Moments of the eigenvalue densities and of the secular coefficients of $\beta$-ensembles

We compute explicit formulae for the moments of the densities of the eigenvalues of the classical $\beta$-ensembles for finite matrix dimension as well as the expectation values of the coefficients of the characteristic polynomials. In particular, the moments are linear combinations of averages of Jack polynomials, whose coefficients are related to specific examples of Jack characters.


Introduction
The density of the eigenvalues is of particular importance in the study of random matrices for its intrinsic theoretical interest as well as its many applications to various areas of physics. One of the main reasons is that the fluctuations of the eigenvalues around the limiting density manifest on a global scale of the spectrum in the properties of their linear statistics (see, e.g. [19,21,30]), which play a primary role whenever a mathematical or physical problem requires a probabilistic analysis.
Recently there has been a surge of interest in computing the expectation value where X is a × N N Hermitian matrix belonging to an appropriate ensemble E. Equation (1) can also be interpreted as the moments of the eigenvalue density. There is an extensive literature on the average (1) as well as on its large N asymptotics when E is one of the Gaussian, Laguerre or Jacobi ensembles of real, complex or quaternion matrices, usually labelled by β = 1, 2, 4. Indeed, the moments of the density of the eigenvalues of the Gaussian unitary ensemble (GUE) have a particular important combinatorial meaning: they count certain graphs embedded on surfaces of a given genus g. The 2gth coefficient in the asymptotic expansion of (1) for large N is the number of pairings of 2g vertices in a regular polygon. This idea was pioneered by Brézin et al [6], and has since played a prominent role in quantum field theory (see, e.g. [5] and references therein). When E is the Jacobi or Laguerre ensemble with β = 1, 2, 4 (1) many important properties of the electrical conductance and the Wigner delay time in chaotic quant um cavities can be extracted from the averages (1) [10,11,37,39,40,43,48].
In this article we shall derive explicit formulae for the moments of the spectral densities of the Gaussian, Laguerre and Jacobi β-ensembles as well as the averages of the secular coefficients of the characteristic polynomials. The joint probability density function (j.p.d.f.) of the eigenvalues is defined by where β can be any strictly positive real number and x x e , , G aussian -ensemble, e , , 1 , L aguerre -ensemble, 1 , 0, 1 , , 1 , Jacobi -ensemble.
The normalization constants β N can be evaluated explicitly using Selberg's integral. (For more details see, e.g. [23, section 4.7].) The β-ensembles were introduced by Dumitriu and Edelman [18], and by Dumitriu [17], who developed tridiagonal matrix models for the Gaussian and Laguerre β-ensembles; Lippert [36], and Killip and Nenciu [33] discovered a sparse matrix model for the Jacobi β-ensemble. Dumitriu et al [20] designed a computer algorithm that calculates the Jack polynomials and their averages in terms of other standard symmetric polynomials; this software can be used to compute the moments (1) iteratively. Recent articles [2-4, 7, 8, 24, 41, 49, 50] have studied the large N expansions of the resolvent of β-ensembles using the loop equations formalism; in turn, this technique leads to a large N expansion of the moments [49,50]. Cunden et al [9] proved a formula for the covariances of the moments of one-cut β-ensembles in the limit → ∞ N . Fyodorov et al [26] used the moments of the Gaussian β-ensemble to compute a phase transition in the distribution of the velocities of a one-dimensional turbulent fluid satisfying the Burger equation. Interestingly, in a recent paper Fyodorov and Le Doussal [25] showed that the moments of the Jacobi β-ensemble play a central role in the theory of the maximum of the GUE characteristic polynomials and log-correlated Gaussian processes, which in the past few years have received much attention.
In this article we will prove that the averages (1) can be expressed as finite sums in terms of averages of Jack polynomials ( ) where λ j is a combinatorial factor which will be defined in section 2, equation (9c). The expectation values in (4) are with respect one of the j.p.d.f.'s (3) with the identification / α β = 2 4 and were computed by Kadell [31], and Baker and Forrester [1]; one of the main contributions of this paper is the derivation of explicit formulae for (4). The sum in (4) is over all the partitions of ( ) The formalism of the theory of symmetric functions is very powerful in studying βensembles. As a corollary of the results on the moments, we compute the secular coefficients of the expectation value of the characteristic polynomial. More precisely, consider We show that is the Pochhammer symbol. Haake et al [28] and later Diaconis and Gamburd [12] computed all the joint moments of the secular coefficients of the characteristic polynomial of a Haar distributed random unitary matrix. It is far from obvious at this stage how to determine all the joint moments of the traces and of the secular coefficients for β-ensembles, as it seems beyond techniques available at present; we discuss the reasons in section 3 and in detail in section 7.
The structure of this paper is the following: section 2 contains the definition and basic properties of the Jack polynomials; in section 3 we introduce the Jack characters; in section 4 we compute the coefficients of the expansion (4); section 5 gives the explicit formulae of the moments for the Laguerre, Jacobi and Gaussian β-ensembles; in section 6 we detail how to compute the negative moments; section 7 presents an alternative derivation of the Jack characters, which illustrates the challenges one encounters when computing the joint moments; in section 8 we compute the secular coefficients of the average of the characteristic polynomials; section 9 ends the paper with concluding remarks and an outlook on future research.

Partitions and Jack polynomials
Before discussing our results we need to introduce some notions from the theory of symmetric functions. 4 This notation is not very common in the random matrix theory literature, but it is more convenient in the theory of symmetric functions. Because of the extensive use of Jack polynomials, we shall adopt the parametrization / α β = 2 throughout the paper.

Partitions and Ferrers diagrams
Let λ be a partition of k and let ( ) λ be the length of λ, i.e. the number of parts of λ different from zero. It is sometimes convenient to represent λ with a Ferrers diagram, which is a table of k boxes arranged in ( ) λ left-justified rows: the first row contains λ 1 boxes, the second λ 2 , and so on. We Formally a Ferrers diagram of λ can be defined as the set of points if the box s = (i, j) belongs to the diagram of λ. The conjugate λ′ of λ is the partition whose Ferrers diagram is the transpose of λ, i.e.
Note that The arm length is the number of boxes to the right of s = (i, j); the leg length is the number of boxes below s. Similarly, the co-arm length ( ) = − ′ λ a s j 1 and co-leg length ( ) = − ′ λ l s i 1 are the number of boxes to the left of and above s, respectively. These definitions are visualized in figure 2 in terms of Ferrers diagrams. The parameter α > 0 is the same that defines the scalar product (21).
It is often convenient to define the following quantities: In this paper we will often use the multivariate generalization of the Pochhammer symbol, We introduce a total ordering in the set of partitions of an integer k by saying that λ µ > whenever λ µ − i i is strictly positive for the first index i such that λ µ ≠ i i . This is known as lexicographical ordering. For example, . Another ordering on the set of partitions of particular relevance to the the theory of the Jack polynomials is the dominance, or natural, ordering. We say that If any of the inequalities is strict we write λ µ ≺ . It is worth emphasizing that both the lexicographical and dominance orderings compare partitions of the same integer. The dominance ordering is a partial ordering as soon as ⩾ λ µ = 6; for example, the partitions (4,1,1) and (3,3) cannot be compared. If ⪯ λ µ then ⩽ λ µ, but the opposite is not necessarily true.

Jack polynomials
In the following we shall adopt the notation . Let λ k and ( ) ⩽ λ = m N, the Jack polynomial ( ) ( ) λ α C x is a symmetric, homogeneous polynomial that satisfies the following properties: (iii) The normalization of ( ) λ α C is fixed by the condition It can be shown that the statements (i)-(iii) define the Jack polynomials uniquely. A nice proof is presented in Muirhead [42, section 7.2.1] for the Zonal polynomials (α = 2), but it can be easily generalized to any α > 0. The operator ( ) ∆ α is (up to a similarity transformation) the Hamiltonian of a Calogero-Sutherland quantum many-body system (see [1] for the details). The polynomials ( ) λ α C are non-degenerate eigenfunctions of ( ) ∆ α with eigenvalues A formula that will be useful in the rest of the paper is  [20, section 2]. The 'C' definition that we adopt is more natural for studying β-ensembles and Selberg-type integrals, as it appears in the theory of the scalar hypergeometric functions of matrix argument. The other two common definitions are the 'P' and 'J' normalizations. In the 'P' definition the coefficients of the monomial of highest weight is required to be one; the 'J' normalization sets the coefficient of the monomial x x k 1 (known as the trailing coefficient) to k !, where λ = k. Their relations to the 'C' definition are A closed formula for the Jack polynomials does not exist, but they can be computed using certain recurrence relations involving the monomial symmetric functions It turns out that The coefficients λσ α u can be calculated recursively and can be used to construct the Jack polynomials explicitly from (19) [20,38].
Traces of matrices are particular cases of power sum symmetric functions, Denote by Λ N k the ring of homogeneous symmetric polynomials of degree k in N variables.
The Jack polynomials and the power sum symmetric functions form two sets of bases in Λ N k .
We can define the scalar product and r j denotes the number of times j appears in the partition λ. The power sum symmetric functions play a prominent role in the theory of the Jack polynomials due to the following orthogonality relation One can show that equations (19) and (22)

Jack characters
Since the Jack polynomials and the power sum symmetric functions both form bases in Λ N k , they are related by a linear transformation. The coefficients that express the Jack polynomials in terms of the power sum symmetric functions are known as Jack characters and play an important role in combinatorics. Using standard notation we write In this paper we are interested in the inverse transformation, namely Writing these maps in the 'C' normalization gives In the context of random matrix theory (RMT) the averages of the µ p 's are the joint moments The RHS of equation (26b) depends on the eigenvalues only through the Jack polynomials λ α C . Therefore, in order to compute their expectation values we need to determine the coeffi- , which are independent of the ensemble. The knowledge of the coefficients ( ) κ α µ λ is equivalent to that of the Jack characters ( ) θ α µ λ , which at present is beyond reach in its full generality. Nevertheless, we can ask if we can compute few particular cases. The first can be obtained trivially from the normalization condition (14) and formula (26b), which give In this paper we will compute two other particular cases. We will find an explicit formula for ( ) ( ) κ α λ k 5 , which allows us to determine the expectation values of In addition, in section 7 we will be able to evaluate the average of Unfortunately, the methods applied to these two examples do not seem to extend to the general case.
In order to understand the difficulties involved in computing (27) (29) we obtain a sum involving the products ( ) ( ) λ α µ α C C . Then, there are two possible approaches. The product ( ) ( ) λ α µ α C C is a homogeneous symmetric polynomial, which can be written in the basis of Jack polynomials; if we can compute the coefficients of this linear combination, we can then average the resulting sum using the formulae in section 5.3. Alternatively, we could attempt to average ( ) ( ) λ α µ α C C directly. In both cases we would have to compute the coefficients of the linear combination or equivalently the scalar product ( ) ( ) ( ) In the special case of the Schur functions, i.e. α = 1, the µν λ c reduce to the Littlewood-Richardson coefficients. It turns out that this problem is equivalent to that of computing the ( ) θ α λ µ 's (see, e.g. Stanley [46]). We discuss the technical challenges encountered in this problem in detail in section 7.
The study of the Jack characters was initiated by Hanlon [29], who conjectured a first combinatorial interpretation; Stanley [46] proved various properties, among which he derived explicit formulae for few specific cases. Recently there has been a surge of interest in the Jack characters in the combinatorics literature [14-16, 22, 32, 34, 35, 47], as they play a central role in in the theory of symmetric functions. In particular, Dołęga and Féray [15] showed that they are polynomials in α with rational coefficients; Kanunnikov and Vassilieva [32] proved a recurrence relation for them. At present a general expression for arbitrary partitions λ µ , and any α > 0 is still lacking.
where are the χ µ λ 's are the characters of the irreducible representations of the symmetric group S m (see, e.g. [38, section I.7]) 6 . This leads to The inverse relation of (33) is When α = 2 the Jack polynomials reduce to the Zonal polynamials and the ( ) θ µ λ 2 are known as Zonal characters. Féray and Śniady [22] expressed the Zonal characters as sums over pair-partitions. A consequence of the invariance of the j.p.d.f.'s (2) under permutations of the eigenvalues is that the theory of symmetric functions appears naturally in many areas of RMT, whenever an algebraic or combinatorial structure of the ensemble can be exploited by the formalism of symmetric functions. For example, formulae (33) and (35), together with the fact that the Schur functions are the characters of the unitary group, were used by Diaconis and Shahshahani [13] to prove that the joint moments of traces of Haar distributed unitary matrices are those of independent standard complex random variables. Notwithstanding their name, when α ≠ 1 the Jack characters are not associated to any group. However, their averages with respect to the j.p.d.f.'s (3) are known [1]; these formulae combined with the theory of the Jack polynomials and of special functions of matrix argument have allowed us to compute explicit formulae for the coefficients ( ) and hence for the moments (1).

The coefficients
Writing in terms of the coordinates in the Ferrers diagram s = (i, j) and using ( ) = − ′ λ a s j 1 and ( ) = − ′λ l s i 1 gives the equivalent formula Consider the generating function 6 The irreducible representation of S m are labelled by the partitions of m. The notation χ µ λ denotes the character of the irreducible representation λ evaluated on permutations of cycle-type μ.
We now write the RHS as and use a generalization of the binomial theorem (see Forrester [23, proposition 13 , and a is a real parameter. The LHS denotes a hypergeometric function of two sets of variables, which can be expressed in terms of the series This is a particular case of the more general hypergeometric function Since the Jack polynomials are homogeneous ( ) ( ) (42) and (40) gives Comparing the coefficients of the powers of t in (39) and (44) leads to The generalised Pochhammer's symbol ( ) − λ α u is a polynomial in u, whose coefficient of the linear term gives the limit (45). Finally, from equation (10) and the fact that for any partition which inserted into (45) gives (36).
In the combinatorics literature, the objects of interest are the coefficients ( ) θ α µ λ in the expansion (16), rather than their inverse ( ) κ α µ λ . The two quantities, however, are connected by a simple relation of proportionality. This follows from the observation that both the λ p 's and ( ) λ α J 's are orthogonal with respect to the scalar product (21) and By definition the ( ) κ α λ µ is the inverse of the transfer matrix of ( ) θ α λ µ , therefore When ( ) λ = k , ( ) λ = 1 and = λ z k; therefore, equation (36) gives can also be found in [46] and in the book by Forrester-equation (12.145) of [23] is the equivalent of (29) with the coefficients ( ) ( ) κ α λ k explicitly given by quantities defined in earlier sections of that reference.

The moments of β-ensembles
In section 4 we computed the coefficients of the expansions of the symmetric polynomials p k in terms of the Jack symmetric functions. In order to compute the moments of the eigenvalue densities we need the expectation values of the Jack polynomials, which were obtained by Kadell [31, theorem 1] for the Jacobi β-ensemble and by Baker and Forrester [1, corollaries 3.2 and 4.1] for the Gaussian and Laguerre β-ensembles. Averages with respect to the measures of the β-ensembles are evaluated by introducing multivariate generalizations of the classical Hermite, Laguerre and Jacobi polynomials. We briefly summarise their basic definitions and properties in appendices A, B and C.

Laguerre β-ensemble
The classical Laguerre polynomials can be generalized to multivariate homogeneous polynomials that are orthogonal with respect to the measure of the Laguerre β-ensemble. The theories of the Jack and of the multivariate Laguerre polynomials are intertwined, since the Laguerre polynomials can be expressed as linear combinations of the Jack polynomials (see equation (A.6), appendix A). It turns out that where ( ) α N L is the normalization constant of the Laguerre ensemble, 0 is the origin in R N and . When N = 1 this average reduces to the classical formula ( ) ( Finally equation (49) gives where the coefficients ( ) ( ) κ α λ k are given in equation (36) and ( ) ( ) λ α C 1 N in equation (16). This is one of the main results of this paper. Formula (50) gives an explicit and self-contained expression for the moments of the LβE.

Gaussian β-ensemble
The calculation of the moments of the Gaussian β-ensemble follows a similar pattern. When λ is even we have where ( ) λ α H x are the multivariate Hermite polynomials (see appendix C). When λ is odd the symmetry of the integrand implies that the average (54) is zero. When N = 1 the expectation values can be written as where the H k (x)'s are the classical Hermite polynomials. The moments are To our knowledge an explicit formula for ( ) λ α H 0 does not exist; however, in appendix D we present a proof by Brian Winn of a particular case, namely This formula will become useful in the next section. For general partition ( ) λ α H 0 can be evaluated using the Maple routine MOPS [20], which computes Jack polynomials and multivariate Laguerre, Jacobi and Hermite polynomials symbolically.

Negative moments of β-ensembles
It was brought to our attention during the writing of this paper that it is also possible to calculate some explicit formulae for the negative integer moments of the Laguerre and Jacobi β-ensembles. Formulae for negative moments of the Jacobi β-ensemble first appeared in [25]. We outline here a simple derivation for calculating negative integer moments via Jack polynomial theory. Denote by 1/x the vector ( / / ) … x x 1 , , 1 N 1 and consider (29) There exists an interesting functional relation between ( )

is an integer and
Equation (59) is the expression in the 'C'-normalization of a formula that can be found in [23, p 643]. The negative moments can be computed by substituting (59) into (58) and calculating the corresponding expectation values.
• Laguerre β-ensemble: The parameter γ is the exponent in the integrand (49); clearly this average exists only if In [25] the authors find that the explicit formulae for negative moments of the Jacobi βensemble do not depend on the choice of t and that t may be set to zero to find a simple final expression.

Higher order correlations: discussion and an example
In section 3 we argued that the computation of the joint moments (27) is beyond our present ability as it involves the knowledge of the complete set of the Jack characters ( ) θ α λ µ . Here we detail a unified proof of three particular examples, which combined with the results in section 5, give the expectation value (1) and This proof is particularly instructive because it gives clear evidence that computing the Jack characters in full generality is beyond the techniques available at present. Consider the map ε Y defined on the ring Λ N k by where Y is an arbitrary parameter.
where ( ) ′ λ a s and ( ) ′ λ l s are the co-arm and co-leg lengths of λ ∈ s , and ( ) λ α t is the generalised Pochhammer symbol (see section 2.1). Substituting equation (24) into the LHS of (65) gives Y a s l s . . For arbitrary integers k there are at least three partitions with this property: , which is consistent with equations (28) and (47); the third partition leads to This formula can also be found in [38, chapter VI.10, p 348]. The derivation in this section starts from formula (65), which is quite subtle and whose proof is far from trivial. It is based on Pieri formulae and on the duality a s l s a s l s 1 .
Pieri formulae gives the product ( ) λ α µ P e as a linear combination of ( ) λ α P , whose coefficients can be explicitly computed in terms of certain combinatorial expressions. It can be thought of as the first step toward calculating the generalization of the Littlewood-Richardson coefficients (32). Other Pieri-type formulae are available, but they are far from leading to an expression for the Littlewood-Richardson coefficients. This in turn means that computing the average (27) or even the expectation values { } E X X Tr Tr j k is beyond the techniques available at the moment.

The secular coefficients
The quantities ( ) X Sc k are the secular coefficients. Note that The secular coefficients of P X (z) are symmetric polynomials of the eigenvalues. More precisely, where the quantities The entries of the transition matrix λµ K are the Kostka numbers, which give the number of semi-standard Young tableaux of shape λ and weight μ 7 .
The first ones to address the problem of computing the averages of the secular coefficients of random unitary matrices were Haake et al [28]. Diaconis and Gamburd [12] used (76) and the fact that the Schur functions are the characters of the irreducible representations of the unitary group to compute all the joint moments of ( ) X Sc k . We may ask what is the generalization of (76) in terms of Jack polynomials; the answer to this question would allow us to solve the analogous problem for characteristic polynomials of β-ensembles. As for the Jack characters at the moment we can only give a partial solution. We first notice that by definition Now, a(s) = 0 for any ( ) ∈ s 1 k ; therefore, Finally, we arrive at 7 A semi-standard Young tableaux is a Ferrers diagram filled with integers that are weakly increasing along the rows and strictly decreasing down the columns; the partition formed by the number of times a given integer appears in the tableaux is the weight.
The explicit expressions for each β-ensemble for the averages (79) can be computed using the expectation values (49), (51) and (54) and the formula We obtain ( ) where in the last line we have used formula (57).

Conclusions
We computed the positive and negative moments of the density of the eigenvalues and the averages of the secular coefficients for the Gaussian, Laguerre and Jacobi β-ensembles for matrices of finite dimensions. Our approach is based on the theory of the Jack polynomials, which are a natural tool in the study of β-ensembles. The Jack polynomials form a basis in the ring of homogeneous symmetric polynomials. As such they can be expressed in terms of other symmetric functions like the power sum, the monomial and the elementary symmetric functions. The coefficients that express the Jack polynomials in terms of power sum symmetric functions are known as Jack characters, which recently have been object of intense study in the combinatorics literature [15,16,22,31,32,34,35,47]. Surprisingly, however, little is known about them and an explicit formula is not available. Since the traces of powers of matrices are particular cases of power sum symmetric functions, they can be expressed as linear combinations of Jack polynomials, whose coefficients are a subset of the inverse of the Jack characters. We were able to compute these coefficients explicitly and hence the moments of the density of states. The expectation values of the secular coefficients can also be expressed in terms of averages of Jack polynomials. It is still an open question how to compute all the joint moments of the density of the eigenvalues of the β-ensembles. Their knowledge is tantamount to having a complete understanding of the Jack characters, which at present is out of reach. Similarly, evaluating all of the joint moments of the secular coefficients is equivalent to knowing the transition matrix from the Jack polynomials to the elementary symmetric functions; the elements of this matrix are generalizations of the Kostka numbers.

Acknowledgments
We are grateful to Pierre Le Doussal and Yan Fyodorov for bringing our attention toward formula (59) and to Brian Winn for writing appendix D with the proof of equation (57). We also thank Marcel Novaes for helpful discussions. FM was partially supported by EPSRC research grant EP/L010305/1. No empirical or experimental data was created during this study.

Appendix A. Multivariate Laguerre polynomials
Here we briefly introduce the generalized Laguerre polynomials and discuss the properties that were used in section 5.1. The exposition in this appendix follows the theory in [1,20].
The classical Laguerre polynomials ( ) where the normalization constant ( ) N L can be computed using Selberg's integral, be homogeneous polynomials in R N and define the scalar product We now present the explicit expression of the first three moments calculated via (50):

Appendix B. Multivariate Jacobi polynomials
The theory of the multivariate Jacobi polynomials follows the same pattern [1,20,31]. They are identified up to a constant as the polynomial part of the eigenfunctions of the operator The multivariate Jacobi polynomials are orthogonal with respect to the measure  where ρ λ α was defined in (15)

Appendix D. A multivariate Hermite polynomial identity (by Winn)
In this appendix we give a proof of the identity (57): Our proof uses two main ingredients. The first is a result of Okounkov [44] (which was conjectured by Goulden and Jackson [27, conjecture 3.4]), which states that the value of the integral in (D.2) is ( ) ( )