Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

We consider the Laguerre partition function, and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers, that can be explicitly computed from our formulae. As a second result we identify the Laguerre partition function with only positive couplings and a special value of the parameter $\alpha=-1/2$ with the modified GUE partition function, which has recently been introduced as a generating function of Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals.


Introduction and Results
The parameter α could be taken as an arbitrary complex number satisfying Re α > −1. Writing α = M − N, a random matrix X distributed according the measure (1.1) is called complex Wishart matrix with parameter M; in particular, when M is an integer there is the equality in law X = 1 N W W † where W is an N ×M random matrix with independent identically distributed Gaussian entries [36].
The proof is given in Section 2.3. Theorem 1.1 generalizes formulae for one-point correlators, since the formulae for the generating series C 1,0 and C 0,1 boil down to the following identities tr X k = A k (N, N + α), tr X −k−1 = A k (N, N + α) (α − k) 2k+1 , k ≥ 0 (1.14) which were already derived in the literature [44,23]. From Theorem 1.1 for example one can deduce compact expressions for correlators of the form tr X k tr X c = kA k (N, N + α), for the derivation see Example 2.14. For general positive moments, see also [48]. The entries A ℓ (N, M), defined in (1.11), are known to satisfy a three term recursion [42,24]. We deduce this recursion together with a similar three term recursion for B ℓ (N, M), in Lemma 2.11. It was pointed out in [23] that the entries A ℓ (N, M) are hypergeometric orthogonal polynomials (in particular suitably normalized Hahn and dual Hahn polynomials [50,23]), a fact which provides another interpretation of the same three term recursion; this interpretation extends to the entries B ℓ (N, M), see Remark 2.12. In Lemma 3.2 we provide an alternative expression for the entries A ℓ (N, M), B ℓ (N, M), which makes clear that they are polynomials in N, M with integer coefficients.
Formulae of the same sort as (1.13) have been considered in [30] for the Gaussian Unitary Ensemble, and already appeared in the Topological Recursion literature, see e.g. [35,20,6,5,34]. Our approach is not directly based on the Matrix Resolvent method [30] or the Topological Recursion [20]; in particular we provide a self-contained proof to Theorem 1.1 via orthogonal polynomials and their Riemann-Hilbert problem [46].
Insertion of negative powers of traces in the correlators and computation of mixed correlators are, to the best of our knowledge, novel aspects; as we shall see shortly, these general correlators have expansions with integer coefficients, a fact which generalizes results of (see e.g. [22]). It would be interesting to implement this method to other invariant ensembles of random matrices [26,36]. With the aid of the formulae of Theorem 1.1 we have computed several LUE connected correlators which are reported in the tables of App. A. Moreover, we can make direct use of the formulae of Theorem 1.1 to prove (details in Section 3) the following result, concerning the formal structure as large N asymptotic series of arbitrary correlators of the LUE in the scaling α = (c − 1)N, (1.16) corresponding to M = cN in terms of the Wishart parameter M.
From this result we infer that when c = 2 (equivalently, α = N) the coefficients of this large N expansion are all integers.
From the tables in App. A one easily conjectures that actually a stronger version of this result holds true, namely that that the asymptotic expansion for N ℓ−2− ℓ i=1 k i tr X k 1 · · · tr X k ℓ c (note the different power of N) as N → ∞ is a series in Z[c, (c − 1) −1 ][[N −2 ]]. Such stronger property holds true when all the k j 's have the same sign, see e.g. [22,24] and the section below.
1.2. Topological expansions and Hurwitz numbers. It has been shown in [33,14] that for matrix models with convex potentials, as in our case, correlators, suitably rescaled by a power of N, as in (1.17), have a topological expansion, by which we mean an asymptotic expansion in non-negative powers of N −2 . As mentioned above, the topological expansion of the LUE correlators in the regime (1.16) was considered in [24,22] where the connection with Hurwitz numbers was made explicit.
From the structure of the formula (1.19) it is clear that when c = 2 (equivalently, α = N) the coefficients in this expansion are all positive integers.
Remark 1.4. The type of Hurwitz numbers apparing in the expansions (1.18) and (1.19) can also be expressed in terms of the (connected) multiparametric weighted Hurwitz numbers H d G (µ), introduced and studied in [45,41,3,10], which depend on a single partition µ and are parametrized by a positive integer d and by a sequence g 1 , g 2 , . . . of complex numbers, the latter being encoded in the series G(z) = 1 + i≥1 g i z i . To make the comparison precise, one has to identify and then we have where z µ := i≥1 (i m i ) m i !, m i being the multiplicity of i in the partition µ, as above.
1.3. Laguerre and modified GUE partition functions and Hodge integrals. Our arguments in the proof of Theorem 1.1 mainly revolve around the following generating function for correlators which we call LUE partition function. Here t + = (t 1 , t 2 , . . . ) and t − = (t −1 , t −2 , . . . ) are two independent infinite vectors of times and α is a complex parameter. For precise analytic details about the definition (1.21) we refer to the beginning of Section 2. Eventually we are interested in the formal expansion as t j → 0; more precisely, logarithmic derivatives of the LUE partition function at t + = t − = 0 recover the connected correlators (1.4) as It is known that Z N (α; t + , t − ) is a Toda lattice tau function [49,2] separately in the times t + and t − ; this point is briefly reviewed in Section 2.1.2. Our second main result is the identification (Theorem 1.5 below) of the LUE partition function (1.21) restricted to t − = 0 with another type of tau function, the modified Gaussian Unitary Ensemble (mGUE) partition function, which has been introduced in [28] as a generating function for Hodge integrals (see below), within the context of the Hodge-GUE correspondence [31,28,29,54,55,51].
The mGUE partition function Z N (s) is defined in [28] starting from the even GUE partition function which is the classical GUE partition function with couplings to odd powers set to zero. It is well known [46] that (1.23) is a tau function of the discrete KdV (also known as Volterra lattice) hierarchy, which is a reduction of the Toda lattice hierarchy (see Section 2.1.2 for a brief discussion of the Toda lattice hierarchy). As far as only formal dependence on N and on the times s is concerned (see Section 4.2 for more details) it is then argued in [28] that the identity uniquely defines a function Z N (s), termed mGUE partition function; in (1.24) and throughout this paper, G(z) is the Barnes G-function, with the particular evaluation G(N + 1) = 1!2! · · · (N − 1)! for any integer N > 0. With respect to the normalizations in [28] we are setting ǫ ≡ 1 for simplicity; the dependence on ǫ can be restored by the scaling N = xǫ. In [29] a new type of tau function for the discrete KdV hierarchy is introduced and the mGUE partition function is identified with a particular tau function of this kind. We have the following interpretation for the mGUE partition function.
Theorem 1.5. The modified GUE partition function Z N (s) in (1.24) is identified with the Laguerre partition function Z N (α; t + , t − ) in (1.21) by the relation where t + , s are related by t k = 2 k s k and C N is an explicit constant depending on N only 1 ; The proof is given in Section 4.3. Identity (1.25) can be recast as the following explicit relation; which is obtained from (1.25) by a change of variable X → X 2 in the LUE partition function. Theorem 1.5 provides a direct and new link (Corollary 1.6 below) between the monotone Hurwitz numbers in the expansion (1.18) and special cubic Hodge integrals. To state this result, let us denote M g,n the Deligne-Mumford moduli space of stable nodal Riemann surfaces, ψ 1 , . . . , ψ n ∈ H 2 M g,n , Q and κ j ∈ H 2j M g,n , Q (j = 1, 2, . . . ) the Mumford-Morita-Miller classes, and Λ(ξ) := 1 + λ 1 ξ + · · · + λ g ξ g the Chern polynomial of the Hodge bundle, λ i ∈ H 2i M g,n , Q . For the definition of these objects we refer to the literature, see e.g. [56] and references therein. (1.28) The proof is given in Section 4.4. Note that H g,µ in (1.28) is a well defined formal power series in C[[λ − 1]], as for dimensional reasons each coefficient of (λ − 1) m in (1.28) is a finite sum of intersection numbers of Mumford-Morita-Miller and Hodge classes on the moduli spaces of curves.
Matching coefficients in (1.27), we obtain the following partial monotone ELSV-like formulae, valid for all partitions µ = (µ 1 , . . . , µ ℓ ) of length ℓ; In the published version of this preprint the value of the constant C N is incorrect.
in genus zero (see also Example 4.6) and g γ=0 2 4γ s≥1 p≥0  [4] for single monotone Hurwitz numbers and in [13] for orbifold monotone Hurwitz numbers. The relation between Hodge integrals and Hurwitz numbers expressed by Corollary 1.6 is obtained from Theorem 1.5 by re-expanding the topological expansion (1.18). Indeed fixing α = − 1 2 implies that the parameter c in (1.18) is no longer independent of N (soft-edge limit) but actually scales as c = 1 − 1 2N (hard-edge limit). This explains why we cannot derive from the Hodge-GUE correspondence an expression in terms of Hodge integrals for each Hurwitz number in (1.18), but only an expression for a combination of Hurwitz numbers in different genera.
In particular, to obtain the formulae of Corollary 1.6 one has to re-expand the topological expansion (1.18) in N after the substitution c = 1 − 1 2N ; that the result of this re-expansion, namely the right side of (1.27), involves only even powers of ǫ is a consequence of the invariance of positive LUE correlators under the involution (N, α) → (N + α, −α); this symmetry will be described below in Lemma 4.2. More concretely, this symmetry implies the symmetry of the positive LUE correlators under the involution (N, c) → (Nc, c −1 ) which in view of (1.18) is equivalent to the identity (1.31) The above identity implies that the small ǫ expansion on the right side of (1.27) contains only even powers of ǫ. It is also possible to check the symmetry (1.31) by purely combinatorial arguments, see Rem. 4.3.
Remark 1.7. It is known that special cubic Hodge integrals are related to a q-deformation of the representation theory of the symmetric group [53]; it would be interesting to directly provide a link to the monotone Hurwitz numbers under consideration here.
Organization of the paper. In Section 2 we prove Theorem 1.1; a summary of the proof is given in the beginning of that section. In Section 3 we analyze the formulae of Theorem 1.1 to prove Proposition 1.2. In Section 4 we prove the identification of the mGUE and LUE partition functions, namely Theorem 1.5; then we recall the Hodge-GUE correspondence [28] and we deduce Corollary 1.6. Finally, in the tables of App. A we collect several connected correlators and weighted monotone double Hurwitz numbers, computed applying the formulae of Theorem 1.1.

Proof of Theorem 1.1
In this section we prove our first main result, Theorem 1.1. The proof combines two main ingredients; on one side the interpretation of the matrix integral (1.21) as an isomonodromic tau function [9] and on the other side some algebraic manipulations of residue formulae introduced in [7]. More in detail, we first introduce the relevant family of monic orthogonal polynomials and derive a compatible system of (monodromy-preserving) ODEs in the parameters t (Proposition 2.1); throughout this section, in the interest of lighter notations, we set Such orthogonal polynomials reduce to monic Laguerre polynomials for t = 0. With the aid of this system of deformations we then compute arbitrary derivatives of the LUE partition function (1.21) in terms of formal residues of expressions that do not contain any derivative in t (Propositions 2.4, 2.6 and 2.7). Finally, the formulae of Theorem 1.1 are found by evaluation of these residues at t = 0; the latter task is then to compute the asymptotic expansions of Cauchy transforms of Laguerre polynomials at zero and infinity (Propositions 2.9 and 2.13). It is worth stressing at this point that the two formal series R ± of (1.9)-(1.10) in Theorem 1.1 are actually asymptotic expansions of the same analytic function at two different points.
As a preliminary to the proof, let us comment on the definition (1.21) of the LUE partition function. Even though a formal approach is sufficient to make sense of the LUE partition function as a generating function, we shall also regard it as genuine analytic function of the times t. In this respect let us point out that to make strict non-formal sense of (1.21) one can assume that the vector of times is finite, namely that and then, to ensure convergence of the matrix integral, that Re t K − < 0 for K − < 0 and Re t K + < δ K + ,1 for K + > 0.
Though we have to assume in our computations that we have chosen such an arbitrary truncation of the times, this is inconsequential in establishing the formulae of Theorem 1.1. More precisely, such truncation implies that (1.22) holds true only as long as K + , K − are large enough, and the formal generating functions C r + ,r − (as it follows from our arguments, see Section 2.3) are manifestly independent of K ± and are therefore obtained by a well-defined Moreover, in (1.21) the parameter α has to satisfy Re α > −1; even worse, in (1.22) we have to assume that Re α > − r i=1 k i − 1 to enforce convergence of the matrix integral at X = 0. This restriction can be lifted, if α is not an integer, by taking a suitable deformation of the contour of integration. This caveat is crucial to us, as we shall need the formal expansion of the matrix R(x) at all orders near x = 0, compare with (1.10); the coefficients of this expansion are in general ill-defined for integer α (although truncated expansions are well defined if α is confined to suitable right half-planes). It is clear how to overcome these issues by the aforementioned analytic continuation, hence we do not dwell further on this point.
For t = 0 they essentially reduce to the generalized Laguerre polynomials L (α) ℓ (x); more precisely, denoting π Using Rodrigues formula and integration by parts we obtain Hence the orthogonality property (2.2) for t = 0 reads as where h ℓ = h ℓ (t = 0). For general t instead, the monic orthogonal polynomials π (α) is non-degenerate. In the present case, their existence is ensured for real t by the fact that the moment matrix (m i+j ) L−1 i,j=0 is positive definite. By standard computations we have the following identity where h ℓ (t) are defined by (2.2).

Connection with Toda lattice hierarchy.
It is well known that the monic orthogonal polynomials π . That is, the orthogonal polynomials are eigenvectors of the second order difference operator It is a standard fact that L, and therefore the coefficients v α n (t) and w α n (t) evolve with respect to positive times t + = (t 1 , t 2 , . . . ), for any fixed t − = (t −1 , t −2 , . . . ), according to the Toda lattice hierarchy [27,49,2,30,19] ∂L ∂t k = L k + , L where for any matrix P , P + denotes the lower triangular part of P , i.e. the matrix with entries where P ij are the entries of P . Setting t − = 0, we can also write the initial data of the Toda hierarchy as v α ℓ (t + = t − = 0) = 2ℓ + 1 + α, w α ℓ (t + = t − = 0) = ℓ(ℓ + α) that are the recurrence coefficients for the monic generalized Laguerre polynomials (2.4). Moreover, it is well known, see loc. cit., that Z N (α; t + , t − = 0) is the Toda lattice tau function corresponding to this solution.
It can be observed that the evolution with respect to the negative times t − = (t −1 , t −2 , . . . ) is also described by a Toda lattice hierarchy and a simple shift in α. More precisely, we claim that Z N (α − 2N, t + = 0, t − ) is also a Toda lattice tau function, with a different initial datum; namely, it is associated with the tri-diagonal matrix L satisfying the Toda hierarchy constructed as above from the three term recurrence of monic orthogonal polynomials, this time with respect to the measure where we perform the change of variable X = X −1 , which is a diffeomorphism of H + N . The Lebesgue measure (1.2) can be rewritten (on the full-measure set of semisimple matrices) as where dU is a suitably normalized Haar measure on U(N)/(U(1)) N and x 1 , . . . , x N are the eigenvalues of X. Therefore the measure transforms as Summarizing, we have and the standard arguments of loc. cit. now apply to the matrix integral Z N (α−2N; t + = 0, t − ) to show that it is indeed the Toda lattice tau function associated with the solution L. For our purposes, we need to describe the simultaneous dependence on t + and t − ; this is achieved by the zero-curvature condition (2.16) of the system of compatible ODEs (2.14) which we now turn our attention to.

Cauchy transform and deformation equations. Let us denote by
the Cauchy transforms of the orthogonal polynomials π (α) ℓ (x; t). Then, for fixed N introduce the following 2 × 2 matrix where, for the interest of clarity, we drop the dependence on N, α. The matrix Y (x; t) was introduced in the seminal paper [46] to study the general connection between orthogonal polynomials and random matrix models. The rest of this section follows from [46]. The matrix (2.9) solves the following Riemann-Hilbert problem for orthogonal polynomials; it is analytic for x ∈ C\[0, ∞) and continuous up to the boundary (0, ∞) where it satisfies the jump condition within the sector 0 < arg x < 2π; the matrix G 0 (t) in (2.12) is independent of x and it is invertible (actually it has unit determinant, as we now explain). The jump matrix in (2.10) has unit determinant, hence det Y (x; t) is analytic for all complex x but possibly for isolated singularities at x = 0, ∞; however, det Y (x; t) ∼ 1 when x → ∞, see (2.11), and is bounded as x → 0, see (2.12); therefore we conclude by the Liouville theorem that det Y (x; t) ≡ 1 identically.
Introduce the 2 × 2 matrix Here we choose the branch of the logarithm appearing in Proposition 2.1. The matrix Ψ in (2.13) satisfies a compatible system of linear 2 × 2 matrix ODEs with rational coefficients; (2.14) In particular, for k > 0, the matrices Ω k (x; t) are polynomials in x of degree k, whilst for k < 0 they are polynomials in x −1 of degree |k| without constant term; more precisely, they admit the representations where res Proof. We note that (2.10) implies the following jump condition for the matrix Ψ, with a constant jump matrix; Here Ψ ± (x; t) = lim ǫ→0 + Ψ(x ± iǫ; t); to prove this relation we observe that the branch of the logarithm we are using satisfies log + (x) = log − (x) − 2πi for x ∈ (0, ∞) and so V α,+ (x; t) = V α,− (x; t) + 2iπα, with a similar notation for the ±-boundary values along (0, ∞). Hence all derivatives of Ψ satisfy the same jump condition, with the same jump matrix. It follows that the ratios A := ∂Ψ ∂x Ψ −1 and Ω k := ∂Ψ ∂t k Ψ −1 (for all k = 0) are regular along the positive real axis; however they may have isolated singularities at x = 0 and at x = ∞. Let us start from Ω k for k > 0. In such case, it follows from (2.11) and (2.12) that Ω k has a polynomial growth at x = ∞ and it is regular at x = 0: From the Liouville theorem we conclude that Ω k for k > 0 is a polynomial, which therefore equals the polynomial part of its expansion at x = ∞, which is computed as in (2.15), since at does not contribute to the polynomial part of the expansion. The statement for Ω k for k < 0 follows along similar lines. Likewise, A(x; t) in (2.14) has a polynomial growth at x = ∞ and a pole at x = 0 and therefore it is a Laurent polynomial.
The compatibility of (2.14) is ensured by the existence of the solution Ψ(x; t). In particular this implies the zero curvature equations Since the determinants of Y (x; t) and Ψ(x; t) are identically equal to 1, it follows that Ω k (x; t) and A(x; t), introduced in (2.14), are traceless.
We end this paragraph by considering the restriction which has a Fuchsian singularity at x = 0 and an irregular singularity of Poincaré rank 1 at x = ∞.
Remark 2.3. The Frobenius indices of (2.17) at x = 0 are ± α 2 , and so the Fuchsian singularity x = 0 is non-resonant if and only if α is not an integer. It is worth pointing out that the monodromy matrix α 2 σ 3 at x = 0 is preserved under the t-deformation (2.14). 2.2. Residue formulae for correlators.

2.2.1.
One-point correlators. The general type of formulae of Proposition 2.4 below first appeared in [9], where the authors consider a very general case. Such formulae identify the LUE partition function with the isomonodromic tau function [47] of the monodromy-preserving deformation system (2.14). The starting point for the following considerations is the representation (2.7) for the LUE partition function (1.21).
Proposition 2.4. Logarithmic derivatives of the LUE partition function admit the following expression in terms of formal residues; where the symbol res Proof. For the proof we follow the lines of [21]. First, differentiate the orthogonality relation x k e −Vα(x;t) dx and recall the confluent Christoffel-Darboux formula for orthogonal polynomials where in the last step one uses det Y (x; t) ≡ 1. Now, omitting the dependence on x, t in the rest of the proof for the sake of brevity, it can be checked that the jump relation (2.10) implies Therefore, starting from (2.7), we compute Such an integral of a jump can be performed by a residue computation. First of all, note that despite is not analytic at x = ∞, it has a large x asymptotic expansion given by 2 . Then, recalling our choice for the branch of the logarithm and using contour deformation, we can express (2.18) as the residues being intended in the formal sense explained above. Finally, the proof is complete by noting that for k > 0 (resp. k < 0) the formal residue at x = 0 (resp. x = ∞) vanishes.
For later convenience let us slightly rewrite the result of the above proposition. To this end introduce the matrix denoting E 11 := 1 0 0 0 from now on.
where R(x; t) is introduced in (2.19) and again res Proof. We have from (2.13) and (2.14) where in the last step we have used that where the first equality follows from tr A(x; t) = 0 and the second one from the cyclic property of the trace and the definition (2.19).

Multipoint connected correlators.
We first consider two-point connected correlators.
Proposition 2.6. For every nonzero integers k 1 , k 2 we have where the symbol res Let us take one more time-derivative ) and note that, using (2.16) and Now let us write Ω k 2 from (2.15) as (2.23) Finally, the identity holds true irrespectively of the sign of k 2 , and the proof is completed by inserting (2.22), (2.23) and (2.24) in (2.21), along with tr R(x; t) ≡ 1.
To compute higher order logarithmic derivatives of the LUE partition function, let us introduce the functions where, as explained in the statement of Theorem 1.1, the sum extends over cyclic permutations of {1, . . . , r}. Due to the cyclic invariance of the trace and of the polynomial ( The following proof is reported for the sake of completeness; it has appeared in the literature several times, e.g. see [7,30,8,11]. The only slight difference here is that we consider two different set of times and correspondingly the residues are taken at two different points.
where, as above, the symbol res where we have used (2.15) and res ξ denotes the formal residue at ξ = ∞ if k > 0 or the formal appears twice, but with different denominators; collecting these terms gives − res where the index j in the internal summation is taken mod r, namely i 0 := i r . Summarizing, we have shown that for all r ≥ 2 ∂S r (x 1 , . . . , x r ; t) ∂t k = − res ξ S r+1 (x 1 , . . . , x r , ξ)ξ k dξ and the proof now follows by induction on r ≥ 2, the base r = 2 being established in Proposition 2.6.
Remark 2.8. The functions S r (x 1 , . . . , x r ) are regular along the diagonals x i = x j . In the case r = 2 this can be seen from the fact that hence the function tr (R(x 1 ; t)R(x 2 ; t)) − 1 is symmetric in x 1 and x 2 and vanishes for x 1 = x 2 . Therefore the zero on the diagonal x 1 = x 2 is of order at least 2 and so S 2 (x 1 , x 2 ) is regular at x 1 = x 2 . For r ≥ 3 instead we can reason as follows; since S r is symmetric, we can focus on the case x r−1 = x r , and the only addends in S r which are singular at x r−1 = x r are those coming from the r-cycles (i 1 , . . . , i r−2 , r − 1, r) and (i 1 , . . . , i r−2 , r, r − 1), namely the terms (i 1 ,...,i r−2 ,r−1,r) and this expression is manifestly regular at x r−1 = x r .
In particular, the order in which residues are carried out in (2.26) is immaterial.
Finally we remark that it would be interesting to extend the above formulation to other matrix ensembles like the GOE, see e.g. [26]. where R(x; t) is introduced in (2.19) and compute explicitly series expansions as x → ∞, 0. We start with the expansion as x → ∞.
Proposition 2.9. The matrix R(x) admits the asymptotic expansion uniformly within the sector 0 < arg x < 2π. Here R + is the formal series introduced in the beginning of this paper, see (1.9), and T is defined as Remark 2.10. The matrix T is independent of x and is introduced for convenience as it simplifies the coefficients in the expansions. This simplification does not affect the residue formulae of the previous paragraph, as it involves a constant conjugation of R(x).
Proof. First off, we recall that where the polynomials π ℓ (x) have been given in (2.4) and (2.8) respectively, while h ℓ is in (2.6). We can expand π where we have used the orthogonality property to shift the sum in the first place, then Rodrigues formula (2.5) and integration by parts. The expansion (2.29) is formal; however, it has an analytic meaning of asymptotic expansion as x → ∞. Indeed, for any J ≥ 0 the difference between the Cauchy transform and its truncated formal expansion is where the last step holds as x → ∞, uniformly in C \ [0, +∞). Hence, using (2.4) and (2.29), from which the expansion at x = ∞ can be computed as follows. For the (1, 1)-entry we have and noting a trivial simplification of rising factorials (1.11). In a similar way we compute the (1, 2)-entry where in the second relation we use a similar version of (2.31), and therefore from the above relation and (2.30) we conclude that (1.11). Finally, the (2, 1)-entry of the expansion of T R(x)T −1 is computed in a similar way as  Proof. Introduce the matrices where we use that tr R ≡ 1; hereafter we omit the dependence on x for brevity. Recalling the first equation in (2.14) we infer that and writing using (2.17), we deduce from (2.34) the system of linear ODEs which in turn implies the following decoupled third order equations for ∂ x r 3 , r + , r − , (2.37) Finally, using the Wishart parameter M = N + α, we substitute the expansion at x = ∞ given by (1.9) into the ODEs (2.36) and (2.37) to obtain the claimed recursion relations.
Remark 2.12. Let us remark that the recursion for A ℓ (N, M) in Lemma 2.11 is also deduced, by different means, in [42]. In [23] it is pointed out that such three term recursion is a manifestation of the fact that A ℓ (N, M) is expressible in terms of hypergeometric orthogonal polynomials; this property extends to the entries B ℓ (N, M), as we now show. Introducing the generalized hypergeometric function 3 F 2 we can rewrite the coefficients A ℓ (N, M) and B ℓ (N, M) in the form Alternatively, introducing the Hahn and dual Hahn polynomials [50,23] Q j (x; µ, ν, k) : the coefficients A ℓ (N, M) and B ℓ (N, M) can be rewritten in the form Let us note that the first differential equation in (2.35) implies, at the level of the coefficients of the power series r 3 , r − , r + , the following relation

38)
which will be used in Example 2.14 to prove formulae(1.15).
Let us now consider the asymptotic expansion as x → 0.
Proposition 2.13. The matrix R(x) admits the asymptotic expansion Here R − is the formal series introduced in the beginning of this paper, see (1.10), and T is defined in (2.28).
Proof. First we observe that by arguments which are entirely analogous to those employed in the proof of Proposition 2.9, the matrices Y (x) and (consequently) R(x) possess asymptotic expansions in integer powers of x as x → 0, which are uniform in C \ [0, +∞). The first coefficients of these expansions at x = 0 can be computed from where the former is found directly from (2.4) and the latter by a computation analogous to (2.29); hence recalling the definition (2.9) we have as x → 0 within 0 < arg x < 2π; this implies that in the same regime we have Therefore, our goal is just to show that the coefficients of the latter expansion are related to those of the expansion at x = ∞ as stated in the formulae (1.9) and (1.10). To this end let us write, in terms of the decomposition (2.33),    Indeed from (2.20) we see that for all k = 0 we have Indeed we have ∂ x S 1 (x) = tr ((∂ x A(x))R(x)) + tr (A(x)(∂ x R(x))) . (2.46) and noting the following identities we can rewrite (2.46) as

and (2.45) follows noting tr (A(x)[A(x), R(x)]) = tr ([A(x), A(x)R(x)])
= 0 and 1 2 tr (σ 3 R(x)) = tr (E 11 R(x)) − 1 2 tr R(x) = R 11 (x) − 1 2 as tr R(x) ≡ 1. Hence, inserting (2.45) into (2.44) we obtain, irrespectively of the sign of k, At the level of generating functions, for C 1,0 (x) we have which, after integration, is the formula in the statement of Theorem 1.1; in the last step of the last chain of equalities, we have to observe that (R + ) 11 Similarly, for C 0,1 (x) we have which, after integration, is the formula in the statement of Theorem 1.1. Here we have noted that (R(x)) 11 = (T R(x)T −1 ) 11 since T is diagonal, see (2.28). The formulae for r ≥ 2 are proven instead by the following computation; r ∂ r log Z N (α; t) ∂t σ 1 k 1 · · · ∂t σr kr t=0 where we have noted that the transformation R → T RT −1 leaves the expression S r invariant, and therefore we are free to use the expansions R ± of Propositions 2.9 and 2.13; the signs σ i are those defined in (1.8). The proof is complete.
Example 2.14. As an application of Theorem 1.1, let us show how to prove formulae (1.15).
Combining (1.6) and (1.12) gives tr X k tr X c = res Let us write the matrix R + (x) as and expand the denominator in 1/(x 1 − x 2 ) 2 as a geometric series (the order we carry out the expansions in x 1 , x 2 is irrelevant, as explained in Remark 2.8) to rewrite the right side of (2.48) Finally, the residues extract the coefficient in front of x −k−1 1 x −2 2 , yielding tr X k tr X c = tr (E 11 R + k ) = kA k (N, M). In a similar way, from the relation tr X −k tr X −1 c = res we obtain The last equality follows from the recursion relations (2.32) and the formula (2.38). The computations of tr X k tr X −1 c and tr X −k tr X 1 c follow in a similar way.

Proof of Proposition 1.2
In this section we prove Proposition 1.2 by means of the explicit formulae for the matrices R ± (x) of Theorem 1.1. The proof follows from two main lemmas; the first one explains why rescaled correlators can be written as series in even powers of N only. We recall that we are working in the regime α = (c − 1)N, i.e. M = cN, with c independent of N. From (1.7) we can write generating functions for the rescaled correlators appearing in (1.17) as where we use the signs in (1.8).
Let us preliminarily observe two properties of the formula (1.13), which are crucial to our proof of Proposition 1.2. First, such formula is invariant under replacing the matrices R ± (x) with GR ± (x)G −1 for some constant non-degenerate matrix G, and second it is invariant (up to a simple modification for the two-point function) under replacing R ± (x) with R ± (x) + γ1 for any constant γ ∈ C. While the first property is trivial, the second one requires few lines of explanation. When r = 2 one can exploit the fact that tr R ± (x) ≡ 1 to write When r ≥ 3 instead we reason as follows. Let us write every r-cycle (i 1 , . . . , i r ) with i r = r, namely (i 1 ,...,ir)∈Cr where for the purpose of this explanation we adopt a short notation R i := R σ(i) (x i ); we point out that the role of the "fixed" matrix R r is completely arbitrary, as the function (1.13) is symmetric.
Let us now show that this expression is invariant under the transformation R r → R r +γ1; indeed the difference between the two expressions is computed from the last formula to be proportional to (i 1 ,...,i r−1 ,r)∈Cr It follows that in (1.13) one may inductively substitute all R i 's by R i + γ1 (in principle, even with different γ's for each R i , but we do not need such freedom) without affecting the formula (1.13).
Proof. Using formula (1.13) in Theorem 1.1 we have After the considerations exposed just before this lemma, it is clear that we are done if we find a matrix G such that the matrices GR ± (Nx)G −1 − 1 2 1 are both odd in N. We claim that the matrix 1 serves this purpose. The proof of this claim is a computation that we now perform; we have (N, cN)) . Therefore our claim is equivalent to the statement that D ℓ , E ℓ , F ℓ are odd functions of N. This is easily seen from the linear recursions of Lemma 2.11. For the coefficients E ℓ the initial datum of the recursion is E 0 (c, N) = −2N, E 1 (c, N) = −2cN, and the recursion reads The same claim for R − (Nx) is proven exactly in the same way, as we have and, since α = (c − 1)N, which is even in N.
The second lemma regards integrality of the coefficients.
Proof. Using the identity we rewrite (1.11), for ℓ ≥ 1, as and then we rewrite this expression, by a change of variable 1 + ξ = x, 1 + η = y, as Similarly, for all ℓ ≥ 0 we have and the proof is complete.
The expression (3.5) is also derived, in a different way, in [42]. It can be checked that the coefficients (3.5) are integers within the range of summation a, b ≥ 0, a + b ≤ ℓ − 1; indeed if a + b ≤ ℓ − 2 one can write such coefficient as which is manifestly integer, while if a + b = ℓ − 1 the same coefficient is written as which is also manifestly integer since a ≤ ℓ − 1. Similarly, the coefficients (3.6) are integers within the range of summation a, b ≥ 0, a + b ≤ ℓ.
Proof of Proposition 1.2. Lemma 3.2 implies that A ℓ (N, cN) and B ℓ (N, cN) are polynomials in N and c with integer coefficients. Then the dependence on N 2 follows from Lemma 3.1 and the expansion of (3.4) as series in N and (c − 1) with integer coefficients as provided ℓ < N(c − 1). Finally we note, e.g. from the recursions, that refers to the behavior as N → ∞. We conclude that (3.1) is O(1) as N → ∞, and has the same parity in N → −N as r (Lemma 3.1), completing the proof.

Example 3.3.
Here we obtain the formulae of Theorem 1.1 in genus zero for one-and two-point correlators. In these cases, formulae of the same kind have already appeared in the literature [37,54,25].
The above relations follow from Lemma 3.2 and the trivial estimate N k ∼ N k k! . In particular due to (1.14), in the regime N → ∞ with α = N(c − 1) we have  [36]. From the one-point function we obtain the weighted strictly monotone and weakly monotone double Hurwitz numbers of genus zero with partition µ = (k) and ν of length s as Similarly, for all two-point generating functions, we obtain the planar limit g = 0 as lim N →∞ The two-point planar limit is strictly related [35] to the so called canonical symmetric bidifferential (called also Bergman kernel) associated to the spectral curve where V (x) is an arbitrary potential for which the polynomials are well defined. The following lemma is elementary and the proof can be found in [21]. ( 1 2 ) n (x 2 ) and (4.1) Next we recall the relation between matrix integrals and the norming constants of the above orthogonal polynomials where Vol(N) is defined in (1.24).
Using the above relations and (4.1) in the case V (x) = x 2 − k≥1 s k x k , we obtain the following identity between the GUE partition function Z even 2N (s) in (1.23) and the Laguerre partition where Z even N (0) is given in (4.4) and Z N ± 1 2 ; 0 in (1.3). There is a similar, slightly more involved, factorization for the matrix model Z even 2N +1 , but we do not need its formulation for our present purposes.

4.2.
Formal matrix models and mGUE partition function. In this section we review the definition of mGUE partition function. First, the logarithm of the even GUE partition function can be considered as a formal Taylor expansion for small s k as log Z even N (s) := log Z even N (0) + r≥1 k 1 ,...,kr≥1 s k 1 · · · s kr r! tr X k 1 · · · tr X kr even c (4.3) where the connected even GUE correlators are introduced as in (1.22) tr X k 1 · · · tr X kr even c := ∂ r log Z even N (s) ∂s k 1 · · · ∂s kr s=0 and the normalizing constant Z even introducing the grading deg s k := k, the latter algebra is obtained taking the inductive limit K → ∞ from the algebras of polynomials in s of degree < K, with coefficients in C[N, α].
Equivalently, this grading can be encoded, up to an inessential shift, by a (small) variable ǫ via the transformation s k → ǫ k−1 s k , which is the same as considering the matrix model For simplicity we have preferred to avoid the explicit ǫ-dependence, even though we shall restore it for the statement of the Hodge-GUE/LUE correspondence (Theorem 4.4, Corollary 4.5).
It must be stressed that (4.3) makes sense for any complex N, and not just for positive integers as it would be required by the genuine matrix integral interpretation; indeed the correlators are polynomials in N.
For the purposes of this section it is convenient to apply the same arguments to the Laguerre partition function (with t − = 0) and similarly identify the latter with the formal series log Z N (α; t + ) = log Z N (α; 0) + r≥1 k 1 ,...,kr≥1 t k 1 · · · t kr r! tr X k 1 · · · tr X kr c (4.5) where Z N (α; 0) is given in (1.3) and the correlators are as in (1.22); using the last expression provided in (1.3) and the fact that the correlators are polynomials in N, α the expression (4.5) makes sense also for N complex. This remark is crucial for a correct understanding of formulae (4.7) and (4.8) below. Let us finally recall from the introduction and [28] that the mGUE partition function is introduced by (1.24), the left side of which being interpreted formally as in (4.3). Of course in the identification of Theorem 1.5, the right side must be interpreted formally as in (4.5).

4.3.
Proof of Theorem 1.5. The proof of Theorem 1.5 relies on two main ingredients; on one side the factorization property (4.2), and a symmetry property of the formal positive LUE partition function (4.5), which we now describe.
Lemma 4.2. The LUE connected correlator tr X k 1 · · · tr X kr c with k 1 , . . . , k r > 0 is a polynomial in N, α, and it is invariant under the involution (N, α) → (N + α, −α).  Such elements commute among themselves and are related by From this relation we deduce that for every partition µ of d of length ℓ we have From the definition of Hurwitz numbers, recalled in Section 1.2, the coefficient in front of C [1 d ] y d−s z 2g−2+ℓ+s on the left side of (4.6) is H > g (µ; s) (up to the normalization factor zµ d! ); the coefficient of the same element C [1 d ] y d−s z 2g−2+ℓ+s on the right side of (4.6) is H > g (µ; 2 − 2g + d − ℓ − s) (up to the same normalization factor zµ d! ), yielding (1.31). Let us restate Lemma 4.2, in view of the formal expansion (4.5), as the following identity The following special case (α = 1 2 ) of (4.7) plays a key role in the proof of Theorem 1.5, which we are now ready to give.
Proof of Theorem 1.5. We use the uniqueness of the decomposition (1.24) which defines the mGUE partition function; rewriting it under the substitution N → 2N we have On the other hand, from (4.2) we have where here and below we are identifying t k = 2 k s k . The proportionality constant D N is explicitly evaluated from (1.3) and (4.4) as .

(4.11)
It is then enough to show that the two factorizations (4.9) and (4.10) are consistent once we identify Z 2N − 1 2 (s) = C N Z N − 1 2 ; t + with C N a constant depending on N only. Such consistency follows from the chain of equalities where we have used the symmetry property (4.8). This shows that the two factorizations (4.9) and (4.10) are consistent, provided we also identify the proportionality constants (4.11) and (4.12) where in the last step we use the duplication formula for the Barnes G-function in the form Equation (4.13) fixes the constant to be as stated in (1.26).
We conclude this section with a couple of remarks. First, the identification of the mGUE and LUE partition functions is manifest also from the Virasoro constraints of the two models. Indeed, Virasoro constraints for the modified GUE partition function have been derived in [28], directly from those of the GUE partition function, and they assume the form L n Z N (s) = 0, for n ≥ 0, where (4.14) On the other hand, it is well known [43,1] that the LUE partition function with only positive couplings t + satisfies the Virasoro constraints L (α) under the identification t k = 2 k s k , in agreement with Theorem 1.5. Second, in [29] formulae of similar nature as those of Theorem 1.1 are derived for the modified GUE partition function. It can be checked that such formulae match with those of Theorem 1.1 restricted to α = − 1 2 under the identifications of times made explicit in the statement of Theorem 1.5.

4.4.
Proof of Corollary 1.6. From Theorem 1.5 and the Hodge-GUE correspondence of [28], which we now recall, we are able to deduce a Hodge-LUE correspondence; to state this result (Corollary 4.5) let us introduce the generating function for special cubic Hodge integrals (with the standard notations recalled before the statement of Corollary 1.6); here p = (p 0 , p 1 , . . . ).
Theorem 4.4 (Hodge-GUE correspondence [28]). Introduce the formal series where we identify and A(λ, s) is defined in (4.16), p(λ, s) is defined in (4.17), and C(N, ǫ) is a constant depending on N and ǫ only.
It would be interesting to construct the Double Ramification hierarchy (see [15,16]) for cubic Hodge integrals, and then check in this case the conjecture formulated in [17] by which the logarithm of the corresponding tau function should coincide with the LUE partition function, after the change of variables described in [18].
Proof of Corollary 1.6. We apply , for ℓ > 0, on both sides of (4.18). On the right side we get, in view of Theorem 1.5 where in the last step we have used (1.18); we also note that the substitutions 2N − 1 2 = λ ǫ , α = − 1 2 , from Theorem 1.5, yield N = On the other side we get A(λ, s).