Fluctuation Moments for Regular Functions of Wigner Matrices

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of Male et al. (Random Matrices Theory Appl. 11(2):2250015, 2022), showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem given in the recent companion paper (Reker in Preprint, arXiv:2204.03419, 2023). and thus allow identifying the fluctuation around the thermal value in certain thermalization problems.


Introduction
In his seminal work [32], Wigner established that the empirical spectral measure of certain random matrix ensembles converges, as the dimension goes to infinity, to the semicircle distribution. Since then, many variations and extensions of this result have been considered, yielding a variety of asymptotic phenomena for a wide range of random matrix models. One particular example is the fact that the resolvent G(z) = (W − z) −1 of a large Hermitian random matrix W tends to concentrate around a deterministic matrix M = M (z) for spectral parameters z ∈ C even just slightly away from the real axis (see, e.g., [5] and references therein for a collection of recent results). It was recently shown (see [6,5]) that a similar concentration holds for alternating products of the form F [1,k] (1.1) Here, A 1 , . . . , A k are bounded deterministic matrices and f 1 , . . . , f k are regular test functions, allowing in particular for f j (W ) = G(z j ). Products of the form (1.1) with f j (W ) replaced by (polynomials of) the random matrix itself play a key role in free probability theory, as they characterize the joint non-commutative probability distribution of Wigner and deterministic matrices.
We remind the reader that a (tracial first-order ) non-commutative probability space is a pair (A, ϕ 1 ) consisting of a complex unital algebra A and a tracial linear functional ϕ 1 : A → C with ϕ 1 (1 A ) = 1, where 1 A is the unit element of the algebra. One particular example is the space (A, contains all random variables with all finite moments, and · denotes the normalized trace. Note that this definition includes deterministic and Wigner matrices. In this context, the non-commutative probability distribution of a ∈ A is characterized in terms of its moments (ϕ 1 (a k )) k with the joint distribution of multiple elements of A being defined analogously. Recent work by Cipolloni, Erdős and Schröder [6] established that the structure of the limit of E F [1,k] as in (1.1) matches the formulas obtained in free probability, and reproduces known results for the alternating moments E W 1 D 1 . . . W k D k of a finite family of independent Wigner matrices (W j ) j and a finite family of deterministic matrices (D j ) j (see, e.g., [22,Sect. 4.4]) in the case f j (x) = x. More precisely, in the large N limit, the leading-order term m[F [1,k] ] of E F [1,k] is of the form m[F [1,k] ] := π∈N CP ([k]) B∈π j∈B . . , f k ), (1.2) where N CP ([k]) denotes the non-crossing partitions of the cyclically ordered set {1, . . . , k} and the functions Φ (1) π only depend on f 1 , . . . , f k and π ∈ N CP ([k]). Hence, the righthand side of (1.2) is a sum of terms that factorize into a contribution of the deterministic matrices resp. the test functions appearing in the product (1.1) with the underlying combinatorics matching the results obtained for the case f j (x) = x in free probability theory. Note, however, that resolvents and functions with an N -dependent mesoscopic scaling are typically not accessible in free probability as many of the standard techniques rely on explicit moment computations for polynomials. The results in [6] thus show that the underlying combinatorics continue to apply in a more general context.
After considering the concentration of (1.1), the next natural step is to study the fluctuations around the deterministic value. It is well-known that the linear statistics Tr f (W ) = N j=1 f (λ j ) with a regular test function f : R → R have a variance of order one (first observed in [14]) and, in fact, satisfy a central limit theorem (CLT) with a Gaussian limit, as shown, e.g., in [15] for the Wigner case and in [13] for invariant ensembles. By now, the statistics Tr f (W ) are well-studied on both macroscopic and mesoscopic scales (see, e.g., [11,1,19,18,29,31,3,30,12,2,16] for the Wigner case and [7,27] for further references on previous results for Wigner matrices and other models). However, while the fluctuations of Tr[f (W )A] are known for general regular functions f (see [17] and [7]), traces of products of the form (1.1) for k ≥ 2 have so far only been studied for f j being polynomials in the context of second-order freeness (see, e.g., [22,Ch. 5] or [8,20,21]).
We remind the reader that a second-order non-commutative probability space is a triplet (A, ϕ 1 , ϕ 2 ), where the functional ϕ 2 : A × A → C is bilinear, tracial in both arguments, symmetric under the interchanging of its arguments, and satisfies ϕ 2 (a, 1 A ) = ϕ 2 (1 A , a) = 0 for all a ∈ A. The second-order probability distribution of a ∈ A is characterized in terms of (ϕ 2 (a k , a ℓ )) k,ℓ , called the fluctuation moments, with the joint moments of multiple elements again being defined analogously. As a canonical example, we remark that M N ×N (L ∞− (Ω, P)) may be endowed with the functional ϕ 2 (·, ·) = Cov(Tr(·), Tr(·)), to make it a second-order probability space. In contrast to the first-order structure, the fluctuation moments are sensitive to the symmetry class of the underlying Wigner matrix and explicitly involve the fourth cumulant of the entry distribution (see [21,Thm. 6], as well as [25,26,7]). In particular, we observe a breaking of universality compared to the first-order problem of computing E · . The joint fluctuation moments of Wigner and deterministic matrices are explicitly known (cf. [22,Thm. 13 of Ch. 5] for the GUE case and [21,Thm. 6] for general Wigner matrices).
A functional CLT for traces of products of the form (1.1) has recently been established in the companion paper [27] and the limiting covariance is derived using a recursion. In the present paper, we supply the combinatorial argument necessary to obtain the solution to the recursion and compute the limiting covariance explicitly. More precisely, we show that if W is a GUE matrix, the leading order term m[F [1,k] |F [k+1,k+ℓ] ] of the covariance of Tr(F [1,k] ) and Tr(F [k+1,k+ℓ] ) (with F [k+1,k+ℓ] = f k+1 A k+1 . . . f k+ℓ A k+ℓ of the same build as (1.1)) is given by m[F [1,k] π 1 ×π 2 (f 1 , . . . , f k+ℓ ).
Here, − −− → N CP (k, ℓ) denotes the non-crossing permutations of the (k, ℓ)-annulus and the functions Φ (2) π resp. Φ (2) π 1 ×π 2 only depend on f 1 , . . . , f k+ℓ and the underlying permutation resp. partition. Similar to (1.2), we thus obtain a sum of terms that factorize into a contribution of the deterministic matrices resp. the test functions appearing in the product (1.1) with the underlying combinatorics again matching the results obtained for the case f j (x) = x in free probability theory (see [24]). Moreover, we show that the overall structure of (1. 3) continues to hold if W is chosen to be a Wigner matrix with W ij d = N −1/2 χ od for i < j and W jj d = N −1/2 χ d for general entry distributions χ od and χ d . In the general case, however, the sum in the first line of the right-hand side of (1.3) splits into four summands Φ of the Wigner matrix W . A similar decomposition is also observed for Φ (2) π 1 ×π 2 in (1.3). In particular, we find that the closed expression obtained from solving the recursion in [27] has the same overall structure as the formulas in [21,Thm. 6]. This shows that the analogies [6] established in the first-order setting have a counterpart for the second-order structures. Our combinatorial approach further allows us to give the functions in (1.3) in a closed form, thus yielding a fully explicit formula for the limiting covariance in the GUE case.
We remark that the main results of the present paper, i.e., combinatorial formulas for m[F [1,k] |F [k+1,k+ℓ] ] such as (1.3), are applied to obtain an explicit limiting covariance structure for the multi-point functional CLT [27,Thm. 2.7]. Here, replacing the recursive definition of the limiting variance by a closed formula allows for an easier application of the theorem, e.g., to thermalization problems in physics (cf. [27,Cor. 2.12]). We emphasize that the main results in the companion paper [27] are of analytic nature and that their main technical difficulty lies in including functions with a mesoscopic scaling of the form where g j is a regular N -independent function, E ∈ R lies in the bulk of the limiting spectrum of W , and N −γ is larger than the typical eigenvalue spacing around E. In contrast, we assume all test functions to be N -independent in the present paper and focus on the combinatorial structures arising in the multi-point functional CLT. While an extension to the functions in (1.5) is possible using the techniques from [27], restricting to the macroscopic regime allows for a cleaner presentation of the results. It further facilitates working with more general assumptions on the Wigner matrix W . Note that [27,As. 1.1] corresponds to setting σ = ω 2 = 0 in (1.4), while Assumption 1.1 below matches the setting of [21,7] with general σ ∈ [−1, 1] and ω 2 ≥ −2, thus generalizing the formulas from [27].
We conclude the section with a brief overview of the paper. After introducing some commonly used notations, the assumptions on the Wigner matrix W are given in Assumption 1.1. We then give a brief overview of the combinatorics needed to identify the deterministic approximation of T 1 . . . T k where T j := G(z j )A j , and the multi-resolvent local laws needed for the analysis of the fluctuations (Section 1.2) as well as the definitions from free probability that are used to characterize the limiting covariance of T 1 . . . T k −E T 1 . . . T k and T k+1 . . . T k+ℓ − E T k+1 . . . T k+ℓ (Section 1.3). To prepare for the statements of our main results, we give a CLT for the case that all functions f j are resolvents (Theorem 2.3). The role of the limiting covariance in the theorem is played by a recursively defined set function m[·|·] (Definition 2.1), which is the main object of interest in the present paper.
We study the recursion in detail in Section 2.2 and obtain explicit combinatorial formulas for its solution (Theorems 2.4, 2.6 -2.8). In Section 2.3, we extend the CLT to more general test functions (Theorem 2.9 and Corollary 2.10) to discuss the connection to free probability theory in detail. In particular, we apply the results to the case f j (x) = x and show that the limiting covariance in the functional CLT reduces to the formula for the joint fluctuation moments of GUE and deterministic matrices (Corollary 2.11) as given in [21]. Lastly, the proofs are given in Sections 3 and 4. To keep the presentation concise, some routine calculations are deferred to the appendix.
Acknowledgements: I am very grateful to László Erdős for suggesting the topic and many valuable discussions during my work on the project. Partially supported by ERC Advanced Grant "RMTBeyond" No. 101020331.

General Notation
We start by introducing some notation used throughout the paper. For two positive quantities f, g, we write f g and f ∼ g whenever there exist (deterministic, N -independent) constants c, C > 0 such that f ≤ Cg and cg ≤ f ≤ Cg, respectively. We denote the Hermitian conjugate of a matrix A by A * and the complex conjugate of a scalar z ∈ C by z. Moreover, · denotes the operator norm, Tr(·) is the usual trace and · = N −1 Tr(·). We further denote the covariance of two complex random variables X 1 , X 2 by Cov(X 1 , X 2 ) and follow the convention i.e., the covariance is linear in the first and anti-linear in the second entry. For k, a, b ∈ N with a ≤ b, we set [k] = {1, . . . , k} and adopt the interval notation [a, b] = {a, a+ 1, . . . , b}. We further write a, b] or [a, b to indicate that a or b are excluded from the interval, respectively. Ordered sets are denoted by (. Given a matrix A ∈ C N ×N , the traceless part of A is denoted byÅ := A − A Id where Id denotes the identity matrix. Further, a := diag(A) denotes the diagonal matrix obtained from extracting only the diagonal entries of A and A 1 ⊙ A 2 denotes the entry-wise (or Hadamard) product of two matrices A 1 and A 2 . For a Hermitian matrix W and z 1 , . . . , z k ∈ C \ R, we write the corresponding resolvents as G j = G(z j ) := (W − z j ) −1 and index products of resolvents using the interval notation Recalling that angled brackets indicate that an edge point of the interval is excluded, we write G a,b] and G [a,b to exclude G a or G b from the product, respectively. Moreover, G ∅ is interpreted as zero. Note that this notation differs slightly from [6,5]. As we often consider alternating products of resolvents with deterministic matrices A 1 , . . . , A k , define T j := G j A j and apply the same interval notation as above to write Again, angled brackets are used to exclude T a or T b from the product, respectively, and T ∅ is interpreted as zero. We call a product of the type (1.6) resolvent chain of length k.
Throughout the paper, we assume W to be an N × N real or complex Wigner matrix satisfying the following assumptions.
Assumption 1.1. The matrix elements of W are independent up to Hermitian symmetry W ij = W ji and we assume identical distribution in the sense that there is a centered real random variable χ d and a centered real or complex random variable χ od such that W ij d = N −1/2 χ od for i < j and W jj d = N −1/2 χ d , respectively. We further assume that E|χ od | 2 = 1 as well as the existence of all moments of χ d and χ od , i.e., there exist constants C p > 0 for any p ∈ N such that We remark that Assumption 1.1 matches the model considered in [7] and [21]. Compared to the conditions Eχ 2 od = 0 and Eχ 2 d = 1 in [27], we allow for arbitrary values of the parameters σ = Eχ 2 od ∈ [−1, 1] and ω 2 = Eχ 2 d ≥ 0. This description includes real symmetric Wigner ensembles such as GOE (σ = 1) as well as matrices of the form W = D + iS where D is a diagonal matrix and S is skew-symmetric (σ = −1). We further introduce the notation for the normalized fourth cumulant of the off-diagonal entries as well as The eigenvalue density profile of W is described by the semicircle law which mainly enters our analysis in the form of its Stieltjes transform We remind the reader that m(z) is the unique solution of the Dyson equation and that its derivative satisfies (1.12) Given fixed z 1 , . . . , z k ∈ C \ R, set m j = m(z j ) and m ′ j = m ′ (z j ), respectively. We further introduce possibly setting q j,j = m ′ j whenever i = j.

Preliminaries Part 1: First-Order Quantities
In this section, we briefly summarize the definitions and results from [5,6] which are needed to characterize the deterministic approximation of T [1,k] . Definition 1.2 (Non-crossing partitions). Let S be a finite (cyclically) ordered set of integers. We call a partition π of the set S crossing if there exist blocks B = B ′ in π with a, b ∈ B, c, d ∈ B ′ , and a < c < b < d, otherwise we call it non-crossing. The set of non-crossing partitions is denoted by N CP (S) and we abbreviate N CP (k) := N CP ([k]).
Recall that non-crossing partitions have an alternative geometrical definition: Arrange the elements of S equidistantly in clockwise order on the circle and for each π ∈ B consider the convex hull P B of the points s ∈ B. Then π is non-crossing if and only if the polygons {P B |B ∈ π} are pair-wise disjoint. Because of this, we also call the elements of N CP (k) disk non-crossing to distinguish them from their annulus analog defined below. We further recall the definition of the Kreweras complement (see Fig. 1 for an example). Definition 1.3 (Kreweras complement, disk case). Let S ⊂ N be a finite set of integers equidistantly arranged in clockwise order on the circle and label the midpoints of the arcs between the points s ∈ S also by the elements of S. We arrange the new labels such that the arc s follows the point s in clockwise order. Let π ∈ N CP (S). Then the (disk) Kreweras complement of π, denoted by K(π), is the element of N CP (S) such that r, s belong to the same block of K(π) if and only if the arcs labeled r, s are in the same connected component in the complement D \ ∪ B∈π P B of the polygons {P B |B ∈ π} in the labeled disk D. Observe that D \ ∪ B∈π P B has |S| − |π| + 1 connected components, hence |π| + |K(π)| = |S| + 1. Further, K 2 = K • K recovers π up to a rotation of D, i.e., K 2 (π) is the partition where for S = {s 1 , . . . , s k } the elements in each block of π are shifted by s 1 → s 2 → · · · → s k → s 1 . In particular, taking the Kreweras complement is invertible as a map on N CP (S We emphasize that Definition 1.4 does not require f to have any particular symmetries. However, in the free probability literature, f usually arises from tracial functional and is hence symmetric under the cyclic permutation of its entries (cf., e.g., [22,Ch. 2]). The implicit relation in (1.14) can be recursively turned into an explicit definition of f • . Alternatively, we may also invert (1.14) explicitly using the Möbius function associated with the lattice of non-crossing partitions. Recall that N CP (S) is a lattice with respect to the refinement order, i.e., the partial order in which π ≤ ν if and only if for each B ∈ π there exists B ′ ∈ ν with B ⊂ B ′ . Moreover, there are unique maximal and minimal elements given by 0 S := {{s}|s ∈ S} and 1 S := {S}, respectively. The free cumulant function can then be written as using the Möbius function µ : {(π, ν)|π ≤ ν ∈ N CP (S)} → Z that is recursively defined by (1.15). We remark that µ(π, 1 S ) can be given in a closed form using the Catalan numbers (see, e.g., [6,Lem. 2.16]).
The following choice for the function f is of particular interest.
whenever there are two distinct z 1 = z k among z 1 , . . . , z k and otherwise  (12) and (1)(2) such that The quantities m and m • were studied in detail in [6], yielding a close connection to noncrossing graphs. We recall the definition and give an example in Fig. 2 below. These graphs are planar. For later convenience, we use a slightly more general notion of planar graphs throughout the paper than the standard literature by allowing for self-connections (loops) and multi-edges.   Emphasizing that Definition 1.7 lives on a disk is important, as we later introduce a noncrossing property on the annulus. Whenever both definitions are used together, we use the specifications disk non-crossing and annular non-crossing to distinguish the underlying geometry. By construction, every Γ ∈ N CG(S) induces a non-crossing partition with blocks representing the vertices in the connected components of Γ. Further, any connected component of Γ is itself a (disk) non-crossing graph. A key technical tool in the proof of our main results is the optimal multi-resolvent local law [5,Thm. 2.5]. As we only work on macroscopic scales, i.e., with N -independent spectral parameters, in the present paper, we state the result in the form of a global law and omit the dependence on η * = min |ℑz j |. Recall the commonly used definition of stochastic domination.
Definition 1.8 (Stochastic domination). Let be two families of non-negative random variables that are indexed by N and possibly some other parameter u. We say that X is stochastically dominated by Y , denoted by X ≺ Y or X = O ≺ (Y ), if, for all ε, C > 0 we have Theorem 1.9 (Macroscopic version of [5,Thm. 2.5]). Fix k ∈ N and pick spectral parameters z 1 , . . . , z k ∈ C \ R with |ℑz j | 1 and max j |z j | ≤ N 100 as well as deterministic matrices where B(k) is the block in K(π) that contains k. Recalling that T j = G j A j , we have the averaged local law and for x, y ∈ C N with x , y 1 we have the isotropic local law (1.20) As we frequently encounter M [k] A k in the following sections, we introduce the notation We further need a generalization of the averaged local law (1.19) that includes transposes. Theorem 1.10 (Global law for resolvent chains with transposes). Let k ∈ N and pick spectral parameters z 1 , . . . , z k with |ℑ(z j )| 1 and max j |z j | ≤ N 100 as well as deterministic matrices A 1 , . . . , A k with A j 1. Moreover, let G ♯ j denote either the resolvent G j = G(z j ) or its transpose G t j and denote by # the binary vector that has a one in j-th position if G ♯ j = G t j and a zero otherwise. Then, where m #,σ • [·] denotes the free cumulants associated with the set function m #,σ [·] by Definition 1.4. Here, m #,σ [·] is defined to satisfy m #,σ [∅] = 0 as well as the recursion i.e., either both G 1 and G k occur as transposes in the product G ♯ 1 . . . G ♯ k or neither of them, and q ♯ 1,k = σm 1 m k 1−σm 1 m k otherwise. Similarly, c 1,j = 1 whenever # 1 = # j and c 1,j = σ otherwise. Recall that σ = Eχ 2 od where χ od is the real or complex random variable that specifies the distribution of the off-diagonal entries of the Wigner matrix W .
The proof of Theorem 1.10 is, modulo careful bookkeeping of the transposes, similar to the proof of the averaged local law in [6,Thm. 3.4]. For the convenience of the reader, a brief sketch of the argument is included in Appendix A.1. We remark that the same result may be obtained on mesoscopic scales with optimal error bounds following the strategy of [5]  Note that σ = 1 implies that the matrix W is real and its resolvent satisfies G t j = G j . Hence, the statement of Theorem 1.10 reduces to that of an averaged global law for real symmetric Wigner matrices in this case. Due to the structural similarity between (1.22) and (1.23), we will slightly abuse notation and write the right-hand side of (1.23) as

Preliminaries Part 2: Second-Order Quantities
In this section, we give an overview of the definitions from free probability that are used in later sections as well as some related quantities appearing in the CLTs.
Recall that the key picture for describing the expectation of T [1,k] is a disk with the labels 1, . . . , k organized in clockwise order along its boundary. In a very similar spirit, the key picture for describing the corresponding second-order object, i.e., the covariance of T [1,k] and T [k+1,k+ℓ] , consists of two concentric labeled circles. Let k, ℓ ∈ N and arrange the numbers 1, . . . , k equidistantly in clockwise order on the outer circle and the numbers k + 1, . . . , k + ℓ equidistantly in counter-clockwise order on the inner circle. We refer to the planar domain between these two circles together with the labeled points on its boundary as the (k, ℓ)-annulus (see Fig. 3 below). The labeled points will often serve as vertices of a graph. In this case, any edges connecting two points are drawn inside the annulus. Definition 1.11 (Annular non-crossing permutations). Let k, ℓ ∈ N. We call a permutation of [k + ℓ] an annular non-crossing permutation if we can draw its cycles 3 on the (k, ℓ)-annulus such that the following conditions (see [22,Def. 5 in Ch. 5]) are satisfied: (i) Non-crossing property: The cycles do not cross.
(ii) Standardness: Each cycle encloses a region in the annulus that is homeomorphic to the disk with boundary oriented clockwise (in particular, the cycles follow the orientation of the numbering of the circles).
(iii) Connectedness: At least one cycle connects both circles.
The set of annular non-crossing permutations is denoted by − −− → N CP (k, ℓ). Any cycle that connects both circles is referred to as connecting cycle.
We remark that − −− → N CP (k, ℓ) can be fully characterized by the avoidance of certain crossing patterns (cf. analogous geometric characterization of N CP (k) below Definition 1.2) and an algebraic analog of the standardness condition. This equivalent definition is discussed, e.g., in [23,Sect. 3], but we will not use it here. Definition 1.12 (Annular non-crossing partitions). Let k, ℓ ∈ N. We call the partitions induced by the cycles of − −− → N CP (k, ℓ) annular non-crossing partitions. The set of annular non-crossing partitions is denoted by N CP (k, ℓ). A block that arises from a connecting cycle is referred to as connecting block.
While there is a one-to-one correspondence between the non-crossing partitions of the disk in Definition 1.2 and the permutations of [k] avoiding the same crossing pattern, there is a crucial difference between non-crossing partitions and permutations on the (k, ℓ)annulus. In particular, there is no bijective mapping between a permutation in − −− → N CP (k, ℓ) and the partition of [k + ℓ] induced by its cycles, as, e.g., both permutations (123) and (132) correspond to the partition {{1, 2, 3}}, but give rise to different pictures due to the orientation induced by Definition 1.11(ii) (see Fig. 4 below). In general, a permutation always uniquely determines the underlying partition, but a partition can be obtained from more than one permutation. This happens if and only if there is exactly one connecting block (cf. [23,Sect. 4]). Lastly, we consider partitions arising from permutations that respect the non-crossing property and standardness condition but do not have a connecting cycle. In this case, we may consider the permutation restricted to each circle separately, i.e., as an element of N CP (k) × N CP (ℓ), and introduce an artificial connection by marking one block on each circle. Definition 1.13 (Marked non-crossing partition). Consider π ∈ N CP (k) × N CP (ℓ) that naturally splits into π = π 1 × π 2 with π 1 ∈ N CP (k), π 2 ∈ N CP (ℓ). We pick one block of π 1 and one of π 2 , respectively, and mark them by underlining. The resulting object is referred to as a marked non-crossing partition.
Marking a block on each circle allows us to artificially introduce a connecting block by considering the union of the two marked blocks. As a consequence, any marked noncrossing partition can be associated with a unique element of N CP (k, ℓ). We further note that there are |π 1 |·|π 2 | possibilities to mark the blocks of π = π 1 ×π 2 ∈ N CP (k)×N CP (ℓ). (1.26) for any finite S 1 , S 2 . Here, f • is the first-order free cumulant function introduced in Definition 1.4.
Note that we use a set function f that is symmetric under the interchanging of its arguments instead of its skew-symmetric version Although we will not rely on Möbius inversion to express f •• , we remark that it is possible to include both − −− → N CP (k, ℓ) and the marked elements of N CP (k) × N CP (ℓ) into one common definition, the non-crossing partitioned permutations, which can be endowed with a partial ordering and hence render it suitable for Möbius inversion. This would allow to rewrite the right-hand side of (1.26) to a structure similar to (1.14) and obtain a closed formula similar to (1.15). We refer to Sections 4 and 5 of [8] for the full construction.
Similar to (1.14) above, the relation (1.26) is applied for one particular choice of f built up from the Stieltjes transform m(z). We will later see that the set function m[·|·] defined below arises as the deterministic approximation of the (appropriately scaled) covariance of G [1,k] and G [k+1,k+ℓ] in a similar way that the divided differences m[·] arise for the expectation of G [1,k] . In particular, m[·|·] satisfies the symmetry and cyclicity assumption in Definition 1.14. We give a recursive definition of m[·|·] for now, however, closed formulas are later obtained in Section 2.2. (i) Symmetry: m[·|·] is symmetric under the interchanging of its arguments, i.e., for any sets (ii) Initial condition: For any sets where the source terms in the last line are given by Here, we wrote out the underlying multi-set in the definition of s κ to indicate that it evaluates to m[s, s] instead of m s if t = s and the vector # ∈ {0, 1} k+ℓ+1 is given by # 1 = · · · = # k = 0 and # k+1 = · · · = # k+ℓ+1 = 1. Recall that m[·] denotes the divided differences as introduced in Definition 1.5 and that m #,σ [·] was introduced in Theorem 1.10.
Note that the recursion for m[·|·] is linear with different types of source terms in the last line of (1.28). Therefore, we may introduce the decomposition , and ω 2 m ω [·|·] satisfy (1.28) with s κ , s σ , and s ω as the only source term, respectively.
Note that the right-hand side of (1.28) only contains divided differences and m[ As an example, setting σ = ω 2 = 0 and applying the recursion once gives , seen as a function of (z 1 , z 2 ), is sometimes referred to as the second-order Cauchy transform of the GUE ensemble in the free probability literature (cf. [10]). The corresponding first-order object is −m(z), which is obtained by applying the usual Cauchy transform to the semicircle law. 4 We consider another special case in the following example. Example 1.16. Whenever κ 4 = σ = ω 2 = 0 and one argument of m GU E [·|·] is a singleton set, only the fourth line of (1.28) gives a non-zero contribution. Rewriting this term using (1.16) (cf. Lemma 1.17 below) yields the closed formula For ℓ = 2, we hence obtain .
Note that the right-hand side of (1.31) is fully expressed terms of non-crossing graphs on a labeled disk. This is because − −− → N CP (1, ℓ) and marked elements of N CP (1)×N CP (ℓ) can be reduced to disk non-crossing partitions in this special case. In particular, the orientation of the circles is not relevant for this example.
The proof of (1.31) is immediate from the following combinatorial lemma which may be of independent interest. We give its proof in Appendix B.1. with q i,j as in (1.13).
We also give some examples to illustrate the combinatorial formula (1.26) for the choice Example 1.18 (Second-order free cumulants). Let κ 4 = σ = ω 2 = 0. In the case k = ℓ = 1, the only non-crossing annular permutation is (12) and there is also only one option for Similarly, considering k = 1 and ℓ = 2 yields In the case k = ℓ = 2, there are 18 distinct non-crossing annular permutations (see Fig. 6 below) and 4 elements in N CP (2) × N CP (2). However, the second sum in (1.26) consists of 9 terms in total due to the marking of the blocks as, e.g.,  Definition 1.19 (Kreweras complement, annulus case). Consider the (k, ℓ)-annulus and label the midpoints of the arcs between the points 1, . . . , k + ℓ (black in Fig. 7) also by 1, . . . , k+ℓ (red in Fig. 7). Respecting the orientation of the two circles, we arrange the new labels such that the arc s follows the point s. Let π ∈ − −− → N CP (k, ℓ) be visualized on the (k, ℓ)annulus as in Definition 1.11. The (annular) Kreweras complement K(π) ∈ − −− → N CP (k, ℓ) is defined as the maximal annular non-crossing permutation on [k + ℓ] that can be drawn using only the labels at the midpoints of the arcs and without intersecting the cycles of π. In particular, each cycle of K(π) again encloses a region in the annulus that is homeomorphic to the disk with boundary oriented clockwise. In this context, we consider an annular noncrossing permutation maximal if none of its cycles can be extended (by merging cycles) without inducing a crossing. Note that |π| + |K(π)| = k + ℓ for any π ∈ − −− → N CP (k, ℓ) (see, e.g., [23,Sect. 6]). We remark that while defining the annular Kreweras complement on the level of partitions would also be possible, the resulting map does not have the same properties as in the disk case (see, e.g., [23, Sect. 1] for a discussion). Therefore, we will only consider the annular Kreweras complement for permutations. Note that one can further assign a unique Kreweras complement to any marked non-crossing partition π arising from some element π 1 × π 2 ∈ N CP (k) × N CP (ℓ) by applying Definition 1.3 circle-wise. In this case, we write K(π) = K(π 1 ) × K(π 2 ).

Main Results
The main focus of the present paper lies in determining the limiting covariance structure arising in the CLT for the centered statistics ) is a multi-index containing bounded deterministic matrices A 1 , . . . , A k and either the spectral parameters z 1 , . . . , z k ∈ C \ R with |ℑz j | 1 appearing in the resolvents or the test functions Whenever we need to refer to the number k of resolvents (resp. test functions) in the product X α (resp. Y α ), we carry the parameter k as a superscript and write X for i < j as well as F ∅ = 0.

Resolvent Central Limit Theorem and Recursion
We start by identifying the joint distribution of multiple X (k i ) α i with different k i and α i . To state the limiting covariance structure, we introduce a recursively defined set func-tion m[·|·], which we later identify as the deterministic approximation of the (appropriately scaled) covariance of T [1,k] and T [k+1,k+ℓ] similar to M [k] and m[·] arising for the expectation of T [1,k G k (see Theorem 1.9 as well as (1.21) and (1.22)) 5 . Note that α = ((z 1 , A 1 ), . . . , (z k , A k )) contains the same information on the spectral parameters and deterministic matrices involved as the set of matrices (T j , j ∈ [k]). We will, therefore, occasionally abuse notation and use (z j , A j ) and T j = G j A j interchangeably. In particular, we write where the two multi-indices α and β index the spectral parameters and deterministic matrices in T 1 , . . . , T k and T k+1 , . . . , T k+ℓ , respectively. At this point, we only give a recursive definition for m[·|·], however, explicit formulas are later obtained in Section 2.2. Note that the case σ = ω 2 = 0 of Definition 2.1 was already given in [27].
as the (deterministic) function of pairs of sets S 1 , S 2 with values in C and the following properties: is symmetric under the interchanging of its arguments, i.e., for any sets (ii) Initial condition: For any sets where the source terms s GU E , s κ , s σ , and s ω are given by ..,k+ℓ,k+1,...,s) Recall that ⊙ denotes the Hadamard product, q 1,k was defined in (1.13), and M (... ) was defined in Theorem 1.9. Moreover, recall that m[·] was defined in (1.21) and the notation with transposes was introduced in (1.25).
Note that setting A 1 = · · · = A k+ℓ = Id reduces (2.4) to (1.28), showing that We use the linearity of the recursion and the different types of source terms to introduce the decomposition Recall that we set A 1 ), . . . , (z k , A k )). Before stating the CLT for X α , we note the following definition.
Definition 2.2. Consider two functions of the Wigner matrix W in Assumption 1.1, which we denote as N -dependent random variables X (N ) and Y (N ) . We say that in the sense of moments if for any polynomial P it holds that where the implicit constant in O(·) only depends on the polynomial P and the constants in Assumption 1.1.
We now give a CLT for X α in (2.1). As the main interest of the present paper is the deterministic approximation m[·|·], we restrict the discussion of the CLT to the macroscopic regime for technical simplicity.
for any ε > 0. Here, P air(S) denotes the pairings of a set S and m[·|·] is a set function that satisfies Definition 2.1. Equation (2.10) establishes an asymptotic version of Wick's rule and hence identifies the joint limiting distribution of the random variables (X α j ) j as asymptotically complex Gaussian in the sense of moments in the limit N → ∞.
We remark that m[·|·] is cyclic in the sense that and that further whenever z 1 = z k , A k = Id, and σ = 0. These identities can be obtained from the "meta argument" below [5, Lem. 4.1] (see also [27,Cor. 3.7]) using that the analogous formulas for the original resolvent chains are trivially true by resolvent identities. However, any additional information on m[·|·] has to be obtained from the recursion (2.4) directly.

Solution of the Recursion
with m • and m •• being the first and second-order free cumulant functions as defined in (1.14) and (1.26), respectively.
Observe that the right-hand side of (2.11) reduces to the combinatorial expression in (1.26) if Remark. Note that the right-hand side of (2.11) is symmetric with respect to interchanging of ((z 1 , A 1 ), . . . , (z k , A k )) and ((z k+1 , A k+1 ), . . . , (z k+ℓ , A k+ℓ )), which is consistent with the symmetry of m[·|·] in Definition 2.1(i). We can check this directly from (2.11) by observing that there is a one-to-one correspondence between non-crossing permutations of the (k, ℓ)-annulus and those of the (ℓ, k)-annulus. This follows from drawing the cycles of the permutation as curves on the respective annuli and observing that interchanging the inner and outer circle with a conformal map (e.g., by inversion to a concentric circle between the outer and inner circle) preserves the standardness and non-crossing property of the picture (cf. Definition 1.15). Moreover, this symmetry of uniquely by Definition 1.14.
Example 2.5 (Asymptotics of covariances for GUE). We consider a special case of Theorem 2.3. Let p = 2, k 1 = k 2 = 1, and assume that W is a GUE matrix 6 . By decomposing A 1 and A 2 into a tracial and a traceless part, the deterministic approximation for the covariance follows directly from [7,Thm. 4.1], giving where the last equation follows from the formulas in Examples 1.6 and 1.18. Note that the error bound Ψ/ √ L in (91) of [7] evaluates to O(1/ √ N ) on macroscopic scales. We remark that the deterministic leading term matches the formula for m GU E [T 1 |T 2 ] obtained from applying (2.4) to the initial condition m GU E [T 1 |∅] = 0.
Next, we consider the recursion for m κ [·|·]. We obtain a closed solution similar to Theorem 2.4, i.e., a sum of terms that factorizes into two parts depending only on the deterministic matrices A 1 , . . . , A k+ℓ and the spectral parameters z 1 , . . . , z k+ℓ , respectively. The proof of Theorem 2.6 is given in Section 4.1 below.
Note that despite the obvious structural similarities between (2.11) and (2.12), considering the minimal example We continue by deriving an explicit formula for m σ [·|·]. As the source term s σ in the corresponding recursion is, up to transposes, identical to s GU E , the solution of the recursion is analogous to Theorem 2.4 but uses m #,σ [·] instead of the iterated divided differences m[·]. We give the proof in Section 4.2.

General Test Functions and Applications to Free Probability
We conclude the discussion by comparing the explicit formulas from Section 2.2 to the free probability results in [21].
For the later applications, we note the following formulas for the case κ 4 = σ = ω 2 = 0. Corollary 2.10 below is proven in [27] using the formulas in Section 2.2 as input. with ρ sc as in (1.9), and We remark that the formula (2.21) also appears in [9] and [28] (see also [27,Cor. 3.8]).
Whenever f j (x) = x for all j = 1, . . . , k + ℓ or, more generally, f j is an (N -independent) polynomial 8 , the N → ∞ limit of (2.16) describes the second-order limiting distribution of GUE and deterministic matrices in free probability. It is readily checked that Theorem 2.9 indeed coincides with the free probability literature in this case. The computations to obtain Corollary 2.11 are included in Appendix B.2.
Corollary 2.11. Under the assumptions of Corollary 2.10 let The limit in (2.22) matches [21,Thm. 6] and thus reproduces the well-known result of second-order freeness of GUE and deterministic matrices from [24]. Moreover, the deterministic approximation in Theorem 2.9 mirrors the overall structure of the joint secondorder distribution of Wigner and deterministic matrices described in Equation (3) of [21]. We remark that resolvents and functions with an N -dependent mesoscopic scaling as considered in [27] are usually not accessible in free probability theory as many of the standard techniques rely on explicit moment computations. Theorem 2.9 and [27, Thm. 2.7] thus show that the underlying combinatorics of non-crossing annular permutations and marked partitions are, in fact, more general. We further remark that the parallels between Theorem 2.9 and [21] continue to hold if we consider multiple independent Wigner matrices instead of one matrix W . More precisely, for n independent GUE matrices, the underlying combinatorial structure is given by the so-called non-mixing annular non-crossing permutations resp. non-mixing marked partitions for n colors (see Remark below [ . For this task, we define a new, albeit closely related, multi-set of graphs on the (k, ℓ)-annulus. We start by introducing a transformation that translates between the disk and the annulus picture.
Definition 3.1. Fix k, ℓ ∈ N, 1 ≤ j ≤ ℓ, and consider a disk with the k + ℓ + 1 labels 1, . . . , k, k+j, . . . , k+ℓ, k+1, . . . , k+j equidistantly placed around its boundary in clockwise order. We define a map τ , refered to as mediating map, that takes this picture to the (k, ℓ)annulus as follows: 1.) Use a homeomorphic continuous deformation, e.g., a conformal map, to map the disk and its labels to the (k, ℓ + 1)-annulus with a slit located between 1 and k on the outer circle and the two copies of k + j on the inner circle.
2.) Remove the slit to obtain an annulus.
3.) Merge the two copies of the label k + j.
We visualize τ for an example in Fig. 8 below. The two labels k + j are denoted as 6 and 6' to distinguish between them more easily. The map τ induces a transformation of any graph Γ defined on the labeled disk to a graph defined on the (k, ℓ)-annulus. We denote the resulting annulus graph as τ (Γ). By construction, τ (Γ) is planar whenever Γ is a disk non-crossing graph. Recall that we use a slightly more general notion of planar graphs than the standard literature by allowing for loops and multi-edges. We give an example in Fig. 9 below. For better visibility, the loop arising in the last step is moved from between the two (1, 3) edges to the right. We can now introduce the family of graphs G(k, ℓ) that constitute the key tool in the proof of Theorem 2.4. In analogy to the disk non-crossing graphs in Definition 1.7, we require the elements of G(k, ℓ) to be drawn on the (k, ℓ)-annulus with the vertices placed around the boundary and the edges drawn in the interior of the annulus (see Fig. 9 and Example 3.4 below).
Here, the union Γ ∪ {1} in G 1 is to be understood as adding a separated vertex 1 to Γ while the union Γ 1 ∪ Γ 2 in G 2 and G 3 refers to the graph with the vertex set {1, . . . , k + ℓ} and the edge set given by the union of the edge sets of Γ 1 and Γ 2 , respectively. Further, recall τ from Definition 3.1. We remark that all elements of G 1 , . . . , G 4 are planar graphs. Next, let where the union is to be understood as adding an edge (1, k) to each graph in G ¬(1,k) . Observe that the resulting graphs are again planar. Finally, we define

3)
where any graphs that occur more than once are counted with multiplicity. In particular, whenever the same graph occurs in both G (1,k) and G ¬(1,k) , the multiplicity of Γ ∈ G([k], [k + 1, k + ℓ]) is the total number of occurrences of Γ in both subsets.
(iii) G(S 1 , S 2 ) = G(S 2 , S 1 ) for any S 1 ⊂ [k] and S 2 ⊂ [k + 1, k + ℓ] (in the sense that there is a well-defined bijective mapping that takes each element of G(S 1 , S 2 ) to its counterpart).
The subset of connected good graphs is denoted by G c (k, ℓ). Any edge (i, j) with i ∈ [k] and j ∈ [k + 1, k + ℓ] is referred to as connecting edge.
Remark. The occurrence of multi-edges or loops is inherent to the construction of G(k, ℓ), which can be seen from a simple counting argument. First, note that an element of G(k, ℓ) can have at most 2(k + ℓ) edges by construction. To see this, observe that the elements of G ¬(1,k) with the highest number of edges lie in G 4 and that Γ ∈ G 4 has the same number of edges as the underlying disk non-crossing graph. As any disk non-crossing graph on n vertices has at most 2n − 3 edges (realized by a triangulation), the maximal number of edges for Γ ∈ G 4 is 2(k + ℓ + 1) − 3 = 2(k + ℓ) − 1. As (3.3) may add another edge to Γ, the maximum for G(k, ℓ) is 2(k + ℓ) edges. On the other hand, the maximal number of edges in a planar graph on the (k, ℓ)-annulus without multi-edges or loops is only 2(k + ℓ) if k, ℓ ≥ 3 (again realized by a triangulation). It is readily seen that such a graph has strictly less than 2(k + ℓ) edges if either k or ℓ is one or two. As the construction in Definition 3.2 does not introduce crossings and begins with the cases k, ℓ ≤ 2, the difference between the two maxima must be reflected as multi-edges or loops.
We further remark that G(k, ℓ) is a genuine multi-set, i.e., some graphs appear with multiplicity larger than one, unless k = ℓ = 1 (cf. Lemma 3.5(d) and Fig. 11). The key property shared by all elements of G(k, ℓ) is that each graph arises from some disk non-crossing graph along the recursive construction. Therefore, we may interpret G 4 as a kind of source term. In view of Lemma 3.3 below, the multi-set G(k, ℓ) gives an annulus analog to the disk non-crossing graphs, as it plays the same role in the combinatorial description of m[·|·] as N CG(S) does for m[·].
Noting that the vertex 1 in Γ ∈ G 1 has no adjacent edges, we may write Further, the transformation τ only changes the geometry underlying a graph Γ, but does not influence its edge set. By the definition of G 4 , any Γ ∈ N CG([1, . . . , k + ℓ, k + j]) used in the construction must have at least one edge (1, k + j). As a consequence, the product q Γ always includes the factor q 1,k+j = m 1 m k+j (1 + q 1,k+j ). This yields Note that the identity q 1,k+j = m 1 m j (1 + q 1,k+j ) allows writing summations restricted to graphs with an edge (1, k + j) on the left-hand side into an unrestricted sum over all graphs on the right-hand side. We use the same trick for G 2 and G 3 as q Γ 1 ∪Γ 2 = q Γ 1 q Γ 2 for the union of graphs introduced in Definition 3.2. With these replacements, (3.5) can be written as Multiplying both sides with k+ℓ s=1 m s and noting that 1 + q 1,j = (1 − m 1 m j ) −1 , we see that the right-hand side of (3.4) satisfies (1.28) as claimed. In particular, we readily reobtain (1.30) by evaluating q Γ for every graph in the above list. Note that decomposing (1.30) into the form m 1 m 2 Γ∈G(1,1) q Γ is possible in multiple ways. However, picking graphs that contain a connecting edge yields the set G(1, 1) in Fig. 10 as the smallest possible set.
We state a few properties of the elements of G(k, ℓ). In Lemma 3.5, we focus on general characteristics of Γ ∈ G(k, ℓ) as a planar graph drawn on the (k, ℓ)-annulus. The properties of G(k, ℓ) that are needed for the proof of the combinatorial formula for m •• are given in the separate Lemma 3.6. (a) The connected components of any Γ ∈ G(k, ℓ) give rise to a non-crossing partition of the (k, ℓ)-annulus. In particular, Γ contains a connecting edge if k, ℓ ≥ 1.
(b) Γ ∈ G(k, ℓ) may have at most two loops. Loops only occur at vertices that are adjacent to a connecting edge. Two loops on vertices on the same circle do not occur.
(c) Γ ∈ G(k, ℓ) may have up to k + ℓ − 1 double edges. Double edges are either connecting edges or adjacent to a connecting edge. Edges with a multiplicity higher than two do not occur.
Proof. (a) We use proof by induction. First, note that the elements of G(S 1 , ∅) = ∅ and G(∅, S 2 ) = ∅ with S 1 ⊆ [k] and S 2 ⊆ [k+1, k+ℓ] clearly give rise to an annular non-crossing partition. Moreover, Example 3.4 establishes the claim in the case k = ℓ = 1 and shows that any Γ ∈ G(1, 1) contains at least one connecting edge.
Assume next that the elements of G(S 1 , [k + 1, k + ℓ]) give rise to an annular non-crossing partition for any S 1 ⊂ [k] with |S 1 | ≤ k − 1 and a fixed ℓ ≥ 1. We aim to show that the connected components of any Γ ∈ G(k, l) also correspond to the blocks of some π ∈ N CP (k, ℓ). Due to the symmetry induced by Definition 3.2(iii), this is enough to establish the induction step.
By definition, the vertices 1 and k lie next to each other on the outer circle. Hence, adding an edge (1, k) may connect two connected components, but cannot introduce a crossing in the partition obtained from them. It is, therefore, sufficient to check the claim for elements of G ¬(1,k) or, equivalently, for the sets G 1 , G 2 , G 3 , and G 4 in (3.2). First, note that the transformation τ indeed takes the disk partition induced by the disk non-crossing graph to an annular non-crossing partition. This is due to the continuous homeomorphism used in the definition of τ . Moreover, the edge (1, k + j) prescribed for Γ ∈ N CG({1, . . . , k, k + j, . . . , k + j − 1, k + j}) by the definition is mapped to a connecting edge, ensuring that the resulting partition has a connecting block.
By construction, the graphs in G 1 , G 2 , and G 3 contain an element of G(S 1 , [k+1, k+ℓ]) with some S 1 ⊂ [k] as a subgraph. Applying the induction hypothesis for G([2, k], k + 1, k + ℓ]) and noting that a separate vertex 1 only adds a singleton set to the underlying partition, we can conclude that G 1 , too, behaves as claimed. The argument for G 2 and G 3 is similar. Here, the key observation is that the connected components of the added disk non-crossing graph induce a non-crossing partition of an interval placed along the outer circle. Recalling that all elements of N CP (k, ℓ) have at least one connecting block, the corresponding connected component of Γ ∈ G(k, ℓ) must contain a connecting edge.
(b) It is readily seen that the elements of G(S 1 , ∅) = ∅ and G(∅, S 2 ) = ∅ with S 1 ⊆ [k] and S 2 ⊆ [k + 1, k + ℓ], as well as all Γ ∈ G(1, 1) have the claimed structure. Moreover, adding an edge (1, k) to a graph cannot introduce a loop unless k = 1, so it is again sufficient to establish the induction step for G ¬(1,k) .
By Definition 1.7, a disk non-crossing graph does not contain any loops. Hence, any loops of Γ ∈ G 1 ∪ G 2 ∪ G 3 must occur in the subgraph in G(S 1 , [k + 1, k + ℓ]) with S 1 ⊂ [k] used to construct Γ. As the induction hypothesis applies to this subgraph, we conclude that Γ has at most two loops that satisfy the claimed placement rules, respectively. For Γ ∈ G 4 , note that applying τ to an element of N CG({1, . . . , k, k + j, . . . , k + j − 1, k + j}) yields an annulus graph with a loop if and only if the disk non-crossing graph contains an edge (k + j, k + j). As the latter must also contain an edge (1, k + j) by definition, a loop in Γ is always adjacent to a connecting edge.
We remark that the loops in Γ ∈ G(k, ℓ) are a consequence of the definition of τ rather than an artifact of the recursive construction of G(k, ℓ), i.e., they can be created only once. In particular, Γ can only contain a loop if it arises from G 4 or its analog in a previous iteration. Hence, the construction cannot yield a graph with more than two loops or more than one loop per circle, respectively. In particular, any graph in G(k, ℓ) containing more than one loop is necessarily obtained from an element in G(1, 1) with two loops.
(c) Again, the elements of G(S 1 , ∅) = ∅ and G(∅, S 2 ) = ∅ with S 1 ⊆ [k] and S 2 ⊆ [k+1, k+ℓ], as well as all Γ ∈ G(1, 1) have the claimed structure. We further note that any double edges of Γ ∈ G 1 ∪ G 2 ∪ G 3 must occur in the subgraph in G(S 1 , [k + 1, k + ℓ]) with S 1 ⊂ [k] used to construct Γ and that the induction hypothesis applies to the latter. For Γ ∈ G 4 , a double edge occurs if and only if the corresponding element of N CG({1, . . . , k, k + j, . . . , k + j − 1, k + j}) has a vertex i that shares an edge with both copies of k + j. However, there is at most one such vertex in [k] and [k + 1, k + ℓ], respectively, as having two vertices i, i ′ connected to both copies of k + j in either set induces a crossing. In particular, any double edge shares a vertex with the edge (1, k + j) ∈ Γ prescribed by the definition. Hence, all graphs in G ¬(1,k) have the claimed structure.
It remains to consider G (1,k) , i.e., to add an edge (1, k) to the graphs considered previously.
In the case j = k of G 2 , this doubles an existing (1, k) edge. By part (a), any such graph must also have an edge connecting k with a vertex in [k + 1, k + ℓ]. This shows that the placement rule for double edges is satisfied. Further, at most one doubled edge can be added with each application of the recursion, i.e., there are at most k + ℓ − 1 double edges in total. It is readily checked that two is indeed the highest edge multiplicity possible.
For the following discussion, we introduce the disjoint decomposition G(k, ℓ) = G dec (k, ℓ) ∪ G ¬dec (k, ℓ), (3.6) where G dec (k, ℓ) denotes the graphs in G(k, ℓ) that have double edges or loops and G ¬dec (k, ℓ) denotes the graphs that do not have either. We refer to the elements of G dec (k, ℓ) as decorated graphs.
(a) Whenever Γ ∈ G(k, ℓ) has more than one connected component that contains a connecting edge, every connected component of Γ can be uniquely identified with a disk non-crossing graph.
(b) Whenever Γ ∈ G(k, ℓ) has exactly one connected component with a connecting edge and Γ is not decorated, the same identification as in (a) holds, but it is no longer unique. If Γ 1 denotes the connected component of Γ that contains a connecting edge and U 1 ∪ U 2 with U 1 ⊂ [k] and U 2 ⊂ [k + 1, k + ℓ] is the vertex set of Γ 1 , there are |U 1 | · |U 2 | different ways to identify the connected components of Γ with a disk non-crossing graph.
Lemma 3.6 translates between a graph Γ ∈ G(k, ℓ), the partition induced by its connected components and the cycle structure arising in (2.11). We give a schematic of the construction of the disk graphs in Fig. 12 below. Recall from Definition 1.11 that any cycle of an annular non-crossing permutation encloses a region homeomorphic to the unit disk with the boundary oriented clockwise. Note that the assignment of the orientation in the second step on the left of Fig. 12 is not unique if π Γ has only one connecting block (cf. [23,Prop. 4.6], see Fig. 4 for an example). We remark that (a) and (b) are almost complementary cases and that only decorated graphs with exactly one connected component containing a connecting edge are not covered by Lemma 3.6. The proof of (a) further shows that any Γ ∈ G(k, ℓ) with at least two connected components containing a connecting edge cannot be decorated.
We give the full construction behind the schematics in Fig. 12 below.
Proof of Lemma 3.6. (b) It follows from Lemma 3.5(a) that the connected components of Γ give rise to a partition π Γ ∈ N CP (k, ℓ). By Propositions 4.4. and 4.6 of [23], any element of N CP (k, ℓ) may be identified with an annular non-crossing permutation, i.e., its blocks can be given an orientation (see the left of Fig. 12). This orientation of the individual cycles is naturally induced by the orientation of the inner and outer circle. However, the identification between N CP (k, ℓ) and − −− → N CP (k, ℓ) is only unique whenever the underlying partition has more than one connecting block. If there is only one connecting block U 1 ∪ U 2 with U 1 ⊂ [k], U 2 ⊂ [k + 1, k + ℓ], there are |U 1 | · |U 2 | possibilities to identify π Γ with an annular non-crossing permutation (cf. Fig. 4). We fix one π ′ Γ ∈ − −− → N CP (k, ℓ) that is associated with π Γ in this way.
By definition, the cycles of π ′ Γ can be drawn on the (k, ℓ)-annulus such that each cycle encloses a region between the circles homeomorphic to the disk with boundary oriented clockwise. Adding the elements of the cycle around the boundary yields a labeled disk as in Definition 1.7. We may use the same transformation to map a connected component of Γ to a disk non-crossing graph (see Fig. 13 below). Note that this transformation cannot induce any crossings of the edges of Γ, however, any loops or double edges of the original graph are kept. Hence, we only obtain a disk non-crossing graph in the sense of Definition 1.7 if Γ does not contain any loops or double edges to begin with. (a) Assume next that Γ ∈ G(k, ℓ) has at least two connected components that contain a connecting edge. It follows from Definition 3.2 that this structure can only arise from G 4 in (3.2) or its analog in a previous iteration of the recursive definition. Since adding subgraphs that only live on one circle of the (k, ℓ)-annulus does not interfere with the following argument, assume w.l.o.g. that Γ arises from G 4 directly. To have two connecting blocks, Γ must have at least two connecting edges (i 1 , i 2 ) and (i ′ Note that either (i 1 , i 2 ) or (i ′ 1 , i ′ 2 ) may coincide with the edge (1, k + j) prescribed by the definition.
Let Γ ∈ N CG({1, . . . , k, k + j, . . . , k + j − 1, k + j}) denote a disk non-crossing graph such that τ ( Γ) = Γ. Here, τ denotes the map introduced in Definition 3.1. By construction, Γ also has two edges edges (i 1 , i 2 ) and (i ′ This structure imposes several restrictions on Γ, as can be seen from Fig. 14  First, there cannot be an edge (1, k) ∈ Γ without violating the non-crossing condition. Together with (3.3), this implies that Γ = τ ( Γ) has at most a single edge (1, k). Further, Γ cannot contain a vertex that connects to both copies of k + j. This implies that Γ = τ ( Γ) cannot have any double edges. Lastly, Γ cannot have any loops, as Γ containing an edge (k + j, k + j) would induce a crossing, too. Hence, any Γ ∈ G(k, ℓ) with more than one connected component containing a connecting edge has only single edges and no loops.
So far, we have only considered the non-crossing annular partition induced by an element of G(k, ℓ). The following lemma allows us to partially reverse this relation and explicitly construct a graph that is associated with a given π ∈ N CP (k, ℓ). Lemma 3.7. For every π ∈ N CP (k, ℓ) there is at least one Γ ∈ G(k, ℓ) for which the vertex sets of the connected components coincide with the blocks of π. If π has exactly one connecting block, then there is at least one such graph in G dec (k, ℓ) and one in G ¬dec (k, ℓ), respectively.
We briefly sketch the construction of an element on G(k, ℓ) from a given annular noncrossing partition. For the example, we assume that the sketched connecting block is the only one in the partition. After completing the steps sketched in Fig. 15, the remaining connected components of the graph are readily added using steps corresponding to G 1 , G 2 , and G 3 in Definition 3.2 and the symmetry under interchanging of the inner and outer circle. Proof of Lemma 3.7. Fix π ∈ N CP (k, ℓ) and assume first that π has at least two connecting blocks. W.l.o.g. let 1 and k be assigned to different blocks of π. Indeed, if 1 and k occurred in the same block, we may split it into two disjoint parts that contain 1 and k, respectively, and later add an edge (1, k) to the graph obtained from this modified partition. Further, assume w.l.o.g. that 1 is contained in a connecting block. If i > 1 were the smallest element that occurs in a connecting block, we may further remove any blocks containing {1, . . . , i − 1} from the partition and later add a suitable subgraph. Note that the latter is possible by only using steps corresponding to G 1 and G 2 in Definition 3.2.
Under these assumptions, there exists π ∈ N CP ({1, . . . , k, k + j, . . . , k + j − 1, k + j}) such that the transformation τ in Definition 3.1 maps the blocks of π to the blocks of π. This is readily seen from Figure 8, as 1 and k being in different blocks implies that none of the blocks of π intersects the slit between 1 and k. We pick j such that 1 and k + j are in the same block of π and may further choose π such that one copy of the doubled label k + 1 occurs as a singleton set. Finally, define Γ ∈ N CG({1, . . . , k, k + 1, . . . , k + ℓ, k + 1}) by considering each block B = {i 1 , . . . , i n } ∈ π separately and adding the edges (i 1 , i 2 ), . . . , (i 1 , i n ) to the graph. By construction, the vertex sets of the connected components of Γ coincide with the blocks of π. Considering Γ π = τ ( Γ) now yields the element of G(k, ℓ) with the claimed properties (cf. Fig. 15, where the procedure is sketched for a single block).
Next, consider π ∈ N CP (k, ℓ) that has only one connecting block Similar to the first case, we can construct a disk noncrossing graph Γ U for which τ (Γ U ) ∈ G c (|U 1 |, |U 2 |). As τ (Γ U ) is only required to have one connecting edge, there is also a choice for Γ U containing a (k + j, k + j) edge (cf. Fig. 13). In particular, we obtain at least one graph in G c,¬dec (|U 1 |, |U 2 |) and one graph in G c,dec (|U 1 |, |U 2 |), respectively. The remaining connected subgraphs of Γ π are then added recursively by alternating between adding an isolated vertex (cf. G 1 ) and a disk noncrossing graph (cf. G 2 and G 3 ) to τ (Γ U ). Note that starting from τ (Γ U ) ∈ G c,¬dec (|U 1 |, |U 2 |) yields a graph that is not decorated while starting from τ (Γ U ) ∈ G c,dec (|U 1 |, |U 2 |) yields a decorated graph.
Using the properties of Γ ∈ G(k, ℓ) from Lemmas 3.6 and 3.7, we obtain an explicit nonrecursive combinatorial formula for m •• .
Note that the constants c Γ for Γ ∈ G ¬dec (k, ℓ) are readily obtained from the multiplicity of the corresponding graph in the multi-set, however, their exact values are not needed for the proof of Theorem 2.4.

Proof.
We start by observing that any connected component of Γ ∈ G(k, ℓ) with a connecting edge is itself an annular non-crossing graph. Further, a connected component of Γ that only involves vertices from either [k] or [k + ℓ] cannot contain loops or double edges (cf. (b) and (c) of Lemma 3.5) and we may identify it with a disk non-crossing graph.
To simplify notation, decompose any B ⊆ S 1 ∪ S 2 as a union where N CG c (B) denotes the connected graphs in N CG(B). By Lemma 3.7, none of the sums on the right-hand side are empty. However, there may be multiple permutations in − −− → N CP (k, ℓ) as well as a marked element of N CP (k) × N CP (ℓ) associated with a given annular non-crossing partition π ∈ N CP (k, ℓ). To obtain the same structure as in (1.26), we thus need to decompose the sum over π ∈ N CP (k, ℓ) on the right-hand side of (3.8) further.
Distinguishing by the number of connecting blocks of π yields As an annular non-crossing partition with at least two connecting blocks uniquely corresponds to an annular non-crossing permutation (cf. where we used that m • is invariant under permutation of the spectral parameters (z j ) j .
Applying a similar argument for the partitions with one connecting block U , interpreted as ( and recall from Lemma 3.7 that neither sum on the right-hand side is empty. Note that the split induced by (3.6) also decomposes the right-hand side of (3.11) into two terms.
Next, consider the term corresponding to G c,¬dec (U ) and recall that any Γ ∈ G c,¬dec (U ) can be identified with a disk non-crossing graph by Lemma 3.6(b). However, this identification is not unique.
Then there are |U 1 | · |U 2 | different disk graphs that can be obtained from a given Γ ∈ G c,¬dec (U ). As the resulting graphs only differ in the labeling of the vertices and m • is invariant under permutation of its arguments, the contribution of each graph to the sum is the same.
Recall that the number of annular non-crossing permutations arising from π ∈ N CP (k, ℓ) is equal to |U 1 | · |U 2 | by [23,Prop. 4.6]. We thus write Γ∈G c,¬dec (U ) with suitable constants c Γ ∈ Z. In particular, we do not necessarily have c Γ = 1. This can be seen from considering the element of G(2, 1) that has the edge set {(1, 2), (1, 3), (2, 3)}. Hence, where we set c Γ = 1 for Γ ∈ G c,dec (U ). Note that the contribution of the first term of (3.12) to (3.11) is (3.13) Putting everything together, the right-hand side of (3.9) reads m[1, . . . , k|k + 1, . . . , k (3.14) with c Γ ∈ Z as in (3.12) and U = U 1 ∪ U 2 . The first term on the right-hand side of (3.14) is the sum of (3.10) and (3.13). The second term is obtained from the second term in (3.12) by noting that U is the only connecting block of the partition, i.e., any block of π \ {U } only lives on one of the circles. Observing that this matches the structure in (1.26), we obtain (3.7) by comparing term-by-term.
We finally have all the necessary tools to prove Theorem 2.4.

Part 2: Conclusion
Proof of Theorem 2.4. Let f denote the right-hand side of (2.11). We recall that the initial condition (2.3) is immediate from Definition 3.2(ii) and that the symmetry in Definition 2.1(i) follows from the remark on Theorem 2.4 above. Hence, it only remains to check that f satisfies the recursion (2.4). To simplify notation, we interpret f as a function of the multi-indices α and β.
Similar to the proof of [6,Lem. 4.4], we use (1.17) and (3.7) to write out the first and second-order free cumulant functions in terms of suitable graphs. This yields To collect the terms involving the deterministic matrices A 1 , . . . , A k , we define for any π ∈ N CP (k, ℓ) that has more than one connecting block. Here, π ′ ∈ − −− → N CP (k, ℓ) denotes the unique permutation for which the blocks of π ′ coincide with π. Further, we define for any π ∈ N CP (k, ℓ) that has exactly one connecting block U . Here, π ′′ is the marked partition obtained from π by splitting the connecting block into U 1 = U ∩ [k] and U 2 = U ∩ [k + 1, k + ℓ], and marking U 1 , U 2 on the respective circles.
Next, we decompose the sum over − −− → N CP (k, ℓ) in (3.15) according to the number of connecting cycles in the permutation and rewrite the right-hand side as

F(π)
B∈π Γ∈N CGc(B) Here, π Γ denotes the partition arising from the vertex sets of the connected components of Γ.
Using (3.3) from Definition 3.2, decompose the right-hand side of (3.18) as Recall that Γ ∪ {(1, k)} denotes the graph obtained from adding an edge (1, k) to Γ and that, therefore, q Γ∪{(1,k)} = q 1,k q Γ . Next, apply (3.2) to split the right-hand side of (3.19) into contributions corresponding to G 1 , G 2 , G 3 , and G 4 . This yields where the edges prescribed in the definitions of G 2 , G 3 , and G 4 are added as a subscript to N CG and G, respectively. It remains to compare (3.20) with (2.4).
Let π Γ ∈ N CP (j) × N CP (k − j + 1, ℓ) denote the partition associated with a graph Γ ∈ N CG (1,j) in the second term of (3.20) and let π ′ Γ is be an annular non-crossing permutation with blocks given by π Γ . Since the edge (1, j) must occur in Γ, the vertices 1 and j must be associated with same cycle of π ′ Γ . Hence, the elements in [1, j and [j, k] are in different cycles of K(π ′ Γ ). Moreover, none of 1, . . . , j − 1 can be part of a connecting cycle in K(π ′ Γ ). This implies the decomposition Again, the only difference between F(π Γ ) and F(π Γ∪{(1,k)} ) is the fact that A k must occur as a separate factor in the second case. Thus, we can argue as in (3.22) and evaluate whenever π Γ has more than one connecting block and (3.16) applies.
In the remaining cases, F(π Γ ) is evaluated using (3.17). Here, the first term can be treated similarly to (3.25), leaving only the contribution of the marked partition π ′′ Γ . Recalling that [1, j and [j, k] lie on the same circle and that K(π ′′ Γ ) is evaluated circle-wise, we can argue as in the permutation case. In particular, i.e., F factorizes similar to (3.25  Computation of the fourth line of (3.20): For the term that arises from G 4 , recall that τ only influences the geometry of the graph, but not its edge set. Hence, the summation reduces to the underlying disk non-crossing graphs and we can evaluate it using (1.22) and (1.17). As the only difference between F(π Γ ) and F(π Γ∪{(1,k)} ) is again the fact that A k must occur as a separate factor in the second case, we obtain  Structure of K 1 : By the induction hypothesis, m κ [T 2 , . . . , T k−1 , G k A k A 1 |T k+1 , . . . , T k+ℓ ] factorizes into expressions involving only deterministic matrices or spectral parameters, respectively, and further has the structure specified on the right-hand side of (2.12) in terms of z 2 , . . . , z k+ℓ and the matrices A 2 , . . . , A k−1 , A k A 1 , A k+1 , . . . , A k+ℓ . As a consequence, the matrices A k and A 1 always occur together in the matrix products. On the level of the indices of the deterministic matrices, we may reinterpret this as an element π ∈ − −− → N CP (k, ℓ) with a cycle (. . . k1 . . . ) or an element of π ∈ N CP (k) × N CP (ℓ) with a block {. . . , k, 1, . . . } depending on the rest of the underlying structure. The treament of the two cases is identical and we consider them in parallel. As the indices 1 and k always occur together in K(π), the index 1 must occur separated in K −1 (π), either as a fixed point (1) or a singleton set {1}, to match the structure in (2.12). Note that the spectral parameter z 1 only appears in the prefactor m 1 . Hence, setting the functions ψ K −1 (π),(1) resp. ψ K −1 (π),{1} equal to m(z 1 ) yields the missing contribution. It follows that K 1 matches the structure on the right-hand side of (2.12). Note that all ψ i associated with permutations without a fixed point (1) and partitions without a singleton block {1}, respectively, are equal to zero for K 1 .
Structure of K 2 : We apply the induction hypothesis for m κ [T 1 , . . . , T j−1 , G j |T k+1 , . . . , T k+ℓ ] as well as (1.22) for m[T j , . . . , T k ] to rewrite K 2 as a sum of terms that naturally factorize into expressions involving only z 1 , . . . , z k+ℓ or A 1 , . . . , A k+ℓ , respectively. It remains to check for the structure on the right-hand side of (2.12), i.e., that each summand can be associated with an annular non-crossing permutation π or a marked element π ∈ N CP (k)×N CP (ℓ) such that the terms involving spectral parameters factorize according to cycles resp. blocks of π and the contribution of the deterministic matrices factorizes according to the cycles resp. blocks in the Kreweras complement K(π). As treatment of the cases π ∈ − −− → N CP (k, ℓ) and π ∈ N CP (k) × N CP (ℓ) is identical, we consider them in parallel.
Observe that the induction hypothesis and (1.22) already prescribe the desired complement structure for the indices of the spectral parameters and deterministic matrices occurring in m 1 m κ [T 1 , . . . , T j−1 , G j |T k+1 , . . . , T k+ℓ ] and m[T j , . . . , T k ], respectively. As we may visualize the elements of N CP (j) with their Kreweras complement on an interval by cutting the boundary of the labeled disk (cf. Fig. 1), we can draw both factors of K 2 onto the same annulus. The result is visualized on the left of Fig. 16 below. For simplicity, we omit most of the intermediate labels and only add some of the matrices associated with the vertices on the midpoints of the arcs between the labels in red.
Note that the interval is placed such that the orientation inherited from the disk aligns with the orientation of the underlying annulus. Moreover, Fig. 16 matches the picture of a (k, ℓ)-annulus up to the label j occurring twice and the label on the midpoint of the arch connecting the two copies of j that is associated with the identity matrix. Since this identity matrix does not influence the value of K 2 , we may remove the label corresponding to Id from the picture in Fig. 16. This leaves k + ℓ labels at the midpoints of arches along the annulus, one associated with each matrix A 1 , . . . , A k+ℓ . On the level of the indices of the deterministic matrices, each term in in K 2 can thus be identified with an element π ∈ − −− → N CP (k, ℓ) resp. an element of π ∈ N CP (k) × N CP (ℓ). As the two labels j now occur next to each other, we merge them to obtain the (k, ℓ)-annulus as the structure underlying the indices of the spectral parameters. The result is visualized on the right of Fig. 16 below. Note that the labels now match Definition 1.19 exactly. It is readily checked that the cycle resp. block structure obtained from merging the two labels j matches K −1 (π) and that any cycles resp. blocks that were connected in this step can be interpreted as a cycle of an element in − −− → N CP (k, ℓ) resp. a block of an element of N CP (k) × N CP (ℓ). In particular, the contribution of the spectral parameters factorizes as claimed on the right-hand side of (2.12). Comparing the permutations resp. partitions contributing to each K (j) 2 and noting that the right-hand side of (2.12) is linear in each ψ i , the sum j K (j) 2 also has the desired structure. Structure of K 3 : The treatment of K 3 is analogous to that of K 2 .
Structure of K 4 : From the formula for M (... ) in (1.18), it follows that K 4 can be written as a sum of terms that naturally factorize into expressions involving only z 1 , . . . , z k+ℓ or A 1 , . . . , A k+ℓ , respectively. In particular, the part that involves deterministic matrices always consists of two factors of the form ( j∈I 1 A j )⊙( j∈I 2 A j ) with index sets I 1 ⊆ [k] and I 2 ⊆ [k + 1, . . . , k + ℓ] due to the Hadamard product in K 4 . It remains to check for the structure on the right-hand side of (2.12).
Using the same trick as for K 2 and K 3 , we can visualize the terms involved in K 4 on the same annulus. The result is sketched in Fig. 17 below. Again, we omit most of the labels and add the matrices associated with the vertices on the midpoints of the arcs in red. To avoid overcrowding the labels in the interior of the inner circle, we further visualize the two circles separately. Observe that whenever a label occurs twice, both copies are placed next to one another and the label at the midpoint of the arch connecting them is always associated with the identity matrix. As the identity matrices do not contribute to K 4 , we may argue as before and remove the corresponding labels from the picture in Fig. 17. Note that this leaves k + ℓ labels at the midpoints of arches along the annulus, one associated with each matrix A 1 , . . . , A k+ℓ , such that we can reinterpret the structure on the level of the indices of the deterministic matrices as an element π ∈ − −− → N CP (k, ℓ) resp. π ∈ N CP (k) × N CP (ℓ). Since the two copies of r, s, and t are now placed right next to their counterpart, we merge them to obtain a (k, ℓ)-annulus as the structure underlying the indices of the spectral parameters. It is readily checked that the cycle resp. block structure obtained from this step matches K −1 (π) and that any cycles resp. blocks that were connected by merging two identical labels can be interpreted as a cycle of an element in − −− → N CP (k, ℓ) resp. a block of an element of N CP (k) × N CP (ℓ). In particular, whenever the merging of the labels creates two new cycles, the cycles can be drawn onto the (k, ℓ)-annulus without crossing. It follows that the contribution of the spectral parameters factorizes as claimed on the right-hand side of (2.12). Comparing the permutations resp. partitions contributing to each K (r,s,t) 4 and noting that the right-hand side of (2.12) is linear in each ψ i , summing over r, s, t preserves the structure of the term.
The remaining terms with prefactor q 1,k A k can be rewritten similarly. As the factor A k always occurs separately, the structure underlying the indices of the deterministic matrices is an element π ∈ − −− → N CP (k, ℓ) resp. π ∈ N CP (k) × N CP (ℓ) with a fixed point (k) resp. a singleton set {k}. Hence, K −1 must contain a cycle (. . . k1 . . . ) resp. a block {. . . , k, 1, . . . } in which 1 and k occur together. Recalling that q 1,k = m 1 m k 1−m 1 m k , it is ensured that the part of the term depending on z 1 , . . . , z k contains a factor ψ i (. . . , z k , z 1 , . . . ). With this modification, we can use the same argument as for K 1 , . . . , K 4 to conclude that all terms contributing to the recursion match the structure on the right-hand side of (2.12). Comparing the permutations resp. partitions contributing to each term and recalling that the right-hand side of (2.12) is linear in each ψ i , it follows that the same holds for their sum, i.e., for m κ [α|β]. This concludes the induction step.

Proof of Theorems 2.7 and 2.8
The proofs of Theorems 2.7 and 2.8 are similar to those of Theorems 2.4 and 2.6, respectively. We, therefore, mainly focus on the necessary modifications below.
Proof of Theorem 2.7. The overall argument is analogous to the proof of Theorem 2.4 in Sections 3.1 and 3.2 up to the definition of the graphs appearing in the combinatorial formula for m σ [·|·]. Recalling that the structure of the recursion for m σ [·|·] is similar to the one for m GU E [·|·], the multi-set of graphs can be constructed as described in Definition 3.2. However, since the source term involves m #,σ [·], the resulting multi-set of graphs will carry the same kind of vertex coloring. By replacing G 4 in Definition 3.2 by Γ has edge (1, k + j) .
An example in the current setting is given in Fig. 18 below. Note the different arrangement of the indices on the left compared to Fig. 8, which results from the structure of the source term in the recursion for m σ [·|·]. Further, τ does not influence the coloring of the vertices. is of the form (2.14) and that summing up the contributions on the right-hand side of the recursion (2.4) does not break the structure. We start by noting that each K (j) 5 contains the matrices A 1 , . . . , A k+ℓ exactly once and that the indices involved in the first and second factor of the Hadamard product match the indices on the outer and inner circle of the (k, ℓ)-annulus, respectively. The desired structure now follows from (1.22) and the fact that any annular non-crossing permutation of the (k, ℓ)-annulus can be decomposed uniquely into a composition of two permutations that only act on the inner and outer circle, respectively. Note that the index k + j occurring twice in M (k+j,...,k+ℓ,k+1,...k+j) does not influence the structure of the Kreweras complement. When visualizing the term on a labeled disk, the two vertices labeled k + j lie next to each other and the vertex on the midpoint of the arch connecting the two copies is associated with the identity matrix, i.e., its effect is not visible when the matrix product in M (k+j,...,k+ℓ,k+1,...k+j) is evaluated.

A Proofs for the CLT in the Resolvent Case
A.1 Proof of Theorem 1.10 (Global Law with Transposes) The proof of Theorem 1.10 is, modulo careful bookkeeping of the transposes, similar to the proof of the averaged local law in [6,Thm. 3.4]. In particular, an explicit formula for m #,σ [·] and the associated free cumulants is obtained along the way. For the convenience of the reader, we give a brief overview of the necessary changes. As there is nothing to prove if σ = 1 and the case σ = −1 needs an additional argument, consider σ ∈ (−1, 1) first. We start by introducing a renormalization that captures the general second moment structure of W more precisely (cf. [4] and [7]). Let with W an independent copy of W . This yields, e.g., Note that a simpler renormalization can be obtained by choosing W to be an independent GUE matrix instead of an independent copy of W . For the formulas (A.2) and (A.3) above, the difference between the two renormalizations is negligible. However, it becomes significant if the matrix product involves at least one transpose of a resolvent (cf. [4,Rem. 4.3]). As an example, consider the normalized trace of To obtain (1.23), we rewrite G ♯ 1 A 1 . . . G ♯ k A k in terms of W G ♯ 1 A 1 . . . G ♯ k A k and estimate the resulting terms. We start by considering the case A 1 = · · · = A k = Id. For k = 2, we obtain, e.g., 2), the resolvent identity W G − zG = Id, (1.11), and the local law for G j . Noting that | W G t 1 G 2 | = O ≺ (N −1 ) following the bounds in [4,Thm. 4.1] and that the term with prefactor N −1 is also of lower order (cf. (69) and (70) in [7]), we obtain (1.23) with Note that the only effect of the inclusion of transposes lies in the emergence of the stability factor 1 − σm 1 m 2 . However, as |1 − σm 1 m 2 | > 1 − |σ| for |σ| < 1, the term is bounded from below and does not change the outcome of the estimates. The analog for general k ≥ 2 follows by induction over the number of resolvents. Similar to [6,Thm. 3.4], the above argument allows us to extract a recursion that is satisfied by the deterministic approximation of G ♯ 1 . . . G ♯ k up to an O ≺ (N −1 ) error. By defining m #,σ [·] to satisfy this recursion exactly, i.e., defining it by (1.24), yields the desired statement. The recursion for m[G ♯ 1 A 1 , . . . , G ♯ k A k ] is obtained analogously. The explicit formula (1.23) now follows by solving this recursion. As the treatment of the deterministic matrices is completely analogous to [6,Thm. 3.4], we restrict the discussion to the case A 1 = · · · = A k = Id, i.e., to (1.24). Here, we obtain that m #,σ [·] and the associated free cumulants have a representation in terms of non-crossing graphs that mirror (1. 16) and (1.17).
Definition A.1 (Bicolored NCG). Let k ∈ N and denote by # a binary vector of length k. For every Γ ∈ N CG(k), color the vertex j ∈ {1, . . . , k} red if the j-th entry of # is 1, otherwise color it black. We call the resulting set of graphs bicolored (disk) non-crossing graphs on {1, . . . , k} and denote it by N CG # (k).
Lemma A.2. Let k ∈ N and fix a binary vector # of length k. Then where N CG # c (k) denotes the connected graphs in N CG # (k) and q Γ : with E(Γ) denoting the edge set of Γ. The edge weights q ♯ i,j are such that q ♯ i,j := q i,j = m i m j 1−m i m j whenever the edge (i, j) connects two vertices of the same color and q ♯ i,j := σm i m j 1−σm i m j whenever (i, j) connects a red vertex to a black one.
The proof is analogous to [6, Lem. 5.2] using the recursive structure of N CG(k). Note that the modified edge weights as well as the factors q ♯ 1,k and c 1,j in the recursion account for the stability factor 1 − σm i m j arising whenever the product G ♯ 1 . . . G ♯ k involves both resolvents and their transposes. We illustrate Lemma A.2 with an example.
Example A.3. Let k = 3 and pick spectral parameters z 1 , z 2 , z 3 ∈ C with |ℑz j | 1 as well as A 1 = A 2 = A 3 = Id. By Theorem 1.10, the deterministic approximation of G 1 G t 2 G t 3 is given by m #,σ [1, 2, 3] with # = (0, 1, 1). We visualize the elements of the set N CG # (3) in Fig. 19 below. For a better overview, the edges are drawn as solid or dashed according to their contribution to q # Γ . We use this sketch to compute .
Note that we thus reobtain (49) of [7] on macroscopic scales if z 2 = z 3 . It remains to consider the case σ = −1. Here, the Wigner matrix is of the form W = D+iS with a diagonal matrix D and a skew-symmetric matrix S. Note that whenever the diagonal part is equal to zero, the resolvent R(z j ) = (iS − z j ) −1 satisfies R(z j ) t = −R(−z j ). This allows treating the proof of (1.23) for R(z 1 ) ♯ . . . R(z k ) ♯ with R(z j ) ♯ ∈ {R(z j ), R(z j ) t } analogous to the case G 1 . . . G k . Recall that this requires in particular a local law for R(z j ) , which holds even if the Wigner matrix has zero diagonal. The general case follows from the bound which reduces the proof of (1.23) to the previously considered case D = 0. The proof of (A.6) is given for G 1 G t 2 in [7, App. B]. By expanding G ♯ 3 ,. . . , G ♯ k analogously, the argument readily extends to k ≥ 3 resolvents.

A.2 Proof of Therorem 2.3 (CLT for Resolvents)
We follow the general outline of the proof of [27,Thm. 3.6] to identify the necessary modifications for the general model described by Assumption 1.1. As the formulas in the general case are derived analogously, we assume w.l.o.g. A k = 0 throughout to keep the following equations short. The first step is the analog of [27,Lem. 2.3] that characterizes the difference between T [1,k] − m[T 1 , . . . , T k ] and T [1,k] − E T [1,k] , thus connecting the statistics X α back to the local law (1.19).
Lemma A.4. Let k ∈ N and fix spectral parameters z 1 , . . . , z n with |ℑz j | 1 and max j |z j | ≤ N 100 as well as bounded deterministic matrices A 1 , . . . , A k such that A k = 0. Proof. We note the following modifications to the proof of [27,Lem. 2.3]. Using the renormalization in (A.1) with W being an independent copy of W (compared to the independent GUE matrix W used in [27]), we obtain the relation Recall that the deterministic approximation m[·] is independent of the value of σ and ω 2 (cf. [6,Thm. 3.4]). The terms in the last line of (A.8) involve an additional factor of N −1 . Lemma A.5. Fix k, ℓ ∈ N and let α, β be two multi-indices of length k and ℓ, respectively. Assume that the Wigner matrix W satisfies Assumption 1.1 and pick spectral parameters z 1 , . . . , z k+ℓ with |ℑz j | 1 and max j |z j | ≤ N 100 as well as deterministic matrices A 1 , . . . , A k+ℓ with A j 1. Then, for any ε > 0,  Recalling that m[S] is invariant under any permutation of the elements of S, we pick an ordering in which the two j's occur in two consecutive positions and visualize the corresponding non-crossing graphs by equidistantly arranging the vertices on a circle. In this picture, the edge e connecting both j's cannot be involved in any crossing, even in an arbitrary planar graph on the given vertices (see left of Fig. 20). Hence, for any noncrossing graph with edges {e 1 , . . . , e n } ∋ e, the graph with edge set {e 1 , . . . , e n , e} is also non-crossing. In particular, every non-crossing graph that involves e has a counterpart that does not. Next, note that there is at most one vertex l among 1, . . . , j − 1, j + 1, . . . , k that is connected to both copies of j, as having two distinct vertices l, l ′ with this property results in a crossing (see right of  where the second equality follows from (1.12), we can factor out (1 + m ′ j ) on the righthand side of (B.2) and reduce to summation over N CG(k) without further restriction.