE and CSE, respectively (see, e.g., [11, 22]). The entries of COE and CUE are asymptotically complex normal random variables when the sizes of the matrices are large [17, 16, 14].

Let be an Haar-invariant unitary matrix. As mentioned earlier, is also a CUE; the matrix gives a COE. Furthermore, the matrix gives a CSE when is even; see Chapter 9 from [22]. For the relations among the zonal polynomials, the Schur functions, the Gelfand pairs and the three circular ensembles; see, for example, Chapter VII in [20] or Section 2.7 in [1] for reference.

Now we consider the moments in (1) and (2) for the circular -ensembles. Taking in (3), that is, choosing such that it is an COE, by an elementary check in Lemma .1, we have

(4) |

for all . This suggests that, unlike the right-hand sides of (1) or (2) that are free of , the moments for the general circular -ensemble may depend on for . In fact, by using the Jack functions, we will soon see from (8) below that the second moment in (4) does depend on except , in which case is an CUE.

In this paper, we will first prove some inequalities on the moments in (1) and (2) for the circular -ensembles with arbitrary . In particular, some of our inequalities for recover the equality in (1) by Diaconis and Evans [4]. Further, we evaluate the limiting behavior by letting for the left-hand side in (1) and for the left-hand side in (2), respectively. Their limits exist and look quite similar to the right-hand sides of (1) and (2). Finally, we spend much effort to study the central limit theorems of for two situations: (a) is a polynomial and is arbitrary; (b) has a Fourier expansion and . The key to obtain (b) is the nonasymptotic behavior of for any and , which are analyzed in detail.

The method of the proof is the Jack functions. The main results are obtained by using their orthogonal properties and combinatorial structures.

From the studies in this paper, it is obvious to see the importance of understanding the circular -ensembles through the Jack functions. Realizing that the Jack functions are a special class of the Macdonald polynomials, we have obtained the analogue of the results in this paper in the setting of the Macdonald polynomials. These will be published elsewhere in the future.

The organization of the rest of the paper is as follows. We present the moment inequalities in Section 2 and their proofs are given in Section 4; the nonasymptotic behavior of and the central limit theorems are stated in Section 3 and their proofs are arranged in Section 5. In the Appendix, we prove (4) by two ways different from the method of the Jack functions. Some other explicit formulas of moments are also given in the same section.

## 2 Moment inequalities for circular beta-ensembles

Let be a partition, that is, the sequence is in nonincreasing order and only finite of ’s are nonzero. The weight of is . Denote by the multiplicity of in for each , and the length of : Recall the convention . Set

(5) |

Let be a partition, and

(6) |

for integer and indeterminates ’s. The function is called the power-sum symmetric function. For real number , integers and , define two constants and by

(7) | |||||

where is if , or otherwise. With these notation, we have one of main results as follows.

###### Theorem 1

Let and have density as in (3). Set and . For partitions and , the following hold: {longlist}[(a)]

If , then

If , then . If and , then

There exists a constant depending only on such that for any and , we have

Take in (a) and (b) of Theorem 1, then and . The two results recover the result of Diaconis and Evans in (1). Further, letting in (a) and (b) of Theorem 1, we see that and (depending on ) converge to ; letting in (c) of the theorem, then the last term in (c) goes to . So we obviously have the following results.

###### Corollary 1

Let the conditions be as in Theorem 1. Then, for any ,

Part (b) of the above corollary says that, as , the limit of does not depend on parameter , which is consistent with (2). We further take a careful examination on as and . Some upper bounds of are given in Propositions 1 and 2. By studying and in (7), we have the following corollary from Theorem 1.

###### Corollary 2

Let and be as in (3). Set and . Let and be partitions with and . If , then

The above results are in the forms of inequalities or limits. We actually derive an exact formula in Proposition 3 to compute for every partition . In general, it is not easy to evaluate this quantity for arbitrary , however, we are able to do so when is special. For instance, by using the exact formula we calculate the moment in (4) for any as follows.

For any ,

(8) |

The verification of this formula through Proposition 3 is provided in the Appendix. We also give , and in closed forms in the Appendix.

The main tool used in our proofs is the Jack functions. Diaconis and Evans [4] and Diaconis and Shahshahani [5] use the group representation theory to study (1) and (2) because is a compact Lie group. The situations for the Circular Orthogonal Ensembles () and the Circular Symplectic Ensembles () are different. In fact, the two ensembles are not groups.

The proofs of (1) and (2) involve with the Schur functions. The connection is that the irreducible characters of the unitary groups, when seen as symmetric functions in the eigenvalues, are given by Schur functions. Looking at Figure 1, an Haar-invariant unitary matrix is also a CUE. From the perspective of symmetric functions, the COE is connected to the zonal polynomials, and the CSE to symplectic zonal polynomials. The three functions are special cases of the Jack polynomial with and , respectively, where is a partition. See Section 4.1 for this or [20] for general properties of the Jack polynomials. By using the Jack functions, we are able to prove (a) and (b) in Theorem 1. Part (c) in the theorem is proved by evaluating the expectation/integral with respect to in (3) directly.

## 3 Central limit theorems for circular beta-ensembles

For the sake of precision, we replace appeared earlier with . Specifically, let follow the -circular ensemble with and the density function as in (3). According to our notation in previous sections, for any integer . In the paper, the symbol stands for the complex normal distribution generated by , where an are i.i.d. real random variables with the standard normal distribution . The first result is a CLT for general circular -ensemble.

###### Theorem 2 ((CLT for any -circular ensemble))

Let follow the -circular ensemble. Then, for fixed , the random vector converges weakly to as , where ’s are independent random variables with for each .

An immediate consequence of the theorem is as follows.

###### Corollary 3

Let follow the -circular ensemble. Let with fixed and for all . Set . Then converges weakly to as , where

We next study the central limit theorem when the function is not a polynomial. To avoid a lengthier paper, we only focus on the cases and . A discussion on the general case will be given later in this section. We first need to understand the variance of .

###### Proposition 1 ((Bound of variance on COE))

For all , and , there exists a universal constant such that

###### Proposition 2 ((Bound of variance on CSE))

Let . Then there exists a universal constant such that the following hold: {longlist}[(iii)]

for all and .

for all and .

for all where .

From (ii) and (iii), we see that is of the scale “” when and are not far from each other. It is known from (2) and Proposition 1 that for any , and , where is a universal constant. This together with (b) of Corollary 1 seems to suggest that the second moment for is always bounded by . Proposition 2 tells us a different story. However, (b) of Corollary 1 is indeed consistent with (i).

The proofs of Propositions 1 and 2 are very involved. We use the combinatorial structure (42) to understand the second moments. Major effort is devoted to analyzing (42) through (5) and (5.1).

Another way to calculate above variance is through the covariance of and by symmetry [see (4.2)], which again can be computed by using the two-point correlation function . The explicit form of is given in Proposition 13.2.2 from [11]. It seems very hard to estimate the variance by using the proposition. But it is possible in principle.

###### Theorem 3 ((CLT for COE))

Let follow the circular orthogonal ensemble (). Let satisfy . Then, converges weakly to the law of as , where has the law with

Obviously, if for all , then with , and hence .

###### Theorem 4 ((CLT for CSE))

Let follow the circular symplectic ensemble (). Let satisfy . Set . Then converges weakly to the law of as , where has the law with

Similar to the comment below Theorem 3, if for all , then with , and hence .

Though Proposition 2 says that is of scale “” when and are not far from each other, the variance of the limiting distribution in Theorem 4 is not affected by this fact. The variance is similar to those in the circular orthogonal and unitary ensemble ().

Diaconis and Evans [4] obtains the CLTs for the orthogonal groups, the unitary groups and the symplectic groups. Their tool is the identities in (1) and (2). Reviewing Corollary 2, we no longer have identities for any ; this increases much difficulty to get the corresponding CLTs. It is understandable because after all the three members in the classical compact groups have group structures in addition to their combinatorial ones. So the group representation theory can be possibly used in the paper by Diaconis and Evans. The general circular -ensemble loses the former property and has only the combinatorial structure.

Johansson in [18] further explores the convergence speed of to a normal distribution, where is fixed and is an Haar-invariant orthogonal, unitary or symplectic random matrix. He shows that the convergence rate is exponentially fast.

By Proposition 2, the condition “” in Theorem 4 can be slightly relaxed. For simplicity, we just leave it as it is. Also, the conditions “” and “” can be easily satisfied. For instance, the first condition is satisfied if and are of the order for some , and the second one is satisfied if and are of the order for some .

To study the number of eigenvalues falling in an arc of the unit circle in the complex plane, namely, with being a subset of , one needs to handle the Fourier expansion of the indicator function with . It is known from [4] that the coefficients and in the contexts of Theorems 3 and 4 are of scale . Our theorems do not cover this special case. By using a construction of the circular -ensemble, Killip [19] specifically considers this situation and obtains a CLT. The author does not investigate the general CLTs as treated in our Theorems 2, 3 or 4.

Finally, we provide some examples which satisfy the condition

They are the solutions of some classical partial differential equations. We leave readers for the trivial calculations of the means and the variances of the limiting normal distributions.

Let be defined on and satisfy the Laplace equation

where is a known function and is given. Let . The solution has a Poisson’s formula. It can also be expressed in the following Fourier series:

(9) |

for and , where ’s and ’s are obtained from the Fourier series of so that

See, for example, more details on page 160 from [26]. Clearly, if , then and . And the coefficients and in (9) are bounded by for . Then use the formulas and to transfer in (9) to the form of , where ’s and ’s are complex numbers. Fix . It is easy to see that and as . Theorems 3 and 4 can then be applied to get the CLT for for and , respectively.

Let be a function defined on . Consider the following heat equation with boundary conditions defined by

(10) |

where is a constant. Suppose for all . Then the solution of (10) is given by

See, for example, page 85 from [26]. If , then as . Similar to the previous example, we can write in the form of , where ’s and ’s are complex numbers with as . Theorems 3 and 4 can then be applied to obtain the CLT for with and , respectively.

To get the analogues of Theorems 3 and 4 for any , one needs to get upper bounds for as in Propositions 1 and 2. It will be even more involved because of the lack of classifications of partitions as in (45) for general , particularly for irrational . However, by using our method, it is possible to get upper bounds for any

## 4 Proofs of moment inequalities in Section 2

This section is divided into two parts. In Section 4.1, the necessary background of the Jack functions including their orthogonal properties and combinatorial structures are given. With this preparation, we prove parts (a) and (b) of Theorem 1 and Corollary 2. In Section 4.2, we prove part (c) of Theorem 1 by analysis.

### 4.1 Proofs of (a) and (b) of Theorem 1 and Corollary 2

For a partition , the notation represents the conjugate partition of , whose Young diagram is obtained by transposing the Young diagram of .

Let us review Jack symmetric functions briefly. We do not need the exact definition of Jack functions. In fact, their orthogonal properties are actively used here. For any real number and each integer , we denote by the algebra of symmetric functions of degree over the field . Recall power-sum symmetric function in (6). The family of over partitions of forms a basis on . A scalar product on is defined by

(11) |

for any partitions and of , where is as in (5). Set

(12) |

where runs over all cells of the Young diagram of . By definition, Jack functions form an orthogonal basis on and satisfy

(13) |

Since both power-sum symmetric functions and Jack functions form a basis of , they can be mutually expanded. Let denote the coefficient of in , that is,

(14) |

The ’s are real numbers. Inversely, let be the coefficient of in , that is,

(15) |

The coefficients ’s satisfy the following orthogonality relations ((10.31) and (10.32) from [20]):

(17) | |||||

In other words, if , then is an orthogonal matrix of size for . Here, is the number of partitions of . The following are some special cases of the Jack polynomials.

In other words, if , then is an orthogonal matrix of size for . Here, is the number of partitions of . The following are some special cases of the Jack polynomials.

Let be the Schur polynomial and the character value for the irreducible representation of the symmetric groups. It is well known that with as the hook-length product. Further, by (7.8) from Chapter VI of [20] and (15) that

Let . Then coincides with the zonal polynomial . By (2.13) and (2.16) from Chapter VII of [20], we have

with , where is the hook-length product of and is the value of the zonal spherical function of a Gelfand pair . Here, is the symmetric group and