
==== Front
Math Phys Anal Geom
Math Phys Anal Geom
Mathematical Physics, Analysis, and Geometry
1385-0172
1572-9656
Springer Netherlands Dordrecht

38911699
9483
10.1007/s11040-024-09483-y
Article
Fluctuation Moments for Regular Functions of Wigner Matrices
http://orcid.org/0009-0001-6597-715X
Reker Jana jana.reker@ist.ac.at

https://ror.org/03gnh5541 grid.33565.36 0000 0004 0431 2247 IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
20 6 2024
20 6 2024
2024
27 3 1023 8 2023
3 5 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by/4.0/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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.

Keywords

Wigner matrix
Global fluctuations
Fluctuation moments
Annular non-crossing permutations
Free probability
Mathematics Subject Classification

60B20
15B52
46L54
http://dx.doi.org/10.13039/501100000781 European Research Council 101020331 Institute of Science and Technology (IST Austria)Open access funding provided by Institute of Science and Technology (IST Austria).

issue-copyright-statement© Springer Nature B.V. 2024
==== Body
pmcIntroduction

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 [5, 6]) that a similar concentration holds for alternating products of the form1.1 F[1,k]:=f1(W)A1…fk(W)Ak,

where A1,…,Ak are bounded deterministic matrices and f1,…,fk are regular test functions, allowing in particular for fj(W)=G(zj). Apart from resolvents, two choices of fj are of special interest in this setting. First, consider the case fj(x)=eitjx with tj∈R. Interpreting the Wigner matrix W as the Hamiltonian of a mean-field quantum system and the deterministic bounded matrix A as an observable, the quantityA(t):=eitWAe-itW

describes the Heisenberg time evolution of A. For k≥2, (1.1) hence describes the time evolution of multiple observables, possibly to different times, in the same quantum system. Considering the regime where the differences between the individual times are large links the matrix product to thermalization problems in mathematical physics (see [6, Sect. 2.1] and [5, Rem. 2.8]). Second, products of the form (1.1) with fj(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,φ1) consisting of a complex unital algebra A and a tracial linear functional φ1:A→C with φ1(1A)=1, where 1A is the unit element of the algebra. One particular example is the space (A,φ1)=(MN×N(L∞-(Ω,P)),E⟨·⟩) of N×N random matrices, where (Ω,P) is a classical probability space, MN×N(S) denotes the N×N-matrices with entries in S, the spaceL∞-(Ω,P):=⋂1≤p<∞Lp(Ω,P)

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 (φ1(ak))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⟨W1D1…WkDk⟩ of a finite family of independent Wigner matrices (Wj)j and a finite family of deterministic matrices (Dj)j (see, e.g., [22, Sect. 4.4]) in the case fj(x)=x. More precisely, in the large N limit, the leading-order term m1[F[1,k]] of E⟨F[1,k]⟩ is of the form1.2 m1[F[1,k]]:=∑π∈NCP([k])(∏B∈π〈∏j∈BAj〉)Φπ(1)(f1,…,fk),

where NCP([k]) denotes the non-crossing partitions of the cyclically ordered set {1,…,k} and the functions Φπ(1) only depend on f1,…,fk and π∈NCP([k]). Hence, the right-hand 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 fj(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. In particular, evaluating (1.2) for fj(x)=eitjx with tj∈R implies a closed formula for the thermal value of quantities such as ⟨eitWA1e-itWA2⟩, which allows for a direct analysis of the t≫1 regime relevant for thermalization (see, e.g., [6, Cor. 2.9]).

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 Trf(W)=∑j=1Nf(λj) 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 Trf(W) are well-studied on both macroscopic and mesoscopic scales (see, e.g., [1–3, 11, 12, 16, 18, 19, 29–31] 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 fj being polynomials in the context of second-order freeness (see, e.g., [22, Chap. 5] or [8, 20, 21]).

We remind the reader that a second-order non-commutative probability space is a triplet (A,φ1,φ2), where the functional φ2:A×A→C is bilinear, tracial in both arguments, symmetric under the interchanging of its arguments, and satisfies φ2(a,1A)=φ2(1A,a)=0 for all a∈A. The second-order probability distribution of a∈A is characterized in terms of (φ2(ak,aℓ))k,ℓ, called the fluctuation moments, with the joint moments of multiple elements again being defined analogously. As a canonical example, we remark that MN×N(L∞-(Ω,P)) may be endowed with the functional φ2(·,·)=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 [7, 25, 26]). 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. In particular, we show that if W is a GUE matrix, the leading order term m2[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]=fk+1Ak+1…fk+ℓAk+ℓ of the same build as (1.1)) is given by1.3 m2[F[1,k]|F[k+1,k+ℓ]]=∑π∈NCP→(k,ℓ)(∏B∈π〈∏j∈BAj〉)Φπ(2)(f1,…,fk+ℓ)+∑π1×π2∈NCP(k)×NCP(ℓ)(∏B1∈π1,B2∈π2〈∏j∈B1Aj〉〈∏j∈B2Aj〉)×Φπ1×π2(2)(f1,…,fk+ℓ).

Here, NCP→(k,ℓ) denotes the non-crossing permutations of the (k,ℓ)-annulus and the functions Φπ(2) resp. Φπ1×π2(2) only depend on f1,…,fk+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 fj(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 Wij=dN-1/2χod for i<j and Wjj=dN-1/2χd for general entry distributions χod and χd. In the general case, however, the sum in the first line of the right-hand side of (1.3) splits into four summands Φπ(GUE), κ4Φπ(κ), σΦπ(σ), and ω~2Φπ(ω) which have different prefactors in terms of the deterministic matrices A1,…,Ak+ℓ. Here, Φπ(GUE) corresponds to the GUE case in (1.3) and the remaining contributions are associated with the parameters1.4 κ4=E|χod|4-2,σ=Eχod2,ω~2=Eχd2-1-σ

of the Wigner matrix W. A similar decomposition is also observed for Φπ1×π2(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 m2[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. Although the form of the limiting variance in the k=2 case considered in [27, Cor. 2.12] is sufficiently simple and could be derived without additional combinatorial tools, characterizing the fluctuation for general k≥2 does require them to obtain an explicit formula for the variance. We further 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 form1.5 fj(x)=gj(Nγ(x-E))

where gj 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 matches the setting of [7, 21] 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 ⟨T1…Tk⟩ where Tj:=G(zj)Aj, and the multi-resolvent local laws needed for the analysis of the fluctuations (Sect. 1.2) as well as the definitions from free probability that are used to characterize the limiting covariance of ⟨T1…Tk⟩-E⟨T1…Tk⟩ and ⟨Tk+1…Tk+ℓ⟩-E⟨Tk+1…Tk+ℓ⟩ (Sect. 1.3). To prepare for the statements of our main results, we give a CLT for the case that all functions fj are resolvents (Theorem 2.3). The role of the limiting covariance in the theorem is played by a recursively defined set function m2[·|·] (Definition 2.1), which is the main object of interest in the present paper. We study the recursion in detail in Sect. 2.2 and obtain the main results, explicit combinatorial formulas for its solution (Theorems 2.4, 2.6 - 2.8). In Sect. 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 fj(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 Sects. 3 and 4. To keep the presentation concise, some routine calculations are deferred to the appendix. This includes the proof of a multi-resolvent global law with transposes (Appendix A.1) and the proof of the resolvent CLT in Theorem 2.3 (Appendix A.2) which are similar to previous results in [6] and [27], respectively.

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-1Tr(·). We further denote the covariance of two complex random variables X1,X2 by Cov(X1,X2) and follow the conventionCov(Y1,Y2)=E(Y1-EY1)(Y2-EY2)¯,

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 (…) instead of {…}. Sets of the form [k] and intervals are treated as ordered sets.

Given a matrix A∈CN×N, the traceless part of A is denoted by A˚:=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 A1⊙A2 denotes the entry-wise (or Hadamard) product of two matrices A1 and A2. For a Hermitian matrix W and spectral parameters z1,…,zk∈C\R, we write the corresponding resolvents as Gj=G(zj):=(W-zj)-1 and index products of resolvents using the interval notationG[a,b]:=GaGa+1…Gb

for a,b∈N with a≤b. 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 Ga or Gb from the product, respectively. Moreover, G∅ is interpreted as zero. Note that this notation differs slightly from [5, 6]. As we often consider alternating products of resolvents with deterministic matrices A1,…,Ak, define Tj:=GjAj and apply the same interval notation as above to write1.6 T[k]:=T1…Tk=G1A1…GkAk,T[a,b]:=TaTa+1…Tb.

Again, angled brackets are used to exclude Ta or Tb 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 Wij=Wji¯ and we assume identical distribution in the sense that there is a centered real random variable χd and a centered real or complex random variable χod such that Wij=dN-1/2χod for i<j and Wjj=dN-1/2χd, respectively. We further assume that E|χod|2=1 as well as the existence of all moments of χd and χod, i.e., there exist constants Cp>0 for any p∈N such thatE|χd|p+E|χod|p≤Cp.

We remark that Assumption 1.1 matches the model considered in [7] and [21]. Compared to the conditions Eχod2=0 and Eχd2=1 in [27], we allow for arbitrary values of the parameters σ=Eχod2∈[-1,1] and ω2=Eχd2≥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 notation1.7 κ4:=E|χod|4-2

for the normalized fourth cumulant of the off-diagonal entries as well as1.8 ω2~:=ω2-1-σ.

The eigenvalue density profile of W is described by the semicircle law1.9 ρsc(x):=4-x22π1[-2,2](x)

which mainly enters our analysis in the form of its Stieltjes transform1.10 m(z):=∫ρsc(x)x-zdx,z∈C\R.

We remind the reader that m(z) is the unique solution of the Dyson equation1.11 -1m(z)=m(z)+z,ℑzℑ(m(z))>0

and that its derivative satisfies1.12 m′(z)=m(z)21-m(z)2.

Given fixed z1,…,zk∈C\R, set mj=m(zj) and mj′=m′(zj), respectively. We further introduce1.13 qi,j=mimj1-mimj,

and remark that qj,j=mj′ 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 NCP(S) and we abbreviate NCP(k):=NCP([k]). For each non-crossing partition π={B1,…,Bn}, set |π|:=n for the number of blocks in the partition.

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 PB of the points s∈B. Then π is non-crossing if and only if the polygons {PB|B∈π} are pair-wise disjoint. Because of this, we also call the elements of NCP(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∈NCP(S). Then the (disk) Kreweras complement of π, denoted by K(π), is the element of NCP(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∈πPB of the polygons {PB|B∈π} in the labeled disk D.

Fig. 1 The non-crossing partition π={{1,3,4},{2},{5,8,9},{6,7},{10}} (black) and its Kreweras complement K(π)={{1,2},{3},{4,9,10},{5,7},{6},{8}} (red). Singleton sets are indicated by a black or red dot, respectively. (Color figure online)

Observe that D\∪B∈πPB has |S|-|π|+1 connected components, hence |π|+|K(π)|=|S|+1. Further, K2=K∘K recovers π up to a rotation of D, i.e., K2(π) is the partition where for S={s1,…,sk} the elements in each block of π are shifted by s1↦s2↦⋯↦sk↦s1. In particular, taking the Kreweras complement is invertible as a map on NCP(S).

Definition 1.4

(Free cumulant function) Fix k∈N, denote the power set1 of [k] by P([k]) and let (fn)n be a family of functions such that fn maps n-element sets in P([k]) to C. We define the (first-order) free cumulants (f∘,n)n associated with the sequence (fn)n through the relation1.14 f|S|[S]=∑π∈NCP(S)∏B∈πf∘,|B|[B]

for any S⊆[k]. For simplicity. we associate the sequences (fn)n (resp. (f∘,n)n) with a single function f (resp. f∘) by setting f[s1,…,sn]:=fn[{s1,…,sn}] (resp. f∘[s1,…,sn]:=f∘,n[{s1,…,sn}]) and call f∘ the (first-order) free cumulant function associated to f.

We emphasize that Definition 1.4 does not require the functions fn to have any particular symmetries. However, in the free probability literature, f usually arises from a family of tracial functionals 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 NCP(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 0S:={{s}|s∈S} and 1S:={S}, respectively. The free cumulant function can then be written as1.15 f∘[S]=∑π∈NPC[S]μ(π,1S)∏B∈πf[B],μ(π,ν):=1,π=ν,-∑π<τ≤νμ(τ,ν),π<ν,

using the Möbius function μ:{(π,ν)|π≤ν∈NCP(S)}→Z that is recursively defined by (1.15). We remark that μ(π,1S) 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. Recall that m denotes the Stieltjes transform of the semicircle law (1.9).

Definition 1.5

(Divided differences) For finite sets {z1,…,zk}⊂C\R we recursively definem[z1,…,zk]:=m[z2,…,zk]-m[z1,…,zk-1]zk-z1.

The definition readily extends to multi-sets {z1,…,zk}⊂C\R by settingm[zj,…,zj⏟ktimes]:=m(k-1)(zj)(k-1)!

whenever an element zj occurs with a multiplicity greater than one. Here, m(k-1) denotes the (k-1)th derivative of the function m in (1.10). Note that m[·] is well-defined in the sense that m[z1,…,zk] is independent of the ordering of the multi-set {z1,…,zk}. We abbreviate m[1,…,k]:=m[z1,…,zk].

We emphasize that m[·], and hence m∘[·], have full permutation symmetry, which is much more than what was assumed for f in Definition 1.4. The following example illustrates the combinatorial formulas (1.14) and (1.15) for f=m[·].

Example 1.6

(First-order free cumulants) In the case k=1 we simply have m[1]=m(z1). For k=2, the only non-crossing partitions are (12) and (1)(2) such thatm∘[1,2]=m[1,2]-m1m2,mj:=m[j]=m[zj]

while for k=3 we havem∘[1,2,3]=m[1,2,3]-m1m[2,3]-m2m[1,3]-m3m[1,2]+2m1m2m3.

The quantities m and m∘ were studied in detail in [6], yielding a close connection to non-crossing graphs. We recall the definition and give an example in Fig. 2. 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.

Definition 1.7

(Disk non-crossing graphs) Let S⊂N be a finite (cyclically) ordered set of integers equidistantly arranged in clockwise order on the circle. We call an undirected planar graph (S, E) on the vertex set S without loops or multi-edges (disk) crossing if there exist two edges (a,b),(c,d)∈E with a<c<b<d, otherwise we call it (disk) non-crossing.2 The set of all (disk) non-crossing graphs with vertex set S is denoted by NCG(S) and we denote the subset of connected graphs as NCGc(S). Whenever S=[k], abbreviate NCG(k):=NCG([k]).

Fig. 2 An element of NCG(10) and its connected subgraphs

Emphasizing that Definition 1.7 lives on a disk is important, as we later introduce a non-crossing 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∈NCG(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.

Lemma 5.2 of [6] proves the representations1.16 m[S]=(∏s∈Sms)∑Γ∈NCG(S)∏(i,j)∈E(Γ)qi,j,

1.17 m∘[S]=(∏s∈Sms)∑Γ∈NCGc(S)∏(i,j)∈E(Γ)qi,j

in terms of the weights qi,j in (1.13). Here, E(Γ) is the edge set of the graph Γ. Note that (1.16) and (1.17) are still well-defined if S is an ordered multi-set, i.e., if some elements are repeated. In this case, we consider NCG(|S|) instead of NCG(S) and use the one-to-one correspondence between the (possibly repeated) labels {s|s∈S} and {1,…,|S|} to obtain a uniquely defined right-hand side (see Fig. 3).Fig. 3 The graphs contributing to the sum on the right-hand side of (1.16) (top and bottom row) and (1.17) (bottom row only) for the multi-set S=(1,1,2). Note that the prefactor is given by ∏s∈Sms=m12m2

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|ℑzj|. Recall the commonly used definition of stochastic domination.

Definition 1.8

(Stochastic domination) LetX={X(N)(u)|N∈N,u∈U(N)}andY={Y(N)(u)|N∈N,u∈U(N)}

be two families of non-negative random variables that are indexed by N and possibly some other parameter u in some (possibly N-dependent) domain U(N). We say that X is stochastically dominated by Y, denoted by X≺Y or X=O≺(Y), if, for all ε,C>0 we havesupu∈U(N)P(X(N)(u)>NεY(N)(u))≤N-C

for large enough N≥N0(ε,C).

Theorem 1.9

(Macroscopic version of [5, Thm. 2.5]) Fix k∈N and pick spectral parameters z1,…,zk∈C\R with |ℑzj|≳1 and maxj|zj|≤N100 as well as deterministic matrices A1,…,Ak∈CN×N with ‖Ai‖≲1. Define31.18 M[k]:=∑π∈NCP(k)(∏B∈K(π),k∉B〈∏j∈BAj〉∏i∈B(k)\{k}Ai)(∏B∈πm∘[B]),

where B(k) is the block in K(π) that contains k. Recalling that Tj=GjAj, we have the averaged,4 local law1.19 ⟨T[1,k]⟩=⟨M[k]Ak⟩+O≺(1N),

and for x,y∈CN with ‖x‖,‖y‖≲1 we have the isotropic local law1.20 ⟨x,T[1,k⟩Gky⟩=⟨x,M[k]y⟩+O≺(1N).

As we frequently encounter ⟨M[k]Ak⟩ in the following sections, we introduce the notation1.21 m1[T1,…,Tk]=m1[z1,A1,…,zk,Ak]:=⟨M[k]Ak⟩.

In particular,1.22 m1[T1,…,Tk]=∑π∈NCP(k)(∏B∈K(π)〈∏j∈BAj〉)(∏B∈πm∘[B])

and we have m1[G1,…,Gk]=m[1,…,k] as a consequence of (1.14). Comparing (1.22) with the formulas in [22, Sect. 4.4], the set functions m and m∘ may be interpreted as the moments resp. free cumulants characterizing the limiting non-commutative probability distribution of resolvents (Gj)j while m1[·] describes the limiting joint distribution of resolvents (Gj)j and deterministic matrices (Aj)j.

We further apply (1.19) or (1.20) for a product Ts1…Tsk-1Gsk that is indexed by a (cyclically) ordered set S=(s1,…,sk) instead of an interval. In this case, the deterministic approximation is denoted asMS=M(s1,…,sk)

with the same definition as in (1.18).

Remark

The quantities m[1,…,k], m∘[1,…,k], m1[T1,…,Tk], (M[k])ij, and ‖M[k]‖ are of order one for any k∈N and i,j∈[N] in the macroscopic regime (cf. Lemma 2.4 and Appendix A of [5]). Theorem 1.9 asserts that the deterministic M[k] is the leading order approximation of T[1,k⟩Gk. In particular, the error terms in (1.19) and (1.20) are smaller than the natural upper bound on their leading term by a factor of 1/N and 1/N, respectively.

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 z1,…,zk with |ℑ(zj)|≳1 and maxj|zj|≤N100 as well as deterministic matrices A1,…,Ak with ‖Aj‖≲1. Moreover, let Gj♯ denote either the resolvent Gj=G(zj) or its transpose Gjt and denote by # the binary vector that has a one in jth position if Gj♯=Gjt and a zero otherwise. Then,1.23 ⟨G1♯A1…Gk♯Ak⟩=∑π∈NCP(k)(∏B∈K(π)〈∏j∈BAj〉)(∏B∈πm∘#,σ[B])+O≺(1N)

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 recursion1.24 m#,σ[1,…,k]=m1(1+q1,k♯)×(m#,σ[2,…,k]+∑j=2kc1,jm#,σ[1,…,j]m#,σ[j,…,k])

with q1,k♯=q1,k=m1mk1-m1mk whenever #1=#k, i.e., either both G1 and Gk occur as transposes in the product G1♯…Gk♯ or neither of them, and q1,k♯=σm1mk1-σm1mk otherwise. Similarly, c1,j=1 whenever #1=#j and c1,j=σ otherwise. Recall that σ=Eχod2 where χod 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] (cf. [5, Rem. 2.2]) and that several examples in the cases k∈{2,3} are considered in Proposition 3.4 and Remark 3.5 of [4] as well as in Propositions 3.3 and 3.4 of [7].

Note that σ=1 implies that the matrix W is real and its resolvent satisfies Gjt=Gj. 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) as1.25 m1[G1♯A1,…,Gk♯Ak]:=∑π∈NCP(k)(∏B∈K(π)〈∏j∈BAj〉)(∏B∈πm∘#,σ[B]).

Moreover, by Definition 1.4, we have m1[G1♯,…,Gk♯]=m#,σ[1,…,k] and (1.24) reduces to the divided differences in Definition 1.5 whenever # is the zero vector.

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 (roughly following the notation of [22, Ch. 5]) 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. 4). 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 cycles5 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 NCP→(k,ℓ). Any cycle that connects both circles is referred to as connecting cycle.

We remark that NCP→(k,ℓ) can be fully characterized by the avoidance of certain crossing patterns (cf. analogous geometric characterization of NCP(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.Fig. 4 The labels of the (4, 3)-annulus

Definition 1.12

(Annular non-crossing partitions) Let k,ℓ∈N. We call the partitions induced by the cycles of NCP→(k,ℓ) annular non-crossing partitions. The set of annular non-crossing partitions is denoted by NCP(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 NCP→(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. 5). 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]).Fig. 5 The non-crossing permutations (123) and (132) on the (1, 2)-annulus. They are different as permutations, but their cycles induce the same non-crossing partition

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 NCP(k)×NCP(ℓ), and introduce an artificial connection by marking one block on each circle.

Definition 1.13

(Marked non-crossing partition) Consider π∈NCP(k)×NCP(ℓ) that naturally splits into π=π1×π2 with π1∈NCP(k), π2∈NCP(ℓ). We pick one block of π1 and one of π2, respectively, and mark them by underlining. The resulting object is referred to as a marked non-crossing partition.

Fig. 6 A visualization of {{1_},{2}}×{{3_}} and {{1},{2_}}×{{3_}}. Marking is indicated by a dashed line

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 non-crossing partition can be associated with a unique element of NCP(k,ℓ). We further note that there are |π1|·|π2| possibilities to mark the blocks of π=π1×π2∈NCP(k)×NCP(ℓ). For example, {{1_},{2}}×{{3_}} and {{1},{2_}}×{{3_}} are considered different marked partitions although both arise from {{1},{2}}×{{3}}∈NCP(2)×NCP(1) (see Fig. 6). We further recall from [22, Ch. 5] that there is a second-order analog of Definition 1.4.

Definition 1.14

(Second-order free cumulant function) Let (fm,n)m,n be a sequence of functions such that fm,n maps tuples (S1,S2) of two finite (cyclically) ordered sets of integers with |S1|=m and |S2|=n to C. Assume further that the functions are symmetric under interchanging of the two arguments in the sense that fm,n[S1|S2]=fn,m[S2|S1] and cyclic, i.e., fm,n[S1|{s1,…,sn}]=fm,n[S1|{s2,…,sn,s1}]. Moreover, we assume that fm,0[S1|∅]=f0,n[∅|S2]=0. Similar to Definition 1.4, we associate the sequence (fm,n)m,n with a single function f by settingf[s1,…,sm|sm+1,…,sm+n]:=fm,n[{s1,…,sm}|{sm+1,…,sm+n}].

We implicitly define the second-order free cumulant function of f as the unique map f∘∘ defined on pairs of finite (cyclically) ordered sets (U1,U2) that satisfies1.26 f[S1|S2]=∑π∈NCP→(|S1|,|S2|)∏B∈πf∘[B]+∑π1×π2∈NCP(|S1|)×NCP(|S2|),U1∈π1,U2∈π2markedf∘∘[U1|U2]∏B∈π1\U1∪π2\U2f∘[B]

for any finite S1,S2. 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 f[S1|S2]=f[S2|S1]¯ typically used in the free probability literature to mimic the covariance functional (cf. [22, Ch. 5]). This choice will simplify the computations by reducing the number of complex conjugates arising in the intermediate steps.

Similar to (1.14), the implicit relation (1.26) may be turned into an explicit definition of f∘∘ by recursion. Note that the term f∘∘[[k]|[k+1,k+ℓ]] in formula (1.26) with f[[k]|[k+1,k+ℓ]] on the left-hand side only occurs for the marked partition {{1,…,k_}}×{{k+1,…,k+ℓ_}} and hence always has coefficient one, so we can express it in terms of f, f∘, and the previously identified values of f∘∘. This shows that f∘∘ is well-defined. Although we will not rely on Möbius inversion to express f∘∘, we remark that it is possible to include both NCP→(k,ℓ) and the marked elements of NCP(k)×NCP(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 Sects. 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 Sect. 2.2.

Definition 1.15

Let S1=(z1,…,zk′)⊂C\R and S2=(zk′+1,…,zk′+ℓ′)⊂C\R be two finite ordered multi-sets. We define m~[·|·] to be the set function taking values in C with the properties (i)–(iii) listed below. Similar to m[·] in Definition 1.5, we interpret m~[·|·] as a function of the indices of the spectral parameters. (i) Symmetry: m~[·|·] is symmetric under the interchanging of its arguments, i.e., for any sets B1⊆S1,B2⊆S2 we have m~[(i,zi∈B1)|(j,zj∈B2)]=m~[(j,zj∈B2)|(i,zi∈B1)].

(ii) Initial condition: For any sets B1⊆S1,B2⊆S2 we have 1.27 m~[(i,zi∈B1)|∅]=m~[∅|(j,j∈B2)]=0.

(iii) Recursion: Let B1⊆S1 and B2⊆S2 be ordered subsets with |B1|=k≤k′ and |B2|=ℓ≤ℓ′ elements, respectively. For simplicity, we index them by [k] and [k+1,k+ℓ]. The function m~[·|·] satisfies the following linear recursion 1.28 m~[1,…,k|k+1,…,k+ℓ]=m11-m1mk(m~[2,…,k|k+1,…,k+ℓ]+∑j=1k-1m~[1,…,j|k+1,…,k+ℓ]m[j,…,k]+∑j=2km[1,…,j]m~[j,…,k|k+1,…,k+ℓ]+sGUE+sκ+sσ+sω)

where the source terms in the last line are given by sGUE:=∑j=1ℓm[1,…,k,k+j,…,k+ℓ,k+1,…,k+j]sκ:=κ4∑r=1k∑k+1≤s≤t≤k+ℓm[1,…,r]×m[r,…,k]m[s,…,t]m[t,…,k+ℓ,k+1,…,s]sσ:=σ∑j=1ℓm#,σ[1,…,k,k+j,…,k+ℓ,k+1,…,k+j]sω:=ω2~∑j=1ℓm[1,…,k]m[k+j,…,k+ℓ,k+1,…k+j].

Here, we wrote out the underlying multi-set in the definition of sκ to indicate that it evaluates to m[s, s] instead of ms 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 decomposition1.29 m~[·|·]=m~GUE[·|·]+κ4m~κ[·|·]+σm~σ[·|·]+ω~2m~ω[·|·]

where m~GUE[·|·] satisfies (1.28) for κ4=σ=ω~2=0 and κ4m~κ[·|·], σm~σ[·|·], and ω~2m~ω[·|·] satisfy (1.28) with sκ, sσ, and sω as the only source term, respectively.

Note that the right-hand side of (1.28) only contains divided differences and m~[B1|B2] for |B1|+|B2|<k+ℓ, so (1.28) indeed defines m~[·|·] recursively. The symmetry assumption in (i) then extends (1.28) to the second entry of m~[·|·]. Moreover, all source terms in the last line of (1.28) are fully expressable as a function of m1,…,mk+ℓ by (1.16), making m~[·|·] eventually a function of m1,…,mk+ℓ as well.

As an example, setting σ=ω~2=0 and applying the recursion once gives1.30 m~[1|2]=m12m22(1-m12)(1-m22)(1-m1m2)2+κ4m13m23(1-m12)(1-m22)=m1′m2′(1-m1m2)2+κ4m1m1′m2m2′

with mi′=m′(zi). We remark that m~GUE[1|2], seen as a function of (z1,z2), 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.6

We consider another special case in the following example.

Example 1.16

Whenever κ4=σ=ω~2=0 and one argument of m~GUE[·|·] 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) yields the closed formula1.31 m~GUE[1|2,…,ℓ+1]=m11-m12∑j=2ℓ+1m[1,…,j,…,ℓ+1,j]=(∏s=1ℓ+1ms)(∑Γ∈NCG([ℓ+1])∏(a,b)∈E(Γ)qa,b)×∑j=2ℓ+1m1′m1mj′mj(1+∑i≠jqi,j).

For ℓ=2, we hence obtainm~GUE[1|2,3]=m1′(m2′m3(1-m1m3)+m2m3′(1-m1m2))(1-m1m2)2(1-m1m3)2(1-m2m3).

Note that the right-hand side of (1.31) is fully expressed terms of non-crossing graphs on a labeled disk. This is because NCP→(1,ℓ) and marked elements of NCP(1)×NCP(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.

Lemma 1.17

For j∈{1,…,k}, k≥1, we have1.32 m[1,…,j,…,k,j]=m[1,…,k](1+∑l∈[k]\{j}qj,l)mj′mj

with qi,j as in (1.13).

We also give some examples to illustrate the combinatorial formula (1.26) for the choice f[·|·]=m~GUE[·|·].

Example 1.18

(Second-order free cumulants) Let κ4=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 marking {{1}}×{{2}}, namely {{1_}}×{{2_}}. Rearranging (1.26), we thus getm∘∘[1|2]=m~[1|2]-m[1,2]+m1m2.

Similarly, considering k=1 and ℓ=2 yieldsm∘∘[1|2,3]=m~[1|2,3]-m~[1|2]m3-m~[1|3]m2-2m[1,2,3]+2m1m[2,3]+2m2m[1,3]+2m3m[1,2]-4m1m2m3.

In the case k=ℓ=2, there are 18 distinct non-crossing annular permutations (see Fig. 7) and 4 elements in NCP(2)×NCP(2). However, the second sum in (1.26) consists of 9 terms in total due to the marking of the blocks as, e.g., {{1_},{2}}×{{3,4_}} and {{1},{2_}}×{{3,4_}} correspond to m∘∘[1|3,4]m2 and m∘∘[2|3,4]m1, respectively, which do not need to coincide. In total, the formula defining m∘∘[1,2|3,4] has 27 terms on the right-hand side of (1.26).

Fig. 7 The 18 elements of NCP→(2,2)

We conclude this section by recalling the Kreweras complement for annular non-crossing permutations from [23] (see Fig. 8 for an example). Similarly to the disk case, taking the Kreweras complement is an invertible map on NCP→(k,ℓ).

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. 8) also by 1,…,k+ℓ (red in Fig. 8). Respecting the orientation of the two circles, we arrange the new labels such that the arc s follows the point s. Let π∈NCP→(k,ℓ) be visualized on the (k,ℓ)-annulus as in Definition 1.11. The (annular) Kreweras complement K(π)∈NCP→(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 non-crossing permutation maximal if none of its cycles can be extended (by merging cycles) without inducing a crossing.

Fig. 8 The annular non-crossing permutation π=(1275)(34)(6) in black and its Kreweras complement K(π)=(1)(2456)(3)(7) in red. (Color figure online)

Note that |π|+|K(π)|=k+ℓ for any π∈NCP→(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 π1×π2∈NCP(k)×NCP(ℓ) by applying Definition 1.3 circle-wise. In this case, we write K(π)=K(π1)×K(π2).

Main Results

The main focus of the present paper lies in determining the limiting covariance structure arising in the CLT for the centered statistics2.1 Xα:=⟨T[1,k]⟩-E⟨T[1,k]⟩=⟨G1A1…GkAk⟩-E⟨G1A1…GkAk⟩,

2.2 Yα:=⟨f1(W)A1…fk(W)Ak⟩-E⟨f1(W)A1…fk(W)Ak⟩.

Here, α=((z1,A1),…,(zk,Ak)) resp. α:=((f1,A1),…,(fk,Ak)) is a multi-index containing bounded deterministic matrices A1,…,Ak and either the spectral parameters z1,…,zk∈C\R with |ℑzj|≳1 appearing in the resolvents or the test functions f1,…,fk∈Hk+1(R) with ‖fj‖≲1. Whenever we need to refer to the number k of resolvents (resp. test functions) in the product Xα (resp. Yα), we carry the parameter k as a superscript and write Xα(k) (resp. Yα(k)). Recall that we set Tj=GjAj=G(zj)Aj as well as T[i,j]=TiTi+1…Tj. Similarly, we introduce Fj:=fj(W)Aj and use the interval notationF[i,j]:=fi(W)Ai…fj(W)Aj

for i<j as well as F∅=0.

Resolvent Central Limit Theorem and Recursion

We start by identifying the joint distribution of multiple Xi(ki) with different ki and αi. To state the limiting covariance structure, we introduce a recursively defined set function m2[·|·], 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 m1[·] arising for the expectation of T[1,k⟩Gk (see Theorem 1.9 as well as (1.21) and (1.22)).7 Note that α=((z1,A1),…,(zk,Ak)) contains the same information on the spectral parameters and deterministic matrices involved as the set of matrices (Tj,j∈[k]). We will, therefore, occasionally abuse notation and use (zj,Aj) and Tj=GjAj interchangeably. In particular, we writem2[α|β]=m2[T1,…,Tk|Tk+1,…,Tk+ℓ]

where the two multi-indices α and β index the spectral parameters and deterministic matrices in T1,…,Tk and Tk+1,…,Tk+ℓ, respectively. At this point, we only give a recursive definition for m2[·|·], however, explicit formulas are later obtained in Sect. 2.2. Note that the case σ=ω~2=0 of Definition 2.1 was already given in [27].

Definition 2.1

Let S1=(T1,…,Tk′) and S2=(Tk′+1,…,Tk′+ℓ′) be two (ordered) finite sets of complex N×N-matrices of the form Tj=GjAj. We define m2[·|·] as the (deterministic) function of pairs of sets S1,S2 with values in C and the following properties: (i) Symmetry: m2[·|·] is symmetric under the interchanging of its arguments, i.e., for any sets B1⊆S1,B2⊆S2 we have m2[(Ti,i∈B1)|(Tj,j∈B2)]=m2[(Tj,j∈B2)|(Ti,i∈B1)].

(ii) Initial condition: For any sets B1⊆S1,B2⊆S2 we have 2.3 m2[(Ti,i∈B1)|∅]=m2[∅|(Tj,j∈B2)]=0.

(iii) Recursion: Let B1⊆S1 and B2⊆S2 be ordered subsets with |B1|=k≤k′ and |B2|=ℓ≤ℓ′ elements, respectively. We index the matrices in B1 by [k] and the matrices in B2 by [k+1,k+ℓ]. The function m2[·|·] satisfies the following linear recursion 2.4 m2[T1,…,Tk|Tk+1,…,Tk+ℓ]=m1(m2[T2,…,Tk-1,GkAkA1|Tk+1,…,Tk+ℓ]+q1,km2[T2,…,Tk-1,GkA1|Tk+1,…,Tk+ℓ]⟨Ak⟩+∑j=1k-1m2[T1,…,Tj-1,Gj|Tk+1,…,Tk+ℓ]+(m1[Tj,…,Tk]+q1,km1[Tj,…,Tk-1,Gk]⟨Ak⟩)+∑j=2km1[T1,…,Tj-1,Gj](m2[Tj,…,Tk|Tk+1,…,Tk+ℓ]+q1,km2[Tj,…,Tk-1,Gk|Tk+1,…,Tk+ℓ]⟨Ak⟩)+sGUE+sκ+sσ+sω)

where the source terms sGUE, sκ, sσ, and sω are given by 2.5 sGUE:=∑j=1ℓ(m1[T1,…,Tk,Tk+j,…,Tk+j-1,Gk+j]+q1,km1[T1,…,Tk-1,Gk,Tk+j,…,Tk+j-1,Gk+j]⟨Ak⟩)

2.6 sκ:=κ4∑r=1k∑s=k+1k+ℓ(∑t=k+1s⟨M[r]⊙M(s,…,k+ℓ,k+1,…,t)⟩⟨(M[r,k]Ak)⊙M[t,s]⟩+∑t=sk+ℓ⟨M[r]⊙M[s,t]⟩⟨(M[r,k]Ak)⊙M(t,…,k+ℓ,k+1,…,s)⟩)+κ4q1,k∑r=1k∑s=k+1k+ℓ(∑t=k+1s⟨M[r]⊙M(s,…,k+ℓ,k+1,…,t)⟩⟨M[r,k]⊙M[t,s]⟩+∑t=sk+ℓ⟨M[r]⊙M[s,t]⟩⟨M[r,k]⊙M(t,…,k+ℓ,k+1,…,s)⟩)⟨Ak⟩.

2.7 sσ:=σ∑j=1ℓm1[T1…,Tk,Gk+jtAk+j-1t,…,Gk+ℓtAk+1t,…,Gk+j-1tAk+jt,Gk+j]+q1,kσ∑j=1ℓm1[T1…,Tk-1,Gk,Gk+jtAk+j-1t,…,Gk+ℓtAk+1t,…,Gk+j-1tAk+jt,Gk+j]⟨Ak⟩

2.8 sω:=ω2~∑j=1ℓ⟨(M[k]Ak)⊙M(k+j,…,k+ℓ,k+1,…k+j)⟩+q1,kω2~∑j=1ℓ⟨M[k]⊙M(k+j,…,k+ℓ,k+1,…k+j)⟩⟨Ak⟩

Recall that ⊙ denotes the Hadamard product, q1,k was defined in (1.13), and M(…) was defined in Theorem 1.9. Moreover, recall that m1[·] was defined in (1.21) and the notation with transposes was introduced in (1.25).

Note that setting A1=⋯=Ak+ℓ=Id reduces (2.4) to (1.28), showing thatm2[G1,…,Gk|Gk+1,…,Gk+ℓ]=m[1,…,k|k+1,…,k+ℓ].

We use the linearity of the recursion and the different types of source terms to introduce the decomposition2.9 m2[·|·]=mGUE[·|·]+κ4mκ[·|·]+σmσ[·|·]+ω~2mω[·|·],

where mGUE[·|·] satisfies (2.4) for κ4=σ=ω~2=0, and κ4mκ[·|·], σmσ[·|·] resp. ω~2mω[·|·] satisfy (2.4) with sκ, sσ resp. sω as only source term. Note that sGUE+sκ+sσ+sω in (2.4) is fully expressible as a function of A1,…,Ak+ℓ and m1,…,mk+ℓ by (1.22), (1.17) and Lemma A.2, eventually making m2[·|·] a function of the same quantities.

Recall that we set Xα=⟨T1…Tk⟩-E⟨T1…Tk⟩ with α=((z1,A1),…,(zk,Ak)). Before stating the CLT for Xα, 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 X(N)=Y(N)+O(N-c) with c>0 in the sense of moments if for any polynomial P it holds thatEP(X(N))=EP(Y(N))+O(N-c+ε),

for any smal ε>0, 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 Xα in (2.1). As the main interest of the present paper is the deterministic approximation m2[·|·], we restrict the discussion of the CLT to the macroscopic regime (|ℑzi|≳1) for technical simplicity. Note that this implies in particular that m2[·|·] as well as its components mGUE[·|·], mκ[·|·], mσ[·|·], and mω[·|·] are of order one. The proof of Theorem 2.3 is analogous to that of [27, Thm. 3.6]. For the convenience of the reader, we include the necessary modifications for adapting the proof in [27] to the generalized model in Assumption 1.1 in Appendix A.2.

Theorem 2.3

(Macroscopic CLT for resolvents) Fix p∈N, let α1,…,αp be multi-indices, and let W be a Wigner matrix satisfying Assumption 1.1. For each j=1,…,p pick a set of spectral parameters z1(j),…,zkj(j) with |ℑzi(j)|≳1 and maxj|zj|≤N100 as well as deterministic matrices A1(j),…,Akj(j) with ‖Ai(j)‖≲1. Then,2.10 NpE(∏j=1pXαj)=∑Q∈Pair([p])∏{i,j}∈Qm2[αi|αj]+O(NεN)

for any ε>0. Here, Pair(S) denotes the pairings of a set S and m2[·|·] 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 (Xαj)j as asymptotically complex Gaussian in the sense of moments in the limit N→∞.

By (2.9), Theorem 2.3 implies in particular thatlimN→∞N2E(XαXβ)=mGUE[α|β]+κ4mκ[α|β]+σmσ[α|β]+ω~2mω[α|β].

We remark that m2[·|·] is cyclic in the sense thatm2[(Tj,j∈S1)|T1,…,Tk]=m2[(Tj,j∈S1)|T2,…,Tk,T1]

and that furtherm2[(Tj,j∈S1)|T1,…,Tk-1,Gk]=m2[(Tj,j∈S1)|T2,…,Tk-1,GkA1]-m2[(Tj,j∈S1)|T1,…,Tk-1]zk-z1.

whenever z1≠zk, Ak=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 m2[·|·] has to be obtained from the recursion (2.4) directly.

Solution of the Recursion

After identifying m2[α|β] as the deterministic approximation of E[XαXβ], we consider Definition 2.1 in detail. In this section, we derive the main result, which is a solution to the (deterministic) recursion (2.4). This characterizes the overall structure of the function m2[·|·] and yields explicit combinatorial formulas to replace the recursive definition in applications. Making use of the linearity of the recursion and the decomposition (2.9), it is sufficient to consider the components mGUE[·|·], mκ[·|·],mσ[·|·], and mω[·|·] separately. We start by studying mGUE[·|·]. The proof consists of two steps that are carried out in Sect. 3.

Theorem 2.4

Let α=((z1,A1),…,(zk,Ak)) and β=((zk+1,Ak+1),…,(zk+ℓ,Ak+ℓ)) for some k,ℓ∈N. Then,2.11 mGUE[α|β]=∑π∈NCP→(k,ℓ)(∏B∈K(π)〈∏j∈BAj〉)∏B∈πm∘[B]+∑π1×π2∈NCP(k)×NCP(ℓ),U1∈π1,U2∈π2marked(∏B1∈K(π1),B2∈K(π2)〈∏j∈B1Aj〉〈∏j∈B2Aj〉)m∘∘[U1|U2]×∏B1∈π1\U1,B2∈π2\U2m∘[B1]m∘[B2]

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 A1=⋯=Ak+ℓ=Id.

Remark

Note that the right-hand side of (2.11) is symmetric with respect to interchanging of ((z1,A1),…,(zk,Ak)) and ((zk+1,Ak+1),…,(zk+ℓ,Ak+ℓ)), which is consistent with the symmetry of m2[·|·] 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 m~GUE[·|·] implies that m∘∘[·|·] is also invariant under interchanging ((z1,A1),…,(zk,Ak)) and ((zk+1,Ak+1),…,(zk+ℓ,Ak+ℓ)) since m~GUE[·|·] determines m∘∘[·|·] 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, k1=k2=1, and assume that W is a GUE matrix.8 By decomposing A1 and A2 into a tracial and a traceless part, the deterministic approximation for the covariance follows directly from [7, Thm. 4.1], givingN2E(⟨T1⟩-E⟨T1⟩)(⟨T2⟩-E⟨T2⟩)=⟨A1A2⟩m12m22(1-m1m2)+⟨A1⟩⟨A2⟩(m1′m2′(1-m1m2)2-m12m22(1-m1m2))+O(NεN)=⟨A1A2⟩m∘[1,2]+⟨A1⟩⟨A2⟩m∘∘[1|2]+O(NεN),

where the last equation follows from the formulas in Examples 1.6 and 1.18. Note that the error bound Ψ/L in [7, Eq. (91)] evaluates to O(1/N) on macroscopic scales. We remark that the deterministic leading term matches the formula for mGUE[T1|T2] obtained from applying (2.4) to the initial condition mGUE[T1|∅]=0.

Next, we consider the recursion for mκ[·|·]. 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 A1,…,Ak+ℓ and the spectral parameters z1,…,zk+ℓ, respectively. The proof of Theorem 2.6 is given in Sect. 4.1.

Theorem 2.6

Let α=((z1,A1),…,(zk,Ak)) and β=((zk+1,Ak+1),…,(zk+ℓ, Ak+ℓ)) for some k,ℓ∈N. Then there exist (i) a family (ψπ,B)B∈π of functions ψB,π:C|B|→C for every π∈NCP→(k,ℓ) and

(ii) a family (ψπ,U1,U2)U1⊂[k],U2⊂[k+1,k+ℓ] of functions ψπ,U1,U2:C|U1|×C|U2|→C that are invariant under interchanging of the two arguments as well as functions (ψπ1,B1)B1∈π1\U1 and (ψπ2,B2)B2∈π2\U2 with ψπi,Bi:C|Bi|→C for every π=π1×π2∈NCP(k)×NCP(ℓ) with marked blocks U1∈π1 and U2∈π2

such that2.12 mκ[α|β]=∑π∈NCP→(k,ℓ)∏B∈K(π)〈(∏j∈B∩[k]Aj)⊙(∏j∈B∩[k+1,k+ℓ]Aj)〉×∏B∈πψπ,B(zj|j∈B)+∑π=π1×π2∈NCP(k)×NCP(ℓ),U1∈π1,U2∈π2marked(∏B1∈K(π1),B2∈K(π2)〈∏j∈B1Aj〉〈∏j∈B2Aj〉)×ψπ,U1,U2(zj|j∈U1∪U2)∏B1∈π1\U1,B2∈π2\U2ψπ1,B1(zj|j∈B1)ψπ2,B2(zj|j∈B2),

where ⊙ denotes the Hadamard product.

Theorem 2.6 is a purely structural result which shows that mκ[·|·] can be written in terms of non-crossing permutations and partitions similar to mGUE[·|·]. Despite the obvious similarities between (2.11) and (2.12), considering the minimal examplem2[T1|T2]=mGUE[T1|T2]+κ4mκ[T1|T2]=⟨A1A2⟩m12m22(1-m1m2)+⟨A1⟩⟨A2⟩(m1′m2′(1-m1m2)2-m12m22(1-m1m2))+κ4(⟨a1a2⟩m13m23+⟨A1⟩⟨A2⟩(2m1m1′m2m2′-m13m23))

with σ=ω~2=0 already shows that the functions ψi describing the dependence on the spectral parameters do not coincide with the free cumulant functions m∘[·] and m∘∘[·|·] in general. However, (2.4) implies that the functions ψi themselves satisfy a recursion. Theorem 2.6 hence reduces the computation of mκ[·|·] from iterating (2.4) to an inductive computation of the part that only depends on the spectral parameters.

We continue by deriving an explicit formula for mσ[·|·]. As the source term sσ in the corresponding recursion is, up to transposes, identical to sGUE, 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 Sect. 4.2.

Theorem 2.7

Let α=((z1,A1),…,(zk,Ak)), β=((zk+1,Ak+1),…,(zk+ℓ,Ak+ℓ)) for some k,ℓ∈N and abbreviatem~σ[1,…,k|k+1,…,k+ℓ]:=mσ[G1,…Gk|Gk+1,…,Gk+ℓ]

in the special case A1=⋯=Ak+ℓ. If B∈π is a connecting cycle of π∈NCP→(k,ℓ) decomposed as B=(i1,…,ir)∘(j1,…,js) with i1,…,ir⊂[k] and j1,…,js⊂[k+1,k+ℓ], we introduce the notationBσ:=(i1,…,ir)∘(js,…,j1).

Then,2.13 mσ[α|β]=∑π∈NCP→(k,ℓ)(∏B∈K(π)〈∏j∈B∩[k]Aj(∏j∈B∩[k+1,k+j]Aj)t〉)∏B∈πm∘#,σ[Bσ]+∑π1×π2∈NCP(k)×NCP(ℓ),U1∈π1,U2∈π2marked(∏B1∈K(π1),B2∈K(π2)〈∏j∈B1Aj〉〈∏j∈B2Aj〉)×(mσ)∘∘[U1|U2]∏B1∈π1\U1,B2∈π2\U2m∘#,σ[B1]m∘#,σ[B2]

where #=(0,…,0,1,…,1) with the number of zeros and ones matching the number of labels on the inner and outer circle involved in B, respectively. Moreover, m∘#,σ[·] denotes the free cumulant function associated with m#,σ[·] via (1.14) and (mσ)∘∘[·|·] denotes the second-order free cumulant function associated to m~σ[·|·] and m#,σ[·] via (1.26), respectively.

Note that the set function mσ[·|·] satisfies the same factorization property as mGUE[·|·] and mκ[·|·]. For k=ℓ=1, it readily follows thatmσ[T1|T2]=⟨A1A2t⟩m12m221-σm1m2+⟨A1⟩⟨A2⟩(m1′m2′(1-σm1m2)2-m12m221-σm1m2).

We further remark that (2.9) evaluates to m2[·|·]=mGUE[·|·]+mσ[·|·] in the case of GOE matrices (σ=1 and κ4=ω~2=0).

It remains to consider mω[·|·]. The proof of Theorem 2.8 is given in Sect. 4.2.

Theorem 2.8

Let α=((z1,A1),…,(zk,Ak)) and β=((zk+1,Ak+1),…,(zk+ℓ, Ak+ℓ)) for some k,ℓ∈N. Then there exist (i) a family (Ψπ,B)B∈π of functions ΨB,π:C|B|→C for every π∈NCP→(k,ℓ) and

(ii) a family (Ψπ,U1,U2)U1⊂[k],U2⊂[k+1,k+ℓ] of functions Ψπ,U1,U2:C|U1|×C|U2|→C that are invariant under interchanging of the two arguments as well as functions (Ψπ1,B1)B1∈π1\U1 and (Ψπ2,B2)B2∈π2\U2 with Ψπi,Bi:C|Bi|→C for every π=π1×π2∈NCP(k)×NCP(ℓ) with marked blocks U1∈π1 and U2∈π2

such that2.14 mω[α|β]=∑π∈NCP→(k,ℓ)∏B∈K(π)〈(∏j∈B∩[k]Aj)⊙(∏j∈B∩[k+1,k+ℓ]Aj)〉×∏B∈πΨB,π(zj|j∈B)+∑π=π1×π2∈NCP(k)×NCP(ℓ),U1∈π1,U2∈π2marked(∏B1∈K(π1),B2∈K(π2)〈∏j∈B1Aj〉〈∏j∈B2Aj〉)×Ψπ,U1,U2(zj|j∈U1∪U2)∏B1∈π1\U1,B2∈π2\U2Ψπ1,B1(zj|j∈B1)Ψπ2,B2(zj|j∈B2).

Similar to Theorem 2.6, Theorem 2.8 is a pure structural result. It shows that the last contribution mω[·|·] satisfies the same factorization property as mGUE[·|·], mκ[·|·], and mσ[·|·]. In the case k=ℓ=1, we have the formulamω[T1|T2]=⟨a1a2⟩m12m22+⟨A1⟩⟨A2⟩(m1′m2′-m12m22).

It further follows from (2.4) that the functions Ψi themselves satisfy a recursion. Theorem 2.8 thus reduces the computation of mω[·|·] to the inductive computation of the part that only depends on the spectral parameters.

General Test Functions and Applications to Free Probability

We conclude the discussion by comparing the explicit formulas from Sect. 2.2 to the free probability results in [21]. To do so, we generalize the CLT for resolvents in Theorem 2.3 to a full multi-point functional CLT for (N-independent) test functions f1,…,fk, i.e., a CLT for the statistics Yα in (2.2). Given Theorem 2.3 and the formulas fron Sect. 2.2, the proof follows from a classical application of the Helffer–Sjöstrand formula (analogous to the proof of [27, Thm. 2.7]) and is hence omitted. Note that we restrict Theorem 2.9 to real-valued test functions only for simplicity. Extending the results in this section to complex-valued test functions only requires minor modifications to the argument.

Theorem 2.9

(Macroscopic multi-point functional CLT) Let k∈N and pick deterministic matrices A1,…,Ak∈CN×N with ‖Aj‖≲1. Let further W be a Wigner matrix satisfying Assumption 1.1 and let f1,…,fk∈Hk+1(R) be real-valued compactly supported test functions with ‖fj‖≲1. Then, for any ε>0, the centered statistics (2.2) are approximately distributed (in the sense of moments) as2.15 NYα(k,a)=ξ(α)+O(NεN)

with a centered (N-dependent) Gaussian process ξ(α) satisfying2.16 Eξ(α)ξ(β)=∑π∈NCP→(k,ℓ)(∏B∈K(π)〈∏j∈BAj〉)Φπ(GUE)(f1,…,fk+ℓ)+κ4∑π∈NCP→(k,ℓ)∏B∈K(π)〈(∏j∈B∩[k]Aj)⊙(∏j∈B∩[k+1,k+ℓ]Aj)〉×Φπ(κ)(f1,…,fk+ℓ)+σ∑π∈NCP→(k,ℓ)(∏B∈K(π)〈∏j∈B∩[k]Aj(∏j∈B∩[k+1,k+j]Aj)t〉)×Φπ(σ)(f1,…,fk+ℓ)+ω~2∑π∈NCP→(k,ℓ)∏B∈K(π)〈(∏j∈B∩[k]Aj)⊙(∏j∈B∩[k+1,k+ℓ]Aj)〉×Φπ(ω)(f1,…,fk+ℓ)+∑π1×π2∈NCP(k)×NCP(ℓ),U1∈π1,U2∈π2marked(∏B1∈K(π1),B2∈K(π2)〈∏j∈B1Aj〉〈∏j∈B2Aj〉)×Φπ1×π2,U1×U2(f1,…,fk+ℓ).

Here, β denotes another multi-index of length ℓ containing the deterministic matrices Ak+1,…,Ak+ℓ satisfying ‖Aj‖≲1 and the test functions fk+1,…,fk+ℓ∈Hℓ+1(R). The functions Φπ(·) and Φπ1×π2,U1×U2 in (2.16) can be computed recursively and only depend on the underlying permutation resp. marked partition, the functions f1,…,fk+ℓ and the model parameters κ4, σ, and ω~2.

For the later applications, we note the following formulas for the case κ4=σ=ω~2=0. Corollary 2.10 is proven in [27] by explicitly evaluating the integrals that are obtained from the Helffer–Sjöstrand formula in the proof of the functional CLT.

Corollary 2.10

(Cor. 2.9 in [27]) Consider Theorem 2.9 for a GUE matrix9W. In this case, we have2.17 Φπ(GUE)(f1,…,fk+ℓ):=∏B∈πsc∘[B],

where sc∘[·] denotes the free cumulant function associated with2.18 sc[i1,…,in]:=∫-22[∏j=1nfij(x)]ρsc(x)dx,

with ρsc as in (1.9), and2.19 Φπ1×π2,U1×U2(f1,…,fk+ℓ):=sc∘∘[U1|U2]∏B1∈π1\U1,B2∈π2\U2sc∘[B1]sc∘[B2],

where sc∘∘[·|·] denotes the second-order free cumulants associated with sc[·] in (2.18) and2.20 sc[i1,…,in|in+1,…,in+m]:=12∫-22∫-22(∏j=1nfij(x))′(∏j=1mfin+j(y))′u(x,y)dxdy

by Definition 1.14. The kernel u:[-2,2]×[-2,2]→R is given by2.21 u(x,y):=14π2ln[(4-x2+4-y2)2(xy+4-4-x24-y2)(4-x2-4-y2)2(xy+4+4-x24-y2)].

We remark that the formula (2.21) also appears in [9] and [28] (see also [27, Cor. 3.8]).

Whenever fj(x)=x for all j=1,…,k+ℓ or, more generally, fj is an (N-independent) polynomial,10 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 f1(x)=⋯=fk+ℓ(x)=x, i.e., Yα(k,a)=⟨WA1…WAk⟩-E⟨…⟩ and Yβ(ℓ,b)=⟨WAk+1…WAk+ℓ⟩-E⟨…⟩. Then,2.22 limN→∞N2E(Yα(k,a)Yβ(ℓ,b))=∑π∈NCP→2(k,ℓ)(∏B∈K(π)〈∏j∈BAj〉)

where NCP→2(k,ℓ) denotes the pairings in NCP→(k,ℓ).

We remark that the limit in (2.22) reproduces the well-known result of second-order freeness of GUE and deterministic matrices from [24]. Moreover, computations similar to Corollaries 2.10 and 2.11 for the GOE (σ=1 and κ4=ω~2=0) yieldlimN→∞N2E(Yα(k,a)Yβ(ℓ,b))=∑π∈NCP→2(k,ℓ)(∏B∈K(π)〈∏j∈BAj〉)+∑π∈NCP→2(k,ℓ)(∏B∈K(π)〈∏j∈B∩[k]Aj(∏j∈B∩[k+1,k+j]Aj)t〉)

which is the (real) second-order freeness of GOE and deterministic matrices established in [26]. Apart from these two special cases, the deterministic approximation in Theorem 2.9 mirrors the overall structure of the joint second-order distribution of Wigner and deterministic matrices described in [21, Eq. (3)]. We remark that resolvents and functions with an N-dependent mesoscopic scaling (1.5) 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 its mesoscopic analog [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 (or GOE) 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 (cf. Remark [27, Cor. 2.11]).

Proof of Theorem 2.4 (Formula for mGUE[·|·])

The proof of Theorem 2.4 consists of two main steps: Solve the recursion for mGUE[·|·] in the case A1=⋯=Ak+ℓ=Id. As m[·] can be expressed in terms of non-crossing graphs on the disk, we can rewrite the coefficients and the source term of the recursion in terms of graphs. By defining a suitable family of graphs on the (k,ℓ)-annulus, we extend this representation to obtain an ansatz for m~GUE[·|·] and, by Definition 1.14, also an expression for the second-order free cumulant function.

Using the formula for m~GUE[·|·] obtained in Step 1, write out the right-hand side of (2.11) and show that it satisfies the recursion for mGUE. Checking that this ansatz also satisfies the remaining properties in Definition 2.1 yields equality between the two sides of (2.11).

We address the two steps in Sects. 3.1 and 3.2, respectively.

Step 1: Graphs

As we only consider m~GUE throughout this section, let κ4=σ=ω~=0 and thus m~GUE[·|·]=m~[·|·]. Recall from (1.16) and (1.17) that both m[·] and m∘[·] are expressable in terms of disk non-crossing graphs. In this section, we give analogous combinatorial formulas for m~[·|·] and m∘∘[·|·]. 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 (cf. [22, Thm. 8 in Ch. 5]).

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: 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.

Remove the slit to obtain an annulus.

Merge the two copies of the label k+j.

We visualize τ for an example in Fig. 9. The two labels k+j are denoted as 6 and 6’ to distinguish between them more easily.Fig. 9 The geometry of the transformation τ for k=4, ℓ=3, and j=2. (Color figure online)

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. 10. 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. 10 and Example 3.4).Fig. 10 Construction of τ(Γ) for a graph Γ∈NCG({1,2,3,3′})

Definition 3.2

For k,ℓ∈N we define G([k],[k+1,k+ℓ]) to be the multi-set11 of undirected, planar graphs on the (k,ℓ)-annulus with vertex set {1,…,k+ℓ} and possible loops or double edges that is obtained from the following recursive construction: (i) For any S1⊂[k] and S2⊂[k+1,k+ℓ] we have 3.1 G(S1,∅)=G(∅,S2)=∅.

(ii) The multi-set G([k],[k+1,k+ℓ]) can be constructed from the multi-sets G(S1,[k+1,k+ℓ]) with S1⊊[k] as follows: We define 3.2 G¬(1,k):=G1∪G2∪G3∪G4,

as the disjoint union of the sets G1:={Γ∪{1}|Γ∈G([2,k],[k+1,k+ℓ])}G2:=⋃j=2k{Γ=Γ1∪Γ2|Γ1∈NCG([1,j])with edge(1,j),Γ2∈G([j,k],[k+1,k+ℓ])},G3:=⋃j=1k-1{Γ=Γ1∪Γ2|Γ1∈G([1,j],[k+1,k+ℓ])with edge(1,j),Γ2∈NCG([j,k])},G4:=⋃j=1ℓτ({Γ∈NCG1,…,k,k+j,…,k+ℓ,k+1,…,k+j-1,k+j|Γhas edge(1,k+j)}).

Here, the union Γ∪{1} in G1 is to be understood as adding a separated vertex 1 to Γ while the union Γ1∪Γ2 in G2 and G3 refers to the graph with the vertex set {1,…,k+ℓ} and the edge set given by the union of the edge sets of Γ1 and Γ2, respectively. Further, recall τ from Definition 3.1. We remark that all elements of G1,…,G4 are planar graphs. Next, let G(1,k):={Γ∪{(1,k)}|Γ∈G¬(1,k)},

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 3.3 G([k],[k+1,k+ℓ]):=G(1,k)∪G¬(1,k),

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(S1,S2)=G(S2,S1) for any S1⊂[k] and S2⊂[k+1,k+ℓ] (in the sense that there is a well-defined bijective mapping that takes each element of G(S1,S2) to its counterpart).

We abbreviate G(k,ℓ):=G([k],[k+1,k+ℓ]) and refer to its elements as good graphs. The subset of connected good graphs is denoted by Gc(k,ℓ). Any edge (i, j) with i∈[k] and j∈[k+1,k+ℓ] is referred to as connecting edge.

We give a schematic overview of the elements in G(k,ℓ) in Fig. 11. To avoid overcrowding, labels and edges that are not prescribed by the definition are left out.Fig. 11 Schematic visualization of the elements of G1, G2, G3, and G4 (left to right). Different subgraphs are indicated by different line styles

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 G4 and that Γ∈G4 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∈G4 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. 13). 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 G4 as a kind of source term. In view of Lemma 3.3, 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 NCG(S) does for m[·].

Lemma 3.3

For fixed k,ℓ∈N, we have3.4 m~[1,…,k|k+1,…,k+ℓ]=(∏s=1k+ℓms)∑Γ∈G(k,ℓ)∏(i,j)∈E(Γ)qi,j.

Proof

As qi,j=qj,i by (1.13) and G(k,ℓ)=G(ℓ,k) by Definition 3.2(iii), it readily follows that the right-hand side of (3.4) is symmetric under the interchanging [k] and [k+1,k+ℓ]. Similar to the proof of [6, Lem. 5.2], we use the combinatorial formula (3.4) as an ansatz to solve the recursion given in (1.27) and (1.28). First, observe that(∏s=1kms)∑Γ∈G(S1,∅)∏(i,j)∈E(Γ)qi,j=(∏s=k+1k+ℓms)∑Γ∈G(∅,S2)∏(i,j)∈E(Γ)qi,j=0

for any S1⊂[k] and S2⊂[k+1,k+ℓ] due to the sums being empty. Hence, the initial condition (1.27) is satisfied.

It remains to check (1.28). We introduce the notationqΓ:=∏(i,j)∈E(Γ)qi,j

and conclude from the decompositions (3.3) and (3.2) that3.5 ∑Γ∈G(k,ℓ)qΓ=(1+q1,k)∑Γ∈G¬(1,k)qΓ=(1+q1,k)(∑Γ∈G1qΓ+∑Γ∈G2qΓ+∑Γ∈G3qΓ+∑Γ∈G4qΓ).

Noting that the vertex 1 in Γ∈G1 has no adjacent edges, we may write∑Γ∈G1qΓ=∑Γ∈G([2,k],[k+1,k+ℓ])qΓ.

Further, the transformation τ only changes the geometry underlying a graph Γ, but does not influence its edge set. By the definition of G4, any Γ∈NCG([1,…,k+ℓ,k+j]) used in the construction must have at least one edge (1,k+j). As a consequence, the product qΓ always includes the factor q1,k+j=m1mk+j(1+q1,k+j). This yields∑Γ∈G4qΓ=∑j=1ℓ(m1mk+j∑Γ∈NCG([1,…,k+ℓ,k+j])qΓ).

Note that the identity q1,k+j=m1mj(1+q1,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 G2 and G3 as qΓ1∪Γ2=qΓ1qΓ2 for the union of graphs introduced in Definition 3.2. With these replacements, (3.5) can be written as11+q1,k∑Γ∈G(k,ℓ)qΓ=∑Γ∈G([2,k],[k+1,k+ℓ])qΓ+∑j=2km1mj(∑Γ∈NCG([1,j])qΓ)(∑Γ∈G([j,k],[k+1,k+ℓ])qΓ)+∑j=1k-1m1mj(∑Γ∈G([1,j],[k+1,k+ℓ])qΓ)(∑Γ∈NCG([j,k])qΓ)+∑j=1ℓm1mk+j(∑Γ∈NCG([1,…,k+ℓ,k+j])qΓ).

Multiplying both sides with ∏s=1k+ℓms and noting that 1+q1,j=(1-m1mj)-1, we see that the right-hand side of (3.4) satisfies (1.28) as claimed.

Example 3.4

We have |G(1,1)|=8. The graphs are visualized in Fig. 12.Fig. 12 The elements of G(1,1)

In particular, we readily reobtain (1.30) by evaluating qΓ for every graph in the above list. Note that decomposing (1.30) into the form m1m2∑Γ∈G(1,1)qΓ is possible in multiple ways. However, picking graphs that contain a connecting edge yields the set G(1,1) in Fig. 12 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.

Lemma 3.5

Let k,ℓ∈N. 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.

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.

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.

G(k,ℓ) is a genuine multi-set unless k=ℓ=1.

Proof

(a) We use proof by induction. First, note that the elements of G(S1,∅)=∅ and G(∅,S2)=∅ with S1⊆[k] and S2⊆[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(S1,[k+1,k+ℓ]) give rise to an annular non-crossing partition for any S1⊂[k] with |S1|≤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∈NCP(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 G1, G2, G3, and G4 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∈NCG({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 G1, G2, and G3 contain an element of G(S1,[k+1,k+ℓ]) with some S1⊂[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 G1, too, behaves as claimed. The argument for G2 and G3 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 NCP(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(S1,∅)=∅ and G(∅,S2)=∅ with S1⊆[k] and S2⊆[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∈G1∪G2∪G3 must occur in the subgraph in G(S1,[k+1,k+ℓ]) with S1⊂[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∈G4, note that applying τ to an element of NCG({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 G4 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(S1,∅)=∅ and G(∅,S2)=∅ with S1⊆[k] and S2⊆[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∈G1∪G2∪G3 must occur in the subgraph in G(S1,[k+1,k+ℓ]) with S1⊂[k] used to construct Γ and that the induction hypothesis applies to the latter. For Γ∈G4, a double edge occurs if and only if the corresponding element of NCG({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 G2, 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.

(d) As the case k=ℓ=1 has already been discussed in Example 3.4, let k=2, ℓ=1, and consider τ(NCG({1,2,3,3})). We relabel the vertices as 1,2,3,3′ to distinguish between the two copies of the doubled vertex 3 more easily. Observe that the graphs Γ1 and Γ2 with edge sets {(1,3′),(2,3′)} and {(1,3′),(2,3)}, respectively, give rise to the same element of G(2,1), namely the annulus graph with edge set {(1,3),(2,3)} (see Fig. 13).

Hence, G(1,2) is indeed a multi-set. By Definition 3.2, either G(1,2) or G(2,1) is used in the construction of G(k,ℓ) for k,ℓ>2, i.e., the construction yields again a multi-set.

Fig. 13 Two disk non-crossing graphs (left) that give rise to the same annulus graph (right)

For the following discussion, we introduce the disjoint decomposition3.6 G(k,ℓ)=Gdec(k,ℓ)∪G¬dec(k,ℓ),

where Gdec(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 Gdec(k,ℓ) as decorated graphs.

Lemma 3.6

Let k,ℓ∈N. 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.

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 Γ1 denotes the connected component of Γ that contains a connecting edge and U1∪U2 with U1⊂[k] and U2⊂[k+1,k+ℓ] is the vertex set of Γ1, there are |U1|·|U2| 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. 14. 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.Fig. 14 A subgraph of Γ∈G(k,ℓ) with the induced partition πΓ and a possible permutation πΓ′ (left) as well as the disk non-crossing graph obtained for a connected component of Γ (right)

Note that the assignment of the orientation in the second step on the left of Fig. 14 is not unique if πΓ has only one connecting block (cf. [23, Prop. 4.6], see Fig. 5 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. 14.

Proof of Lemma 3.6

(b) It follows from Lemma 3.5(a) that the connected components of Γ give rise to a partition πΓ∈NCP(k,ℓ). By Propositions 4.4. and 4.6 of [23], any element of NCP(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. 14). This orientation of the individual cycles is naturally induced by the orientation of the inner and outer circle. However, the identification between NCP(k,ℓ) and NCP→(k,ℓ) is only unique whenever the underlying partition has more than one connecting block. If there is only one connecting block U1∪U2 with U1⊂[k], U2⊂[k+1,k+ℓ], there are |U1|·|U2| possibilities to identify πΓ with an annular non-crossing permutation (cf. Fig. 5). We fix one πΓ′∈NCP→(k,ℓ) that is associated with πΓ in this way.

By definition, the cycles of πΓ′ 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. 15). 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.Fig. 15 Transformation of a cycle of πΓ′∈NCP→(k,ℓ) to a circle (left) and the induced transformation of a connected component of Γ∈G(k,ℓ) into a disk graph (right)

(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 G4 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 G4 directly. To have two connecting blocks, Γ must have at least two connecting edges (i1,i2) and (i1′,i2′) where i1,i1′∈[k], i2,i2′∈[k+1,k+ℓ] and i1≠i1′, i2≠i2′. Note that either (i1,i2) or (i1′,i2′) may coincide with the edge (1,k+j) prescribed by the definition.

Let Γ~∈NCG({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 (i1,i2) and (i1′,i2′) with i1,i1′∈[k], i2,i2′∈[k+1,k+ℓ] and i1≠i1′, i2≠i2′. This structure imposes several restrictions on Γ~, as can be seen from Fig. 16.Fig. 16 A schematic visualization of Γ~ (left) and its mapping to the original graph Γ=τ(Γ~) (right). (Color figure online)

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∈NCP(k,ℓ).

Lemma 3.7

For every π∈NCP(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 Gdec(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 non-crossing 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. 17, the remaining connected components of the graph are readily added using steps corresponding to G1, G2, and G3 in Definition 3.2 and the symmetry under interchanging of the inner and outer circle.Fig. 17 The construction of an element of G(2,1) (right) from the connecting block of an annular non-crossing partition (left)

Proof of Lemma 3.7

Fix π∈NCP(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 G1 and G2 in Definition 3.2.

Under these assumptions, there exists π~∈NCP({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 Fig. 9, 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~∈NCG({1,…,k,k+1,…,k+ℓ,k+1}) by considering each block B={i1,…,in}∈π~ separately and adding the edges (i1,i2),…,(i1,in) to the graph. By construction, the vertex sets of the connected components of Γ~ coincide with the blocks of π~. Considering Γπ=τ(Γ~) now yields the element of G(k,ℓ) with the claimed properties (cf. Fig. 17, where the procedure is sketched for a single block).

Next, consider π∈NCP(k,ℓ) that has only one connecting block U=U1∪U2 with U1⊆[k], U2⊆[k+1,k+ℓ]. Similar to the first case, we can construct a disk non-crossing graph ΓU for which τ(ΓU)∈Gc(|U1|,|U2|). As τ(ΓU) is only required to have one connecting edge, there is also a choice for ΓU containing a (k+j,k+j) edge (cf. Fig. 15). In particular, we obtain at least one graph in Gc,¬dec(|U1|,|U2|) and one graph in Gc,dec(|U1|,|U2|), respectively. The remaining connected subgraphs of Γπ are then added recursively by alternating between adding an isolated vertex (cf. G1) and a disk non-crossing graph (cf. G2 and G3) to τ(ΓU). Note that starting from τ(ΓU)∈Gc,¬dec(|U1|,|U2|) yields a graph that is not decorated while starting from τ(ΓU)∈Gc,dec(|U1|,|U2|) yields a decorated graph.

Using the properties of Γ∈G(k,ℓ) from Lemmas 3.6 and 3.7, we obtain an explicit non-recursive combinatorial formula for m∘∘.

Lemma 3.8

Let k,ℓ∈N. Then,3.7 m∘∘[1,…,k|k+1,…,k+ℓ]=(∏s=1k+ℓms)∑Γ∈Gc(k,ℓ)cΓqΓ,

with suitable constants cΓ∈Z. In particular, cΓ=1 for Γ∈Gdec(k,ℓ).

Note that the constants cΓ 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⊆S1∪S2 as a union B1∪B2 with B1⊆[k], B2⊆[k+1,k+ℓ] and associate it with a tuple (B1,B2) if neither of the subsets is empty. This allows us to use the common notation NCG(B) for both disk and annular graphs by setting NCG(B)=G(B1,B2) whenever B contains elements from both [k] and [k+1,k+ℓ], and NCG(B)=NCG(B1) resp. NCG(B)=NCG(B2) if it does not. Splitting the sum in (3.4) according to the underlying partition, we can write3.8 m~[1,…,k|k+1,…,k+ℓ]=(∏s=1k+ℓms)∑Γ∈G(k,ℓ)qΓ=∑π∈NCP(k,ℓ)∏B∈π[(∏s∈Bms)∑Γ∈NCGc(B)qΓ]

where NCGc(B) denotes the connected graphs in NCG(B). By Lemma 3.7, none of the sums on the right-hand side are empty. However, there may be multiple permutations in NCP→(k,ℓ) as well as a marked element of NCP(k)×NCP(ℓ) associated with a given annular non-crossing partition π∈NCP(k,ℓ). To obtain the same structure as in (1.26), we thus need to decompose the sum over π∈NCP(k,ℓ) on the right-hand side of (3.8) further.

Distinguishing by the number of connecting blocks of π yields3.9 ∑π∈NCP(k,ℓ)∏B∈π[(∏s∈Bms)∑Γ∈NCGc(B)qΓ]=∑π∈NCP(k,ℓ),1conn.block∏B∈π[(∏s∈Bms)∑Γ∈NCGc(B)qΓ]+∑π∈NCP(k,ℓ),≥2conn.blocks∏B∈π[(∏s∈Bms)∑Γ∈NCGc(B)qΓ].

As an annular non-crossing partition with at least two connecting blocks uniquely corresponds to an annular non-crossing permutation (cf. [23, Prop. 4.4]), we may replace NCP(k,ℓ) by NCP→(k,ℓ) in the summation if we interpret a cycle as an ordered set. Together with Lemma 3.6 and (1.17), it follows that3.10 ∑π∈NCP(k,ℓ),≥2conn.blocks∏B∈π[(∏s∈Bms)∑Γ∈NCGc(B)qΓ]=∑π∈NCP→(k,ℓ),≥2conn.cycles∏B∈π[(∏s∈B1∪B2ms)∑Γ∈NCGc(B1∪B2)qΓ]=∑π∈NCP→(k,ℓ),≥2conn.cycles∏B∈πm∘[B1∪B2].

where we used that m∘ is invariant under permutation of the spectral parameters (zj)j.

Applying a similar argument for the partitions with one connecting block U, interpreted as (U1,U2) with U1=U∩[k],U2=U∩[k+1,k+ℓ], yields3.11 ∑π∈NCP(k,ℓ),1conn.block∏B∈π[(∏s∈Bms)∑Γ∈NCGc(B)qΓ]=∑π∈NCP(k,ℓ),1conn.block[(∏s∈Ums)∑Γ∈Gc(U)qΓ]∏B∈π\{U}[(∏s∈Bms)∑Γ∈NCGc(B)qΓ]=∑π∈NCP(k,ℓ),1conn.block[(∏s∈Ums)∑Γ∈Gc(U)qΓ]∏B∈π\{U}m∘[B]

by (1.17). Note that B≠U cannot involve both sets [k] and [k+1,k+ℓ], as U is the only connecting block. Using (3.6), we decompose∑Γ∈Gc(U)qΓ=∑Γ∈Gc,dec(U)qΓ+∑Γ∈Gc,¬dec(U)qΓ

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 Gc,¬dec(U) and recall that any Γ∈Gc,¬dec(U) can be identified with a disk non-crossing graph by Lemma 3.6(b). However, this identification is not unique. Decompose U into U1=U∩[k],U2=U∩[k+1,k+ℓ]. Then there are |U1|·|U2| different disk graphs that can be obtained from a given Γ∈Gc,¬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∈NCP(k,ℓ) is equal to |U1|·|U2| by [23, Prop. 4.6]. We thus write∑Γ∈Gc,¬dec(U)qΓ=|U1|·|U2|∑Γ∈NCGc(U1∪U2)qΓ+∑Γ∈Gc,¬dec(U)cΓqΓ.

with suitable constants cΓ∈Z. In particular, we do not necessarily have cΓ=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,3.12 ∑Γ∈Gc(U)qΓ=|U1|·|U2|∑Γ∈NCGc(U1∪U2)qΓ+∑Γ∈Gc(U)cΓqΓ.

where we set cΓ=1 for Γ∈Gc,dec(U). Note that the contribution of the first term of (3.12) to (3.11) is3.13 ∑π∈NCP(k,ℓ),1conn.blockU[(∏s∈Ums)·|U1|·|U2|∑Γ∈NCGc(U1∪U2)qΓ]∏B∈π\{U}m∘[B]=∑π∈NCP→(k,ℓ),1conn.block∏B∈πm∘[B1∪B2].

Putting everything together, the right-hand side of (3.9) reads3.14 m~[1,…,k|k+1,…,k+ℓ]=∑π∈NCP→(k,ℓ)∏B∈πm∘[B]+∑π∈NCP(k)×NCP(ℓ),U1,U2marked∏B∈π\{U1,U2}m∘[B1∪B2]×[(∏s∈Ums)∑Γ∈Gc(U)cΓqΓ].

with cΓ∈Z as in (3.12) and U=U1∪U2. 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.

Step 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 yields3.15 f[α|β]m1…mk+ℓ=∑π∈NCP→(k,ℓ)(∏B∈K(π)〈∏j∈BAj〉)∏B∈π(∑Γ∈NCGc(B1∪B2)qΓ)+∑π∈NCP(k)×NCP(ℓ),U1,U2marked(∏B∈K(π)〈∏j∈BAj〉)(∑Γ∈NCGc(U1,U2)cΓqΓ)×∏B∈π\{U1,U2}(∑Γ∈NCGc(B1∪B2)qΓ).

To collect the terms involving the deterministic matrices A1,…,Ak, we define3.16 F(π):=∏B∈K(π′)⟨∏j∈BAj⟩

for any π∈NCP(k,ℓ) that has more than one connecting block. Here, π′∈NCP→(k,ℓ) denotes the unique permutation for which the blocks of π′ coincide with π. Further, we define3.17 F(π):=∑π′∈NCP→(k,ℓ)blocks(π′)=π(∏B∈K(π′)〈∏j∈BAj〉)+∏B∈K(π′′)〈∏j∈BAj〉

for any π∈NCP(k,ℓ) that has exactly one connecting block U. Here, π′′ is the marked partition obtained from π by splitting the connecting block into U1=U∩[k] and U2=U∩[k+1,k+ℓ], and marking U1,U2 on the respective circles.

Next, we decompose the sum over NCP→(k,ℓ) in (3.15) according to the number of connecting cycles in the permutation and rewrite the right-hand side asf[α|β]m1…mk+ℓ=∑π∈NCP(k,ℓ)F(π)∏B∈π(∑Γ∈NCGc(B)qΓ)

using (3.16) and (3.17). Recall that any connected component of Γ∈G(k,ℓ) can either be identified with a disk non-crossing graph or itself satisfies Definition 3.2. We can thus interpret NCGc(B) as a connected component of a bigger graph and rewrite3.18 ∑π∈NCP(k,ℓ)F(π)∏B∈π(∑Γ∈NCGc(B)qΓ)=∑Γ∈G(k,ℓ)F(πΓ)qΓ.

Here, πΓ 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) as3.19 ∑Γ∈G(k,ℓ)F(πΓ)qΓ=∑Γ∈G¬(1,k)qΓ(F(πΓ)+q1,kF(πΓ∪{(1,k)})).

Recall that Γ∪{(1,k)} denotes the graph obtained from adding an edge (1, k) to Γ and that, therefore, qΓ∪{(1,k)}=q1,kqΓ. Next, apply (3.2) to split the right-hand side of (3.19) into contributions corresponding to G1, G2, G3, and G4. This yields3.20 ∑Γ∈G(k,ℓ)F(πΓ)qΓ=∑Γ∈G([2,k],[k+1,k+ℓ])qΓ(F(πΓ)+q1,kF(πΓ∪{(1,k)}))+∑j=2k∑Γ∈NCG(1,j)([1,j])×G([j,k],[k+1,k+ℓ])qΓ(F(πΓ)+q1,kF(πΓ∪{(1,k)}))+∑j=1k-1∑Γ∈G(1,j)([1,j],[k+1,k+ℓ])×NCG([j,k)]qΓ(F(πΓ)+q1,kF(πΓ∪{(1,k)}))+∑j=1ℓ∑Γ∈NCG(1,k+j)({1,k,k+j,…,k+ℓ,…,k+j})qΓ(F(πΓ)+q1,kF(πΓ∪{(1,k)})),

where the edges prescribed in the definitions of G2, G3, and G4 are added as a subscript to NCG and G, respectively. It remains to compare (3.20) with (2.4).

Computation of the first line of (3.20): Recall that ∑Γ∈G1qΓ=∑Γ∈G([2,k],[k+1,k+ℓ])qΓ. As any Γ∈G1 has an isolated vertex 1, the set {1} appears as a singleton block of the underlying partition πΓ. Let πΓ′ denote an annular non-crossing permutation with blocks given by πΓ. Then (…k1…)∈K(πΓ′), i.e., 1 and k appear in the same cycle of the Kreweras complement. This gives3.21 ∏B∈K(πΓ′)〈∏j∈BAj〉=∏B∈K(πΓ′)|(1,k]∪[k+1,k+ℓ]〈∏j∈BAj′〉

where Aj′=Aj for j=2,…k-1 and j∈[k+1,k+ℓ], but Ak′=AkA1. Considering F(πΓ∪(1,k)), note that adding an edge (1, k) to the graph Γ implies that (k)∈K(πΓ∪{(1,k)}), i.e., ⟨Ak⟩ always occurs as a separate factor in F. Hence, whenever πΓ has more than one connecting block we can use (3.16) to evaluate3.22 F(πΓ)+q1,kF(πΓ∪{(1,k)})=∏B∈K(πΓ′)|⟨1,k]∪[k+1,k+ℓ]〈∏j∈BAj′〉+q1,k⟨Ak⟩∏B∈K(πΓ′)|(1,k)∪[k+1,k+ℓ]〈∏j∈BAj〉

with Aj′ as in (3.21). In the remaining cases, πΓ has only one connecting block U, and F is a sum of two terms. As the first term of (3.17) can be evaluated similarly to (3.22), we only consider the second term. Let πΓ′′ denote the element of NCP(k)×NCP(ℓ) in which the blocks U1=U∩[k] and U2=U∩[k+1,k+ℓ] are marked. Recall that the marking does not influence the Kreweras complement, which is taken for both circles separately here. As 1 and k lie on the same circle, we can argue as in the permutation case. It follows that3.23 (∏s=1k+ℓms)∑Γ∈G([2,k],[k+1,k+ℓ])qΓ(F(πΓ)+q1,kF(πΓ∪{(1,k)}))=m1(F[T2,…,Tk-1,GkAkA1|Tk+1,…,Tk+ℓ]+q1,kF[T2,…,Tk-1GkA1|Tk+1,…,Tk+ℓ]⟨Ak⟩).

Computation of the second line of (3.20): For the contribution arising from G2, recall that Γ=Γ1∪Γ2 with Γ1∈NCG(1,j)([1,j]) and Γ2∈G([j,k],[k+1,k+ℓ]) implies qΓ=qΓ1qΓ2, i.e., the weights factorize. Further, the term involving Γ1 evaluates to3.24 ∑Γ1∈NCG(1,j)([1,j])qΓ1=q1,j1+q1,j∑Γ1∈NCG[1,j])qΓ1=m1mj∑Γ1∈NCG[1,j])qΓ1

as qΓ1 always includes a factor q1,j if Γ1∈NCG(1,j)([1,j]).

Let πΓ∈NCP(j)×NCP(k-j+1,ℓ) denote the partition associated with a graph Γ∈NCG(1,j)([1,j])×G([j,k],[k+1,k+ℓ]) in the second term of (3.20) and let πΓ′ is be an annular non-crossing permutation with blocks given by πΓ. Since the edge (1, j) must occur in Γ, the vertices 1 and j must be associated with same cycle of πΓ′. Hence, the elements in [1,j⟩ and [j, k] are in different cycles of K(πΓ′). Moreover, none of 1,…,j-1 can be part of a connecting cycle in K(πΓ′). This implies the decomposition∏B∈K(πΓ′)〈∏j∈BAj〉=(∏B∈K(πΓ′|[1,j])〈∏i∈B\{j}Ai〉)(∏B∈K(πΓ′|[j,k]∪[k+1,k+ℓ])〈∏i∈BAi〉).

Again, the only difference between F(πΓ) and F(πΓ∪{(1,k)}) is the fact that ⟨Ak⟩ must occur as a separate factor in the second case. Thus, we can argue as in (3.22) and evaluate3.25 F(πΓ)+q1,kF(πΓ∪{(1,k)})=(∏B∈K(πΓ|[1,j])〈∏i∈B\{j}Ai〉)(∏B∈K(πΓ′|[j,k]∪[k+1,k+ℓ])〈∏i∈BAi〉)+q1,k⟨Ak⟩(∏B∈K(πΓ|[1,j])〈∏i∈B\{j}Aj〉)×(∏B∈K(πΓ′|[j,k]∪[k+1,k+ℓ])〈∏i∈B\{k}Ai〉)

whenever πΓ has more than one connecting block and (3.16) applies.

In the remaining cases, F(πΓ) is evaluated using (3.17). Here, the first term can be treated similarly to (3.25), leaving only the contribution of the marked partition πΓ′′. Recalling that [1,j⟩ and [j, k] lie on the same circle and that K(πΓ′′) is evaluated circle-wise, we can argue as in the permutation case. In particular,F(πΓ)=∑πΓ′∈NCP→(k,ℓ)blocks(πΓ′)=πΓ(∏B∈K(πΓ′|[1,j])〈∏i∈B\{j}Ai〉)(∏B∈K(πΓ′|[j,k]∪[k+1,k+ℓ])〈∏i∈BAi〉)+(∏B∈K(πΓ′′|[1,j])〈∏i∈B\{j}Ai〉)(∏B∈K(πΓ′′|[j,k]∪[k+1,k+ℓ])〈∏i∈BAi〉)=(∏B∈K(πΓ|[1,j])〈∏i∈B\{j}Ai〉)F(πΓ2)

i.e., F factorizes similar to (3.25). Here, Γ2 denotes the subgraph of Γ∈G2 that lies in G([j,k],[k+1,k+ℓ]). Using (1.22) and (1.17), we evaluate3.26 (∏s=1k+ℓms)∑Γ∈NCG(1,j)([1,j])×G([j,k],[k+1,k+ℓ])qΓ(F(πΓ)+q1,kF(πΓ∪{(1,k)}))=m1m1[T1,…,Tj-1,Gj](F[Tj,…,Tk|Tk+1,…,Tk+ℓ]+q1,kF[Tj,…,Tk-1,Gk|Tk+1,…,Tk+ℓ]⟨Ak⟩).

Computation of the third line of (3.20): The contribution from G3 can be treated similarly to the second line of (3.20).

Computation of the fourth line of (3.20): For the term that arises from G4, 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(πΓ) and F(πΓ∪{(1,k)}) is again the fact that ⟨Ak⟩ must occur as a separate factor in the second case, we obtain3.27 (∏s=1k+ℓms)∑Γ∈NCG(1,k+j)({1,k,k+j,…,k+ℓ,…,k+j})qΓ(F(πΓ)+q1,kF(πΓ∪{(1,k)}))=m1(m1[T1,…,Tk,Tk+j,…,Tk+j-1,Gk+j]+q1,km1[T1,…,Tk-1,Gk,Tk+j,…,Tk+j-1,Gk+j]⟨Ak⟩).

Similar to (3.24), Γ containing an edge (1,k+j) ensures that the contribution has the prefactor m1.

Putting (3.23), (3.26), and (3.27) together, we see that (3.20) is equivalent to (2.4) with f[·|·] in place of m2[·|·]. We conclude that f[·|·] satisfies the same symmetry and initial condition as m2[·|·], as well as the same recursion. Recall that these three properties uniquely identify f[α|β] and m2[α|β] for any multi-indices α,β. It now readily follows by induction that f[α|β] and m2[α|β] indeed coincide for all α,β, i.e., that f[·|·]=m2[·|·], as claimed.

Proof of the Formulas for mκ[·|·], mσ[·|·], and mω[·|·]

Proof of Theorem 2.6

We use proof by induction over the length of the multi-indices and the recursion for mκ[·|·] from (2.4). First, recall that by definition of NCP(k), NCP(ℓ), and NCP→(k,ℓ) both sums on the right-hand side of (2.12) are empty whenever either α or β is empty. Observing that Definition 2.1(i) together with Theorem 2.4 implies that mκ[S1|∅]=mκ[∅|S2]=0, the base case is established. As the formula on the right-hand side of (2.12) is further symmetric under the interchanging α and β (cf. Remark Theorem 2.4), it is sufficient to only carry out the induction step for one of the arguments in mκ[·|·].

Fix k,ℓ∈N and assume that (2.12) holds for multi-indices α of length 1,…,k-1 and β of length ℓ. Recalling that κ4mκ[·|·] satisfies (2.4) with sκ in (2.6) as the only source term, we can use the recursion to express mκ[α|β] in terms of m1[·] and mκ[α′|β] with multi-indices α′ of length 1,…,k-1. The claim thus follows by applying the induction hypothesis and showing that the expression obtained from the recursion can be rewritten to match the structure on the right-hand side of (2.12). We start by considering the summands on the right-hand side of (2.4) separately and then check that their sum, i.e., mκ[·|·], is of the same form. To facilitate keeping track of the individual contributions, we start with the terms on the right-hand side of (2.4) that do not contain the prefactor q1,k and abbreviateK1:=m1mκ[T2,…,Tk-1,GkAkA1|Tk+1,…,Tk+ℓ]K2(j):=m1mκ[T1,…,Tj-1,Gj|Tk+1,…,Tk+ℓ]m1[Tj,…,Tk],j∈[k-1]K3(j):=m1m1[T1,…,Tj-1,Gj]mκ[Tj,…,Tk|Tk+1,…,Tk+ℓ],j∈[2,k]K4(r,s,t):=m1⟨M[r]⊙M[s,t]⟩⟨(M[r,k]Ak)⊙M(t,…,k+ℓ,k+1,…,s)⟩,r∈[k],k+1≤s≤t≤k+ℓ.

As the argument is similar, we fix j resp. r, s, t and omit the superscripts for the following discussion.

Structure of K1: By the induction hypothesis, mκ[T2,…,Tk-1,GkAkA1|Tk+1,…,Tk+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 z2,…,zk+ℓ and the matrices A2,…,Ak-1,AkA1,Ak+1,…,Ak+ℓ. As a consequence, the matrices Ak and A1 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∈NCP→(k,ℓ) with a cycle (…k1…) or an element of π∈NCP(k)×NCP(ℓ) with a block {…,k,1,…} depending on the rest of the underlying structure. The treatment 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 z1 only appears in the prefactor m1. Hence, setting the functions ψK-1(π),(1) resp. ψK-1(π),{1} equal to m(z1) yields the missing contribution. It follows that K1 matches the structure on the right-hand side of (2.12). Note that all ψi associated with permutations without a fixed point (1) and partitions without a singleton block {1}, respectively, are equal to zero for K1.

Structure of K2: We apply the induction hypothesis for mκ[T1,…,Tj-1,Gj|Tk+1,…,Tk+ℓ] as well as (1.22) for m1[Tj,…,Tk] to rewrite K2 as a sum of terms that naturally factorize into expressions involving only z1,…,zk+ℓ or A1,…,Ak+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∈NCP(k)×NCP(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∈NCP→(k,ℓ) and π∈NCP(k)×NCP(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 m1mκ[T1,…,Tj-1,Gj|Tk+1,…,Tk+ℓ] and m1[Tj,…,Tk], respectively. As we may visualize the elements of NCP(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 K2 onto the same annulus. The result is visualized on the left of Fig. 18. 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. 18 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 K2, we may remove the label corresponding to Id from the picture in Fig. 18. This leaves k+ℓ labels at the midpoints of arches along the annulus, one associated with each matrix A1,…,Ak+ℓ. On the level of the indices of the deterministic matrices, each term in in K2 can thus be identified with an element π∈NCP→(k,ℓ) resp. an element of π∈NCP(k)×NCP(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. 18. Note that the labels now match Definition 1.19 exactly.Fig. 18 Visualization of the indices in m1mκ[T1,…,Tj-1,Gj|Tk+1,…,Tk+ℓ] (dashed) and in m1[Tj,…,Tk] (dotted) before the rewriting (left) and the indices after merging the two labels j (right). (Color figure online)

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 NCP→(k,ℓ) resp. a block of an element of NCP(k)×NCP(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 K2(j) and noting that the right-hand side of (2.12) is linear in each ψi, the sum ∑jK2(j) also has the desired structure.

Structure of K3: The treatment of K3 is analogous to that of K2.

Structure of K4: From the formula for M(…) in (1.18), it follows that K4 can be written as a sum of terms that naturally factorize into expressions involving only z1,…,zk+ℓ or A1,…,Ak+ℓ, respectively. In particular, the part that involves deterministic matrices always consists of two factors of the form ⟨(∏j∈I1Aj)⊙(∏j∈I2Aj)⟩ with index sets I1⊆[k] and I2⊆[k+1,…,k+ℓ] due to the Hadamard product in K4. It remains to check for the structure on the right-hand side of (2.12).

Using the same trick as for K2 and K3, we can visualize the terms involved in K4 on the same annulus. The result is sketched in Fig. 19. 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.Fig. 19 Visualization of the indices in m1⟨M[r]⊙M[s,t]⟩ (dashed) and ⟨(M[r,k]Ak)⊙M(t,…,k+ℓ,k+1,…,s)⟩ (dotted) on the outer (left) and inner circle (right) of the (k,ℓ)-annulus (pictured separately). (Color figure online)

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 K4, we may argue as before and remove the corresponding labels from the picture in Fig. 19. Note that this leaves k+ℓ labels at the midpoints of arches along the annulus, one associated with each matrix A1,…,Ak+ℓ, such that we can reinterpret the structure on the level of the indices of the deterministic matrices as an element π∈NCP→(k,ℓ) resp. π∈NCP(k)×NCP(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 NCP→(k,ℓ) resp. a block of an element of NCP(k)×NCP(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 K4(r,s,t) and noting that the right-hand side of (2.12) is linear in each ψi, summing over r, s, t preserves the structure of the term.

The remaining terms with prefactor q1,k⟨Ak⟩ can be rewritten similarly. As the factor ⟨Ak⟩ always occurs separately, the structure underlying the indices of the deterministic matrices is an element π∈NCP→(k,ℓ) resp. π∈NCP(k)×NCP(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 q1,k=m1mk1-m1mk, it is ensured that the part of the term depending on z1,…,zk contains a factor ψi(…,zk,z1,…). With this modification, we can use the same argument as for K1,…,K4 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 ψi, it follows that the same holds for their sum, i.e., for mκ[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 Sects. 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~GUE[·|·], 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 G4 in Definition 3.2 byG4σ:=⋃j=1ℓτ({Γ∈NCG#({1,…,k,k+j,k+j-1,…,k+1,k+ℓ,k+j})|Γhas edge(1,k+j)}).

where #=(0,…,0,1,…,1) with k zeros and k+ℓ+1 ones, we obtain a family Gσ(k,ℓ) of graphs with the desired properties. Recall that map τ was introduced in Definition 3.1. An example in the current setting is given in Fig. 20. Note the different arrangement of the indices on the left compared to Fig. 9, 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.Fig. 20 The geometry of the transformation τ for k=4, ℓ=3, and j=2. (Color figure online)

Similar to the proof of Lemma 3.3, we obtainm~σ[1,…,k|k+1,…,k+ℓ]=(∏s=1k+ℓms)∑Γ∈Gσ(k,ℓ)∏(i,j)∈E(Γ)qi,j♯

where qi,j♯ is as in Lemma A.2, i.e., qi,j♯=qi,j=mimj1-mimj whenever the edge (i, j) connects two vertices of the same color and qi,j♯=σmimj1-σmimj otherwise. From here, the remaining steps are carried out as in the proof of Theorem 2.4.

Proof of Theorem 2.8

We use again proof by induction. As the recursions for mω[·|·] and mκ[·|·] are the same up to the source term, it only remains to show thatK5(j):=⟨(M[k]Ak)⊙M(k+j,…,k+ℓ,k+1,…k+j)⟩,j∈[ℓ]

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 K5(j) contains the matrices A1,…,Ak+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.

Proofs for the CLT in the Resolvent Case

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]). LetA.1 Wf(W)_:=Wf(W)-E~W~(∂W~f)(W)

with W~ an independent copy of W. This yields, e.g.,A.2 WG1_=WG1+⟨G1⟩G1+σNG1tG1+ω2~Ndiag(G1)G1,

A.3 WT1…Tk_=WG1_A1T[2,k]+∑j=2k(⟨T[1,j⟩Gj⟩T[j,k]+σN(T[1,j⟩Gj)tT[j,k]+ω2~Ndiag(T[1,j⟩Gj)T[j,k]).

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 ofWG1t_=WG1t+1NG1G1t+σ⟨G1⟩G1t+ω2~Ndiag(G1)G1t

where σ⟨G1⟩2∼1.

To obtain (1.23), we rewrite G1♯A1…Gk♯Ak in terms of WG1♯A1…Gk♯Ak_ and estimate the resulting terms. We start by considering the case A1=⋯=Ak=Id. For k=2, we obtain, e.g.,WG1G2t_=WG1_G2t+1N(G1G2t)tG2t+σ⟨G1G2t⟩G2t+ω2~Ndiag(G1G2t)G2t

which yields(1-σm1m2+O≺(1N))⟨G1G2t⟩=m1m2-m1WG1G2t_+m1N(σ⟨G1tG1G2t⟩+ω2~⟨diag(G1)G1G2t+diag(G1G2t)G2t⟩)+O≺(1N)

by (A.2), the resolvent identity WG-zG=Id, (1.11), and the local law for ⟨Gj⟩. Noting that |⟨WG1tG2_⟩|=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. [7, Eq. (69)–(70)]), we obtain (1.23) withm1[G1,G2t]=m(0,1),σ[1,2]=m1m21-σm1m2.

Note that the only effect of the inclusion of transposes lies in the emergence of the stability factor 1-σm1m2. However, as |1-σm1m2|>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 ⟨G1♯…Gk♯⟩ 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 m1[G1♯A1,…,Gk♯Ak] 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 A1=⋯=Ak=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∈NCG(k), color the vertex j∈{1,…,k} red if the jth 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 NCG#(k).

Lemma A.2

Let k∈N and fix a binary vector # of length k. ThenA.4 m#,σ[1,…,k]=(∏s=1kms)∑Γ∈NCG#(k)qΓ#

A.5 m∘#,σ[1,…,k]=(∏s=1kms)∑Γ∈NCGc#(k)qΓ#

where NCGc#(k) denotes the connected graphs in NCG#(k) and qΓ:=∏(i,j)∈E(Γ)qi,j♯ with E(Γ) denoting the edge set of Γ. The edge weights qi,j♯ are such that qi,j♯:=qi,j=mimj1-mimj whenever the edge (i, j) connects two vertices of the same color and qi,j♯:=σmimj1-σmimj 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 NCG(k). Note that the modified edge weights as well as the factors q1,k♯ and c1,j in the recursion account for the stability factor 1-σmimj arising whenever the product G1♯…Gk♯ involves both resolvents and their transposes. We illustrate Lemma A.2 with an example.

Example A.3

Let k=3 and pick spectral parameters z1,z2,z3∈C with |ℑzj|≳1 as well as A1=A2=A3=Id. By Theorem 1.10, the deterministic approximation of ⟨G1G2tG3t⟩ is given by m#,σ[1,2,3] with #=(0,1,1). We visualize the elements of the set NCG#(3) in Fig. 21. For a better overview, the edges are drawn as solid or dashed according to their contribution to qΓ#. We use this sketch to computem#,σ[1,2,3]=m1m2m3∑Γ∈NCG#=m1m2m3(1-σm1m2)(1-σm1m3)(1-m2m3).

Note that we thus reobtain [7, Eq. (49)] on macroscopic scales if z2=z3.

Fig. 21 The elements of NCG#(3) for #=(0,1,1). The solid edges contribute a factor of mimj1-mimj to qΓ# while dashed edges contribute a factor σmimj1-σmimj. (Color figure online)

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(zj)=(iS-zj)-1 satisfies R(zj)t=-R(-zj). This allows treating the proof of (1.23) for ⟨R(z1)♯…R(zk)♯⟩ with R(zj)♯∈{R(zj),R(zj)t} analogous to the case ⟨G1…Gk⟩. Recall that this requires in particular a local law for ⟨R(zj)⟩, which holds even if the Wigner matrix has zero diagonal. The general case follows from the boundA.6 ⟨G1♯…Gk♯⟩=⟨R(z1)♯…R(zk)♯⟩+O≺(1N)

which reduces the proof of (1.23) to the previously considered case D=0. The proof of (A.6) is given for ⟨G1G2t⟩ in [7, App. B]. By expanding G3♯,..., Gk♯ analogously, the argument readily extends to k≥3 resolvents. □

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. ⟨Ak⟩=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]⟩-m1[T1,…,Tk] and ⟨T[1,k]⟩-E⟨T[1,k]⟩, thus connecting the statistics Xα back to the local law (1.19).

Lemma A.4

Let k∈N and fix spectral parameters z1,…,zn with |ℑzj|≳1 and maxj|zj|≤N100 as well as bounded deterministic matrices A1,…,Ak such that ⟨Ak⟩=0. Then,A.7 E⟨T1…Tk⟩=m1[T1,…,Tk]+1NE[T1,…,Tk]+O(NεN3/2)

with m1[·] as in (1.21) and a set function E[·] that satisfies E[∅]=0 as well as the recursionE[T1,…,Tk]=m1(E[T2,…,Tj-1,GkAkA1]+∑j=1k-1E[T1,…,Tj-1,Gj]m1[Tj,…,Tk]+∑j=2km1[T1,…,Tj-1,Gj]E[Tj,…,Tk]+ω2~N∑j=1k⟨M[j]⊙(M[j,k]Ak)⟩+σN∑j=1km1[GjtAj-1t,…,G2tA1t,G1t,Tj,…,Tk]+κ4∑1≤r≤s≤t≤k⟨M[r]⊙M[s,t]⟩⟨M[r,s]⊙(M[t,k]Ak)⟩),

where ⊙ denotes the Hadamard product and m2[·|·] is interpreted as in (1.25).

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 relationA.8 ⟨T[1,k]⟩-m1[T1,…,Tk]=m1(-⟨WT[1,k]_⟩+(⟨T[2,k⟩GkAkA1⟩-m1[T2,…,Tk-1,GkAkA1])+∑j=1k-1(⟨T[1,j⟩Gj⟩-m1[T1,…,Tj-1,Gj])m1[Tj,…,Tk]+∑j=2km1[T1,…,Tj-1,Gj](⟨T[j,k]⟩-m1[Tj,…,Tk])+σN∑j=1k⟨(T[1,j⟩Gj)tT[j,k]⟩+ω2~N∑j=1k⟨diag(T[1,j⟩Gj)T[j,k]⟩)+O≺(1N2).

Recall that the deterministic approximation m1[·] 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. Applying Theorem 1.10 as well as the isotropic local law (1.20) thus yields⟨(T[1,j⟩Gj)tT[j,k]⟩=m1[GjtAj-1t,…,G2tA1t,G1t,Tj,…,Tk]+O≺(1N)⟨diag(T[1,j⟩Gj)T[j,k]⟩=1N∑r=1N(T[1,j⟩Gj)rr(T[j,k])rr=1N∑r=1N(M[j])rr(M[j,k]Ak)rr+O≺(1N)

and we can replace the respective terms in (A.8) by their deterministic approximation. Note that this also changes the error term to O≺(N-3/2). Computing the expectation of (A.8) is now analogous to [27, Lem. 2.3]. In particular, the cumulant expansionNE⟨WT[1,k]_⟩=∑n≥2∑x,y∈[N]∑ν∈{xy,yx}nκ(xy,ν)n!E∂ν(T[1,k])yx

yields the same result, since the term does not involve any second moments of the entries of W.

Next, we identify the limiting covariance of two modes Xα and Xβ. This yields an analog of [27, Lem. 2.5] for the model in Assumption 1.1 on macroscopic scales and thus constitutes the base case for the induction argument in the proof of Theorem 2.3.

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 z1,…,zk+ℓ with |ℑzj|≳1 and maxj|zj|≤N100 as well as deterministic matrices A1,…,Ak+ℓ with ‖Aj‖≲1. Then, for any ε>0,N2EXα(k,a)Xβ(ℓ,b)=m2[α|β]+O(NεN),

with m2[·|·] as introduced in Definition 2.1.

Proof of Lemma A.5

The proof is analogous to that of [27, Lem. 2.5] with one key difference. When evaluating the cumulant expansionN2E(⟨WT[1,k]_⟩Xβ(ℓ))=N∑n≥1∑x,y∈[N]∑ν∈{xy,yx}nκ(xy,ν)n!E∂ν((T[1,k])yxXβ(ℓ)),

the second moment structure of the model in Assumption 1.1 has to be taken into account for evaluating the n=1 term. We obtainN∑x,y∈[N]∑ν∈{xy,yx}κ(xy,ν)E∂ν((T[1,k])yxXβ(ℓ))=-∑j=1ℓE(⟨T[1,k]T[k+j,k+ℓ]T[1,k+j⟩Gk+j⟩+σ⟨T[1,k](T[k+j,k+ℓ]T[1,k+j⟩Gk+j)t⟩+ω2~⟨diag(T[1,k])diag(T[k+j,k+ℓ]T[1,k+j⟩Gk+j)⟩),

which, compared to the computation in the proof of [27, Lem. 2.5], additionally involves σ and ω2~. The deterministic approximation of the terms follow from (1.19), Theorem 1.10 and  (1.20), respectively, which yields the source terms sGUE, sσ, and sω in (2.4). The rest of the expansion evaluates exactly as its counterpart in [27] since the term does not involve any second moments of the entries of W.

The rest of the proof of Theorem 2.3 is analogous to that of [27, Thm. 3.6], i.e., we apply induction on the number of factors in (2.10) using EXα=0 and Lemma A.5 as base cases. □

Additional Proofs and Computations

Proof of Lemma 1.17

Using (1.16) for the ordered multi-set S={z1,…,zj,…,zk,zj}, rewrite the left-hand side of (1.32) asB.1 m[1,…,j,…,k,j]=m1…mj-1mj2mj+1…mk∑Γ∈NCG[{1,…,k,j}]∏(a,b)∈E(Γ)qa,b.

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. 22). Hence, for any non-crossing graph with edges {e1,…,en}∌e, the graph with edge set {e1,…,en,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 Fig. 22).

Consider the disjoint decompositionNCG({1,…,j,…,k,j})=S1∪(⋃l∈[k]\{j}S2(l))

where S2(l) contains all Γ∈NCG({1,…,j,…,k,j}) for which the vertex l∈[k]\{j} is connected to both copies of j and S1 contains any remaining Γ. This impliesB.2 ∑Γ∈NCG[{1,…,k,j}]∏(a,b)∈E(Γ)qa,b=∑Γ∈S1∏(a,b)∈E(Γ)qa,b+∑l∈[k]\{j}∑Γ∈S2(l)∏(a,b)∈E(Γ)qa,b.

Next, merge both vertices with the label j. The resulting graphs are either an element of NCG(k) or arise from an element of NCG(k) by adding a loop (j, j). Recalling thatB.3 qj,j=mj21-mj2=mj′

where the second equality follows from (1.12), we can factor out (1+mj′) on the right-hand side of (B.2) and reduce to summation over NCG(k) without further restriction. Note that this also results in the edge (l, j) doubling whenever Γ∈S2(l), i.e., we obtain an extra factor of qj,l in this case. Hence,Fig. 22 An edge between two consecutive vertices never results in a crossing (left), but connecting two distinct vertices to both copies of j always does (right). (Color figure online)

∑Γ∈NCG[{1,…,k,j}]∏(a,b)∈E(Γ)qa,b=(1+mj′)(∑Γ∈NCG[{1,…,k}]∏(a,b)∈E(Γ)qa,b)(1+∑l∈[k]\{j}qj,l).

Noting that (B.3) is equivalent to mj(1+mj′)=mj′mj and applying (1.16) to recover m[1,…,k] yields (1.32). □

Proof of Corollary 2.11

We start by evaluating sc[i1,…,in] and sc[i1,⋯,in|in+1,…,in+m] before considering the associated free cumulant functions. First, note thatsc[i1,…,in]=∫-22xnρsc(x)dx=0,ifn odd,Cn/2,ifneven,

where C0,C1,C2,… denote the Catalan numbers. In particular, sc[i1,…,in] coincides with the number of non-crossing pairings of the set [n]. It readily follows that sc∘[i1,i2]=1 and that sc∘[i1,…,in]=0 whenever n is odd. Moreover, sc∘[i1,…,in]=0 for any even n≥4. The latter follows inductively from (1.14) by writingB.4 sc∘[i1,…,in]=sc[i1,…,in]-∑π∈NCP(n)\{[n]}∏B∈πsc∘[B],

and observing that only pairings contribute to the sum on the right-hand side of (B.4), i.e., the two terms cancel.

Next, we compute sc[i1,…,in|in+1,…,in+m]. Observe that by (1.26) and Theorem 2.9,sc[i1,…,in|in+1,…,in+m]=limN→∞E[(TrWn-EWn)(TrWm-EWm)].

The limit on the right-hand side is well-known in the free probability literature (see, e.g., [21]) and hence readily identified as the number of non-crossing pairings of the (n, m)-annulus. Solving (1.26) for sc∘∘[i1,…,in|in+1,…,in+m], it follows inductively that sc∘∘[i1,…,in|in+1,…,in+m]=0 for any n, m. Hence, Φπ1×π2,U1×U2(f1,…,fk+ℓ)=0 andΦπ(f1,…,fk+ℓ)=1,ifπ∈NCP→2(k,ℓ),0,otherwise,

which is the claim. Note that the error in (2.16) evaluates to O(Nε-1/2) as ‖fi‖Hp for i=1,…,k resp. ‖fj‖Hq for j=k+1,…,k+ℓ are N-independent constants in the macroscopic regime. □

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.

Funding

Open access funding provided by Institute of Science and Technology (IST Austria).

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The author has no conflict of interest to declare that are relevant to the content of this article.

1 Recall that [k] is treated as an ordered set. Hence, the subsets in P([k]) carry a natural ordering inherited from [k].

2 The edges of a disk non-crossing graph Γ can be drawn in the interior of the disk without intersecting, i.e., Γ is a planar graph drawn inside a labeled disk.

3 The noncommutative product ∏j∈BAj for B=(j1,…,jr) is defined as Aj1…Ajr in the order inherited from B⊆S.

4 Recall that ⟨·⟩ carries a N-1 normalization factor, but ⟨·,·⟩ does not. For A:=|y⟩⟨x| with two unit vectors x,y∈CN, we hence have ⟨xG(z)y⟩=Tr[G(z)A] and ⟨G(z)A⟩=N-1⟨x,G(z)y⟩.

5 Recall that every permutation has a unique cycle decomposition. We represent a cycle (abc…x) by an oriented graph with edge set {(a,b),(b,c),…,(x,a)}.

6 The Cauchy transform of a probability measure μ is given by c(z)=∫Rμ(dx)z-x and hence only differs from the Stieltjes transform by a sign.

7 Note the similarity between the notations m1[·] and m2[·|·], which take one and two resolvent chains as arguments, respectively.

8 Note that we are only using that W satisfies Assumption 1.1 and κ4=σ=ω~2=0.

9 We only use that W satisfies Assumption 1.1 and κ4=σ=ω~2=0.

10 We implicitly assume fj to be compactly supported by setting f~j(x)=fj(x)χ(x), where fj is a polynomial supported on all of R and χ is a smooth cutoff function that is equal to one on [-5/2,5/2] and equal to zero on [-3,3]c. Since fj(W)=f~j(W) with high probability by eigenvalue rigidity, we may use fj and f~j interchangeably here.

11 The graph Γ may be obtained in several different ways from the recursive construction. This is reflected by the multiplicity of Γ in the multi-set G([k],[k+1,k+ℓ]).

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
==== Refs
References

1. Bai ZD Yao J On the convergence of the spectral empirical process of Wigner matrices Bernoulli 2005 11 1059 1092 10.3150/bj/1137421640
Bai, Z.D., Yao, J.: On the convergence of the spectral empirical process of Wigner matrices. Bernoulli 11, 1059–1092 (2005)
2. Bao Z He Y Quantitative CLT for linear eigenvalue statistics of Wigner matrices Ann. Appl. Probab. 2023 33 6 5171 5207
Bao, Z., He, Y.: Quantitative CLT for linear eigenvalue statistics of Wigner matrices. Ann. Appl. Probab. 33(6), 5171–5207 (2023)
3. Borot G Guionnet A Asymptotic expansion of β matrix models in the one-cut regime Commun. Math. Phys. 2013 317 447 483 10.1007/s00220-012-1619-4
Borot, G., Guionnet, A.: Asymptotic expansion of matrix models in the one-cut regime. Commun. Math. Phys. 317, 447–483 (2013)
4. Cipolloni G Erdős L Schröder D Eigenstate thermalization hypothesis for Wigner matrices Commun. Math. Phys. 2021 388 2 1005 1048 10.1007/s00220-021-04239-z
Cipolloni, G., Erdős, L., Schröder, D.: Eigenstate thermalization hypothesis for Wigner matrices. Commun. Math. Phys. 388(2), 1005–1048 (2021)
5. Cipolloni G Erdős L Schröder D Optimal multi-resolvent local laws for Wigner matrices Electron. J. Probab. 2022 27 1 38 10.1214/22-EJP838
Cipolloni, G., Erdős, L., Schröder, D.: Optimal multi-resolvent local laws for Wigner matrices. Electron. J. Probab. 27, 1–38 (2022)
6. Cipolloni G Erdős L Schröder D Thermalization for Wigner matrices J. Funct. Anal. 2022 282 8 109394 10.1016/j.jfa.2022.109394
Cipolloni, G., Erdős, L., Schröder, D.: Thermalization for Wigner matrices. J. Funct. Anal. 282(8), 109394 (2022)
7. Cipolloni G Erdős L Schröder D Functional central limit theorems for Wigner matrices Ann. Appl. Probab. 2023 33 1 447 489 10.1214/22-AAP1820
Cipolloni, G., Erdős, L., Schröder, D.: Functional central limit theorems for Wigner matrices. Ann. Appl. Probab. 33(1), 447–489 (2023)
8. Collins B Mingo J Śniady P Speicher R Second order freeness and fluctuations of random matrices: III. Higher order freeness and free cumulants Doc. Math. 2007 12 1 70 10.4171/dm/220
Collins, B., Mingo, J., Śniady, P., Speicher, R.: Second order freeness and fluctuations of random matrices: III. Higher order freeness and free cumulants. Doc. Math. 12, 1–70 (2007)
9. Diaz M Jaramillo A Pardo JC Fluctuations for matrix-valued Gaussian processes Ann. Henri Poincaré 2022 58 4 2216 2249
Diaz, M., Jaramillo, A., Pardo, J.C.: Fluctuations for matrix-valued Gaussian processes. Ann. Henri Poincaré 58(4), 2216–2249 (2022)
10. Diaz M Mingo JA On the analytic structure of second-order non-commutative probability spaces and functions of bounded Fréchet variation Random Matrices Theory Appl. 2022 12 1 2250044 10.1142/S2010326322500447
Diaz, M., Mingo, J.A.: On the analytic structure of second-order non-commutative probability spaces and functions of bounded Fréchet variation. Random Matrices Theory Appl. 12(1), 2250044 (2022)
11. Guionnet A Large deviation upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices Ann. Inst. H. Poincaré Probab. Stat. 2002 38 341 384 10.1016/S0246-0203(01)01093-7
Guionnet, A.: Large deviation upper bounds and central limit theorems for non-commutative functionals of Gaussian large random matrices. Ann. Inst. H. Poincaré Probab. Stat. 38, 341–384 (2002)
12. He Y Knowles A Mesoscopic eigenvalue density correlations of Wigner matrices Probab. Theory Relat. Fields 2020 177 147 216 10.1007/s00440-019-00946-w
He, Y., Knowles, A.: Mesoscopic eigenvalue density correlations of Wigner matrices. Probab. Theory Relat. Fields 177, 147–216 (2020)
13. Johansson K On fluctuations of eigenvalues of random Hermitian matrices Duke Math. J. 1998 91 1 151 204 10.1215/S0012-7094-98-09108-6
Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)
14. Khorunzhi AM Khoruzhenko BA Pastur LA On the 1/N corrections to the Green functions of random matrices with independent entries J. Phys. A Math. Gen. 1995 28 L31 10.1088/0305-4470/28/1/006
Khorunzhi, A.M., Khoruzhenko, B.A., Pastur, L.A.: On the 1/N corrections to the Green functions of random matrices with independent entries. J. Phys. A Math. Gen. 28, L31 (1995)
15. Khorunzhi AM Khoruzhenko BA Pastur LA Asymptotic properties of large random matrices with independent entries J. Math. Phys. 1996 37 5033 5060 10.1063/1.531589
Khorunzhi, A.M., Khoruzhenko, B.A., Pastur, L.A.: Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37, 5033–5060 (1996)
16. Landon, B., Sosoe, P.: Almost-optimal bulk regularity conditions in the CLT for Wigner matrices. Preprint (2022). arXiv:2204.03419
17. Lytova A On non-Gaussian limiting laws for certain statistics of Wigner matrices Zh. Mat. Fiz. Anal. Geom. 2013 9 536 581
Lytova, A.: On non-Gaussian limiting laws for certain statistics of Wigner matrices. Zh. Mat. Fiz. Anal. Geom. 9, 536–581 (2013)
18. Lytova A Pastur L Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices Metrika 2009 69 153 172 10.1007/s00184-008-0212-5
Lytova, A., Pastur, L.: Central limit theorem for linear eigenvalue statistics of the Wigner and the sample covariance random matrices. Metrika 69, 153–172 (2009)
19. Lytova A Pastur L Fluctuations of matrix elements of regular functions of Gaussian random matrices J. Stat. Phys. 2009 134 147 159 10.1007/s10955-008-9665-1
Lytova, A., Pastur, L.: Fluctuations of matrix elements of regular functions of Gaussian random matrices. J. Stat. Phys. 134, 147–159 (2009)
20. Male C Freeness over the diagonal and global fluctuations of complex Wigner matrices J. Oper. Theory 2021 85 1 153 181 10.7900/jot.2019oct09.2287
Male, C.: Freeness over the diagonal and global fluctuations of complex Wigner matrices. J. Oper. Theory 85(1), 153–181 (2021)
21. Male C Mingo JA Peché S Speicher R Joint global fluctuations of complex Wigner and deterministic matrices Random Matrices Theory Appl. 2022 11 2 2250015 10.1142/S2010326322500150
Male, C., Mingo, J.A., Peché, S., Speicher, R.: Joint global fluctuations of complex Wigner and deterministic matrices. Random Matrices Theory Appl. 11(2), 2250015 (2022)
22. Mingo, J.A., Speicher, R.: Free Probability and Random Matrices, vol. 35. Fields Institute Research Monographs. Springer, New York (2017)
23. Mingo JA Nica A Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices Int. Math. Res. Not. 2004 2004 28 1413 1460 10.1155/S1073792804133023
Mingo, J.A., Nica, A.: Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices. Int. Math. Res. Not. 2004(28), 1413–1460 (2004)
24. Mingo JA Speicher R Second order freeness and fluctuations of random matrices I: Gaussian and Wishart random matrices and cyclic Fock spaces J. Funct. Anal. 2006 235 1 226 270 10.1016/j.jfa.2005.10.007
Mingo, J.A., Speicher, R.: Second order freeness and fluctuations of random matrices I: Gaussian and Wishart random matrices and cyclic Fock spaces. J. Funct. Anal. 235(1), 226–270 (2006)
25. Redelmeier, C.E.I.: Real and quaternionic second-order free cumulants and connections to matrix cumulants. Preprint (2018). arXiv:1808.10589v2
26. Redelmeier, C.E.I.: Real second-order freeness and the asymptotic real second-order freeness of several real matrix models. Int. Math. Res. Not. 2014(12), 3353–3395 (2014)
27. Reker, J.: Multi-point functional central limit theorem for Wigner matrices. Preprint (2023). arXiv:2307.11028
28. Riabov, V.: Mesoscopic eigenvalue statistics for Wigner-type matrices. Preprint (2023). arXiv:2301.01712
29. Shcherbina M Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices Zh. Mat. Fiz. Anal. Geom. 2011 7 176 192
Shcherbina, M.: Central limit theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices. Zh. Mat. Fiz. Anal. Geom. 7, 176–192 (2011)
30. Shcherbina M Fluctuations of linear eigenvalue statistics of β matrix models in the multi-cut regime J. Stat. Phys. 2013 151 1004 1034 10.1007/s10955-013-0740-x
Shcherbina, M.: Fluctuations of linear eigenvalue statistics of matrix models in the multi-cut regime. J. Stat. Phys. 151, 1004–1034 (2013)
31. Sosoe P Wong P Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices Adv. Math. 2013 249 37 87 10.1016/j.aim.2013.09.004
Sosoe, P., Wong, P.: Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices. Adv. Math. 249, 37–87 (2013)
32. Wigner E Characteristic vectors of bordered matrices with infinite dimensions Ann. Math. 1955 62 548 564 10.2307/1970079
Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955)
