A Free Algebraic Model for Averaged Z-Weighted Wick Functionals

Abstract

This paper constructs a universal algebraic realization model for Hermitian parameter matrices whose entries have modulus at most one and whose diagonal entries are normalized. The entries of the parameter matrix are used as weights of oriented crossings in averaged Wick-type moment formulas. The construction separates each crossing parameter into its unit complex factor and its modulus. We first construct a universal free algebraic unimodular factor model whose moments are defined by a balanced pair-oriented crossing formula. The modulus factors are then encoded by an auxiliary commutative algebra and recombined with the unimodular factors by tensorization. The resulting normalized sums converge in joint algebraic moments to the averaged weighted Wick moment functional, whose moments are given by a fully averaged pair-partition formula. In the general complex Hermitian case, the construction is purely algebraic, and no positivity, traciality, operator boundedness, or operator algebraic realization is claimed. In the real symmetric specialization, the averaged oriented formula reduces to the standard mixed Gaussian pair-partition formula with color-dependent crossing parameters, so the construction contains the known Fock-representable mixed Gaussian cases as positive Fock-space examples. Moreover, the averaged functional satisfies a uniform Gaussian-type moment growth bound.

1. Introduction

The connection between operator algebras and random matrix theory has been shaped by noncommutative analogues of classical probabilistic limit theorems, beginning with Voiculescu’s introduction of free probability [1,2]. In this framework, the semicircular law plays the role of the Gaussian distribution, and its moments admit Wick-type expansions indexed by noncrossing pair partitions [3,4,5]. More generally, pair partition formulas provide a flexible combinatorial framework for many Gaussian-type limits in noncommutative probability, including deformations of Gaussian and semicircular systems. The use of crossing weights in such formulas is closely related to Speicher’s noncommutative central limit theorem [6].

Beyond the free case, a wide range of deformed and related models has been studied. Examples include q-Gaussian systems [7], random unitary models and z-circular systems [8], second-order freeness and fluctuations of random matrices [9], and broader random matrix/free-probability frameworks [10,11]. Furthermore, research has also explored different types of algebraic and deformed central limit theorems, for example, [12,13,14].

This paper discusses multivariate algebraic moment functions defined by a Hermitian parameter matrix.

$$Z={\left(z_{r,t}\right)}_{1\le r,t\le s},z_{r,r}=1,z_{t,r}=\overline{z_{r,t}},\left|z_{r,t}\right|\le 1.$$

The entries of Z are not interpreted as covariance coefficients. Rather, they appear as weights attached to oriented crossings of pair partitions in Wick-type expansions. The purpose of this paper is to realize this prescribed crossing-weighted formula as the large N moment limit of explicit normalized sums in a relation-free algebraic model.

The core strategy is to separate the unimodular component from the modulus by writing

$$z_{r,t}={\omega }_{r,t}{\rho }_{r,t},{\rho }_{r,t}:=\left|z_{r,t}\right|,$$

where ${\omega }_{r,t}={\displaystyle \frac{z_{r,t}}{{\rho }_{r,t}}}$ if ${\rho }_{r,t}>0$, and ${\omega }_{r,t}=1$ if ${\rho }_{r,t}=0$. In earlier scalar unimodular models, unimodular factors arise from concrete $\omega$-sequences of partial isometries and oriented crossings of directed ordered pair partitions; see [15] and compare [8]. For a general matrix of unimodular parameters $\left({\omega }_{r,t}\right)$, however, we do not assume an operator algebraic realization by partial isometries. Instead, we construct a universal algebraic unimodular factor model: a free ∗-algebra equipped with a unital linear functional whose moments are defined directly by the balanced pair-oriented crossing formula.

The present construction is naturally related to the z-semicircular systems introduced in Chapter 5 of [15]. In the scalar case, the z-semicircular moment formula is obtained by averaging over ordered pair partitions and balanced direction maps, with weights coming from positively and negatively oriented crossings. The averaged Z-weighted functional studied here extends this idea by replacing the single scalar parameter z with a Hermitian parameter matrix $Z=\left(z_{r,t}\right)$. In this manner, the two colors involved in the crossing determine the color-dependent weight assigned to each oriented crossing. Thus the present paper can be viewed as a formal algebraic matrix parameter extension of the z-semicircular moment formula. The main difference is that, in the present generality, we do not assume a concrete $\omega$-sequence model or a positive operator algebraic model. The construction is carried out at the level of algebraic moment functionals.

The modulus factors ${\rho }_{r,t}$ are then inserted by an auxiliary commutative modulus algebra. Tensoring the universal unimodular factor model with this auxiliary modulus algebra recombines each unimodular factor ${\omega }_{r,t}$ with the corresponding modulus factor ${\rho }_{r,t}$ to produce $z_{r,t}$. This yields a limit theorem for normalized sums in a purely algebraic tensor product model.

The main contribution is therefore a universal algebraic moment realization of the averaged Z-weighted Wick functional. Consequently, the limiting object is a moment functional on a free ∗-algebra, not necessarily a noncommutative distribution in a $C^{\ast }$-probability space.

The present formula involves all pair partitions together with an averaging over block orderings. Therefore, in the one variable case, the fourth moment is 3, whereas a standard semicircular variable of variance 1 has a fourth moment 2. Thus, the terminology used here is descriptive: “Wick” refers to the pair-partition expansion, “Z-weighted” refers to the crossing weights determined by the Hermitian matrix Z, and “averaged” refers to the full averaging over block orderings.

The paper is organized as follows: Section 2 introduces the combinatorial notation and conventions used throughout the paper, including pair partitions, block orderings, balanced maps, and oriented crossings. Section 3 constructs the universal algebraic unimodular model and proves its unimodular limiting theorem. Section 4 defines the averaged Z-weighted Wick moment functional, proves a uniform Gaussian-type moment-growth bound, and records the real symmetric specialization, where the formula reduces to the mixed $q_{ij}$-Gaussian pair-partition formula in the known Fock-representable cases. Section 5 constructs the auxiliary modulus algebra and the tensorized variables. Section 6 proves the universal algebraic realization theorem. Finally, Section 7 summarizes the construction and discusses positivity and realization questions left for future work.

• Notation.

We write $\left[m\right]=\{1,\dots ,m\}$ and $P_2\left(2k\right)$ for the set of pair partitions of $\left[2k\right]$. $Ker\left(i\right)$ denotes the partition of $\left[m\right]$ into the level sets of i, where $i\left[m\right]\to \left[s\right]$ is a color map. For a pair partition $\pi$, the notation $\pi \le Ker\left(i\right)$ means that each block of $\pi$ has a single color.

2. Preliminaries

A pair partition of $\left[2k\right]$ for some $k\ge 1$ is a partition $\pi =\{V_1,\dots ,V_k\}$ where $|V_j| =2$ for every j. $P_2\left(2k\right)$ denotes the set of pair partitions of $\left[2k\right]$.

A color map is a function $i:\left[2k\right]\to \left[s\right]$. If for a block $V=\{a,b\}\in \pi$, $i\left(a\right)=i\left(b\right)$, then we denote this color by $i\left(V\right)$.

Definition 1.

Let $\pi \in P_2\left(2k\right)$. A map

$$\epsilon :\left[2k\right]\to \{1,\ast \}$$

is called $\pi$-balanced, denoted by $B\left(\pi \right)$, if for every block $\{a,b\}\in \pi$ one has

$$\epsilon \left(a\right)\ne \epsilon \left(b\right).$$

The moment formulas use pair partitions together with an ordering of their blocks. If $\pi =\{V_1,\dots ,V_k\}$ is listed in standard order, namely,

$$min{V}_1<\cdots <min{V}_k,$$

then a permutation $\sigma \in S_k$ determines the ordered list

$$(V_{\sigma \left(1\right)},\dots ,V_{\sigma \left(k\right)}).$$

V precedes W in the $\sigma$-ordering if $V=V_{\sigma \left(i\right)}$ and $W=V_{\sigma \left(j\right)}$ for some $i<j$. Given two blocks $V=a<b$ and $W=c<d$ of a pair partition, these two blocks form a crossing if the endpoints of V and W alternate, meaning that either $a<c<b<d$ or $c<a<d<b$.

Definition 2.

Let $\pi \in P_2\left(2k\right)$. We write $I_N^{distinct}\left(\pi \right)$ for the set of maps $n:\left[2k\right]\to \left[N\right]$ that are constant on each block of π and assign distinct values to distinct blocks.

The following elementary counting lemma explains why the averaging over block orderings appears in the limiting moment formula.

Lemma 1.

Fix $\pi \in P_2\left(2k\right)$ and list its blocks in standard order as $\pi =\{V_1,\dots ,V_k\}$. For $n\in I_N^{distinct}\left(\pi \right)$, define $\sigma \left(n\right)\in S_k$ by

$$n\left(V_{\sigma \left(n\right)\left(1\right)}\right)<n\left(V_{\sigma \left(n\right)\left(2\right)}\right)<\cdots <n\left(V_{\sigma \left(n\right)\left(k\right)}\right).$$

Then, for every $\sigma \in S_k$,

$$\#\{n\in I_N^{distinct}\left(\pi \right):\sigma \left(n\right)=\sigma \}=\left({\displaystyle \genfrac{}{}{0pt}{}{N}{k}}\right).$$

Proof. 

Choose an increasing k-tuple $c_1<\cdots <c_k$ from $\left[N\right]$, and then set $n\left(V_{\sigma \left(j\right)}\right)=c_j$. This gives all assignments with ordering $\sigma$, and there are exactly $\left({\displaystyle \genfrac{}{}{0pt}{}{N}{k}}\right)$ choices. □

If $\epsilon \in B\left(\pi \right)$ and $V=\{a<b\}\in \pi$, define

$${ori}_{\epsilon }\left(V\right):=\left\{\begin{array}{cc}+1,\hfill & \left(\epsilon \right(a),\epsilon (b\left)\right)=(1,\ast ),\hfill \\ -1,\hfill & \left(\epsilon \right(a),\epsilon (b\left)\right)=(\ast ,1).\hfill \end{array}\right.$$

For two crossing blocks $V=\{a<b\}$ and $W=\{c<d\}$, define the positional sign

$$pos(V,W):=\left\{\begin{array}{cc}+1,\hfill & a<c<b<d,\hfill \\ -1,\hfill & c<a<d<b.\hfill \end{array}\right.$$

Finally, if V precedes W in the chosen block ordering, define the oriented crossing sign by

$${sgn}_{\pi ,\epsilon ,\sigma }(V,W):={ori}_{\epsilon }\left(V\right){ori}_{\epsilon }\left(W\right)\mathrm{pos}(V,W)\in \{+1,-1\}.$$

This convention is a combinatorial encoding of the usual right-hand-rule sign for directed ordered crossings.

Definition 3.

Let $\pi \in P_2\left(2k\right)$, let $i:\left[2k\right]\to \left[s\right]$ satisfy $\pi \le Ker\left(i\right)$, let $\epsilon \in B\left(\pi \right)$, and let $\sigma \in S_k$. Define ${\mathrm{Cr}}^{+}(\pi ,i,\epsilon ,\sigma )$ to be the multiset of ordered color pairs $\left(i\right(V),i(W\left)\right)$ over all crossing pairs of blocks $(V,W)$ such that V precedes W in the σ-ordering and

$${sgn}_{\pi ,\epsilon ,\sigma }(V,W)=+1.$$

Similarly, define ${\mathrm{Cr}}^{-}(\pi ,i,\epsilon ,\sigma )$ to be the corresponding multiset over those ordered crossing pairs for which

$${sgn}_{\pi ,\epsilon ,\sigma }(V,W)=-1.$$

The unordered set of crossing pairs of blocks of π is denoted by ${\mathrm{Cr}}_{un}\left(\pi \right)$.

Definition 4.

Let $\omega =\left({\omega }_{r,t}\right)$ be Hermitian unimodular. We define

$${\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma ):=\prod _{(r,t)\in {\mathrm{Cr}}^{+}(\pi ,i,\epsilon ,\sigma )}{\omega }_{r,t}\prod _{(r,t)\in {\mathrm{Cr}}^{-}(\pi ,i,\epsilon ,\sigma )}\overline{{\omega }_{r,t}}.$$

Likewise, for a Hermitian matrix $Z=\left(z_{r,t}\right)$, we define

$${\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma ):=\prod _{(r,t)\in {\mathrm{Cr}}^{+}(\pi ,i,\epsilon ,\sigma )}{z}_{r,t}\prod _{(r,t)\in {\mathrm{Cr}}^{-}(\pi ,i,\epsilon ,\sigma )}\overline{z_{r,t}}.$$

The empty products are interpreted as 1.

Remark 1.

The parameter $\overline{z_{r,t}}$ can be written as $z_{t,r}$ since Z is Hermitian. In the scalar case $z_{r,t}\equiv z$, the weighting factor reads $z^{{cr}^{+}}{\overline{z}}^{{cr}^{-}}.$

3. The Universal Algebraic Unimodular Model

Let $\omega ={\left({\omega }_{r,t}\right)}_{1\le r,t\le s}$ be a Hermitian unimodular matrix, i.e.,

$${\omega }_{r,r}=1,{\omega }_{t,r}=\overline{{\omega }_{r,t}},\mathrm{and}\left|{\omega }_{r,t}\right|=1.$$

Definition 5.

Let ${\mathcal{A}}_{\omega }$ be the free unital ∗-algebra generated by the symbols

$$\{a_{r,n}:r\in \left[s\right],n\in \mathbb{N}\}.$$

The involution is the unique conjugate linear anti-automorphism determined by ${\left(a_{r,n}\right)}^{\ast }=a_{r,n}^{\ast }$ and ${\left(a_{r,n}^{\ast }\right)}^{\ast }=a_{r,n}$.

Remark 2.

There are no commutation or local partial isometry relations involved, the absence of which is deliberate. ${\mathcal{A}}_{\omega }$ serves as a relation-free algebraic carrier for a given moment functional, thereby bypassing the separate, usually non-trivial question of whether a unimodular matrix can be realized by concrete partial isometries.

Definition 6.

A monomial $W=a_{i\left(1\right),n\left(1\right)}^{\epsilon \left(1\right)}{a}_{i\left(2\right),n\left(2\right)}^{\epsilon \left(2\right)}\cdots a_{i\left(m\right),n\left(m\right)}^{\epsilon \left(m\right)}$ in ${\mathcal{A}}_{\omega }$, where $i:\left[m\right]\to \left[s\right]$, $n:\left[m\right]\to \mathbb{N}$ and $\epsilon :\left[m\right]\to \{1,\ast \}$, is called balanced by sites if $m=2k$ is even and $Ker\left(n\right)$ is a pair partition $\pi \in P_2\left(2k\right)$ such that $\pi \le Ker\left(i\right)$ and $\epsilon \in B\left(\pi \right)$. We then write $\pi \left(W\right):=Ker\left(n\right)$.

If $\pi =\{V_1,\dots ,V_k\}$ is listed in standard order and n is constant on the blocks of $\pi$ and injective on the set of blocks, define $\sigma \left(n\right)\in S_k$ by

$$n\left(V_{\sigma \left(n\right)\left(1\right)}\right)<n\left(V_{\sigma \left(n\right)\left(2\right)}\right)<\cdots <n\left(V_{\sigma \left(n\right)\left(k\right)}\right),$$

where $n\left(V\right)$ denotes the common value of n on V.

Definition 7.

A linear functional ${\Phi }_{\omega }:{\mathcal{A}}_{\omega }\to \mathbb{C}$ is defined by ${\Phi }_{\omega }\left(\mathbb{1}\right)=1$ and if $W=a_{i\left(1\right),n\left(1\right)}^{\epsilon \left(1\right)}\cdots a_{i\left(m\right),n\left(m\right)}^{\epsilon \left(m\right)}$ is a non-empty monomial, we define ${\Phi }_{\omega }\left(W\right)=0$ unless W is balanced by sites, in which case we let $\pi =Ker\left(n\right)\in P_2\left(2k\right)$ and we set ${\Phi }_{\omega }\left(W\right)=2^{-k}{\mathrm{wt}}_{\omega }(\pi ,i,\epsilon ,\sigma \left(n\right))$. The definition is extended linearly to ${\mathcal{A}}_{\omega }$.

Proposition 1.

Let ${\mathcal{A}}_{\omega }$ be the free unital ∗-algebra generated by $\{a_{r,n}:r\in \left[s\right],n\in \mathbb{N}\}$. There exists a unique unital linear functional ${\Phi }_{\omega }:{\mathcal{A}}_{\omega }\to \mathbb{C}$ satisfying the balanced site moment rule of Definition 7. Consequently, $({\mathcal{A}}_{\omega },{\Phi }_{\omega })$ is universal in the relation-free algebraic sense: no relations among the generators are imposed, and the prescribed moment values determine a unique unital linear functional on the free ∗-algebra.

Proof. 

The set of words in $a_{r,n}$ and $a_{r,n}^{\ast }$ is a vector space basis for the free unital ∗-algebra ${\mathcal{A}}_{\omega }$. Definition 7 provides a complex number for each basis term and 1 is assigned to the empty word. We obtain a unital linear functional by extending this assignment by linearity. A linear functional on a vector space is uniquely determined by its values on a basis. □

Proposition 2.

The functional ${\Phi }_{\omega }$ is unital and satisfies:

(i) 

every odd monomial has zero ${\Phi }_{\omega }$-moment;

(ii) 

every monomial which is not balanced by sites has zero ${\Phi }_{\omega }$-moment;

(iii) 

if W is balanced by sites and $\pi =Ker\left(n\right)\in P_2\left(2k\right)$, then

$${\Phi }_{\omega }\left(W\right)=2^{-k}{\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma \left(n\right));$$

(iv) 

${\Phi }_{\omega }\left(P^{\ast }\right)=\overline{{\Phi }_{\omega }\left(P\right)}$ for every $P\in {\mathcal{A}}_{\omega }$.

Proof. 

Definition 7 directly leads to the first three claims. It remains to prove (iv). The claim about monomials can be verified by linearity. Let

$$W=a_{i\left(1\right),n\left(1\right)}^{\epsilon \left(1\right)}\cdots a_{i\left(2k\right),n\left(2k\right)}^{\epsilon \left(2k\right)}.$$

If W is not balanced by sites, then $W^{\ast }$ is also not balanced by sites, and hence both moments are zero. Thus assume that W is balanced by sites. Let

$$\pi =Ker\left(n\right)\in P_2\left(2k\right).$$

The adjoint word is

$$W^{\ast }=a_{i\left(2k\right),n\left(2k\right)}^{\epsilon {\left(2k\right)}^{\ast }}\cdots a_{i\left(1\right),n\left(1\right)}^{\epsilon {\left(1\right)}^{\ast }},$$

where $1^{\ast }=\ast$ and ${\ast }^{\ast }=1$. Equivalently, if $r\left(j\right)=2k+1-j$, then the data of $W^{\ast }$ at position j are

$$i^{\prime }\left(j\right)=i\left(r\left(j\right)\right),n^{\prime }\left(j\right)=n\left(r\left(j\right)\right),{\epsilon }^{\prime }\left(j\right)=\epsilon {\left(r\left(j\right)\right)}^{\ast }.$$

The map r sends the site pairing $\pi$ of W bijectively onto the site pairing

$${\pi }^{\prime }=Ker\left(n^{\prime }\right)$$

of $W^{\ast }$. If $V=\{a<b\}\in \pi$, write

$$V^{\prime }=\{r\left(b\right),r\left(a\right)\}=\{2k+1-b,2k+1-a\}\in {\pi }^{\prime },$$

listed in increasing order. Since n is constant on V, $n^{\prime }$ is constant on $V^{\prime }$, and the common site value of $V^{\prime }$ is the same as the common site value of V. Therefore, the ordering of blocks by site values is preserved under the correspondence $V\mapsto V^{\prime }$. Specifically, $V^{\prime }$ comes before $W_0^{\prime }$ in the $\sigma \left(n^{\prime }\right)$ ordering if V comes before $W_0$ in the $\sigma \left(n\right)$ ordering. We next compare the oriented crossing signs. Let $V=\{a<b\}$ be a block of $\pi$. Since W is balanced, either

$$\left(\epsilon \right(a),\epsilon (b\left)\right)=(1,\ast )\mathrm{or}\left(\epsilon \right(a),\epsilon (b\left)\right)=(\ast ,1).$$

For the corresponding block $V^{\prime }=\{r\left(b\right)<r\left(a\right)\}$ in the adjoint word, we have

$$({\epsilon }^{\prime }\left(r\left(b\right)\right),{\epsilon }^{\prime }\left(r\left(a\right)\right))=(\epsilon {\left(b\right)}^{\ast },\epsilon {\left(a\right)}^{\ast }).$$

Thus, if $\left(\epsilon \right(a),\epsilon (b\left)\right)=(1,\ast )$, then

$$({\epsilon }^{\prime }\left(r\left(b\right)\right),{\epsilon }^{\prime }\left(r\left(a\right)\right))=(1,\ast ),$$

and if $\left(\epsilon \right(a),\epsilon (b\left)\right)=(\ast ,1)$, we have

$$({\epsilon }^{\prime }\left(r\left(b\right)\right),{\epsilon }^{\prime }\left(r\left(a\right)\right))=(\ast ,1).$$

Hence

$${ori}_{{\epsilon }^{\prime }}\left(V^{\prime }\right)={ori}_{\epsilon }\left(V\right).$$

Now let $V=\{a<b\}$ and $W_0=\{c<d\}$ be two crossing blocks of $\pi$. If

$$a<c<b<d,$$

then the corresponding blocks in ${\pi }^{\prime }$ satisfy

$$r\left(d\right)<r\left(b\right)<r\left(c\right)<r\left(a\right),$$

so their positional sign is reversed. Similarly, if

$$c<a<d<b,$$

then the positional sign is again reversed after applying r. Therefore

$$pos(V^{\prime },W_0^{\prime })=-pos(V,W_0).$$

Combining this with the preservation of the block orientations gives

$${sgn}_{{\pi }^{\prime },{\epsilon }^{\prime },\sigma \left(n^{\prime }\right)}(V^{\prime },W_0^{\prime })=-{sgn}_{\pi ,\epsilon ,\sigma \left(n\right)}(V,W_0).$$

Finally, the color of a block is unchanged by the correspondence $V\mapsto V^{\prime }$, because $i^{\prime }\left(V^{\prime }\right)=i\left(V\right).$ Moreover, as observed above, the ordering of the blocks by site values is preserved. Hence every positive oriented crossing of W with ordered color pair $(i\left(V\right),i\left(W_0\right))$ becomes a negative oriented crossing of $W^{\ast }$ with the same ordered color pair, and every negative oriented crossing of W becomes a positive oriented crossing of $W^{\ast }$ with the same ordered color pair. Therefore,

$${\mathrm{wt}}_{\omega }({\pi }^{\prime },i^{\prime };{\epsilon }^{\prime },\sigma \left(n^{\prime }\right))=\prod _{(r,t)\in {\mathrm{Cr}}^{-}(\pi ,i,\epsilon ,\sigma \left(n\right))}{\omega }_{r,t}\prod _{(r,t)\in {\mathrm{Cr}}^{+}(\pi ,i,\epsilon ,\sigma \left(n\right))}\overline{{\omega }_{r,t}}.$$

Since

$${\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma \left(n\right))=\prod _{(r,t)\in {\mathrm{Cr}}^{+}(\pi ,i,\epsilon ,\sigma \left(n\right))}{\omega }_{r,t}\prod _{(r,t)\in {\mathrm{Cr}}^{-}(\pi ,i,\epsilon ,\sigma \left(n\right))}\overline{{\omega }_{r,t}},$$

we obtain

$${\mathrm{wt}}_{\omega }({\pi }^{\prime },i^{\prime };{\epsilon }^{\prime },\sigma \left(n^{\prime }\right))=\overline{{\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma \left(n\right))}.$$

The scalar factor $2^{-k}$ is real, and hence

$${\Phi }_{\omega }\left(W^{\ast }\right)=2^{-k}{\mathrm{wt}}_{\omega }({\pi }^{\prime },i^{\prime };{\epsilon }^{\prime },\sigma \left(n^{\prime }\right))=\overline{2^{-k}{\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma \left(n\right))}=\overline{{\Phi }_{\omega }\left(W\right)}.$$

This proves (iv). □

Definition 8.

For $r\in \left[s\right]$ and $N\in \mathbb{N}$, define

$$X_r^{\left(N\right)}:={\displaystyle \frac{1}{\sqrt{N}}}{\displaystyle \sum _{n=1}^N}(a_{r,n}+a_{r,n}^{\ast })\in {\mathcal{A}}_{\omega }.$$

Proposition 3.

For every $k\ge 1$ and every color map $i:[2k-1]\to \left[s\right]$, we have

$${\Phi }_{\omega }\left[X_{i\left(1\right)}^{\left(N\right)}\cdots X_{i(2k-1)}^{\left(N\right)}\right]=0.$$

Moreover, for every color map $i:\left[2k\right]\to \left[s\right]$, we have

$$\underset{N\to \infty }{lim}{\Phi }_{\omega }\left[X_{i\left(1\right)}^{\left(N\right)}\cdots X_{i\left(2k\right)}^{\left(N\right)}\right]={\displaystyle \frac{1}{k!2^k}}\sum _{\sigma \in S_k}\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\sum _{\epsilon \in B\left(\pi \right)}{\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma ).$$

Proof. 

Odd moments vanish because every monomial appearing after expansion has odd length and therefore cannot be balanced by sites. For $m=2k$, expansion gives

$${\Phi }_{\omega }\left[X_{i\left(1\right)}^{\left(N\right)}\cdots X_{i\left(2k\right)}^{\left(N\right)}\right]={\displaystyle \frac{1}{N^k}}{\displaystyle \sum _{n_1,\dots ,n_{2k}=1}^N}\sum _{\epsilon \in {\{1,\ast \}}^{2k}}{\Phi }_{\omega }\left[{\displaystyle \prod _{j=1}^{2k}}{a}_{i\left(j\right),n_j}^{\epsilon \left(j\right)}\right].$$

By Definition 7, a summand is nonzero only if $Ker\left(n\right)$ is a pair partition $\pi \in P_2\left(2k\right)$, $\pi \le Ker\left(i\right)$, and $\epsilon \in B\left(\pi \right)$. Therefore

$${\Phi }_{\omega }\left[X_{i\left(1\right)}^{\left(N\right)}\cdots X_{i\left(2k\right)}^{\left(N\right)}\right]={\displaystyle \frac{1}{N^k}}\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\sum _{\epsilon \in B\left(\pi \right)}\sum _{n\in I_N^{distinct}\left(\pi \right)}{2}^{-k}{\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma \left(n\right)),$$

where $I_N^{distinct}\left(\pi \right)$ denotes the assignments from blocks of $\pi$ to pairwise distinct elements of $\left[N\right]$. For fixed $\pi$, each block ordering $\sigma \in S_k$ occurs exactly $\left({\displaystyle \genfrac{}{}{0pt}{}{N}{k}}\right)$ times among these assignments. Hence

$${\displaystyle \frac{1}{N^k}}\sum _{n\in I_N^{distinct}\left(\pi \right)}{\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma \left(n\right))={\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{N}{k}\right)}{N^k}}\sum _{\sigma \in S_k}{\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma ).$$

Since $\left({\displaystyle \genfrac{}{}{0pt}{}{N}{k}}\right)/N^k\to 1/k!$, the formula follows. □

Remark 3.

When ${\omega }_{r,t}\equiv \omega \in \mathbb{T}$, the formula reduces to the scalar oriented crossing CLT formula for ω-sequences of partial isometries obtained in [15]. Thus, the novelty of Proposition 3 lies in its relation-free universal formulation and in the use of a color dependent unimodular matrix of crossing parameters.

4. The Averaged $\mathit{Z}$-Weighted Wick Moment Functional

Let

$$Z={\left(z_{r,t}\right)}_{1\le r,t\le s},z_{r,r}=1,z_{t,r}=\overline{z_{r,t}},\mathrm{and}\left|z_{r,t}\right|\le 1.$$

Definition 9.

The averaged Z-weighted Wick moment functional is the unital linear functional defined on self-adjoint generators $x_1,\dots ,x_s$ with moment rules: All odd moments vanish. For each $k\ge 1$ and each color map $i:\left[2k\right]\to \left[s\right]$, we set

$${\phi }_Z(x_{i\left(1\right)}\cdots x_{i\left(2k\right)}):={\displaystyle \frac{1}{k!2^k}}\sum _{\sigma \in S_k}\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\sum _{\epsilon \in B\left(\pi \right)}{\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma ).$$

Proposition 4.

Let $Z={\left(z_{r,t}\right)}_{1\le r,t\le s}$ be Hermitian such that $z_{r,r}=1$, $z_{t,r}=\overline{z_{r,t}}$, and $|z_{r,t}|\le 1$. Then, for every $k\ge 1$ and every color map $i:\left[2k\right]\to \left[s\right]$, we have

$$\left|{\phi }_Z(x_{i\left(1\right)}\cdots x_{i\left(2k\right)})\right|\le (2k-1)!!.$$

All odd moments vanish.

Proof. 

By Definition 9,

$${\phi }_Z(x_{i\left(1\right)}\cdots x_{i\left(2k\right)})={\displaystyle \frac{1}{k!2^k}}\sum _{\sigma \in S_k}\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\sum _{\epsilon \in B\left(\pi \right)}{\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma ).$$

Since $|z_{r,t}|\le 1$, every factor appearing in ${\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma )$ has absolute value at most 1. Hence

$$|{\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma )|\le 1.$$

Therefore,

$$\left|{\phi }_Z(x_{i\left(1\right)}\cdots x_{i\left(2k\right)})\right|\le {\displaystyle \frac{1}{k!2^k}}\sum _{\sigma \in S_k}\sum _{\pi \in P_2\left(2k\right)}\sum _{\epsilon \in B\left(\pi \right)}1.$$

Now

$$|S_k|=k!,|B\left(\pi \right)|=2^k,\left|P_2\left(2k\right)\right|=(2k-1)!!.$$

Thus

$$\left|{\phi }_Z(x_{i\left(1\right)}\cdots x_{i\left(2k\right)})\right|\le {\displaystyle \frac{1}{k!2^k}}k!(2k-1)!!2^k=(2k-1)!!.$$

The odd moments vanish by Definition 9. □

Remark 4.

Proposition 4 is a moment growth estimate. It does not imply that the generators $x_1,\dots ,x_s$ are bounded operators, nor does it imply positivity of ${\phi }_Z$. Rather, it shows that the averaged Z-weighted Wick functional has Gaussian type moment growth uniformly in the parameter matrix Z.

Remark 5.

The terminology is descriptive rather than standard. The word “Wick” refers to the pair-partition expansion, while “Z-weighted” indicates the crossing weights determined by the Hermitian matrix Z. The adjective “averaged” refers to the full averaging over block orderings ${\displaystyle \frac{1}{k!}}{\sum }_{\sigma \in S_k}$. This terminology is chosen to distinguish the present functional from the usual semicircular law, whose moments are indexed by noncrossing pair partitions.

Remark 6.

If $z_{r,t}\equiv z$ is scalar, the above formula coincides with the oriented crossing averaging structure appearing in the z-semicircular systems introduced in [15], where moments are obtained by averaging over block orderings and balanced orientation maps. These systems are closely related to the z-circular framework of Mingo and Nica [8]. If $|z_{r,t}|=1$, the modulus contribution becomes trivial and the formula reduces to the universal unimodular formula of Proposition 3.

The Real Symmetric Specialization and Positivity

The preceding construction was formulated for a general Hermitian matrix Z with complex entries. In this level of generality, the functional ${\phi }_Z$ is an algebraic moment functional, and positivity is not asserted in full generality. There is, however, an important specialization in which the present averaged oriented formula reduces to a standard positive model from noncommutative probability.

Let

$$Q={\left(q_{r,t}\right)}_{1\le r,t\le s}$$

be real symmetric, with

$$q_{r,r}=1,q_{r,t}=q_{t,r}\in [-1,1].$$

In this case the crossing weight is independent of the orientation sign and is also independent of the order in which the two crossing blocks are listed.

Proposition 5.

Let $Q={\left(q_{r,t}\right)}_{1\le r,t\le s}$ be real symmetric with $q_{r,r}=1$ and $q_{r,t}=q_{t,r}\in [-1,1]$. Then, for every $k\ge 1$ and every color map $i:\left[2k\right]\to \left[s\right]$, the averaged Q-weighted Wick functional satisfies

$${\phi }_Q(x_{i\left(1\right)}\cdots x_{i\left(2k\right)})=\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\prod _{\{V,W\}\in {\mathrm{Cr}}_{un}\left(\pi \right)}{q}_{i\left(V\right),i\left(W\right)}.$$

All odd moments vanish.

Proof. 

The odd moments vanish by Definition 9. Let $m=2k$, and fix $\pi \in P_2\left(2k\right)$ such that $\pi \le Ker\left(i\right)$. Let $\epsilon \in B\left(\pi \right)$ and $\sigma \in S_k$.

For each crossing pair of blocks $V,W$, the oriented formula contributes either $q_{i\left(V\right),i\left(W\right)}$ or $\overline{q_{i\left(V\right),i\left(W\right)}}$, with the ordered pair depending on the chosen block ordering. Since Q is real symmetric, we have

$$\overline{q_{r,t}}=q_{r,t}=q_{t,r}.$$

Therefore the contribution of a crossing is simply $q_{i\left(V\right),i\left(W\right)}$, independently of whether the oriented crossing sign is positive or negative, and independently of which of the two crossing blocks precedes the other in the $\sigma$-ordering. Hence

$${\mathrm{wt}}_Q(\pi ,i;\epsilon ,\sigma )=\prod _{\{V,W\}\in {\mathrm{Cr}}_{un}\left(\pi \right)}{q}_{i\left(V\right),i\left(W\right)}.$$

The right-hand side is independent of both $\epsilon \in B\left(\pi \right)$ and $\sigma \in S_k$. Since

$$\left|B\left(\pi \right)\right|=2^k\mathrm{and}\left|S_k\right|=k!,$$

the normalizing factor ${(k!2^k)}^{-1}$ in Definition 9 cancels the two averaging sums. Therefore

$$\begin{array}{cc}\hfill {\phi }_Q(x_{i\left(1\right)}\cdots x_{i\left(2k\right)})& ={\displaystyle \frac{1}{k!2^k}}\sum _{\sigma \in S_k}\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\sum _{\epsilon \in B\left(\pi \right)}{\mathrm{wt}}_Q(\pi ,i;\epsilon ,\sigma )\hfill \\ & =\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\prod _{\{V,W\}\in {\mathrm{Cr}}_{un}\left(\pi \right)}{q}_{i\left(V\right),i\left(W\right)}.\hfill \end{array}$$

This proves the claim. □

Example 1.

Let $s=2$ and let

$$Q=\left(\begin{array}{cc}1& q\\ q& 1\end{array}\right),q\in [-1,1].$$

Consider the fourth mixed moment

$${\phi }_Q\left(x_1{x}_2{x}_1{x}_2\right).$$

The color map is given by

$$i\left(1\right)=i\left(3\right)=1,i\left(2\right)=i\left(4\right)=2.$$

Among the three pair partitions of $\left[4\right]$, only

$$\pi =\left\{\right\{1,3\},\{2,4\left\}\right\}$$

is compatible with the color map. This pairing is crossing, and its two blocks have colors 1 and 2. By Proposition 5, we therefore obtain

$${\phi }_Q\left(x_1{x}_2{x}_1{x}_2\right)=q.$$

In contrast, for the word $x_1{x}_1{x}_2{x}_2$, the compatible pairing is

$$\pi =\left\{\right\{1,2\},\{3,4\left\}\right\},$$

which has no crossing. Hence

$${\phi }_Q\left(x_1{x}_1{x}_2{x}_2\right)=1.$$

This illustrates that, in the real symmetric specialization, the parameters $q_{r,t}$ appear only through crossings of color-compatible pairings.

Corollary 1.

Assume that $Q={\left(q_{r,t}\right)}_{1\le r,t\le s}$ is real symmetric, satisfies $q_{r,r}=1$, and lies in a known Fock-representable mixed $q_{ij}$-Gaussian regime. Then ${\phi }_Q$ is positive. More precisely, its moments are realized as vacuum moments of the corresponding mixed $q_{ij}$-Gaussian field

$$G_r=a_r+a_r^{\ast },r=1,\dots ,s.$$

In particular, the real symmetric specialization admits a concrete Fock-space realization in these regimes.

Proof. 

By Proposition 5, the moments of ${\phi }_Q$ agree with the mixed $q_{ij}$-Gaussian pair-partition formula associated with Q. In the known Fock-representable regimes for mixed $q_{ij}$-Gaussian systems, the variables $G_r=a_r+a_r^{\ast }$ are realized as operators on the corresponding Fock space, and the vacuum expectation is a positive linear functional, see for example [7,16,17]. Hence ${\phi }_Q$ is positive in these regimes. □

Remark 7.

Corollary 1 is not a positivity theorem for the full complex Hermitian parameter matrix Z. Rather, it records that the present framework contains the known Fock-representable mixed $q_{ij}$-Gaussian models as a positive real symmetric subclass. Determining the positivity for general complex Hermitian Z remains open.

Example 2.

Let $s=1$ and $q_{1,1}=1$. Then every crossing contributes the factor 1, and Proposition 5 gives

$${\phi }_Q\left(x^{2k}\right)=\left|P_2\left(2k\right)\right|=(2k-1)!!.$$

Thus the one-variable specialization with parameter 1 has the classical Gaussian even moments. In particular,

$${\phi }_Q\left(x^4\right)=3.$$

This explains why the present averaged formula is not the standard semicircular moment formula, whose fourth moment is 2.

5. The Auxiliary Modulus Algebra and Tensorization

Fix $Z=\left(z_{r,t}\right)$ as above and set ${\rho }_{r,t}:=\left|z_{r,t}\right|.$ Then let ${\omega }_{r,t}:={\displaystyle \frac{z_{r,t}}{{\rho }_{r,t}}}\in \mathbb{T}$ if ${\rho }_{r,t}>0$ and if ${\rho }_{r,t}=0$, let ${\omega }_{r,t}:=1$. Then

$${\omega }_{r,r}=1,{\omega }_{t,r}=\overline{{\omega }_{r,t}},{\rho }_{t,r}={\rho }_{r,t},z_{r,t}={\omega }_{r,t}{\rho }_{r,t}.$$

5.1. The Auxiliary Modulus Algebra

Definition 10.

Let $C_R$ be the commutative polynomial ∗-algebra

$$C_R:=\mathbb{C}[R_{r,t}:1\le r<t\le s],$$

with involution determined by $R_{r,t}^{\ast }=R_{r,t}$. We also set

$$R_{t,r}:=R_{r,t},R_{r,r}:=\mathbb{1}.$$

Definition 11.

Let ${\mathcal{F}}_{mod}$ be the free unital ∗-algebra generated by

$$\{U_{r,n}:r\in \left[s\right],n\in \mathbb{N}\}.$$

Define the auxiliary modulus algebra by

$$B:=C_R\otimes {\mathcal{F}}_{mod}.$$

Lemma 2.

The algebra B is a unital ∗-algebra. Moreover, the elementary tensors

$$R^{\alpha }\otimes W,$$

where $R^{\alpha }$ is a monomial in the commuting variables $R_{r,t}$ and W is a word in the alphabet $\{U_{r,n},U_{r,n}^{\ast }:r\in \left[s\right],n\in \mathbb{N}\}$, form a vector space basis of B.

Proof. 

This is the standard basis of an algebraic tensor product of the polynomial algebra $C_R$ and the free unital ∗-algebra ${\mathcal{F}}_{mod}$. □

5.2. Admissible Modulus Words

Definition 12.

Let

$$W=U_{i\left(1\right),n\left(1\right)}^{\epsilon \left(1\right)}\cdots U_{i\left(m\right),n\left(m\right)}^{\epsilon \left(m\right)}$$

be a basis word of ${\mathcal{F}}_{mod}$. We call W admissible if $m=2k$ is even and $Ker\left(n\right)$ is a pair partition $\pi \in P_2\left(2k\right)$ such that

$$\pi \le Ker\left(i\right),\epsilon \in B\left(\pi \right).$$

We also regard the empty word $\mathbb{1}$ as admissible, with empty pairing $P_2\left(0\right)=\{\varnothing \}$, and set $M\left(\mathbb{1}\right)=\mathbb{1}$. If W is a non-empty admissible word, define $\pi \left(W\right):=Ker\left(n\right)$. The crossing monomial of W is

$$M\left(W\right):=\prod _{\{V,W^{\prime }\}\in {\mathrm{Cr}}_{un}\left(\pi \left(W\right)\right)}{R}_{i\left(V\right),i\left(W^{\prime }\right)}.$$

If W is not admissible, set $M\left(W\right):=0$.

Remark 8.

The monomial $M\left(W\right)$ counts one modulus symbol for each unordered crossing pair. Since $R_{r,t}=R_{t,r}$, the order of the two colors is irrelevant.

Lemma 3.

For every basis word W of ${\mathcal{F}}_{mod}$,

$$M\left(W^{\ast }\right)=M\left(W\right).$$

Proof. 

If W is not admissible, then $W^{\ast }$ is not admissible. If $W=\mathbb{1}$, the claim is immediate from $M\left(\mathbb{1}\right)=\mathbb{1}$. If W is a non-empty admissible word, then taking adjoints reverses the word and flips the directions, but it preserves the underlying equal-site pairing, the colors of the blocks, and the unordered crossing relation. Hence the crossing monomial is unchanged. □

Definition 13.

Let ${ev}_{\rho }:C_R\to \mathbb{C}$ be the unique unital ∗-homomorphism such that

$${ev}_{\rho }\left(R_{r,t}\right)={\rho }_{r,t}(1\le r<t\le s).$$

Definition 14.

Let $\psi :B\to \mathbb{C}$ on basis elements by

$$\psi (R^{\alpha }\otimes W):={ev}_{\rho }\left(R^{\alpha }M\left(W\right)\right),$$

and extend linearly.

Proposition 6.

The functional ψ is well defined and satisfies:

(i) 

$\psi \left(\mathbb{1}\right)=1$;

(ii) 

$\psi \left(b^{\ast }\right)=\overline{\psi \left(b\right)}$ for every $b\in B$;

(iii) 

for every $c\in C_R$ and $b\in B$,

$$\psi \left(cb\right)={ev}_{\rho }\left(c\right)\psi \left(b\right);$$

(iv) 

$\psi \left(W\right)=0$ for every inadmissible basis word W of ${\mathcal{F}}_{mod}$.

Proof. 

Well-definedness follows from Lemma 2. Since $M\left(\mathbb{1}\right)=\mathbb{1}$, we have $\psi \left(\mathbb{1}\right)=1$. The third and fourth claims are immediate from the definition. The second one follows from Lemma 3 and from the fact that ${ev}_{\rho }$ is a ∗-homomorphism with real values on the self-adjoint variables $R_{r,t}$. □

5.3. Tensorized Variables

Definition 15.

Let

$${\tilde{\mathcal{A}}}_Z:={\mathcal{A}}_{\omega }\odot B,{\tilde{\Phi }}_Z:={\Phi }_{\omega }\otimes \psi .$$

For $r\in \left[s\right]$ and $n\in \mathbb{N}$, define

$${\tilde{a}}_{r,n}:=a_{r,n}\otimes U_{r,n},{\tilde{a}}_{r,n}^{\ast }:=a_{r,n}^{\ast }\otimes U_{r,n}^{\ast }.$$

For $N\in \mathbb{N}$, define the self-adjoint normalized sums

$${\tilde{X}}_r^{\left(N\right)}:={\displaystyle \frac{1}{\sqrt{N}}}{\displaystyle \sum _{n=1}^N}\left({\tilde{a}}_{r,n}+{\tilde{a}}_{r,n}^{\ast }\right).$$

Remark 9.

The pair $({\tilde{\mathcal{A}}}_Z,{\tilde{\Phi }}_Z)$ is an algebraic moment model.

Lemma 4.

Let $\pi \in P_2\left(2k\right)$, let $i:\left[2k\right]\to \left[s\right]$ satisfy $\pi \le Ker\left(i\right)$, let $\epsilon \in B\left(\pi \right)$, and let $n:\left[2k\right]\to \mathbb{N}$ be constant on the blocks of π and injective on the set of blocks. Then

$$\psi \left[{\displaystyle \prod _{j=1}^{2k}}{U}_{i\left(j\right),n\left(j\right)}^{\epsilon \left(j\right)}\right]=\prod _{\{V,W\}\in {\mathrm{Cr}}_{un}\left(\pi \right)}{\rho }_{i\left(V\right),i\left(W\right)}.$$

Proof. 

The equal site pairing of the word is exactly $\pi$. Since $\pi \le Ker\left(i\right)$ and $\epsilon \in B\left(\pi \right)$, the word is admissible. Its crossing monomial is therefore

$$\prod _{\{V,W\}\in {\mathrm{Cr}}_{un}\left(\pi \right)}{R}_{i\left(V\right),i\left(W\right)}.$$

Applying ${ev}_{\rho }$ gives the formula. □

Lemma 5.

Let $\pi \in P_2\left(2k\right)$, let $i:\left[2k\right]\to \left[s\right]$ satisfy $\pi \le Ker\left(i\right)$, let $\epsilon \in B\left(\pi \right)$, and let $n:\left[2k\right]\to \mathbb{N}$ be constant on the blocks of π and injective on the set of blocks. Then

$${\tilde{\Phi }}_Z\left[{\displaystyle \prod _{j=1}^{2k}}{\tilde{a}}_{i\left(j\right),n\left(j\right)}^{\epsilon \left(j\right)}\right]=2^{-k}{\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma \left(n\right)).$$

Proof. 

By Definition 7,

$${\Phi }_{\omega }\left[{\displaystyle \prod _{j=1}^{2k}}{a}_{i\left(j\right),n\left(j\right)}^{\epsilon \left(j\right)}\right]=2^{-k}{\mathrm{wt}}_{\omega }(\pi ,i;\epsilon ,\sigma \left(n\right)).$$

By Lemma 4, the modulus factor is

$$\prod _{\{V,W\}\in {\mathrm{Cr}}_{un}\left(\pi \right)}{\rho }_{i\left(V\right),i\left(W\right)}.$$

Each oriented crossing contributes one unimodular factor ${\omega }_{r,t}$ or its conjugate and one modulus factor ${\rho }_{r,t}$. Since $z_{r,t}={\omega }_{r,t}{\rho }_{r,t}$ and $\overline{z_{r,t}}=\overline{{\omega }_{r,t}}{\rho }_{r,t}$, the product is precisely ${\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma \left(n\right))$. □

Example 3.

Let $n\ne m$. Consider

$${\tilde{a}}_{r,n}{\tilde{a}}_{t,m}{\tilde{a}}_{r,n}^{\ast }{\tilde{a}}_{t,m}^{\ast }.$$

The site pairing is

$$\pi =\left\{\right\{1,3\},\{2,4\left\}\right\},$$

which is crossing, and both blocks have orientation $+1$. If the block with site n precedes the block with site m, the crossing contributes $z_{r,t}$. If the block with site m precedes the block with site n, then the ordered color pair is reversed and the crossing contributes $\overline{z_{t,r}}=z_{r,t}$. Hence in either case

$${\tilde{\Phi }}_Z\left({\tilde{a}}_{r,n}{\tilde{a}}_{t,m}{\tilde{a}}_{r,n}^{\ast }{\tilde{a}}_{t,m}^{\ast }\right)={\displaystyle \frac{1}{4}}{z}_{r,t}.$$

The unimodular part supplies the corresponding factor involving ω, while the modulus algebra supplies the factor ${\rho }_{r,t}$.

6. Universal Algebraic Realization Theorem

We are now ready to combine the unimodular factor model with the auxiliary modulus algebra and prove the algebraic realization of the averaged Z-weighted Wick functional.

Theorem 1.

Let $Z={\left(z_{r,t}\right)}_{1\le r,t\le s}$ be Hermitian such that

$$z_{r,r}=1,z_{t,r}=\overline{z_{r,t}},\left|z_{r,t}\right|\le 1.$$

Let $({\tilde{\mathcal{A}}}_Z,{\tilde{\Phi }}_Z)$ and the self-adjoint sums ${\tilde{X}}_r^{\left(N\right)}$ be as in Definition 15. Then all odd joint moments vanish:

$${\tilde{\Phi }}_Z\left[{\tilde{X}}_{i\left(1\right)}^{\left(N\right)}\cdots {\tilde{X}}_{i(2k-1)}^{\left(N\right)}\right]=0$$

for every $k\ge 1$ and every $i:[2k-1]\to \left[s\right]$. Moreover, for every $i:\left[2k\right]\to \left[s\right]$,

$$\underset{N\to \infty }{lim}{\tilde{\Phi }}_Z\left[{\tilde{X}}_{i\left(1\right)}^{\left(N\right)}\cdots {\tilde{X}}_{i\left(2k\right)}^{\left(N\right)}\right]={\displaystyle \frac{1}{k!2^k}}\sum _{\sigma \in S_k}\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\sum _{\epsilon \in B\left(\pi \right)}{\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma ).$$

Thus the normalized tensorized sums converge in joint algebraic ∗-moments to the averaged Z-weighted Wick moment functional of Definition 9.

Proof. 

The odd case follows immediately from the definitions of ${\Phi }_{\omega }$ and $\psi$, since no balanced pair partition exists on an odd number of positions.

Let $m=2k$. Expanding the normalized sums gives

$${\tilde{\Phi }}_Z\left[{\tilde{X}}_{i\left(1\right)}^{\left(N\right)}\cdots {\tilde{X}}_{i\left(2k\right)}^{\left(N\right)}\right]={\displaystyle \frac{1}{N^k}}{\displaystyle \sum _{n_1,\dots ,n_{2k}=1}^N}\sum _{\epsilon \in {\{1,\ast \}}^{2k}}{\tilde{\Phi }}_Z\left[{\displaystyle \prod _{j=1}^{2k}}{\tilde{a}}_{i\left(j\right),n_j}^{\epsilon \left(j\right)}\right].$$

A summand is nonzero only if the site kernel $Ker\left(n\right)$ is a pair partition $\pi \in P_2\left(2k\right)$, if $\pi \le Ker\left(i\right)$, and if $\epsilon \in B\left(\pi \right)$. Hence

$${\tilde{\Phi }}_Z\left[{\tilde{X}}_{i\left(1\right)}^{\left(N\right)}\cdots {\tilde{X}}_{i\left(2k\right)}^{\left(N\right)}\right]={\displaystyle \frac{1}{N^k}}\sum _{\begin{array}{c}\pi \in P_2\left(2k\right)\\ \pi \le Ker\left(i\right)\end{array}}\sum _{\epsilon \in B\left(\pi \right)}\sum _{n\in I_N^{distinct}\left(\pi \right)}{2}^{-k}{\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma \left(n\right)),$$

where Lemma 5 was used in the last step. By Lemma 1, each $\sigma \in S_k$ appears exactly $\left({\displaystyle \genfrac{}{}{0pt}{}{N}{k}}\right)$ times. Therefore

$${\displaystyle \frac{1}{N^k}}\sum _{n\in I_N^{distinct}\left(\pi \right)}{\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma \left(n\right))={\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{N}{k}\right)}{N^k}}\sum _{\sigma \in S_k}{\mathrm{wt}}_Z(\pi ,i;\epsilon ,\sigma ).$$

Since $\left({\displaystyle \genfrac{}{}{0pt}{}{N}{k}}\right)/N^k\to 1/k!$, the desired formula follows. □

7. Conclusions

We have constructed a relation-free algebraic model whose normalized sums converged, in joint algebraic moments, to the averaged Z-weighted Wick functional. The construction used the decomposition $z_{r,t}={\omega }_{r,t}{\rho }_{r,t}$. The unimodular factor ${\omega }_{r,t}$ was recorded by the oriented crossing functional ${\Phi }_{\omega }$, while the modulus ${\rho }_{r,t}$ was supplied by the auxiliary commutative algebra generated by the variables $R_{r,t}$. After tensorization, each oriented crossing contributed exactly one factor $z_{r,t}$ or $\overline{z_{r,t}}$. In the general complex Hermitian case, the construction remained an algebraic moment model. The Gaussian-type moment bound proved above has shown that the limiting functional has controlled moment growth, uniformly in Z. We also recorded a positive subclass in the real symmetric specialization: in the known Fock-representable mixed $q_{ij}$-Gaussian regimes, the averaged oriented formula reduced to the standard mixed $q_{ij}$-Gaussian pair-partition formula and, hence, was realized by vacuum moments on the corresponding Fock space. The general complex case has remained open from the point of view of positivity and operator realization. A natural problem has been to find conditions on Z under which ${\phi }_Z$ is positive and to determine whether such positive cases can be realized by operators or random matrices.