Symmetric Function Theory and Unitary Invariant Ensembles

Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices drawn from the Circular Unitary Ensemble and other Circular Ensembles related to the classical compact groups. The reason is that they enable the derivation of exact formulae, which then provide a route to calculating the large-matrix asymptotics of these quantities. We develop a parallel theory for the Gaussian Unitary Ensemble of random matrices, and other related unitary invariant matrix ensembles. This allows us to write down exact formulae in these cases for the joint moments of the traces and the joint moments of the characteristic polynomials in terms of appropriately defined symmetric functions. As an example of an application, for the joint moments of the traces we derive explicit asymptotic formulae for the rate of convergence of the moments of polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.


Introduction
Many important quantities in Random Matrix Theory, such as joint moments of traces and joint moments of characteristic polynomials, can be calculated exactly for matrices drawn from the Circular Unitary Ensemble and the other Circular Ensembles related to the classical compact groups using representation theory and the theory of symmetric polynomials. In the case of joint moments of the traces, this approach has proved highly successful, as in, for example, the work of Diaconis and Shahshahani [19]. Similarly, the joint moments of characteristic polynomials were calculated exactly in terms of Schur polynomials by Bump and Gamburd [14], leading to expressions equivalent to those obtained using the Selberg integral and related techniques [2,42,41,16]. Lee and Oh [47] extended the work of Bump and Gamburd [14] and computed the correlation functions of characteristic polynomials in the Sato-Tato groups as sums of characters of irreducible characters of the symplectic group Sp(N). Our aim here is to develop a parallel theory for the classical unitary invariant Hermitian ensembles of random matrices, in particular for the Gaussian (GUE), Laguerre (LUE), and Jacobi ensembles (JUE).
Diaconis and Shashahani [19] used group-theoretic arguments and symmetric functions to calculate joint moments of traces of matrices for classical compact groups. Here, using multivariate orthogonal polynomials, we develop a similar approach to calculate joint moments of traces for Hermitian ensembles, leading to closed form expressions using combinatorial and symmetric-function-theoretic methods.
Moments of Hermitian ensembles and their correlators have recently received considerable attention. Cunden et al. [18] showed that as a function of their order, the moments are hypergeometric orthogonal polynomials. Cunden, Dahlqvist and O'Connell [17] showed that the cumulants of the Laguerre ensemble admit an asymptotic expansion in inverse powers of N of whose coefficients are the Hurwitz numbers. Dubrovin and Yang [20] computed the cumulant generating function for the GUE, while Gisonni, Grava and Ruzza calculated the generating function of the cumulants of the LUE in [33] and the JUE in [34].
If M is drawn at random from the classical compact groups U(N), O(N), Sp(N) equipped with Haar measure, then TrM k , k ∈ N, converges to a complex normal random variable as N → ∞. Johansson [38] was the first to prove a central limit theorem when M belongs to an ensemble of Hermitian matrices invariant under unitary conjugation. In this case the analogue of TrM k is played by Tr T k (M), where T k is the Chebyshev polynomial of the first kind; see also [10,63,55,26,27,56,6,52,51,45,5,43] and references therein. As an example of an application of the general approach we take here, we apply our results to establish explicit asymptotic formulae for the rate of convergence of the moments and cumulants of Chebyshev-polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity. In a companion article [39], we use the techniques developed in this paper to investigate the moments of the characteristic polynomials in the GUE, uncovering structure that had been overlooked in previous studies.
The theory of symmetric functions has been applied to orthogonal polynomials also outside the context of random matrices. In a spirit similar to that in this paper, generalizations of classical combinatorial identities like the Cauchy formula have played a fundamental role. The Schur functions arise naturally in the representation theory of the Heisenberg algebra. Lam [44] showed that by choosing a particular representation, the Schur polynomials are replaced by a new class of symmetric functions that obey Pieri and Cauchy-like identities. In turn, these can be applied to the theory of Hall-Littlewood and Macdonald polynomials. More recently, Borodin [8] studied a one-parameter family of rational symmetric functions that generalize the Hall-Littlewood polynomials. He also showed that such symmetric functions satisfy Pieri and Cauchy-like identities. This paper is structured as follows. In Sec. 2 we introduce our main results. The preliminaries, multivariate orthogonal polynomials and their properties are discussed in Sec. 3. The correlation functions of characteristic polynomials are calculated in Sec. 4. We discuss the change of basis among different symmetric functions in Sec. 5 and prove there the results for the moments of characteristic polynomials and the joint moments of traces for different ensembles. Finally, in Sec. 6, by way of an example, we apply the results to derive explicit asymptotic formulae for the rate of convergence of the moments and cumulants of Chebyshev-polynomial functions of GUE matrices to those of a standard normal distribution when the matrix size tends to infinity.

Statements and results
For the classical compact groups, Schur polynomials and their generalisations are the characters of U(N), O(N) and Sp (N). In this context they have been used extensively to calculate correlation functions of characteristic polynomials, joint moments of the traces, see [19,14]. Although group theoretic tools are not available for the set of Hermitian matrices, multivariate orthogonal polynomials play the role of Schur functions for the GUE, LUE and JUE and can be used to study fundamental quantities like moments of traces and characteristic polynomials.
For a partition µ, let Φ µ be the multivariate symmetric polynomials with leading coefficient equal to 1 that obey the orthogonality relation for a weight function w. Here the lengths of the partitions µ and ν are less than or equal to the number of variables N, and C µ is a constant which depends on N. We prove the following lemma, which is a generalization of the dual Cauchy identity.
Lemma 2.1. Let Φ µ be multivariate polynomials given in (2.1). Let p, q ∈ N and for λ ⊆ (q p ) ≡ (q, . . . , q Here partition λ = (λ 1 , . . . , λ l ) such that λ 1 ≥ · · · ≥ λ l is a sub-partition of the partition (q p ), denoted by λ ⊆ (q p ). (See Sec. 3.1). This lemma appears in [24, p.625] for the Jacobi multivariate polynomials for arbitrary β. Here we present a different proof for β = 2, which holds for the Hermite and Laguerre polynomials, too. A key difference in our approach is that we have closed-form expressions for multivariate polynomials as determinants of univariate classical orthogonal polynomials, while in the previous literature their construction was based on recurrence relations. This means that in this paper formula (2.2) becomes a powerful tool and plays a role analogous to that of the classical dual Cauchy identity for U(N). It is worth noting that the identity in Lemma 2.1 is independent of the weight w. This is a consequence of the row and column operations on determinants (see Eqs. (4.4) and (4.5)) that are central to the proof. It is the analogue of the well known equality for Vandermonde determinants, where ϕ i , i = 1, 2 . . . , is any sequence of monic orthogonal polynomials. We focus in particular on when w(x) in (2.1) is a Gaussian, Laguerre and Jacobi weight: The classical polynomials orthogonal with respect to these weights satisfy The identity in (2.2) gives a compact way to calculate the correlation functions and moments of characteristic polynomials of unitary ensembles using symmetric functions. The results are as follows.
Here the subscripts (H), (L), (J) indicate Hermite, Laguerre and Jacobi, respectively, and are multivariate polynomials orthogonal with respect to the generalised weights in (2.1).
Similar to the case of the classical compact groups, correlations of traces of Hermitian ensembles can be calculated using the theory of symmetric functions. For a partition λ = (λ 1 , λ 2 , . . . , λ N ), j λ j ≤ N, define (2.7) The constants C λ (N) and G λ (N, γ) have several interesting combinatorial interpretations which are discussed in Sec. 5.1.

Theorem 2.2.
Let M be an N × N GUE, LUE or JUE matrix and let µ = (µ 1 , . . . , µ l ) be a partition such that |µ| = l j=1 µ l ≤ N. Then which is a polynomial in N.
In the above equations χ λ µ are the characters of the symmetric group S m , m = |λ| = |µ|, associated to the λ th irreducible representation on the µ th conjugacy class.
Next, we now focus our attention on the GUE with rescaled matrices M R = M/ √ 2N. Define the random variables (2.12) Here T k is the Chebyshev polynomial of degree k. Johansson proved the following multidimensional central limit theorem for X k [38]: where r j are independent standard normal random variables and d = ⇒ means convergence in distribution [38]. Define 14) The formalism that we developed to study moments of traces allow us to derive explicit estimates for the error E n,k as a function of matrix size N. For rescaled Gaussian matrices, the correlators of traces are Laurant polynomials in N. This fact can be seen from (2.8) when applied to rescaled matrices. Consequently, the moments of polynomial test functions are also Laurant polynomials in N.
Let f and g be real or complex valued functions defined on some subset of R. In what follows we will use the notations f (x) g(x) and )) if and only if there exist a x 0 and a positive constant K such that More in general, if t belongs to the extended real line, then We have the following theorem for Chebyshev polynomials.
Theorem 2.3. Fix k ∈ N and let kn ≤ N. With the notation introduced above the following statements hold as N → ∞.
1. For k odd and k > 1, 18) as n → ∞ and k fixed.
Along with the moments we also give an estimate for the cumulants of random variables X k . Computing cumulants from (2.8) is not straightforward. Instead, we employ the well established connection between correlators of traces and the enumeration of ribbon graphs to estimate the cumulants. The results are elaborated in Section 6.2.
To summarise, for a fixed n and k, we show that the n th moment of X k converges to the n th moment of independent scaled Gaussian variable as N −1 or N −2 depending on the parity of n; and the n th cumulant of X k converges to 0 as N n−2 for n > 2. Theorem 2.3 provides explicit asymptotic estimates for the rate of convergence of the moments.

Background
Symmetric polynomials arise naturally in random matrix theory because the joint eigenvalue probability density function remains invariant under the action of the symmetric group. There has been a considerable focus on symmetric functions to study moments in various ensembles [19,14,50,21]. Here we define some symmetric functions that will play a central role in our calculations and state some of their properties.

Review of symmetric functions
A partition λ is a sequence of non-negative integers such that λ 1 ≥ λ 2 ≥ · · · ≥ λ l > 0. We call the maximum l such that λ l > 0 the length of the partition l(λ) and |λ| = l i=1 λ i the weight. A partition can be represented with a Young diagram which is a left adjusted table of |λ| boxes and l(λ) rows such that the first row contains λ 1 boxes, the second row contains λ 2 boxes, and so on. The conjugate partition λ ′ is defined by transposing the Young diagram of λ.
Another way to represent a partition is as follows: if λ has b 1 1's, b 2 2's and so on, then λ = (1 b 1 2 b 2 . . . k b k ). In this representation, the weight |λ| = k j=1 jb j and length l(λ) = k j=1 b j . In the rest of the paper, we use both notations interchangeably and we do not distinguish partitions that differ only by a sequence of zeros; for example, (4,2,2,1) and (4,2,2,1,0,0) are the same partitions. We denote the empty partition by λ = 0 or λ = ().
The elementary symmetric functions e r (x 1 , . . . , x N ) are defined by and the complete symmetric functions h r (x 1 , . . . , x N ) by Given a partition λ, we define The Schur polynomials are symmetric polynomials indexed by partitions. Given a partition λ such that l(λ) ≤ N, we write where ∆(x) is the Vandermonde determinant: If l(λ) > N, then S λ = 0. The Jacobi-Trudi identities express Schur polynomials in terms of elementary and complete symmetric functions: . (3.7) The power sum is defined by Power sum and Schur functions are bases in the space of homogeneous symmetric polynomials and they are related by where χ λ µ are the characters of the symmetric group S m , m = |λ| = |µ| and z µ is the size of the centraliser of an element of conjugacy class µ. Proposition 3.1 (Cauchy Identity [48]). Let t 1 , t 2 , . . . and x 1 , x 2 , . . . be two finite or infinite sequences of independent variables. Then, (3.11) When the sequences t i and x j are finite, (3.12) where λ runs over all partitions of length l(λ) ≤ min(p, q). We also have the dual Cauchy identity [48]: Since S λ = 0 or S λ ′ = 0 unless l(λ) ≤ p or l(λ ′ ) ≤ q, λ runs over a finite number of partitions such that the Young diagram of λ fits inside a p × q rectangle. Changing x j → x −1 j in the dual Cauchy identity and simplifying the fractions gives Since the Schur polynomials are homogeneous, we have Thus (3.14) becomes The Cauchy and dual Cauchy identities, combined with the fact that the Schur polynomials are the characters of U(N), were essentials tools in the proofs of the ratios of characteristic polynomials by Bump and Gamburd [14]. In order to prove Thm. 2.1, we use a similar approach, in which (3.16) is replaced by the generalized Cauchy dual identity (2.2).

Multivariate orthogonal polynomials
Multivariate orthogonal polynomials can be defined by the determinant formula [54] , (3.17) where l(µ) ≤ N and ϕ i , i = 0, 1, . . . , are a sequence of polynomials orthogonal with respect to the weight w(x). One can check by straightforward substitution that, up-to a constant, the multivariate polynomials (3.17) coincide with those in (2.1). When ϕ j in (3.17) are the Hermite, Laguerre and Jacobi polynomials we have the multivariate generalizations H µ , These polynomials can be expressed as a linear combination of Schur polynomials, i.e.
For the Hermite, Laguerre and Jacobi multivariate polynomials we set the leading coefficient κ µµ in consistency with the definitions (2.5) and (3.17), The analogy between multivariate orthogonal polynomials and Schur functions becomes apparent by comparing definitions (3.5) with (3.17). In the literature the polynomials (3.17) are called generalised orthogonal polynomials or multivariate orthogonal polynomials [3] as well as generalised Schur polynomials [54]. The multivariate orthogonal polynomials also satisfy a generalization of the Jacobi-Trudi identities [54] similar to (3.7).
The classical Hermite, Laguerre and Jacobi polynomials satisfy second order Sturm Liouville problems. Similarly, their multivariate generalizations are eigenfunctions of secondorder partial differential operators, known as Calogero-Sutherland Hamiltonians, These generalised orthogonal polynomials obey similar properties to their univariate counterparts [3]. The differential equations in (3.20) are also related to the Dyson Brownian motion.

Correlation functions of characteristic polynomials
The main tool to compute correlations of characteristic polynomials and spectral moments is Lemma. 2.1 which we prove here.

(4.2)
Proposition 4.2. Let λ be a partition such that λ 1 ≤ q and λ ′ 1 ≤ p. Then the p + q numbers Prop. 4.1 is a fact from Linear algebra and the proof of Prop. 4.2 can be found in [48].
Proof of Lemma. 2.1. Let the ϕ j s be monic. Choosing a different normalization would affect Eq. (2.2) by an overall constant. Using the definition of generalised polynomials, Proposition 4.1 and Proposition 4.2, the right-hand side of (2.2) can be written as Now using column operations we arrive at The determinant in (4.5) can be evaluated using the formula for the Vandermonde determinant. We have Combining eqs. (4.4) to (4.6) proves the lemma.
, as for Hermite polynomials, then It follows that (2.2) becomes Proof of Thm. 2.1. Unlike Hermite polynomials, the univariate Laguerre and Jacobi polynomials that obey (2.5) are not monic. This fact is reflected in the normalisation in (3.19) and also in the following formulae, After taking the expectation value, the non-zero contribution comes fromλ = 0 because of (2.1). Thus λ ′ = (p N ) which implies λ = (N p ). Now using proves the result.

Correlations
In this section we calculate moments of traces and characteristic polynomials of N ×N GUE, LUE and JUE matrices. We can write . , x N ) are the eigenvalues of M. The power symmetric functions form a basis in the space of symmetric polynomials of degree |µ|. The main idea is to express them in the basis of the multivariate orthogonal polynomials.

Change of basis between symmetric functions
In this section we give expressions for change of basis between multivariate orthogonal polynomials and other symmetric functions. We mainly focus on the GUE but the same approach can be used for the LUE and the JUE.
Denote by H n (x) the Hermite polynomials normalised according to (2.5a). Given a partition λ with l(λ) ≤ N, the multivariate Hermite polynomials are given by and satisfy the orthogonality relation It turns out that it has a nice interpretation in terms of characters of the symmetric group.
where i runs across the rows from top to bottom and j across the columns from left to right of the Young diagram. For example, if λ = (4, 3, 3, 1), the roots of C λ (N) are It is shown in [40] that The constant z λ is defined in (3.10) and dim V λ is the dimension of the irreducible representation labelled by λ of the symmetric group S |λ| . and Schur polynomials can be expressed in terms of multivariate Hermite polynomials, , |λ| is even, 2j + 1, |λ| is odd. (5.9) The function g(j) takes care of the fact that polynomials of odd and even degree do not mix similar to the one variable case. The first summation in (5.9) running over all lower order partitions takes care of the fact that H λ are, unlike S λ , not homogeneous polynomials. For example, when |λ| is even, the only partitions that appear in (5.9) are those with weight |ν| = |λ| − 2k, k = 0, . . . , |λ| 2 , and ν ⊆ λ. The following proposition gives an explicit expression for the coefficients ψ Proof. Let λ = (λ 1 , . . . , λ L , 0, . . . , 0) and ν = (ν 1 , . . . , ν l , 0, . . . , 0). Here l is the length of ν and N − l is the length of the sequence of zeros added to ν. From (5.4) and the fact that The last N − L and N − l columns in S λ and in H ν , respectively, are written in terms of the Hermite polynomials using column operations. In addition, ψ (H) λν can be expanded as a sum over the permutations of N: Since the integrand is symmetric in x i , every term in the above sum gives the same contribution and it is sufficient to consider only the identity permutation. All the factors can be absorbed into the determinant by multiplying the j th row with x Then using orthogonality of Hermite polynomials (2.5a) for the last N − L rows gives (5.14) Expanding monomials in terms of Hermite polynomials with the formula and using orthogonality leads to (5.10). The determinant D Proof. The skew diagram λ/ν is the set-theoretic difference of the Young diagrams of λ and ν: the set of squares that belong to the diagram of λ but not to that of ν. Using (5.6), Since ν ⊆ λ, the roots of ψ By expanding monomials in terms of Hermite polynomials only the diagonal terms survive.
Proof. Since Hermite polynomials of odd and even degree do not mix, ψ (H) λ0 = 0 when |λ| is odd. When |λ| is even, Using Frobenius formula for characters of the symmetric group After absorbing x 2n i i into the i th row of the determinant, for each n i at most one term in the i th row has the exponent λ i +L−i, say the (i, j) th element x 2n i +L−j i , which implies 2n i = λ i −i+j. For L-tuples {n 1 , . . . , n L } such that there is exactly one term in each row that has the required exponent, the non-zero summands are given by n! sgn(σ) i ((λ i −i+σ(i)/2)!) −1 where σ ∈ S L . Considering all such L-tuples and using Laplace expansion for the determinant proves the proposition.
Therefore the expansion of Schur polynomials in terms of multivariate Hermite polynomials can be written as , |λ| is even, 2j + 1, |λ| is odd, (5.24) where κ (H) Alternatively, where G λ (N, γ) is given in (2.7). The multivariate Laguerre polynomials defined by The polynomials in the determinant (5.28) are normalized according to (2.5b). The Schur polynomials can be expanded in terms of multivariate Laguerre polynomials as where ψ (L) Multivariate Laguerre polynomials can also be expanded in the Schur basis: λν . (5.32) Similar to the Hermite case, D λ0 turns out to be a character of the symmetric group.

Proposition 5.3. We have
Classical Jacobi polynomials are given by and satisfy the orthogonality relation (2.5c). The multivariate Jacobi polynomials are l(λ) ≤ N, and obey the orthogonality relation The expansion of the Schur polynomials in terms of multivariate Jacobi polynomials is . (5.39) When N = 1 (5.38) coincides with the one variable analogue Multivariate Jacobi polynomials can be expanded in Schur polynomials via . (5.42)

Moments of characteristic polynomials
The Cauchy identity can be written as where Φ µ is one of the generalised polynomials H µ , L (γ) µ or J (γ 1 ,γ 2 ) µ . By using orthogonality of multivariate polynomials (5.4), (5.29) and (5.37) we have the following proposition. Let t 1 , . . . , t p and T 1 , . . . , T q be two sets of variables. Then

Proposition 5.4.
hhNote that the RHS is a formal power series in the variables T.
where dim V ν is given in (5.7).
Proof. Let us consider the GUE case. We have proves the statement. Similarly, the Laguerre and Jacobi cases can be computed in an identical way.

Joint moments of traces
Recently the study of moments and joint moments of Hermitian ensembles have attracted considerable interest [17,18,20,33,34]. Here we give new and self contained formulae for the joint moments of unitary ensembles in terms of characters of the symmetric group. We focus on the GUE but exactly the same method applies to the LUE and JUE. Using (3.10) and (5.9), power sum symmetric polynomials can be written in terms of multivariate Hermite polynomials

Proof of Thm. 2.2.
When |µ| is odd P µ is a sum of product of monomials in x i with the degree of at least one x i being odd. Since the generalised weight ∆ 2 When |µ| is even, writing P µ in terms of multivariate Hermite polynomials (5.57) and using orthogonality of the H ν along with (5.19) proves the first line of (2.8).

Remark. When |µ| is even, the orthogonality of characters indicate that E
Thus, The corollary is proved by noticing that the symmetric and anti-symmetric combination of C λ (N) and C λ (−N) is an even and odd polynomial in N, respectively.
Since E (H) N [P µ ] are polynomials in N, the domain of N can be analytically continued from integers to the whole complex plane. In [18], it is shown that E where P (1) k (iN, π/2) is a Meixner-Pollaczek polynomial and 2 F 1 (. . . ) is a terminating hypergeometric series. These polynomials P (λ) Clearly, the zeros of E  [7,59,60]. This connection is briefly discussed in Appendix B. By counting ribbon graphs, it can be easily shown that

is also a polynomial in N.
A few examples of joint moments of traces corresponding to partitions of 6 are given below. Here p j = TrM j .

Moments
Here we focus on the GUE but the Laguerre and Jacobi ensembles can be studied in a similar way. Consider the rescaled GUE matrices The limiting eigenvalue density is Proof. When µ = (1 2n ) in (2.8), using (5.6) and the fact that χ λ Using (3.10) and P ν (1 N The R.H.S. is the 2n th moment of r 1 /2 where r 1 ∼ N (0, 1). This exact equality of moments with the moments of Gaussian normals is special to E  (Tr g(M)) n ] for a well-defined function g. Johansson [38] showed that when g is the Chebyshev polynomial of the first kind of degree k, the random variable (6.6) converges in distribution to the Gaussian variable N (0, k/4). In this section we prove Theorem 2.3, which implies that where η n = 1 if n is even and 0 otherwise, and where m k,n is either 0 or 1, with asymptotic estimates for d(n, k). Results for k = 1 are already discussed in Prop. 6.1. We first consider X 2 and discuss results for general values of k in Sec. 6.1.2.

Second degree
We have that For a fixed n, this can be obtained by substituting in the character values of S 2n in (2.8). Alternatively, a proof by counting topologically invariant ribbon graphs is sketched in App. B. Clearly, The asymptotic expansion for the ratios of Gamma functions is [28] Γ(z + a)  (6.14) Calculating the coefficient of N n−2l for arbitrary n and l is not straightforward because there are no simple expressions for generalised Bernoulli numbers. Though these numbers can be written in terms of Stirling's numbers of first kind, the coefficients can be explicitly computed only for small values of l. It can be shown for a given n that Coef. of N n−2k = 0, for 0 ≤ k < ⌊n/2⌋, (6.15) where η n = 1 if n is even and 0 otherwise. Our goal is not to compute these coefficients more generally, but rather to give an estimate for the sub-leading term in (6.7). Since the Chebyshev polynomials of even and odd degree do not mix, the moments of X k show a similar behaviour as in Corollary. 5.4.
Coefficients d 2 (n, 2) and d 3 (n, 2) can be estimated using (6.11). In the next section, we give an estimate of these coefficients for arbitrary values of n and k.

General degree
The explicit expression for the joint moments of eigenvalues in Thm. 2.2 allows us to obtain Thm. 2.3. Consequently, X k converges to a normal random variable. For a fixed k and n, In reality, the correct bounds in Thm. 2.3 are much more smaller than given. This is due to sequential cancellations in the sum λ χ λ µ χ λ 2 |µ|/2 C λ (N) (6.18) and in the Chebyshev expansion The bounds in Thm. 2.3 are better for smaller moments. To prove Thm. 2.3, we first need to estimate the coefficient of the 1/N term in the Laurent series of E (H) N [P µ ] of rescaled matrices which leads to estimating the characters of the symmetric group.
All the characters of the symmetric group are integers and satisfy (6.20) It turns out that under suitable assumptions, the ratios |χ λ µ |/χ λ (1 |µ| ) are very small, sometimes exponentially and super-exponentially small [23,53]. Particularly useful bounds are where a µ depends on µ.
The frequency representation of a partition µ = (1 b 1 , 2 b 2 , . . . , k b k ) also represents a permutation cycle of an element in S |µ| . The number b j gives the number of j−cycles in µ, 1 ≤ j ≤ k. For example, if b 1 = 0 then are no 1-cycles. In other words, there are no fixed points when b 1 = 0.
The only obstruction to the small character values of |χ λ µ | is when µ has many short cycles [46]. With this information, Proposition 6.2.
We are now ready to prove Thm. 2.3.

Proof of
The odd moments of Tr T k (M R ) are identically zero because E (H) N [P µ ] = 0 when |µ| is odd, see (6.26). When n is even, the leading term is given by the n th moment of √ kr k /2, r k ∼ N (0, 1), according to Szegő's theorem. For n even, l(µ) is always even. Hence the sub-leading term in (6.36) is O(N −2 ) (see Corollary. 5.4. Note that the matrix is now rescaled).
Using asymptotics of Barnes-G functions and Stirling's approximation By combining the previous equations we arrive at 43) as n → ∞. We are interested to find the order of the coefficient of 1/N as n increases for a fixed k. To capture the right behaviour, it is sufficient to approximate (6.43) to (2) When k is even, let We have, According to Szegő's theorem, when n is even the leading order term in the R.H.S. of (6.46) is given by E[( √ kr k /2) n ], r k ∼ N (0, 1). The sub-leading term is d 3 (n, k)N −2 . When n is odd, the leading term in the R.H.S. is given by d 2 (n, k)N −1 . Next we compute the coefficients d 2 (n, k) and d 3 (n, k). Coefficient d 2 (n, k): c k decays as 1/N 2 for k > 2, so we neglect it in (6.46). Note that µ in (6.47) doesn't have any 1-cycles. So µ is fixed-point-free and (6.23) can also be used to estimate characters χ λ µ in Prop. 6.3. Here we just use (6.25) for q = 2 and follow the exact same calculation as k odd. In the limit n → ∞ this leads to Similarly, d 3 (n, k) can be approximated as Since X k converges to independent Gaussian normals, the correlators of X k also converges to random Gaussian variables as N → ∞. For instance, Given below are the moments corresponding to partitions of 6.
In this paper, we do not pursue correlations of X k any further.

Cumulants
In general, the moments and the cumulants are related by the recurrence relation We have is the number of decompositions of a set of n elements into disjoint subsets containing λ 1 , . . . , λ l elements.
In this section, we give an estimate on the cumulants of random variables X k and to do so we rely on the well studied connection between GUE correlators and enumerating ribbon graphs which has been briefly discussed in App. B.
Consider the formal matrix integral over the space of N × N rescaled GUE matrices. By choosing s j to be the coefficients of Chebyshev polynomials in (6.58), we recover the moments of X k . Thus, (6.58) is the moment generating function of X k . For a given k, by fixing s j to be the Chebyshev coefficients in T k , By matching the terms in the L.H.S. and R.H.S. of (6.59) by powers in ξ we recover the moments of X k . The correlators of Tr M j R are connected to the problem of enumerating ribbon graphs. For a brief introduction see App. B and for more details see [22,37] and references within. The trace correlators count ribbon graphs that are connected and also multiplicatively count ribbon graphs that are disconnected. When we have a generating function that counts disconnected objects multiplicatively, taking logarithm counts only the connected objects [36]. Hence, the cumulant generating function, keeps only connected ribbon graphs indicated by subscript c. For µ = (1 n 1 , . . . , k n k ) ≡ (µ 1 , . . . , µ l ), the connected correlators are given by 2 |µ| a g (µ 1 , . . . , µ l )N 2−2g−l , |µ| is even. where Γ is a connected (unlabelled) ribbon graph of genus g with l vertices of valencies µ 1 , . . . , µ l , #Sym(Γ) is the order of the symmetry group of Γ, and the last summation is taken over all such Γ. For the explicit expressions of connected correlators of the GUE see [20].
We are now ready to estimate the cumulants of X k . We treat k even and odd cases separately.
(1) k odd: In this case, the parameters s 2j = 0 for 0 ≤ j ≤ k−1 2 and When k is odd, all the odd moments are zero. Hence all the odd cumulants are also zero. By inserting (6.61) in (6.60), the even cumulants are given by (2) k even: In this case, the parameters s 2j+1 = 0 for 0 ≤ j ≤ k 2 − 1 and Third and higher order cumulants of Gaussian random variable are identically zero. Since X k converges to N (0, k/4) as N → ∞, cumulants of X k , κ n → 0 as N → ∞ for all n ≥ 3. For a fixed n, we see from (6.64) and (6.67) that κ n decay as N −n+2 . Example: The simplest non-trivial example is to calculate the cumulants of X 2 . By mapping the problem to counting ribbon graphs (see App. B), For X 2 , s 0 = − 1 2 , s 2 = 2, and s j = 0 for j = 0, 2. Hence which have the generating function By comparing (5.26) and (A.1), we can see the analogies between classical Hermite polynomials and their multivariate counterparts: the sum over j is replaced by the sum over partitions; the role of monomials is played by Schur polynomials; the factorials are replaced with C λ (N). With this analogy the generating function of H λ is [3] λ 3) The validity of the above formula can be easily verified for lower order partitions (say |λ| = 2, 4) using Pieri's formula but the second factor in (A.3) can be simplified further.  [3]: Several other analogues of properties of the classical Hermite polynomials, including an integral representation, summation, integration and differentiation formulae, are given for β−ensembles in [3]. Note that in [3] C α µ (α ∈ R) is used to denote Schur polynomials with a specific normalisation where as in this work C µ (N) is a constant in N given in (5.6). Laguerre case. When N = 1, L (γ) λ coincides with the classical Laguerre polynomials whose generating function is where J γ is the Bessel function. By comparing (5.32) and (A.7), the generating function for multivariate Laguerre polynomials [3] is (N, 0) , (A.9) or equivalently using (5.33), (A.10)

B Ribbon graphs and matrix integrals
where A is a positive definite symmetric matrix. The inverse is called the propagator.
Correlations of Gaussian random variables can be computed in a combinatorial way using Wick's theorem [62], also known as Isserlis' theorem, which is stated below.

Theorem B.1 (Wick's theorem). The expectation value of product of Gaussian random variables is
if n is odd, pairings of (i 1 ,...,in) pairs (k,l) B i k i l , if n ≥ 2 and even.

(B.3)
For example, Wick's theorem becomes particularly useful when the indices i j are repeated. The problem of computing the expectation values E[x b 1 i 1 . . . x bn in ] can be mapped to counting the number of ways of gluing n vertices with valencies b 1 , . . . b n , whose weights are determined by the propagators that correspond to their edges. Clearly many graphs in (B.5) are topologically identical and have the same weight because of the symmetries among the edges and vertices. Let G be the group of these symmetries, #gluings be the number of gluings of obtaining a graph, and Aut(G) be the automorphism group of the graph. By orbit-stabiliser theorem, #Aut(G) × #gluings = #G, (B.7) where #G is the order of group relabelling. Wick's theorem can be written only in terms of non-equivalent graphs as follows: In the case of Gaussian matrix integrals, Wick's theorem can be applied to compute correlators of traces by studying fat graphs also called ribbon graphs.
Consider the Hermitian Gaussian matrix model with probability measure (B.10) The Wick's propagator is As an example, consider i,j,k l,m,n δ ik δ km δ il δ ml + δ jk δ jm δ kl δ ml + . . . If we attach to each vertex a factor of N, the N dependence of a graph is: There is a factor N per vertex, a factor N −1 per edge, a factor N for each single line when summed over indices. The number of single lines remaining at the end is the number of faces of the graph. So the total N dependency of a graph is N #vertices−#edges+#faces = N χ(G) , (B.14) where χ(G) is the topological invariant of the graph called its Euler-characteristic. This notion of counting ribbon graphs can be extended to compute correlators of the form E where the sum is over non-topologically equivalent ribbon graphs and #Aut(G) is the number of automorphisms of G. There are a total of ( j jb j − 1)!! graphs (counting equivalent and non-equivalent graphs). The total number of vertices is b = j b j with j valencies for each vertex and the total number of edges is ( j jb j )/2.

B.1 Special cases
Here we consider two cases (i) E The next non-trivial contribution comes from choosing any two vertices and gluing their valencies to form an edge between them. There are n 2 ways of choosing two vertices. Let (i p , i p+1 ) and (i q , i q+1 ), 1 ≤ p, q ≤ 2n, be the indices of the valencies of these two vertices. There are two ways to pair (i p , i p+1 ) and (i q , i q+1 ). This gives a contribution of n(n − 1)N 2 /(4N) 2 . The remaining n−2 disconnected graphs multiplicatively gives N 2n−4 /(4N) n−2 . Hence the first two leading terms are for k = 1, . . . , n − 1.