Segal-Bargmann transform: the $q$-deformation

TL;DR

This paper explores the $q$-deformed Segal-Bargmann transform, establishing its connections to classical, free, and matrix models, and providing new identifications and definitions in the context of mixed $q$-Gaussian variables.

Contribution

It introduces the $q$-deformed Segal-Bargmann transform, links it to classical and free transforms, and extends its definition to mixed $q$-Gaussian variables.

Findings

Classical Segal-Bargmann converges to $q$-deformed version in large $N$ limit.

$q$-deformed transform can be obtained as a mixture of classical and free transforms.

Identifications of the $q$-deformed transform with matrix models are established.

Abstract

We give identifications of the $q$-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed $q$-Gaussian variables. We prove that, when defined on the random matrix model of \'Sniady for the $q$-Gaussian variable, the classical Segal-Bargmann transform converges to the $q$-deformed Segal-Bargmann transform in the large $N$ limit. We also show that the $q$-deformed Segal-Bargmann transform can be recovered as a limit of a mixture of classical and free Segal-Bargmann transform.

Equations

$$\mathcal{S}:L^2(H,\gamma )\to \mathcal{H}{L}^2(H^{\mathbb{C}},\mu )$$

$${\nu }_q(dx)=\text{mathbbm}{1}_{|x|\le 2/\sqrt{1-q}}\frac{1}{\pi }\sqrt{1-q}\sin \theta \prod _{n=1}^{\infty }(1-q^n)|1-q^n{e}^{2i\theta }{|}^2\mathrm{d}x,$$

$${\Gamma }_q(x,z)=\prod _{k=0}^{\infty }\frac{1}{1-(1-q)q^{k}zx+(1-q)q^{2k}{z}^2},|x|\le \frac{2}{\sqrt{1-q}},|z|<\frac{1}{\sqrt{1-q}}.$$

$${\mathcal{S}}_q(f):z\mapsto {\int }_{\mathbb{R}}f(x){\Gamma }_q(x,z)\mathrm{d}{\nu }_q(x)$$

$$\text{mathscrbf}{S}_N=\mathcal{S}\otimes I{d}_{{\mathbb{M}}_N}:L^2({\mathbb{H}}_N,{\gamma }_N;{\mathbb{M}}_N)\to \mathcal{H}{L}^2({\mathbb{M}}_N,{\mu }_N;{\mathbb{M}}_N).$$

$$\underset{N\to \infty }{\lim }{\Vert \text{mathscrbf}{S}_N(P)-{\mathcal{S}}_0(P)\Vert }_{\mathcal{H}{L}^2({\mathbb{M}}_N,{\mu }_N;{\mathbb{M}}_N)}=0.$$

$${\int }_{{\mathbb{H}}_{d^N}}\exp (i\mathrm{Tr}(MX))\mathrm{d}{\gamma }_{d^N}^{\sigma }(X)=\exp (-\Vert M{\Vert }_{\sigma }^2/2)$$

$${\int }_{{\mathbb{M}}_{d^N}}\exp (i\mathrm{Tr}(MX))\mathrm{d}{\mu }_{d^N}^{\sigma }(X)=\exp (-\Re \Vert M{\Vert }_{\sigma }^2/4).$$

$$\mathcal{S}:L^2({\mathbb{H}}_{d^N},{\gamma }_{d^N}^{\sigma })=L^2(\mathrm{supp}{\gamma }_{d^N}^{\sigma },{\gamma }_{d^N}^{\sigma })\to \mathcal{H}{L}^2(\mathrm{supp}{\mu }_{d^N}^{\sigma },{\mu }_{d^N}^{\sigma })=\mathcal{H}{L}^2({\mathbb{M}}_{d^N},{\mu }_{d^N}^{\sigma })$$

$$\text{mathscrbf}{S}_{d^N}=\mathcal{S}\otimes I{d}_{{\mathbb{M}}_{d^N}}:L^2({\mathbb{H}}_{d^N},{\gamma }_{d^N}^{\sigma };{\mathbb{M}}_{d^N})\to \mathcal{H}{L}^2({\mathbb{M}}_{d^N},{\mu }_{d^N}^{\sigma };{\mathbb{M}}_{d^N}).$$

$$\underset{N\to \infty }{\lim }\Vert P{\Vert }_{L^2({\mathbb{H}}_{d^N},{\gamma }_{d^N}^{\sigma };{\mathbb{M}}_{d^N})}=\Vert P{\Vert }_{L^2(\mathbb{R},{\nu }_q)},$$

$$\underset{N\to \infty }{\lim }{\Vert \text{mathscrbf}{S}_{d^N}(P)-{\mathcal{S}}_q(P)\Vert }_{\mathcal{H}{L}^2({\mathbb{M}}_{d^N},{\mu }_{d^N}^{\sigma };{\mathbb{M}}_{d^N})}=0.$$

$${\mathcal{S}}_q^{s,t}(f):z\mapsto {\int }_{\mathbb{R}}f(x){\Gamma }_q^{s,t}(x,z){\nu }_q^s(\mathrm{d}x)$$

$$z\mapsto {\int }_{H}f(z-x)\mathrm{d}{\gamma }_t(x),$$

$${\mathcal{S}}^{s,t}:L^2(H,{\gamma }_s)\to \mathcal{H}{L}^2(H^{\mathbb{C}},{\gamma }_{s-t/2,t/2})$$

$${\nu }_q(dx)=\text{mathbbm}{1}_{|x|\le 2/\sqrt{1-q}}\frac{1}{\pi }\sqrt{1-q}\sin \theta \prod _{n=1}^{\infty }(1-q^n)|1-q^n{e}^{2i\theta }{|}^2\mathrm{d}x$$

$$H_{n+1}^{q,t}(x)=x{H}_n^{q,t}(x)-t[n{]}_q{H}_{n-1}^{q,t}(x).$$

$${\Gamma }_q^t(x,z):=\sum _{k=0}^{\infty }\frac{z^k}{[k{]}_q!t^k}{H}_k^{q,t}(x)=\prod _{k=0}^{\infty }\frac{t}{t-(1-q)q^{k}zx+(1-q)q^{2k}{z}^2},$$

$$\tau [X_1\cdots X_n]=\sum _{\pi \in {\mathcal{P}}_2(n)}{q}^{\mathrm{cr}(\pi )}\prod _{\{i,j\}\in \pi }\tau [X_i{X}_j].$$

$$\tau [X^n]=\sum _{\pi \in {\mathcal{P}}_2(n)}{q}^{\mathrm{cr}(\pi )}={\int }_{\mathbb{R}}{x}^n{\nu }_q(\mathrm{d}x).$$

$$X_1\diamond \cdots \diamond X_n=X_1\cdot (X_2\diamond \cdots \diamond X_n)-\sum _{i=2}^n{q}^{i-1}\tau [X_1{X}_i](X_1\diamond \cdots \diamond \widehat{X_i}\diamond \cdots \diamond X_n)$$

$$X_1\diamond \cdots \diamond X_n=\sum _{\gamma \in \mathcal{F}(n)}(-1{)}^{♯\gamma }{q}^{\mathrm{gap}(\gamma )-\mathrm{cr}(\gamma )}\prod _{\{a,b\}\in \gamma }\tau [X_a{X}_b]\overrightarrow{\prod _{\{c\}\in \gamma }}{X}_c.$$

$$\tau\Big{[}(X_{1}\diamond\cdots\diamond X_{n})\cdot(X_{n+1}\diamond\cdots\diamond X_{n+m})\Big{]}=\delta_{n,m}\sum_{\pi\in\mathcal{P}_{2}(n,m)}q^{\mathrm{cr}(\pi)}\prod_{\{i,j\}\in\pi}\tau[X_{i}X_{j}].$$

$$\{X_{i(1)}\diamond \cdots \diamond X_{i(n)}{\}}_{n\ge 0,i(1),\dots ,i(n)\in I}$$

$$c_i^{\ast }{c}_j=q_{ij}{c}_j{c}_i^{\ast }+{\delta }_{ij}1.$$

$$\mathcal{F}(K)=\mathbb{C}\Omega +\underset{n=1}{\overset{\infty }{⨁}}(K{)}^{\otimes n}$$

$$⟨\Omega ,\Omega {⟩}_Q$$

$$⟨e_{i(1)}\otimes \cdots \otimes e_{i(k)},e_{j(1)}\otimes \cdots \otimes e_{j(\ell )}{⟩}_Q$$

$$c_i(\Omega )$$

$$c_i(e_{i(1)}\otimes \cdots \otimes e_{i(k)})$$

$$c_i^{\ast }(\Omega )$$

Full text

1. 1 Introduction

2. 1.1 $q$-Deformed Segal-Bargmann Transform

3. 1.2 Matrix Approximations

4. 1.3 Two Parameter Case

5. 1.4 Multidimensional Case

6. 1.5 Mixture of $q$-Deformed Segal-Bargmann Transform

7. 1.6 Organization of the Paper

8. 2 Preliminaries

9. 2.1 Segal-Bargmann Transform

10. 2.2 $q$-Gaussian Measure

11. 2.3 $q$-Gaussian Variables and Wick Product

12. 2.4 Mixed $q$-Gaussian Variables

13. 2.5 Random Matrix Model of Śniady

14. 3 The Two-Parameter $q$-Deformed Segal-Bargmann Transform

15. 3.1 An Integral Representation

16. 3.2 The Integral Transform

17. 3.3 Segal-Bargmann Transform and Conditional Expectation

18. 3.4 Random Matrix Model

19. 4 Multidimensional $q$-Segal-Bargmann Transform

20. 4.1 Classical Segal-Bargmann Transform in the Infinite-Dimensional Case

21. 4.2 The $q$-Deformation of the Segal-Bargmann Transform

22. 4.3 Large N Limit

23. 5 Mixture of Classical and Free Segal-Bargmann Transform

24. 5.1 The Mixed $q$-Deformed Segal-Bargmann Transform

25. 5.2 The $q$-Segal-Bargmann Transform in the Limit

1 Introduction

Let $H$ be a real finite-dimensional Hilbert space. Let $\gamma$ be the standard Gaussian measure on $H$, whose density with respect to the Lebesgue measure at $h\in H$ is $(2\pi)^{-d/2}\exp(-\|h\|_{H}^{2}/2)$. Let $\mu$ be the Gaussian measure on the complexification $H^{\mathbb{C}}=H+iH$ of $H$ whose density with respect to the Lebesgue measure at $h\in H^{\mathbb{C}}$ is $\pi^{-d}\exp(-\|h\|_{H^{\mathbb{C}}}^{2})$. For all $f\in L^{2}(H,\gamma)$, the map $z\mapsto\int_{H}f(z-x)\,\mathrm{d}\gamma(x),$ admits an analytic continuation $\mathscr{S}(f)$ to $H^{\mathbb{C}}$. Furthermore, the map $\mathscr{S}(f)$ is in the closed subspace of holomorphic functions of $L^{2}(H^{\mathbb{C}},\mu)$, hereafter denoted by $\mathcal{H}L^{2}(H^{\mathbb{C}},\mu)$. The resulting map

[TABLE]

is known as the Segal-Bargmann transform, introduced by Segal [21, 22] and Bargmann [1, 2] in early 1960s.

1.1 $q$-Deformed Segal-Bargmann Transform

In [28], Leeuwen and Massen considered a $q$-deformation of the Segal-Bargmann transform in the one-dimensional case. For all $0\leq q<1$, the measure replacing the Gaussian measure is the $q$-Gaussian measure $\nu_{q}$ on $\mathbb{R}$, whose density with respect to the Lebesgue measure is

[TABLE]

where $\theta\in[0,\pi]$ is such that $x=2\cos(\theta)/\sqrt{1-q}$. The $q$-deformation of the Segal-Bargmann transform is then defined through the kernel

[TABLE]

For all function $f\in L^{2}(\mathbb{R},\nu_{q})$, the function

[TABLE]

is defined on the unit disk of radius $1/\sqrt{1-q}$ and the map $\mathscr{S}_{q}$ is in fact an isomorphism of Hilbert space between $L^{2}(\mathbb{R},\nu_{q})$ and a reproducing kernel Hilbert space of analytic function on the unit disk of radius $1/\sqrt{1-q}$ which plays the role of the complexified version of $\nu_{q}$.

Let us remark that $\lim_{q\to 1}\nu_{q}(\,\mathrm{d}x)=\exp(-x^{2}/2)/\sqrt{2\pi}\,\mathrm{d}x$ and $\lim_{q\to 1}\Gamma_{q}(x,z)=\exp(xz-z^{2}/2)$, which suggest to denote the standard normal distribution by $\nu_{1}$ and the classical Segal-Bargmann transform $\mathscr{S}$ on $L^{2}(\mathbb{R},\nu_{1})$ by $\mathscr{S}_{1}$. The case $q=0$, studied in [3], is of particular interest, since $\nu_{0}(\,\mathrm{d}x)=\mathbbm{1}_{|x|\leq 2}\sqrt{4-x^{2}}\frac{\,\mathrm{d}x}{2\pi}$ is the well-known semicircular law and the so-called free Segal-Bargmann transform $\mathscr{S}_{0}$ maps isometrically $L^{2}(\mathbb{R},\nu_{0})$ to the Hardy space of analytic functions on the unit disc. Although beyond the scope of this article, let us mention also the related work [5], where Blitvić and Kemp define a refinement of the $q$-deformed Segal-Bargmann transform.

1.2 Matrix Approximations

In [3], Biane proves that the free Segal-Bargmann transform $\mathscr{S}_{0}$ is the limit of the classical Segal-Bargmann transform on Hermitian matrices in the following sense: for all $N\geq 1$, let $\mathbb{M}_{N}$ be the space of complex matrices of size $N\times N$, let $\mathbb{H}_{N}$ be the subspace of Hermitian matrices $M=M^{*}$ of size $N\times N$, and let $\mathrm{Tr}$ denote the usual trace. Let $\gamma_{N}$ be the standard Gaussian measure on $\mathbb{H}_{N}$ for the norm $\|M\|^{2}=N\mathrm{Tr}(MM^{*})$, and $\mu_{N}$ be the standard Gaussian measure on $\mathbb{M}_{N}=\mathbb{H}_{N}+i\mathbb{H}_{N}$ for the norm $\|M\|^{2}=2N\Re\mathrm{Tr}(M^{2})$. This way we can consider the Segal-Bargmann transform $\mathscr{S}:L^{2}(\mathbb{H}_{N},\gamma_{N})\to\mathcal{H}L^{2}(\mathbb{M}_{N},\mu_{N})$ defined by (1.1). Biane extends the transform $\mathscr{S}$ to act on $\mathbb{M}_{N}$-valued functions, by applying $\mathscr{S}$ entrywise. More precisely, endowing $\mathbb{M}_{N}$ with the norm $\|M\|^{2}_{\mathbb{M}_{N}}=\mathrm{Tr}(MM^{*})/N$, he considers the Hilbert space tensor products $L^{2}(\mathbb{H}_{N},\gamma_{N};\mathbb{M}_{N})=L^{2}(\mathbb{H}_{N},\gamma_{N})\otimes\mathbb{M}_{N}$ and $\mathcal{H}L^{2}(\mathbb{M}_{N},\mu_{N};\mathbb{M}_{N})=\mathcal{H}L^{2}(\mathbb{M}_{N},\mu_{N})\otimes\mathbb{M}_{N}$, as well as the boosted Segal-Bargmann transform

[TABLE]

Each polynomial can be seen as an element of $L^{2}(\mathbb{H}_{N},\gamma_{N};\mathbb{M}_{N})$ (or of $\mathcal{H}L^{2}(\mathbb{M}_{N},\mu_{N};\mathbb{M}_{N})$) via the polynomial calculus, and Biane proved that, restricted to those polynomial functions, the Segal-Bargmann transform $\mathscrbf{S}_{N}$ converges to the free Segal-Bargmann transform $\mathscr{S}_{0}$ in the following sense: for all polynomial $P$,

[TABLE]

One of the motivation of this article is to prove that the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}$ can also be approximated by the classical one for $0<q\leq 1$. In the model of Biane, $L^{2}(\mathbb{H}_{N},\gamma_{N};\mathbb{M}_{N})$ is an approximation of $L^{2}(\mathbb{R},\nu_{0})$ in the sense that, for all polynomial $P$, $\|P\|_{L^{2}(\mathbb{R},\nu_{0})}=\lim_{N\to\infty}\|P\|_{L^{2}(\mathbb{H}_{N},\gamma_{N};\mathbb{M}_{N})}$. In the case of $0<q\leq 1$, we replace the previous model by a model of Śniady introduced in [23] in order to approximate $L^{2}(\mathbb{R},\nu_{q})$. Let us briefly describe this model.

Let $d\geq 0$. We endow $\mathbb{M}_{d}$ by the inner products, quotient if necessary, $\langle A,B\rangle_{1}=\frac{1}{d}\mathrm{Tr}(AB^{*})$ and $\langle A,B\rangle_{0}=\mathrm{Tr}(A)\mathrm{Tr}(B^{*}).$ For all $S\subset\{1,\ldots,N\}$, we define the inner product $\langle\cdot,\cdot\rangle_{S}$ on $\mathbb{M}_{d^{N}}\simeq\bigotimes_{r=1}^{N}\mathbb{M}_{d}$ to be the inner product of the Hilbert space tensor product $\bigotimes_{r=1}^{N}\left(\mathbb{M}_{d},\langle A,B\rangle_{\mathbbm{1}_{S}(r)}\right)$. Let $\sigma=(\sigma_{S})_{S\subset\{1,\ldots,N\}}$ be a family of real numbers indexed by all subsets of $\{1,\ldots,N\}$. It determines an averaged inner product $\langle A,B\rangle_{\sigma}=\sum_{S\subset\{1,\ldots,n\}}\sigma_{S}^{2}\cdot\langle A,B\rangle_{S}$ on $\mathbb{M}_{d^{N}}$. Let $\gamma_{d^{N}}^{\sigma}$ be the Gaussian measure on $\mathbb{H}_{d^{N}}$ whose characteristic function is given by

[TABLE]

and $\mu_{d^{N}}^{\sigma}$ be the Gaussian measure on $\mathbb{M}_{d^{N}}$ whose characteristic function is given by

[TABLE]

Denoting by $\mathrm{supp}\,\gamma_{d^{N}}^{\sigma}$ the support of $\gamma_{d^{N}}^{\sigma}$, which is a linear subspace of $\mathbb{H}_{d^{N}}$, we have $\mathrm{supp}\,\mu_{d^{N}}^{\sigma}=\mathrm{supp}\,\gamma_{d^{N}}^{\sigma}+i\mathrm{supp}\,\gamma_{d^{N}}^{\sigma}$. The linear space $\mathrm{supp}\,\gamma_{d^{N}}^{\sigma}$ can be endowed with a unique inner product such that $\gamma_{d^{N}}^{\sigma}$ is the standard Gaussian measure on $\mathrm{supp}\,\gamma_{d^{N}}^{\sigma}$, and therefore the Segal-Bargmann transform

[TABLE]

is well-defined as in (1.1). Following the model of Biane, we consider the two following Hilbert space tensor products $L^{2}(\mathbb{H}_{d^{N}},\gamma_{d^{N}}^{\sigma};\mathbb{M}_{d^{N}})=L^{2}(\mathbb{H}_{d^{N}},\gamma_{d^{N}}^{\sigma})\otimes\mathbb{M}_{d^{N}}$ and $\mathcal{H}L^{2}(\mathbb{M}_{d^{N}},\mu_{d^{N}}^{\sigma};\mathbb{M}_{d^{N}})=\mathcal{H}L^{2}(\mathbb{M}_{d^{N}},\mu_{d^{N}}^{\sigma})\otimes\mathbb{M}_{N}$, where $\mathbb{M}_{d^{N}}$ is endowed with the norm $\|M\|^{2}_{\mathbb{M}_{d^{N}}}=\mathrm{Tr}(MM^{*})/d^{N}$. Finally, we consider the boosted Segal-Bargmann transform

[TABLE]

Theorem 1.1** (see Theorem 3.14).**

Let $0\leq q\leq 1$. Under technical assumptions H.1, H.2, H.3 and H.4 on $\sigma$ (see Section 2.5) which ensure that, for all polynomial $P$,

[TABLE]

the Segal-Bargmann transform $\mathscrbf{S}_{d^{N}}$ converges to the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}$ in the following sense: for all polynomial $P$,

[TABLE]

We are able to prove Theorem 1.1 in the two parameter setting and in the multidimensional case.

1.3 Two Parameter Case

A simple scaling of $\mathscr{S}:L^{2}(H,\gamma)\to\mathcal{H}L^{2}(H^{\mathbb{C}},\mu)$ gives us a unitary isomorphism $\mathscr{S}^{t}$ which depends on one parameter $t>0$. It is also possible to consider one scaling for the space $L^{2}(H,\gamma)$ and another scaling for the transform $\mathscr{S}$. It yields to the two-parameter Segal-Bargmann transform $\mathscr{S}^{s,t}$, where $s$ and $t$ are two parameters with $s>\frac{t}{2}>0$, which was defined by Driver and Hall in [11, 13]. In this article, all the definition and results are considered in this two-parameter setting. In particular, we shall generalize the transform $\mathscr{S}_{q}$ of Leeuwen and Massen to a $q$-deformed Segal-Bargmann transform (with $-1<q<1$) given by

[TABLE]

where $\Gamma_{q}^{s,t}$ is a generating function and $\nu_{q}^{s}$ is scaled from $\nu_{q}$ so that it has variance $s$. With this formula, we are able to compute the range of the Segal-Bargmann transform, which is a reproducing kernel Hilbert space of analytic functions in an ellipse. It allows us to prove Theorem 3.14, which is a version of Theorem 1.1 with two parameters $s$ and $t$.

1.4 Multidimensional Case

In [3], Biane extends the free Segal-Bargmann transform $\mathscr{S}_{0}$ to the multidimensional case, replacing $\mathbb{R}$ by an arbitrary real Hilbert space $H$. The space $L^{2}(H,\gamma)$ has to be replaced by a non-commutative generalization of a $L^{2}$-space. More precisely, in the classical case, $L^{2}(H,\gamma)$ can be viewed as the space of square-integrable random variables generated by the Gaussian field on $H$. If $-1\leq q\leq 1$, it is possible to define some $q$-deformations of Gaussian field over $H$ (see Section 4.2). The free Segal-Bargmann transform $\mathscr{S}_{0}$ acts on the space of square-integrable random variables generated by a [math]-deformed Gaussian field on $H$ (called semicircular system in [3]).

In [16], Kemp generalizes Biane’s results and defines a $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}$ acting on the space of square-integrable random variables generated by a $q$-deformed Gaussian field on $H$. In [14], the second author defined the two-parameter free Segal-Bargmann transform $\mathscr{S}_{0}^{s,t}$ acting on the space of square-integrable random variables generated by a [math]-deformed Gaussian field on $H$. In this article, we will follow [3, 14, 16] and define the two-parameter $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ acting on the space of square-integrable random variables generated by a $q$-deformed Gaussian field on $H$. Of course, if we consider $H=\mathbb{C}$, the $q$-Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ is equivalent to the integral transform $\mathscr{S}_{q}^{s,t}$ already defined in (1.2); that is to say the integral transform gives an explicit formula of the $q$-Segal-Bargmann transform in the one dimensional setting (see Corollary 4.7).

Theorem 1.1 is true in the multidimensional case. Indeed, Theorem 4.8 shows that the two-parameter $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ acting on the space of square-integrable random variables generated by a $q$-deformed Gaussian field on $H$ can be approximated by the classical Segal-Bargmann transform.

1.5 Mixture of $q$-Deformed Segal-Bargmann Transform

In fact, it is possible to deform a Gaussian field over $\mathbb{R}^{n}$ in a much more complicated way, where a $q_{ii}$-deformed Gaussian random variable is considered for each direction of the canonical basis of $\mathbb{C}^{n}$, and where the correlation relation between two different variables is determined by some factors $q_{ij}$ ($q_{ij}=1$ yields the classical independence of random variables and $q_{ij}=0$ yields the free independence of random variables).

This deformation, first considered by Speicher in [25], is known as mixed $q$-Gaussian variables, and is uniquely determined by a symmetric matrix $Q=(q_{ij})_{1\leq i,j\leq n}$ with elements in $[-1,1]$. The case of the previous section corresponds to the case where all the elements of $Q$ are equal to a single $-1\leq q\leq 1$. It is also possible in this framework to define a $Q$-deformed Segal-Bargmann transform $\mathscr{S}_{Q}^{s,t}$, and restricted on the one-dimensional directions, $\mathscr{S}_{Q}^{s,t}$ yields to the already defined $\mathscr{S}_{q_{ii}}^{s,t}$. In particular, if all $q_{ii}$ are equal to [math], $\mathscr{S}_{Q}^{s,t}$ can be seen as a noncommutative mixture of the classical Segal-Bargmann transform $\mathscr{S}^{s,t}$ (see Remark 5.7).

In [24], Speicher proves the following central limit theorem: every $q$-deformed Gaussian random variable can be approximated by a normalised sum of mixed $q$-Gaussian variables for some appropriate choice of $Q$ with elements in $\{-1,1\}$. Similarly, Młotkowski proves in [18] that the elements of $Q$ can be chosen in $\{0,1\}$ in the central limit theorem of Speicher.

Our last result, summed up in Theorem 5.6, is the fact that the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ can be approximated by a noncommutative mixture $\mathscr{S}_{Q}^{s,t}$ of the classical Segal-Bargmann transform applied on normalised sum of mixed $q$-Gaussian variables (see Remark 5.7).

1.6 Organization of the Paper

A brief outline of the paper is as follows. In Section 2, we introduce the Segal-Bargmann transform $\mathscr{S}^{s,t}$, continue with a summary of the (mixed) $q$-random variables and end by a description of the random matrix model of Śniady. In Section 3, we introduce the two-parameter $q$-deformed Segal-Bargmann transform $\mathscr{S}^{s,t}_{q}$, and prove Theorem 1.1. In Section 4, we introduce the two-parameter $q$-deformed Segal-Bargmann transform in the multidimensional case, and prove Theorem 4.8, the analogue of Theorem 1.1 in this multidimensional setting. Finally, in Section 5, we introduce the mixture of $q$-deformed Segal-Bargmann transform, and prove Theorem 5.6.

2 Preliminaries

We begin by briefly introduce the already existing objects and results that will be useful for us: the two-parameter Segal-Bargmann transform, the $q$-deformation of the Gaussian measure, the $q$-deformation of independent Gaussian random variables and the model of random matrix of Śniady which allows to approximate those $q$-deformed Gaussian random variables.

2.1 Segal-Bargmann Transform

Let $H$ be a real finite-dimensional Hilbert space of dimension $d\geq 1$. For all $t>0$, we define $\gamma_{t}$ to be a Gaussian measure on $H$ whose density with respect to the Lebesgue measure at $x\in H$ is $(2\pi t)^{-d/2}\exp(-\|x\|^{2}/2t).$ For all $r,s>0$, we define $\gamma_{r,s}$ to be a Gaussian measure on the complexification $H^{\mathbb{C}}=H+iH$ of $H$ whose density with respect to the Lebesgue measure at $x+iy\in H^{\mathbb{C}}$ is $(2\pi\sqrt{rs})^{-d}\exp(-\|x\|^{2}/2r-\|y\|^{2}/2s)$. In other words, identifying $H^{\mathbb{C}}=H+iH$ with $H\times H$, we have $\gamma_{r,s}=\gamma_{r}\otimes\gamma_{s}$: the parameters $r$ and $s$ define the respective scaling of the Gaussian measure on the real and the imaginary part of $H$.

In [11], Driver and Hall introduced a general version of the Segal-Bargmann transform which depends on two parameters $s$ and $t$. Let $s>t/2>0$. For all $f\in L^{2}(H,\gamma_{s})$, the map

[TABLE]

has a unique analytic continuation $\mathscr{S}^{s,t}(f)$ to $H^{\mathbb{C}}$. Furthermore, the map $\mathscr{S}^{s,t}(f)$ is in the closed subspace of holomorphic functions of $L^{2}(H^{\mathbb{C}},\gamma_{s-t/2,t/2})$, denoted in the following by $\mathcal{H}L^{2}(H^{\mathbb{C}},\gamma_{s-t/2,t/2})$. The two parameter Segal-Bargmann transform is the isomorphism of Hilbert space

[TABLE]

The standard case considered by Segal and Bargmann corresponds to the case $s=t$, and the Segal-Bargmann $\mathscr{S}$ considered in the introduction corresponds to the case $s=t=1$.

2.2 $q$-Gaussian Measure

In this section, we will review some facts about $q$-Gaussian measures and $q$-Hermite polynomials. More discussions can be found in [6, 27].

Definition 2.1**.**

Let $-1<q<1$ and $t\geq 0$. The $q$-Gaussian measure $\nu_{q}$ of variance $1$ is defined to be

[TABLE]

where $\theta\in[0,\pi]$ is such that $x=2\cos\theta/\sqrt{1-q}$. The $q$-Gaussian measure $\nu_{q}^{t}$ of variance $t$ is given by $\nu_{q}^{t}(\,\mathrm{d}x)=\nu_{q}(\,\mathrm{d}x/\sqrt{t})$.

Let $-1<q<1$. For all integer $n$, set $[n]_{q}=1+\cdots+q^{n-1}$. For all $t\neq 0$, the $q$-Hermite polynomials $H_{n}^{q,t}$ of parameter $t$ are defined by $H_{0}^{q,t}(x)=1$, $H_{1}^{q}(x)=x$ and the recurrence relation

[TABLE]

They form an orthogonal family with respect to $\nu_{q}$ with norm $[n]_{q}!t^{n}$. Their generating function

[TABLE]

where $[n]_{q}!=\prod_{j=1}^{n}[j]_{q}$, converges whenever $|x|\leq\left|\frac{2\sqrt{t}}{\sqrt{1-q}}\right|$ and $|z|<\sqrt{\frac{t}{1-q}}$.

2.3 $q$-Gaussian Variables and Wick Product

Definition 2.2**.**

A non-commutative probability space $(\mathscr{A},\tau)$ is a unital $*$-algebra with a linear functional $\tau:\mathscr{A}\to\mathbb{C}$ such that $\tau[1_{\mathcal{A}}]=1$ and $\tau[A^{*}A]\geq 0$ for all $A\in\mathscr{A}$. The element of $\mathscr{A}$ are called random variables.

If $\mathcal{X}$ is a subset of $\mathscr{A}$, we denote by $L^{2}(\mathcal{X},\tau)$ the Hilbert space given by the completion of the (quotiented if necessary) space of the $*$-algebra generated by $\mathcal{X}$ with respect to the norm $\|A\|^{2}=\tau[A^{*}A]$, and by $\mathcal{H}L^{2}(\mathcal{X},\tau)$ the Hilbert space given by the completion of the (quotiented if necessary) space of the algebra generated by $\mathcal{X}$ with respect to the same norm.

The following definition of $q$-Gaussian variables can be considered as a $q$-deformation of the Wick formula of Gaussian variables (the classical case corresponds to $q=1$). Let $\mathcal{P}_{2}(n)$ be the set of pairing of $\{1,\ldots,n\}$. Let $\pi$ be a pairing of $\{1,\ldots,n\}$. A quadruplet $1\leq i<j<k<l\leq n$ is called a crossing of $\pi$ if $\{i,k\}\in\pi$ and $\{j,l\}\in\pi$. The number of crossings of the pairing $\pi$ is denoted by $\mathrm{cr}(\pi)$.

Definition 2.3**.**

Let $-1\leq q\leq 1$. A set $\mathcal{X}$ of self-adjoint and centred non-commutative random variables in a non-commutative probability space $(\mathscr{A},\tau)$ is said to be jointly $q$-Gaussian if, for all $X_{1},\ldots,X_{n}\in\mathcal{X}$, we have

[TABLE]

Two sets of jointly $q$-Gaussian variables $\mathcal{X}$ and $\mathcal{Y}$ are called $q$-independent if and only if $\mathcal{X}\cup\mathcal{Y}$ is jointly $q$-Gaussian and the elements of $\mathcal{X}$ are orthogonal with the elements of $\mathcal{Y}$ in $L^{2}(\mathscr{A},\tau)$.

A set $\mathcal{Z}$ of non-commutative centred random variables in a non-commutative probability space $(\mathscr{A},\tau)$ is said to be jointly $q$-Gaussian if $\{\Re Z,\Im Z:Z\in\mathcal{Z}\}$ is jointly $q$-Gaussian. Moreover, if $\tau[(\Re Z)^{2}]=s$ and $\tau[(\Im Z)^{2}]=t$, we say that $Z$ is a $(s,t)$-elliptic $q$-Gaussian variable.

Let $\mathcal{X}$ be any set of jointly $q$-Gaussian variables (not necessarily self-adjoint). Then, by linearity, it follows that the linear span of $\{X,X^{*}:X\in\mathcal{X}\}$ is also jointly $q$-Gaussian. If we take $X_{1}=\cdots=X_{n}=X=X^{*}$ in (2.3) and such that $\tau[X^{2}]=1$, we obtain the formula for the moments of the $q$-Gaussian measure $\nu_{q}$ :

[TABLE]

The $q$-Gaussian measure is called the distribution of the $q$-Gaussian variable $X$.

Remark 2.4*.*

A family of self-adjoint jointly $q$-Gaussian variables $(X_{i})_{i\in I}$ is, up to isomorphism, a $q$-Gaussian process as defined in [6], with covariance $c:I\times I\to\mathbb{R}$ given by $c(i,j)=\tau[X_{i}X_{j}]$. We can view them as operators acting on a $q$-deformation of the Fock space over $H$ (see Section 2.4). In the literature, for example in [6, 7, 10, 12, 16], the $q$-Gaussian variables have often been considered in this particular representation. Since our work only involves the non-commutative distribution of the $q$-Gaussian variables, we found more convenient to forget about the representation of a $q$-Gaussian variables and define it via its non-commutative distribution. This non-commutative distribution is implicitly given in [7, Proposition 2], or alternatively in [12, Corollary 2.1].

Definition 2.5**.**

Let $-1\leq q\leq 1$. Let $n\geq 0$. The Wick product of $n$ jointly $q$-Gaussian variables $X_{1},\ldots,X_{n}$, denoted by $X_{1}\diamond\cdots\diamond X_{n}$, is uniquely defined by the following recursion formula: the empty Wick product is $1$ and

[TABLE]

where the hat means that we omit the corresponding element in the product.

Remark 2.6*.*

The Wick product has been considered in [6] and [12] with different notation. Considering a set of self-adjoint jointly $q$-Gaussian variables $(X_{i})_{i\in I}$ as a $q$-Gaussian process (as defined in [6]) acting on the $q$-deformation of the Fock space over $L^{2}(X_{i},\tau)$, the Wick product $X_{1}\diamond\cdots\diamond X_{n}$ coincides with the quantity denoted by $\Psi(X_{1}\otimes\cdots\otimes X_{n})$ in [6, Definition 2.5] (they satisfy the same recursion formula thanks to [6, Proof of Proposition 2.7]). The Wick product $X_{1}\diamond\cdots\diamond X_{n}$ is denoted by ${:}X_{1}\cdots X_{n}{:}$ in [12].

In [12] is given an explicit formula for the Wick product of jointly $q$-Gaussian variables which are self-adjoint that we will present now. By linearity, the formula is also valid for non-necessarily self-adjoint variables. A Feynman diagram $\gamma$ on $\{1,\ldots,n\}$ is a partition of $\{1,\ldots,n\}$ into one- and two-element sets. The set of Feynman diagrams on $\{1,\ldots,n\}$ is denoted by $\mathcal{F}(n)$, and we have $\mathcal{P}_{2}(n)\subset\mathcal{F}(n)$. We extend naturally the notion of crossing to $\mathcal{F}(n)$: a quadruplet $1\leq i<j<k<l\leq n$ is called a crossing of $\gamma$ if $\{i,k\}\in\gamma$ and $\{j,l\}\in\gamma$. The number of crossings of a Feynman diagram $\gamma$ is denoted by $\mathrm{cr}(\gamma)$. Similarly, a triplet $1\leq i<j<k\leq n$ is called a gap of $\gamma$ if $\{i,k\}\in\gamma$ and $\{j\}\in\gamma$. The number of gaps of a Feynman diagram $\gamma$ is denoted by $\mathrm{gap}(\gamma)$. Finally, the number of pairings of a Feynman diagram $\gamma$ is denoted by $\sharp\gamma$.

Theorem 2.7** (Theorem 3.1 of [12]).**

The Wick product of jointly $q$-Gaussian variables $X_{1},\ldots,X_{n}$ is given by

[TABLE]

In the following proposition, we sum up some properties of the Wick product which can be found in [6] and in [12] for self-adjoint jointly $q$-Gaussian variables. The general case follows by linearity.

Proposition 2.8**.**

The Wick product is multilinear on the linear span of jointly $q$-Gaussian variables. 2. 2.

If $X_{1},\ldots,X_{n}$ is jointly $q$-Gaussian, $(X_{1}\diamond\cdots\diamond X_{n})^{*}=X_{n}^{*}\diamond\cdots\diamond X_{1}^{*}.$ 3. 3.

If $X_{1},\ldots,X_{n+m}$ is jointly $q$-Gaussian (with $n,m\geq 0$),

[TABLE] 4. 4.

If $\{X_{i}\}_{i\in I}$ is a set of jointly $q$-Gaussian variables, the set of Wick products

[TABLE]

is a spanning set of the algebra $\mathbb{C}\langle X_{i}:i\in I\rangle$ generated by $\{X_{i}\}_{i\in I}$.

2.4 Mixed $q$-Gaussian Variables

Let $Q=(q_{ij})_{i,j\in I}$ be a symmetric matrix with elements in $[-1,1]$. We recall now the construction of the mixed $q$-Gaussian variables operators $X_{i}=c_{i}+c_{i}^{*}$, where $c_{i}$ satisfy the commutation relations of the form

[TABLE]

We consider a complex Hilbert space $K$ with an orthonormal basis $\{e_{i}\}_{i\in I}$, and the algebraic full Fock space

[TABLE]

where $\Omega$ is a unit vector called the vacuum. The set of permutations of $\{1,\ldots,n\}$ is denoted by $\mathfrak{S}_{n}$, and a pair $1\leq a<b<l\leq n$ is called an inversion of a permutation $\pi\in\mathfrak{S}_{n}$ if $\pi(a)\geq\pi(b)$. We define the Hermitian form $\langle\cdot,\cdot\rangle_{Q}$ to be the conjugate-linear extension of

[TABLE]

The $Q$-Fock space $\mathcal{F}_{Q}(K)$ is the completion of the quotient of $\mathcal{F}(K)$ by the kernel of $\langle\cdot,\cdot\rangle_{Q}$. For any $i\in I$, define the left creation operator $c_{i}$ on $\mathcal{F}_{Q}(K)$ to extend

[TABLE]

The annihilation operator is its adjoint, which can be computed as

[TABLE]

Finally, we define the mixed $q$-Gaussian variables $X_{i}$ to be $c_{i}+c_{i}^{*}$. We can compute explicitly the mixed moment of those variables with respect to the vector state $\tau[\cdot]=\langle\cdot\Omega,\Omega\rangle_{Q}$.

Proposition 2.9** (Proof of Theorem 4.4 of [8]).**

We have

[TABLE]

As a consequence, the distribution of the variable $X_{i}$ is the $q_{ii}$-Gaussian measure. Let us remark that if all the $q_{ij}$ are equal to a single $q$, the set $\{X_{i}\}_{i\in I}$ is jointly $q$-Gaussian. Finally, let us mention that it is also possible to define some Wick product for mixed $q$-Gaussian variables: see [15, 17].

2.5 Random Matrix Model of Śniady

Let $d\geq 0$. We endow $\mathbb{M}_{d}$ by the inner products $\langle A,B\rangle_{1}=\frac{1}{d}\mathrm{Tr}(AB^{*})$ and $\langle A,B\rangle_{0}=\mathrm{Tr}(A)\mathrm{Tr}(B^{*}).$ For all $S\subset\{1,\ldots,N\}$, we define the inner product $\langle\cdot,\cdot\rangle_{S}$ on $\mathbb{M}_{d^{N}}\simeq\bigotimes_{r=1}^{N}\mathbb{M}_{d}$ to be the inner product of the Hilbert space tensor product $\bigotimes_{r=1}^{N}\left(\mathbb{M}_{d},\langle A,B\rangle_{\mathbbm{1}_{S}(r)}\right)$. Let $\sigma=(\sigma_{S})_{S\subset\{1,\ldots,N\}}$ be a family of real numbers indexed by all subsets of $\{1,\ldots,N\}$. We define the inner product on $\mathbb{M}_{d^{N}}$ given by

[TABLE]

In order to be concrete, let us compute the inner product of elementary matrices. Setting

[TABLE]

and, for all $\mathbf{i}=(i_{1},\ldots,i_{N}),\mathbf{j},\mathbf{k},\mathbf{l}\in\{1,\ldots,d\}^{N}$,

[TABLE]

we have, for all $\mathbf{i}=(i_{1},\ldots,i_{N}),\mathbf{j},\mathbf{k},\mathbf{l}\in\{1,\ldots,d\}^{N}$,

[TABLE]

Theorem 2.10** (Theorem 1 of [23]).**

Let $\mathcal{X}=\{X_{t}\}_{t\in T}$ be a set of self-adjoint variables which are jointly $q$-Gaussian.

For each $N\geq 0$, let $\sigma^{(N)}=(\sigma_{S}^{(N)})_{S\subset\{1,\ldots,N\}}$ be a family of real numbers, and let $\mathcal{X}^{(N)}=\{X_{t}^{(N)}\}_{t\in T}$ be the Gaussian stochastic process on $\mathbb{H}_{d^{N}}$ (indexed by $T$), uniquely defined by the following covariance: for all $M,N\in\mathbb{H}_{d^{N}}$ and all $s,t\in T$, one has

[TABLE]

In other words, the entries of the matrices in $\mathcal{X}^{(N)}$ are centered Gaussian variables with the following covariance: for all $\mathbf{i},\mathbf{j},\mathbf{k},\mathbf{l}\in\{1,\ldots,d\}^{N}$ and all $s,t\in T$, one has

[TABLE]

Under the technical assumptions H.1, H.2, H.3 and H.4, $\mathcal{X}^{(N)}$ converges to $\mathcal{X}$ in noncommutative distribution in the following sense: for all $t_{1},\ldots,t_{n}$, we have

[TABLE]

Before presenting the technical assumptions H.1, H.2, H.3 and H.4, let us present two simple examples of family of real numbers $\sigma^{(N)}=(\sigma_{S}^{(N)})_{S\subset\{1,\ldots,N\}}$ (for $N\geq 0$) fulfilling all assumptions. Those examples are taken from [23, Proposition 1 and 2]: if $q$ can be written as $q=\exp(c^{2}/d^{2}-c^{2})$ for a real number $c>0$, the sequence of functions defined by

[TABLE]

fulfils the assumptions of Theorem 2.10; if $q$ can be written as $q=\exp(c^{2}/d^{2}-c^{2})$ for a real number $c>0$, the sequence of functions defined for $N$ sufficiently large by

[TABLE]

fulfils the assumptions of Theorem 2.10.

Definition 2.11**.**

For each $N\geq 0$, let $\sigma^{(N)}=(\sigma_{S}^{(N)})_{S\subset\{1,\ldots,N\}}$ be a family of real numbers. The assumptions H.1, H.2, H.3 and H.4 are given as follow:

1. H.1

for each $N\in\mathbb{N}$,

[TABLE] 2. H.2

we have

[TABLE] 3. H.3

there exists a sequence $(p_{i})_{i\geq 0}$ of nonnegative real numbers such that $\sum_{i\geq 0}p_{i}=1$, $\sum_{i\geq 0}^{\infty}p_{i}/d^{2i}=q$, and such that, for any $k\in\mathbb{N}$ and any nonnegative integers numbers $(n_{ij})_{1\leq i<j\leq k}$, we have

[TABLE] 4. H.4

for each $k\in\mathbb{N}$,

[TABLE]

3 The Two-Parameter $q$-Deformed Segal-Bargmann Transform

In this section, we define the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ with parameters $s>t/2>0$ and prove Theorem 3.14, which reduces to Theorem 1.1 when $s=t=1$.

3.1 An Integral Representation

The integral representation for the one-parameter and the two-parameter cases are similar. The two-parameter case is in fact a generalization of the one-parameter; we separate here simply to make the presentation of the computations clearer.

One-Parameter Case

Definition 3.1**.**

Let $-1<q<1$ and $t\geq 0$. We define the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{t}$ by

[TABLE]

Observe that $\mathscr{S}_{q}^{t}H_{n}^{t}(z)=z^{n}$ and $\mathscr{S}_{q}^{t}$ is injective (by looking at the Fourier expansion of $L^{2}(\nu_{q}^{t})$).

Remark 3.2*.*

When $t=1$, the transform $\mathscr{S}_{q}^{t}$ coincides with the the transform $W$ from [28]. The method is different; while van Leeuwen and Maassen discovered the integral kernel by solving an eigenvalue equation [28, Equation (8)], we make use of the generating function directly to match the result from the Fock space. The method we present here will give us a two-parameter generalization in later sections.

Theorem 3.3**.**

The transform $\mathscr{S}_{q}^{t}$ is a unitary isomorphism between $L^{2}(\nu_{q}^{t})$ and the reproducing kernel Hilbert space $\mathcal{H}_{q}^{t}$ of analytic functions on the disk $B\left(0,\sqrt{\frac{t}{1-q}}\right)$ generated by the positive-definite sesqui-analytic kenel

[TABLE]

Proof.

Let us denote $\Gamma_{q}^{t}(x,z)$ by $\Gamma_{z}(x)$ for each $z\in B\left(0,\sqrt{\frac{t}{1-q}}\right)$ and $x\in\mathbb{R}$. We also write $K_{\zeta}(z)=K_{q}^{t}(z,\zeta)$ as an analytic function on $B\left(0,\sqrt{\frac{t}{1-q}}\right)$. Observe that

[TABLE]

Define $\mathcal{H}_{q}^{t}\equiv\mathscr{S}_{q}^{t}(L^{2}(\nu_{q}^{t}))$ equipped with the inner product

[TABLE]

which is well-defined since $\mathscr{S}_{q}^{t}$ is injective on $L^{2}(\nu_{q}^{t})$. By construction $\mathscr{S}_{q}^{t}$ is a unitary isomorphism between $L^{2}(\nu_{q}^{t})$ and $\mathcal{H}_{q}^{t}$. Finally, we see that $K_{\zeta}(z)=\mathscr{S}_{q}^{t}\bar{\Gamma}_{\zeta}(z)$ and, for any $F\in\mathcal{H}_{q}^{t}$,

[TABLE]

which shows that $K_{q}^{t}$ is a reproducing kernel for $\mathcal{H}_{q}^{t}$. ∎

Remark 3.4*.*

Since $\mathscr{S}_{q}^{t}$ coincides with $W$ from [28], the reproducing kernel Hilbert space $\mathcal{H}_{q}^{t}$ actually is equal to the space $H^{2}(D_{q},\mu_{q})$ considered in [28].

Analytic Continuation of a Generating Function

In this subsection, we study the analytic continuation on $y$ to the following generating function

[TABLE]

where $x,y\in[-1,1]$, $0<|r|<1$ which is either real or purely imaginary, and $h_{k}^{q}(x)=H_{k}^{q,s}(\frac{x}{2}\sqrt{1-q})$.

This formula is known as the $q$-Mehler formula and has been studied analytically and combinatorially; see e.g. [6, Theorem 1.10] or [19, Equation (24)]. By a standard theorem (see [20, Theorem 15.4]), the analytic continuation on the parameter $y$ of $\Lambda$ is to solve, for a single $0<|r|<1$ and all $x\in[-1,1]$, what $y$ make $1-4rq^{k}xy+2r^{2}q^{2k}(-1+2x+2y)-4r^{3}q^{3k}xy+r^{4}q^{4k}=0$. The equation

[TABLE]

has solution

[TABLE]

It follows that precisely when

[TABLE]

for some $x\in[-1,1]$, $\Lambda(r,x,y)$ has a zero for the particular $y$. Denote $\Omega_{k,r}$ the bounded component, which contains [math], of the complement of the ellipse. Let

[TABLE]

and

[TABLE]

The derivative $\varphi_{1}^{\prime}(u)=(-\log|q|)\left(\frac{1}{|r||q|^{u}}-|r||q|^{u}\right)>0$ implies $\varphi_{1}$ is increasing. Obviously $\varphi_{2}$ is increasing. Therefore $\Omega_{k,r}$ is increasing. Whence the $y$ parameter in $\Lambda(r,x,y)$ can be analytically continued to the ellipse $\Omega_{0,r}$.

Proposition 3.5**.**

The generating function $\Lambda(r,x,z)$ can be analytically continued to $x\in[-1,1]$ and $z\in\Omega_{0,r}$ which is an ellipse with major axis $[-\frac{1}{2}(1/|r|+|r|),\frac{1}{2}(1/|r|+|r|)]$ and minor axis $i[-\frac{1}{2}(1/|r|-|r|),\frac{1}{2}(1/|r|-|r|)]$.

Two-Parameter Case

We intend to define the integral $q$-Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ by

[TABLE]

where

[TABLE]

By [6, Theorem 1.10], this series converges for $|x|,|z|\leq 2\sqrt{t}/\sqrt{1-q}$ for real $x,z$.

**Case $s>t$:

**

It is easy to see that

[TABLE]

By proposition 3.5, $\Gamma_{q}^{s,t}(x,z)$ is defined as an analytic function on the ellipse $E_{s,t}$ with major axis

[TABLE]

and minor axis

[TABLE]

**Case $s<t$:

**

Similarly,

[TABLE]

By proposition 3.5, $\Gamma_{q}^{s,t}(x,z)$ is defined as an analytic function on the ellipse $E_{s,t}$ with major axis on the purely imaginary axis of length

[TABLE]

and minor axis on the real axis of length

[TABLE]

Remark 3.6*.*

When $q=0$, the ellipse coincides with the ellipse where the Brown measure of an elliptic element is distributed; see [4].

3.2 The Integral Transform

Definition 3.7**.**

Let $-1<q<1$ and $s>t/2>0$. We define the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ by

[TABLE]

for all $f\in L^{2}(\nu_{q}^{s})$. $\mathscr{S}_{q}^{s,t}f$ is an analytic function on the ellipse $E_{s,t}$.

Observe that $\mathscr{S}_{q}^{s,t}H_{n}^{s}(z)=H_{n}^{s-t}(z)$. The two-parameter analogue of Theorem 3.3 holds:

Theorem 3.8**.**

The transform $\mathscr{S}_{q}^{s,t}$ is a unitary isomorphism between $L^{2}(\nu_{q}^{s})$ and the reproducing kernel Hilbert space $\mathcal{H}_{q}^{s,t}$ of analytic functions on the ellipse $E_{s,t}$ generated by the positive-definite sesqui-analytic kenel

[TABLE]

3.3 Segal-Bargmann Transform and Conditional Expectation

The goal of this section is to prove Corollary 3.13, showing that the $q$-deformed Segal-Bargmann transform can be written as the action of a ”$q$-deformed heat kernel”. This result is already known for $q=0$, thanks to [9, Theorem 3.1].

Recall that the Wick product $X_{1}\diamond\cdots\diamond X_{n}$ is orthogonal in $L^{2}(\mathscr{A},\tau)$ to all products in $X_{1},\ldots,X_{n}$ of degree strictly less than $n$. Since $X_{1}\cdots X_{n}-X_{1}\diamond\cdots\diamond X_{n}$ is in the span of the products in $X_{1},\ldots,X_{n}$ of degree strictly less than $n$, $X_{1}\cdots X_{n}-X_{1}\diamond\cdots\diamond X_{n}$ can be seen as the orthogonal projection of $X_{1}\cdots X_{n}$ onto the span of the products in $X_{1},\ldots,X_{n}$ of degree strictly less than $n$. Because the Wick product can be seen as some orthogonal projection, the link with the conditional expectation is not surprising.

Definition 3.9**.**

Let $\mathcal{X}$ be a subset of a non-commutative space $(\mathscr{A},\tau)$. The conditional expectation

[TABLE]

is the orthogonal projection of $L^{2}(\mathscr{A},\tau)$ onto $L^{2}(\mathcal{X},\tau)$.

Remark 3.10*.*

If $(\mathscr{A},\tau)$ is a $W^{*}$-probability space, that is to say a von Neumann algebra with an appropriate $\tau$, the conditional expectation $\tau[\cdot|\mathcal{X}]$ maps $\mathscr{A}$ into the von Neumann algebra $W^{*}(\mathcal{X})$ generated by $\mathcal{X}$.

Proposition 3.11**.**

Let $\mathcal{X}$ and $\mathcal{Y}$ be two sets of jointly $q$-Gaussian variables which are $q$-independent. Let $X_{1},\ldots,X_{n}\in\mathcal{X}\cup\mathcal{Y}$. We have

[TABLE]

if one of the $X_{i}$s belongs to $\mathcal{Y}$, and $X_{1}\diamond\cdots\diamond X_{n}$ if all $X_{i}$s are in $\mathcal{X}$.

Proof.

If all $X_{i}$s are in $\mathcal{X}$, $X_{1}\diamond\cdots\diamond X_{n}$ is in $L^{2}(\mathcal{X},\tau)$ and the conditional expectation does not affect $X_{1}\diamond\cdots\diamond X_{n}$. If one of the $X_{i}$s belongs to $\mathcal{Y}$, it is sufficient to verify that $X_{1}\diamond\cdots\diamond X_{n}$ is orthogonal to $L^{2}(\mathcal{X},\tau)$, and it is an immediate consequence of the following fact: for all $X_{n+1},\ldots,X_{n+m}\in\mathcal{X}$,

[TABLE]

Indeed, using of Proposition 2.8, the computation of the trace always involves a factor $\tau[X_{i}X_{j}^{*}]$ between a $X_{i}\in\mathcal{Y}$ and a $X_{j}\in\mathcal{X}$, which vanishes. ∎

Corollary 3.12**.**

Let $\mathcal{X}=\{X_{i}\}_{i\in I}$, $\mathcal{Y}=\{Y_{j}\}_{j\in J}$ and $\mathcal{Z}=\{Z_{j}\}_{j\in J}$ be three sets of jointly $q$-Gaussian variables which are $q$-independent. The conditional expectations $\tau\left[\cdot|\mathcal{X}\cup\mathcal{Z}\right]:L^{2}(\mathscr{A},\tau)\to L^{2}(\mathcal{X}\cup\mathcal{Z},\tau)$ and $\tau\left[\cdot|\mathcal{X}\right]:L^{2}(\mathscr{A},\tau)\to L^{2}(\mathcal{X},\tau)$ coincide on $L^{2}(\mathcal{X}\cup\mathcal{Y},\tau)$.

Proof.

Thanks to Proposition 3.11, the two conditional expectations coincide on the Wick products of elements in $\mathcal{X}\cup\mathcal{Y}$ which is a dense subset of $L^{2}(\mathcal{X}\cup\mathcal{Y},\tau)$ (see Proposition 2.8). ∎

Corollary 3.13**.**

Let $Z$ be a $(s,t)$-elliptic $q$-Gaussian variable in $(\mathscr{A},\tau)$. If $Y$ is a $(t,0)$-elliptic $q$-Gaussian variable which is $q$-independent from $Z$, we have, for all polynomial $P$,

[TABLE]

Proof.

It suffices to prove the theorem for the Hermite polynomials $\{H^{q,s}_{n}\}_{n\geq 0}$. Because $\mathscr{S}_{q}^{s,t}H^{q,s}_{n}=H^{q,s-t}_{n}$ for all $n\geq 0$, we need to prove that, for all $n\geq 0$,

[TABLE]

We compute

[TABLE]

and we deduce the following equalities by induction:

[TABLE]

Let us conclude by the following computation where we use Proposition 3.11:

[TABLE]

∎

3.4 Random Matrix Model

Let $\gamma_{d^{N}}^{\sigma,t}$ be the Gaussian measure on $\mathbb{H}_{d^{N}}$ whose characteristic function is given by

[TABLE]

The measure $\gamma_{d^{N}}^{\sigma}$ is supported on the following vector subspace

[TABLE]

In particular, if $\|\cdot\|^{2}_{\sigma}$ is not faithful, $\gamma_{d^{N}}^{\sigma,t}$ is not absolutely continuous with respect to the Lebesgue measure on $\mathbb{H}_{d^{N}}$. However, $\gamma_{d^{N}}^{\sigma,t}$ is absolutely continuous with respect to the Lebesgue measure on the vector space $K_{\sigma}$. More precisely, using the Riesz representation theorem, let us define the linear map $\phi:K_{\sigma}\to K_{\sigma}$ to be the unique linear map such that, for all $x,y\in K_{\sigma}$, $\mathrm{Tr}(xy)=\langle\phi(x),y\rangle_{\sigma}.$ With respect to the Lebesgue measure on the vector space $K_{\sigma}$, the measure $\gamma_{d^{N}}^{\sigma,t}$ has density proportional to

[TABLE]

The quantity $\|\phi(x)\|_{\sigma}$ is known as the Mahalanobis distance from $x$ to [math], and it is the norm of $K_{\sigma}$ for which $\gamma_{d^{N}}^{\sigma,1}$ is the standard Gaussian measure.

We follows now Section 2.1 in order to define the Segal-Bargmann transform $\mathscr{S}^{s,t}$ on $L^{2}(K_{\sigma},\gamma_{d^{N}}^{\sigma,s})$. First, we consider the Gaussian measure $\mu_{d^{N}}^{\sigma,r,s}$ on $K_{\sigma}+iK_{\sigma}$ which is given by $\gamma_{d^{N}}^{\sigma,r}\otimes\gamma_{d^{N}}^{\sigma,s}$ when identifying $K_{\sigma}+iK_{\sigma}$ with $K_{\sigma}\times K_{\sigma}$. A short computation shows that $\mu_{d^{N}}^{\sigma,r,s}$ is the Gaussian measure on $\mathbb{M}_{d^{N}}$ whose characteristic function is given by

[TABLE]

The Segal-Bargmann transform

[TABLE]

is well-defined as in (2.2).

Following the model of Biane, we consider the two following Hilbert space tensor products

[TABLE]

and

[TABLE]

where $\mathbb{M}_{d^{N}}$ is endowed with the norm $\|M\|^{2}_{\mathbb{M}_{d^{N}}}=\mathrm{Tr}(MM^{*})/d^{N}$. Finally, we consider the boosted Segal-Bargmann transform

[TABLE]

Theorem 3.14**.**

Let $0\leq q<1$. Assuming (H.1), (H.2), (H.3) and (H.4) on $\sigma$ ensures that the Segal-Bargmann transform $\mathscrbf{S}_{d^{N}}^{s,t}$ converges to the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ in the following sense: for all polynomial $P$, we have

[TABLE]

Proof.

Let us denote by $Q$ the polynomial $\mathscr{S}_{q}^{s,t}P$.

For all $z\in\mathbb{H}_{d^{N}}$, we have

[TABLE]

Because both side are analytic in $z$, the equality is valid for all $z\in\mathbb{M}_{d^{N}}$. Thus we can compute

[TABLE]

Considering three independent random matrices $X^{(N)},Y^{(N)}$ and $Z^{(N)}$ of respective distribution $\gamma^{\sigma,t}$, $\gamma^{\sigma,t}$ and $\mu^{\sigma,s-t/2,t/2}$, we can rewrite

[TABLE]

Let $X,Y$ be two $(t,0)$-elliptic $q$-Gaussian random variables and $Z$ be a $(s-t/2,t/2)$-elliptic $q$-Gaussian random variable such that $X,Y$ and $Z$ are $q$-independent. Remark that, for any random Hermitian matrix $X^{(N)}$ distributed according to $\gamma_{d^{N}}^{\sigma,t}$, for all $M,N\in\mathbb{H}_{d^{N}}$, one has

[TABLE]

Moreover, for any random matrix $Z$ distributed according to $\mu_{d^{N}}^{\sigma,s-t/2,t/2}$, $\Re Z$ and $\Im Z$ are two independent Hermitian random matrices distributed according to $\gamma_{d^{N}}^{\sigma,s-t/2}$ and $\gamma_{d^{N}}^{\sigma,t/2}$. Thus, we can apply Theorem 2.10 which says that the Hermitian random matrices $X^{(N)},Y^{(N)},\Re Z^{(N)}$ and $\Im Z^{(N)}$ converge in noncommutative distribution to $X,Y,\Re Z$ and $\Im Z$. In particular, we have the following convergence:

[TABLE]

From Corollary 3.11 and Corollary 3.13, we know that

[TABLE]

Thus the limit $\tau\left[(P(Z+X)-Q(Z))(P(Z+Y)-Q(Z))^{*}\right]$ of $\left\|\mathscrbf{S}_{d^{N}}^{s,t}(P)-Q\right\|_{\mathcal{H}L^{2}(\mathbb{M}_{d^{N}},\mu_{d^{N}}^{\sigma,s-t/2,t/2};\mathbb{M}_{d^{N}})}$ vanishes:

[TABLE]

∎

4 Multidimensional $q$-Segal-Bargmann Transform

In this section, we will extend the definition of the $q$-Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ to a multidimensional setting, and prove Theorem 4.8, which says that Theorem 3.14 is also true in this new setting. In order to understand the multidimensional case for $-1\leq q\leq 1$, we decide first to explain the infinite-dimensional case for the classical Segal-Bargmann transform.

4.1 Classical Segal-Bargmann Transform in the Infinite-Dimensional Case

The content of this section is entirely expository. In Section 4.1, we shall define a version of the Segal-Bargmann transform in a probabilistic framework which allows to consider infinite-dimensional Hilbert spaces. In Section 4.2 and 4.1, we give two alternative descriptions of the Segal-Bargmann transform which are adapted to consider $q$-deformations.

In a probabilistic framework

In order to consider the $q$-deformation of this Segal-Bargmann transform, it is convenient to have a version of the $L^{2}$-spaces with more probabilistic flavor. Let $h\in H$. The continuous linear functional $\langle\cdot,h\rangle\in H^{*}$ can be considered as a random variable defined on the probability space $(H,\mathcal{B},\gamma_{s})$ (where $\mathcal{B}$ is the Borel $\sigma$-field of $H$). Let us denote by $\mathbf{X}(h)$ the linear functional $x\mapsto\langle x,h\rangle$ defined on $H$ and by $\mathbf{Z}(h)$ the linear functional $z\mapsto\langle z,h\rangle$ defined on $H^{\mathbb{C}}$. Because $H$ is finite-dimensional, the $\sigma$-field generated by the random variables $(\mathbf{X}(h))_{h\in H}$ is the Borel $\sigma$-field $\mathcal{B}$ of $H$. Denoting by $L^{2}(\mathbf{X})$ the random variables of $L^{2}(H,\mathcal{B},\gamma_{s})$ which are measurable with respect to the $\sigma$-field generated by the random variables $(\mathbf{X}(h))_{h\in H}$, we have $L^{2}(\mathbf{X})=L^{2}(H,\mathcal{B},\gamma_{s})$. Furthermore, it is well-known that the density in $L^{2}(H,\mathcal{B},\gamma_{s})$ of the algebra $\mathbb{C}[\mathbf{X}(h):h\in H]$ of polynomial variable follows from Hölder inequality. Finally, the three following Hilbert spaces are identical:

[TABLE]

In the same way, denoting by $\mathcal{H}L^{2}(\mathbf{Z})$ the completion of the algebra of random variables $\mathbb{C}[\mathbf{Z}(h):h\in H]$ in $L^{2}(H^{\mathbb{C}},\mathcal{B},\gamma_{s-t/2,t/2})$ we have the equality between the three following Hilbert spaces (where the first equality is a definition):

[TABLE]

The Segal-Bargmann map (2.2) can now be seen as an isomorphism between two spaces of random variables

[TABLE]

From the definition 2.1, the action of $\mathscr{S}^{s,t}$ on $\mathbb{C}[\mathbf{X}(h):h\in H]$ is easily described in the following way. The Hermite polynomials of parameter $s$ are defined by $H_{0}^{s}(x)=1$, $H_{1}^{s}(x)=x$ and the recurrence relation $xH_{n}^{s}(x)=H_{n+1}^{s}(x)+nsH_{n-1}^{s}(x).$ If $h_{1},\ldots,h_{k}$ is an orthonormal family of $H$, the Hermite polynomials $H_{n_{1}}^{s}(\mathbf{X}(h_{1}))\cdots H_{n_{k}}^{s}(\mathbf{X}(h_{k}))$ form an orthonormal family of $L^{2}(\mathbf{X})$ and the action of $\mathscr{S}^{s,t}$ on this basis is

[TABLE]

The formula (4.1) determines $\mathscr{S}^{s,t}$ on $\mathbb{C}[\mathbf{X}(h):h\in H]$ by linearity, and thus (4.1) determines uniquely $\mathscr{S}^{s,t}$ on $L^{2}(\mathbf{X})$ by continuity.

In the infinite dimensional case

The first approach of Section 2.1 can not extend directly to the infinite-dimensional setting because the Gaussian measures $\gamma_{s}$ do not make sense as measures on an infinite-dimensional Hilbert space. The dual point of view of Section 4.1 allows to define the Segal-Bargmann transform on infinite-dimensional Hilbert spaces. Indeed, $\mathbf{X}$ and $\mathbf{Z}$ of last section are particular cases of what we will called Gaussian fields. One has just to replace the underlying probability space $(H,\mathcal{B},\gamma_{s})$, which is not well-defined, by a sufficiently big one $(\Omega,\mathcal{F},\mathbb{P})$. In the following, the underlying probability space $(\Omega,\mathcal{F},\mathbb{P})$ will be completely arbitrary, but in concrete cases, the measure of reference $\mathbb{P}$ is often supported on a space $\Omega$ bigger than $H$. For example, in [11], the measure of reference $\mathbb{P}$ is a Wiener measure on a Wiener space whose Cameron-Martin space is $H$.

Let us fix an underlying probability space $(\Omega,\mathcal{F},\mathbb{P})$ and call random variables the measurable functions on $\Omega$. For all real Hilbert space, a linear map $\mathbf{X}$ from $H$ to the space of real random variables is called a Gaussian field on $H$ if, for all $h\in H$, $\mathbf{X}(h)$ is centered Gaussian with variance $\mathbb{E}[|\mathbf{X}(h)|^{2}]=\|h\|^{2}$. For all $r,s\geq 0$, a linear map $\mathbf{Z}$ from $H$ to the space of complex random variables is called an $(r,s)$-elliptic Gaussian field if it has the same distribution as $\sqrt{r}\mathbf{Z}_{1}+i\sqrt{s}\mathbf{Z}_{2}$, where $\mathbf{Z}_{1}$ and $\mathbf{Z}_{2}$ are two Gaussian fields on $H$ which are independent (in particular, an $(r,0)$-elliptic Gaussian field is real-valued and an $(0,s)$-elliptic Gaussian field is purely imaginary-valued). Let $r,s\geq 0$, and let $\mathbf{Z}$ be an $(r,s)$-elliptic Gaussian field. Following the last section, we define $\mathcal{H}L^{2}(\mathbf{Z})$ to be the completion of the algebra of random variables $\mathbb{C}[\mathbf{Z}(h):h\in H]$ in $L^{2}(\Omega,\mathcal{F},\mathbb{P})$. When $s=0$ or $r=0$, $\mathcal{H}L^{2}(\mathbf{Z})$ coincide with the random variables of $L^{2}(\Omega,\mathcal{F},\mathbb{P})$ which are measurable with respect to the $\sigma$-field generated by the random variables $(\mathbf{Z}(h))_{h\in H}$, and we will simply write $L^{2}(\mathbf{Z})$ instead of $\mathcal{H}L^{2}(\mathbf{Z})$.

Let $s>t/2\geq 0$. In Section 4.1, $\mathbf{X}$ was an $(s,0)$-elliptic Gaussian field and $\mathbf{Z}$ was an $(s-t/2,t/2)$-elliptic Gaussian field on a finite-dimensional Hilbert space $H$. Thanks to Section 4.1, we have the following proposition.

Proposition 4.1**.**

Let $H$ be a (possibly infinite-dimensional) Hilbert space $H$, $\mathbf{X}$ be an $(s,0)$-elliptic Gaussian field on $H$ and $\mathbf{Z}$ be an $(s-t/2,t/2)$-elliptic Gaussian field on $H$. The map given, for all orthonormal family $h_{1},\ldots,h_{k}$ of $H$, by

[TABLE]

is a well-defined isometry from $\mathbb{C}[\mathbf{X}(h):h\in H]$ to $\mathbb{C}[\mathbf{Z}(h):h\in H]$ which extends uniquely to an isomorphism of Hilbert space $\mathscr{S}^{s,t}:L^{2}(\mathbf{X})\to\mathcal{H}L^{2}(\mathbf{Z})$, called in the following the (two-parameter) Segal-Bargmann transform.

Segal-Bargmann transform and Wick products

In order to define $q$-deformation of the Segal-Bargmann transform, we give here a second description of the Gaussian fields and of the Segal-Bargmann transform defined in Section 4.1.

Let $\mathbf{X}$ be a Gaussian field on $H$. The Wick product is the result of the Gram-Schmidt process for the basis of $L^{2}(\mathbf{X})$ given by monomials. More precisely, for all $n\geq 0$ and $h_{1},\ldots,h_{n}\in H$, we define the Wick product $\mathbf{X}(h_{1})\diamond\cdots\diamond\mathbf{X}(h_{n})$ of $\mathbf{X}(h_{1}),\ldots,\mathbf{X}(h_{n})$ as the unique element of

[TABLE]

which is orthogonal to $Span\{\mathbf{X}(k_{1})\cdots\mathbf{X}(k_{m}):m\leq n,k_{1},\ldots,k_{m}\in H\}$, or equivalently, such that

[TABLE]

for all $m<n,k_{1},\ldots,k_{m}\in H$. In certain cases, the Wick product can be computed explicitly. For all $n\geq 0$, $m_{1},\ldots,m_{n}\geq 1$ and $h_{1},\ldots,h_{n}$ an orthonormal family of $H$, we have

[TABLE]

Let $\mathbf{Z}$ be a Gaussian $(s,t)$-elliptic system on $H$. In the same way, for all $n\geq 0$ and $h_{1},\ldots,h_{n}\in H$, we define the Wick product $\mathbf{Z}(h_{1})\diamond\cdots\diamond\mathbf{Z}(h_{n})$ of $\mathbf{Z}(h_{1}),\ldots,\mathbf{Z}(h_{n})$ as the unique element of

[TABLE]

which is orthogonal to $Span\{\mathbf{Z}(k_{1})\cdots\mathbf{Z}(k_{m}):m\leq n,k_{1},\ldots,k_{m}\in H\}$, or equivalently, such that

[TABLE]

for all $m<n,k_{1},\ldots,k_{m}\in H$. By multilinearity and the discussion above, for all $n\geq 0$, $m_{1},\ldots,m_{n}\geq 1$ and $h_{1},\ldots,h_{n}$ an orthonormal family of $H$, we have $\mathbf{Z}(h_{1})^{\diamond m_{1}}\diamond\cdots\diamond\mathbf{Z}(h_{n})^{\diamond m_{n}}=H_{m_{1}}^{s-t}(\mathbf{Z}(h_{1}))\cdots H_{m_{n}}^{s-t}(\mathbf{Z}(h_{n})).$

We are now able to give an alternative description of the Segal-Bargmann transform. Let $\mathbf{X}$ be a $(s,0)$-elliptic Gaussian system, and $\mathbf{Z}$ be a Gaussian $(s,t)$-elliptic system on $H$. From (4.2), we deduce that, for all orthonormal family $h_{1},\ldots,h_{k}$ of $H$, we have

[TABLE]

which can be generalized by multilinearity to the following.

Proposition 4.2**.**

Let $\mathbf{X}$ be a $(s,0)$-elliptic Gaussian system, and $\mathbf{Z}$ be a Gaussian $(s,t)$-elliptic system on $H$. For all $n\geq 0$ and $h_{1},\ldots,h_{n}\in H$, we have

[TABLE]

Segal-Bargmann transform and conditional expectations

In the proof of Theorem 4.8, we will need a third description of the Segal-Bargmann transform, which follows directly from the definition. Let $\mathbf{X}$ be a $(s,0)$-elliptic Gaussian system, and $\mathbf{Z}$ be a Gaussian $(s,t)$-elliptic system on $H$. If $\mathbf{Y}$ is a $(t,0)$-elliptic Gaussian system which is independent from $\mathbf{Z}$, we have, for all $P\in\mathbb{C}[x_{h}:h\in H]$,

[TABLE]

Because the formula only involves finitely many variables $h$ for each $P\in\mathbb{C}[x_{h}:h\in H]$, it is enough to prove the formula for finite-dimensional Hilbert spaces $H$. For convenience, we take the particular case of Section 4.1: $\mathbf{X}(h)$ is the linear functional $x\mapsto\langle x,h\rangle$ defined on $(H,\mathcal{B},\gamma_{s})$ and $\mathbf{Z}(h)$ the linear functional $z\mapsto\langle z,h\rangle$ defined on $(H^{\mathbb{C}},\mathcal{B},\gamma_{s-t/2,t/2})$. Let $P\in\mathbb{C}[x_{h}:h\in H]$. For all $z\in H$,

[TABLE]

The last line is also valid for all $z\in H^{\mathbb{C}}$, since each side is analytic. We recognize the conditioning of two independent set of variables: by enlarging the underlying probability space, we assume that there exists a $(t,0)$-elliptic Gaussian system $\mathbf{Y}$ independent from $\mathbf{Z}$ and rewrite the last equality as follows.

Proposition 4.3**.**

Let $\mathbf{X}$ be a $(s,0)$-elliptic Gaussian system, and $\mathbf{Z}$ be a Gaussian $(s,t)$-elliptic system on $H$. Let us assume that there exists a $(t,0)$-elliptic Gaussian system $\mathbf{Y}$ independent from $\mathbf{Z}$. For all $P\in\mathbb{C}[x_{h}:h\in H]$, we have

[TABLE]

4.2 The $q$-Deformation of the Segal-Bargmann Transform

Definition 4.4**.**

Let $-1\leq q\leq 1$. A $q$-Gaussian field $\mathbf{X}_{q}$ on $H$ is a linear map from $H$ to a non-commutative probability space $(\mathscr{A},\tau)$ which is an isometry for the $L^{2}$-norm and such that $(\mathbf{X}_{q}(h))_{h\in H}$ is jointly $q$-Gaussian.

A $(r,s)$-elliptic $q$-Gaussian field $\mathbf{Z}_{q}$ is a linear map from $H$ to a non-commutative probability space $(\mathscr{A},\tau)$ which can be decomposed as $\sqrt{r}\mathbf{X}_{q}+i\sqrt{s}\mathbf{Y}_{q}$, where $\mathbf{X}_{q}$ and $\mathbf{Y}_{q}$ are two $q$-Gaussian field which are $q$-independent. Elliptic $q$-Gaussian fields are $q$-independent if the previous decomposition holds simultaneously with $q$-Gaussian fields which are all $q$-independent.

The following definition of the Segal-Bargmann transform in the infinite-dimensional case coincide with the classical Segal-Bargmann transform if $q=1$, with the definition of Kemp in [16] if $s=t$, and with the definition of the second author in [14] if $q=0$.

Proposition / Definition 4.5**.**

Let $\mathbf{X}_{q}$ be a $q$-Gaussian $(s,0)$-elliptic system, and $\mathbf{Z}_{q}$ be a $q$-Gaussian $(s-t/2,t/2)$-elliptic system from $H$ to $\mathscr{A}$. The ($q$-deformed) Segal-Bargmann transform $\mathscr{S}^{s,t}_{q}$ is the unique unitary isomorphism from $L^{2}(\mathbf{X}_{q},\tau)$ to $\mathcal{H}L^{2}(\mathbf{Z}_{q},\tau)$ such that, for all $h_{1},\ldots,h_{n}\in H$,

[TABLE]

We will see in Corollary 4.7 that this transform is indeed a generalization of Definition 3.7.

Proof.

The unicity is clear. It remains to prove the existence and the unitarity. Let us first remark that, for all $h,k\in H$, we have

[TABLE]

Combined with Proposition 2.8, it follows that, for all $h_{1},\ldots,h_{n}\in H$ and $k_{1},\ldots,k_{m}\in H$,

[TABLE]

We deduce the existence of the unitary linear map $\mathscr{S}^{s,t}_{q}$ from $\mathbb{C}\langle\mathbf{X}_{q}(h):h\in H\rangle$ to $\mathbb{C}\langle\mathbf{Z}_{q}(h):h\in H\rangle$ given by (4.6), and we extend this map to $\mathscr{S}^{s,t}_{q}:L^{2}(\mathbf{X}_{q},\tau)\to\mathcal{H}L^{2}(\mathbf{Z}_{q},\tau)$ by density. ∎

Here again, the $q$-deformed Segal-Bargmann transform can be seen as the action of a ”$q$-deformed heat kernel”, a result which extends [9, Theorem 3.1] to $-1\leq q\leq 1$.

Theorem 4.6**.**

Let $\mathbf{X}_{q}$ be a $q$-Gaussian $(s,0)$-elliptic system, and $\mathbf{Z}_{q}$ be a $q$-Gaussian $(s,t)$-elliptic system from $H$ to $\mathscr{A}$. If $\mathbf{Y}_{q}$ is a $q$-Gaussian $(t,0)$-elliptic system which is $q$-independent from $\mathbf{Z}_{q}$, we have, for all noncommutative polynomial $P\in\mathbb{C}\langle x_{h}:h\in H\rangle$,

[TABLE]

Proof.

For all $h_{1},\ldots,h_{n}$, we define a polynomial $P_{h_{1},\ldots,h_{n}}\in\mathbb{C}\langle x_{h}:h\in H\rangle$ by the following recursion formula: $P_{\varnothing}=1$ and

[TABLE]

where the hat means that we omit the corresponding element in the product. Since $\{P_{h_{1},\ldots,h_{n}}\}_{n\geq 0,h_{1},\ldots,h_{n}\in H}$ is a spanning set of $\mathbb{C}\langle x_{h}:h\in H\rangle$, it suffices to prove the theorem for those polynomials. Remark that, for all $h,k\in H$, $\tau[\mathbf{X}_{q}(h)\mathbf{X}_{q}(k)]=s\langle h,k\rangle_{H}$. Consequently, the variables $P_{h_{1},\ldots,h_{n}}(\mathbf{X}_{q}(h):h\in H)$ and $\mathbf{X}_{q}(h_{1})\diamond\cdots\diamond\mathbf{X}_{q}(h_{n})$ satisfies the same recursion formula, and we have

[TABLE]

Similarly, we compute

[TABLE]

and we deduce the following equality by induction:

[TABLE]

Let us conclude by the following computation where we use Proposition 3.11 to compute the conditional expectation:

[TABLE]

∎

Combining Theorem 4.6 with Corollary 3.13, we get the following result which relies Definition 3.7 of $\mathscr{S}^{s,t}_{q}P$ and Definition 4.5 of $\mathscr{S}^{s,t}_{q}\left(P(\mathbf{X}_{q}(h))\right)$ for one polynomial $P$.

Corollary 4.7**.**

Let $-1<q<1$. For a unit vector $h$ and a polynomial $P$, we have

[TABLE]

4.3 Large N Limit

Let us construct a boosted version of the Gaussian $(s,0)$-elliptic system on a Hilbert space $H$. Let us consider the tensor product Hilbert space $H\otimes_{\mathbb{R}}\mathbb{H}_{d^{N}}$ of $H$ with $(\mathbb{H}_{d^{N}},\|\cdot\|_{\sigma})$. Let $\mathbf{X}:H\otimes_{\mathbb{R}}\mathbb{H}_{d^{N}}\to L^{2}(\mathbf{X})\subset L^{2}(\Omega,\mathcal{F},\mathbb{P})$ be a Gaussian $(s,0)$-elliptic system. We define

[TABLE]

by duality as the unique linear map $\mathbf{X}^{(N)}$ from $H$ to the random variables with value in $\mathbb{M}_{d^{N}}$ such that $\mathbf{X}(h\otimes M)=\mathrm{Tr}(M\mathbf{X}^{(N)}(h)).$ Each variable $\mathrm{Tr}(M\mathbf{X}^{(N)}(h))$ is Gaussian with the covariance given by

[TABLE]

In other words, if the norm of $h$ is $1$, the distribution of the random matrix $\mathbf{X}^{(N)}(h)$ is the Gaussian distribution $\gamma_{d^{N}}^{\sigma,s}$ of Section 3.4, and if $k$ is another vector orthogonal to $h$, the random matrices $\mathbf{X}^{(N)}(h)$ and $\mathbf{X}^{(N)}(k)$ are independent.

Similarly, let $\mathbf{Z}:H\otimes_{\mathbb{R}}\mathbb{H}_{d^{N}}\to L^{2}(\mathbf{X})\subset L^{2}(\Omega,\mathcal{F},\mathbb{P})$ be a Gaussian $(s-t/2,t/2)$-elliptic system. We define

[TABLE]

by duality as the unique linear map $\mathbf{Z}^{(N)}$ from $H$ to the random variables with value in $\mathbb{M}_{d^{N}}$ such that $\mathbf{Z}(h\otimes M)=\mathrm{Tr}(M\mathbf{Z}^{(N)}(h)).$ The Segal-Bargmann transform $\mathscr{S}^{s,t}:L^{2}(\mathbf{X})\to\mathcal{H}L^{2}(\mathbf{Z})$ is well-defined as in (4.1). Finally, we consider the following boosted Segal-Bargmann transform

[TABLE]

Theorem 4.8**.**

Let $0\leq q\leq 1$. Assuming (H.1), (H.2), (H.3) and (H.4) on $\sigma$ ensures that the Segal-Bargmann transform $\mathscrbf{S}_{d^{N}}^{s,t}$ converges to the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}:L^{2}(\mathbf{X}_{q},\tau)\to\mathcal{H}L^{2}(\mathbf{Z}_{q},\tau)$ in the following sense: for all polynomial $P$ and $Q\in\mathbb{C}\langle x_{h}:h\in H\rangle$ such that

[TABLE]

the norm $\left\|P(\mathbf{X}^{(N)}(h):h\in H)\right\|_{L^{2}(\mathbf{X})\otimes\mathbb{M}_{d^{N}}}=\left\|\mathscrbf{S}_{d^{N}}^{s,t}(P(\mathbf{X}^{(N)}(h):h\in H))\right\|_{\mathcal{H}L^{2}(\mathbf{Z})\otimes\mathbb{M}_{d^{N}}}$ converges, as $N$ tends to $\infty$, to $\left\|P(\mathbf{X}_{q}(h):h\in H)\right\|_{L^{2}(\mathbf{X}_{q},\tau)}=\left\|Q(\mathbf{Z}_{q}(h):h\in H)\right\|_{\mathcal{H}L^{2}(\mathbf{Z}_{q},\tau)}$ and

[TABLE]

Proof.

Remark that, for all $M,N\in\mathbb{H}_{d^{N}}$, and all $h,k\in H$, we have

[TABLE]

We can apply Theorem 2.10 which says that the random matrices $\{\mathbf{X}^{(N)}(h):h\in H\}$ converge in noncommutative distribution to $\{\mathbf{X}_{q}(h):h\in H\}$. In particular, we have the following convergences:

[TABLE]

The proof of the second limit uses the following lemma. Let $\mathbf{Y}:H\otimes_{\mathbb{R}}\mathbb{H}_{d^{N}}\to L^{2}(\mathbf{Y})\subset L^{2}(\Omega,\mathcal{F},\mathbb{P})$ be a Gaussian $(t,0)$-elliptic system independent from $(\mathbf{Z}^{(N)}(h))_{h\in H}$, and define

[TABLE]

by duality as the unique linear map $\mathbf{Y}^{(N)}$ from $H$ to the random variables with value in $\mathbb{M}_{d^{N}}$ such that $\mathbf{Y}(h\otimes M)=\mathrm{Tr}(M\mathbf{Y}^{(N)}(h)).$

Lemma 4.9**.**

For all $P\in\mathbb{C}[x_{h}:h\in H]$, we have

[TABLE]

Proof of Lemma 4.9.

One can apply (4.5) for each coordinate of $P(\mathbf{X}^{(N)}(h):h\in H)$ in any basis of $\mathbb{M}_{d^{N}}$. Alternatively, one can reason as follows.

Because the formula only involves finitely many variables $h$ for each $P\in\mathbb{C}[x_{h}:h\in H]$, it is enough to prove the formula for finite-dimensional Hilbert spaces $H$. Without loss of generality, we take the particular case of Section 4.1: $\mathbf{X}(h\otimes M)$ is the linear functional $x\otimes N\mapsto\langle x,h\rangle_{H}\mathrm{Tr}(NM)$ defined on $(H\otimes\mathbb{H}_{d^{N}},\mathcal{B},\gamma_{s})$, $\mathbf{Y}(h\otimes M)$ is the linear functional $x\otimes N\mapsto\langle x,h\rangle_{H}\mathrm{Tr}(NM)$ defined on $(H\otimes\mathbb{H}_{d^{N}},\mathcal{B},\gamma_{t})$ and $\mathbf{Z}(h\otimes M)$ the linear functional $z\otimes N\mapsto\langle z,h\rangle\mathrm{Tr}(N^{*}M)$ defined on $(H^{\mathbb{C}}\otimes\mathbb{M}_{d^{N}},\mathcal{B},\gamma_{s-t/2,t/2})$. We consider then the matrix-valued random variables $\mathbf{X}^{(N)}(h):x\otimes M\mapsto\langle x,h\rangle_{H}\cdot M$, $\mathbf{Y}^{(N)}(h):x\otimes M\mapsto\langle x,h\rangle_{H}\cdot M$ and $\mathbf{Z}^{(N)}(h):z\otimes M\mapsto\langle z,h\rangle_{H^{\mathbb{C}}}\cdot M$.

Let $P\in\mathbb{C}[X_{h}:h\in H]$. We use here the definition (2.1) of the Segal-Bargmann transform $\mathscr{S}^{s,t}$, which is also valid for $\mathscrbf{S}^{s,t}$ by linearity: for all $z\otimes M\in H\otimes\mathbb{H}_{d^{N}}$,

[TABLE]

[TABLE]

The last line is also valid for all $z\otimes M\in H^{\mathbb{C}}\otimes\mathbb{M}_{d^{N}}$, since each side is analytic in $z\otimes M$. We recognize the wanted conditioning of Lemma 4.9. ∎

Let us consider an independent copy $\mathbf{W}^{(N)}$ of $\mathbf{Y}^{(N)}$. We consider also two $q$-Gaussian $(t,0)$-elliptic system $\mathbf{W}_{q}$ and $\mathbf{Y}_{q}$ which are $q$-independent from each others and from $\mathbf{Z}_{q}$. Remark that, for all $M,N\in\mathbb{H}_{d^{N}}$, and all $h,k\in H$, we have

[TABLE]

where the symbols $\mathbf{U}$ and $\mathbf{V}$ can be replaced by any from the symbols $\mathbf{W}_{q},\mathbf{Y}_{q},\Re\mathbf{Z}_{q}$ and $\Im\mathbf{Z}_{q}$. Thus, we can apply Theorem 2.10 which says that the random matrices $\{\mathbf{W}^{(N)}(h),\mathbf{Y}^{(N)}(h),\Re\mathbf{Z}^{(N)}(h),\Im\mathbf{Z}^{(N)}(h):h\in H\}$ converge in noncommutative distribution to $\{\mathbf{W}_{q}(h),\mathbf{Y}_{q}(h),\Re\mathbf{Z}_{q}(h),\Im\mathbf{Z}_{q}(h):h\in H\}$. In particular, we have the following convergence:

[TABLE]

The last quantity vanishes because Theorem 4.6 tells us that

[TABLE]

∎

5 Mixture of Classical and Free Segal-Bargmann Transform

In this section, we shall define the Segal-Bargmann transform for a mixture of classical and free random variables and then we recover the $q$-Segal Bargmann trasnform in the limit.

5.1 The Mixed $q$-Deformed Segal-Bargmann Transform

Let $Q=(q_{ij})_{i,j\in I}$ be a symmetric matrix with elements in $[-1,1]$. We consider a complex Hilbert space $K$ with an orthonormal basis $\{e_{i}\}_{i\in I}$, the Fock space $\mathcal{F}_{Q}(K)$, and the set of mixed $q$-Gaussian variables $\{X_{i}\}_{i\in I}$ acting on $\mathcal{F}_{Q}(K)$ as defined in Section 2.4. The set $\{\sqrt{s}X_{i}\}_{i\in I}$ are the mixed $q$-Gaussian variables of variance $s$. Remark that the map $A\mapsto A(\Omega)$ extend to a unitary isomorphism from $L^{2}(\{\sqrt{s}X_{i}\}_{i\in I},\tau)$ to $\mathcal{F}_{Q}(K)$.

As in Section 2.3, we will define the mixed $q$-Gaussian $(s,t)$-elliptic variables as a set of variables $\{Z_{i}\}_{i\in I}$ indexed by $I$ such that $\{\Re Z_{i},\Im Z_{i}\}_{i\in I}$ are a set of mixed $q$-Gaussian variables with prescribed variance. The first step is to replace the index set $I$ by the index set $I\times\{1,2\}$, and the matrix $Q$ by the matrix

[TABLE]

We consider the complex Hilbert space $K^{2}$ with an orthonormal basis $\{e_{(i,1)},e_{(i,2)}\}_{i\in I}$. Considering the Fock space $\mathcal{F}_{\tilde{Q}}(K^{2})$, we define the set of mixed $q$-Gaussian variables $\{X_{(i,1)},X_{(i,2)}\}_{i\in I}$ acting on $\mathcal{F}_{\tilde{Q}}(K^{2})$ as defined in Section 2.4. Finally, we set the mixed $q$-Gaussian $(s-t/2,t/2)$-elliptic variables

[TABLE]

Remark that the map $A\mapsto A(\Omega)$ extend to a unitary isomorphism from $\mathcal{H}L^{2}(\{Z_{i}\}_{i\in I},\tau)$ to $\mathcal{F}_{\tilde{Q}}(K^{2})$.

We are ready to define the mixed $q$-deformed Segal-Bargmann transform.

Definition 5.1**.**

The mixed $q$-deformed Segal-Bargmann transform $\mathscr{S}_{Q}^{s,t}$ is the unitary isomorphism so that the following diagram commute:

[TABLE]

where $\delta_{Q}^{s,t}$ is the Fock space extension of $\delta_{Q}^{s,t}(\sqrt{s}e_{i})=\sqrt{s-t/2}e_{(i,1)}+i\sqrt{t/2}e_{(i,2)}$, meaning that

[TABLE]

For all $i\in I$, we have

[TABLE]

Indeed, the definition of the Hermite polynomials is adjusted with the definition (2.6) of the annihilation operator $c^{*}_{i(1)}$ in such a way that, by a direct induction, for all $n\geq 2$, we have

[TABLE]

Similarly, we have

[TABLE]

We deduce the following result, which says that restricted on the different $L^{2}(X_{i},\tau)$, the mixed $q$-deformed Segal-Bargmann transform $\mathscr{S}_{Q}^{s,t}$ coincides with the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q_{ii}}^{s,t}$.

Proposition 5.2**.**

For all $i\in I$ and all polynomial $P$, we have

[TABLE]

5.2 The $q$-Segal-Bargmann Transform in the Limit

Set $I=\mathbb{N}$. We choose $Q=(q_{ij}=q_{ji})_{i,j\in I}$ randomly in $\{-1,+1\}$ or in $\{0,+1\}$, as independent random variables, identically distributed, with $\mathbb{E}[q_{ij}]=q$. We consider the mixed $q$-Gaussian variables $\{\sqrt{s}X_{i}\}_{i\in I}$ of variance $s$ as defined in the previous section.

Remark 5.3*.*

Let us recall first that $q_{ii}=-1,0$ or $1$ means respectively that $X_{i}$ is a Bernoulli variable, a semicircular variable or a Gaussian variable. Secondly, $q_{ii}=0$ or $1$ means respectively that $X_{i}$ and $X_{j}$ are freely independent or classically independent.

Let us consider the sum

[TABLE]

These variables define an approximation of a $q$-Gaussian variable. Speicher’s central limit theorem ([24, Theorem 1]) makes this statement precise whenever $q_{ij}\in\{-1,+1\}$. If $q_{ij}\in\{0,+1\}$, it is not complicated (using for example the characterisation with cumulants of [26]) to prove that we fall in the framework of $\Lambda$-freeness of Młotkowski. More precisely, if we define $\Lambda$ to be the set of $\{i,j\}$ such that $q_{ij}=1$ and $i\neq j$, the algebras generated by the different $X_{i}$ are $\Lambda$-free. We can use Młotkowski’s central limit theorem ([18, Theorem 4]) and we get the following result.

Theorem 5.4** (Theorem 1 of [24] and Theorem 4 of [18]).**

Almost surely, the variable $X^{(n)}$ converges to a $q$-Gaussian variable $X$ of variance $s$ in noncommutative distribution in the following sense: for all polynomial $P$, we have

[TABLE]

We consider now the mixed $q$-Gaussian $(s-t/2,t/2)$-elliptic variables $\{Z_{i}\}_{i\in I}$, where the relations are governed by the matrix $\tilde{Q}$, and we set

[TABLE]

The entries of $\tilde{Q}$ are not any more independent but only block-independent. Nevertheless, as used in [16, Section 4.2], a straightforward modification of Speicher’s proof and of Młotkowski’s proof generalizes the theorem to this case.

Theorem 5.5** (Theorem 1 of [24] and Theorem 4 of [18]).**

Almost surely, the variable $Z^{(n)}$ converges to a $q$-Gaussian $(s-t/2,t/2)$-elliptic variable $Z$ in noncommutative distribution in the following sense: for all polynomial $P$, we have

[TABLE]

The following theorem says that the mixed $q$-Segal-Bargmann transform is also an approximation of the $q$-deformed case.

Theorem 5.6**.**

Set $I=\mathbb{N}$. We choose $Q=(q_{ij}=q_{ji})_{i,j\in I}$ randomly in $\{-1,+1\}$ or in $\{0,+1\}$, as independent random variables, and identically distributed with $\mathbb{E}[q_{ij}]=q$ for $i>j$. We consider the mixed $q$-Gaussian variables $\{\sqrt{s}X_{i}\}_{i\in I}$ of variance $s$, the mixed $q$-Gaussian $(s-t/2,t/2)$-elliptic variables $\{Z_{i}\}_{i\in I}$ and the mixed $q$-Segal-Bargmann transform $\mathscr{S}_{Q}^{s,t}:L^{2}(\{X_{i}\}_{i\in I},\tau)\to\mathcal{H}L^{2}(\{Z_{i}\}_{i\in I},\tau)$.

Almost surely, the Segal-Bargmann transform $\mathscr{S}_{Q}^{s,t}$ converges to the $q$-deformed Segal-Bargmann transform $\mathscr{S}_{q}^{s,t}$ in the following sense: considering the sums

[TABLE]

for all polynomial $P$, we have $\displaystyle\lim_{n\to\infty}\left\|\mathscr{S}_{Q}^{s,t}\left(P(X^{(n)})\right)-\mathscr{S}_{q}^{s,t}P(Z^{(n)})\right\|_{\mathcal{H}L^{2}(\{Z_{i}\}_{i\in I},\tau)}=0.$

Remark 5.7*.*

• •

We can choose $q_{ii}$ arbitrarily. For example, if we choose $q_{ii}=1$, Proposition 5.2 tells us that $\mathscr{S}_{Q}^{s,t}$ restricted to $L^{2}(\sqrt{s}X_{i},\tau)$ is the classical Segal-Bargmann transform $\mathscr{S}_{q_{ii}}^{s,t}$.

• •

Now, assume that $q_{ij}\in\{0,+1\}$. We define $\Lambda$ to be the (random) set of $\{i,j\}$ such that $q_{ij}=1$ and $i\neq j$. The algebras generated by the different $X_{i}$ are $\Lambda$-free in the sense of [18]. Decomposing $L^{2}(\sqrt{s}X_{i},\tau)=L^{\circ}_{i}\oplus\mathbb{C}$ (with $L^{\circ}_{i}$ composed of the operators $A$ such that $\tau[A]=0$), and $L^{2}(Z_{i},\tau)=\mathcal{H}L^{\circ}_{i}\oplus\mathbb{C}$ decomposed similarly, we have the $\Lambda$-free products observed in [18]:

[TABLE]

[TABLE]

where $S(I,\Lambda)$ is the set of reduced words over $I$ modulo the relations $(\ldots,i,j,\ldots)\simeq(\ldots,j,i,\ldots)$ if $\{i,j\}\in\Lambda$ and $(\ldots,i,i,\ldots)\simeq(\ldots,i,\ldots)$, or, more specifically, a set of representatives of minimal length. Finally, $\mathscr{S}_{Q}^{s,t}:L^{2}(\{X_{i}\}_{i\in I},\tau)\to\mathcal{H}L^{2}(\{Z_{i}\}_{i\in I},\tau)$ can be decomposed as

[TABLE]

or as a $\Lambda$-free product of classical Segal-Bargmann transform whenever $q_{ii}=1$ for all $i\in I$.

Proof.

We consider the index set $I\times\{1,2,3,4\}$, and the matrix

[TABLE]

We consider the complex Hilbert space $K^{4}\supset K^{2}$ with an orthonormal basis $\{e_{(i,1)},e_{(i,2)},e_{(i,3)},e_{(i,4)}\}_{i\in I}$ and the Fock space $\mathcal{F}_{R}(K^{4})$. We have the canonical inclusion $\mathcal{F}_{\tilde{Q}}(K^{2})\subset\mathcal{F}_{R}(K^{4})$ given by the natural extension of $K^{2}\subset K^{4}$. We define the set of mixed $q$-Gaussian variables $\{X_{(i,1)},X_{(i,2)},X_{(i,3)},X_{(i,4)}\}_{i\in I}$ acting on $\mathcal{F}_{\tilde{Q}}(K^{4})$ as defined in Section 2.4, which is an extension of the already defined action of $\{X_{(i,1)},X_{(i,2)}\}_{i\in I}$ on $\mathcal{F}_{\tilde{Q}}(K^{2})$. The action of the mixed $q$-Gaussian $(s-t/2,t/2)$-elliptic variables $Z_{i}$ extends to $\mathcal{F}_{R}(K^{4})$ by

[TABLE]

and the action of $Z^{(n)}$ extends to $\mathcal{F}_{R}(K^{4})$ by

[TABLE]

Finally, we define also

[TABLE]

and

[TABLE]

Lemma 5.8**.**

For all polynomial $Q$, we have

[TABLE]

Proof of Lemma.

For all $i(1),\ldots,i(n)\in I$, we define a polynomial $P_{i(1),\ldots,i(n)}\in\mathbb{C}\langle x_{i}:i\in I\rangle$ by the following recursion formula: $P_{\varnothing}=1$ and

[TABLE]

where the hat means that we omit the corresponding index. Since $\{P_{i(1),\ldots,i(n)}\}_{n\geq 0,i(1),\ldots,i(n)\in I}$ is a spanning set of $\mathbb{C}\langle x_{i}:i\in I\rangle$, it suffices to prove the theorem for those polynomials.

We have

[TABLE]

Indeed, the definition of the polynomials $\{P_{i(1),\ldots,i(n)}\}_{n\geq 0,i(1),\ldots,i(n)\in I}$ is adjusted with the definition (2.6) of the annihilation operator $c^{*}_{i(1)}$ in such a way that, by a direct induction, for all $i(1),\ldots,i(n)\in I$, we have

[TABLE]

Similarly, setting $h_{i}=\sqrt{s-t/2}e_{(i(n),1)}+i\sqrt{t/2}e_{(i(n),2)}+\sqrt{s}e_{(i(n),3)}$, it follows from the definition of the polynomials $\{P_{i(1),\ldots,i(n)}\}_{n\geq 0,i(1),\ldots,i(n)\in I}$ that

[TABLE]

Indeed, for all $i(1),\ldots,i(n)\in I$, we have

[TABLE]

which allows to write the induction step

[TABLE]

Setting $k_{i}=\sqrt{s-t/2}e_{(i(n),1)}+i\sqrt{t/2}e_{(i(n),2)}+\sqrt{s}e_{(i(n),4)}$, the same computation yields

[TABLE]

Finally, we have

[TABLE]

Because the $e_{(i,3)}$ occur only in $h_{i(1)}\otimes\cdots\otimes h_{i(n)}$, and the $e_{(i,4)}$ occur only in $k_{i(1)}\otimes\cdots\otimes k_{i(n)}$, they do not contribute to the scalar product, and we can replace $h_{i(1)}\otimes\cdots\otimes h_{i(n)}$ and $k_{i(1)}\otimes\cdots\otimes k_{i(n)}$ by

[TABLE]

which yields

[TABLE]

∎

Let us denote by $Q$ the polynomial $\mathscr{S}_{q}^{s,t}P$. Let $Y,W$ be two $(t,0)$-elliptic $q$-Gaussian random variables and $Z$ be a $(s-t/2,t/2)$-elliptic $q$-Gaussian random variable such that $Y,W$ and $Z$ are $q$-independent. Thanks to the discussion before Theorem 5.6, we know that we can apply [24, Theorem 1] in the case of $q_{ij}\in\{-1,+1\}$ (or [18, Theorem 4] in the case $q_{ij}\in\{0,+1\}$) which says that the mixed $q$-Gaussian random variables $Y^{(n)},W^{(n)},\Re Z^{(n)}$ and $\Im Z^{(n)}$ converge in noncommutative distribution to the $q$-Gaussian random variables $Y,W,\Re Z$ and $\Im Z$. In particular, we have the following convergence:

[TABLE]

From Corollary 3.11 and Corollary 3.13, we know that

[TABLE]

Thus the limit $\tau\left[(P(Z+Y)-Q(Z))^{*}(P(Z+W)-Q(Z))\right]$ of $\left\|\mathscr{S}_{Q}^{s,t}\left(P(X^{(n)})\right)-Q(Z^{(n)})\right\|_{\mathcal{H}L^{2}(Z_{\varepsilon},\tau)}$ vanishes:

[TABLE]

∎

Acknowledgements

The first author was partially funded by the ERC Advanced Grant “NCDFP” held by Roland Speicher. The second author was funded by the same ERC Advanced Grant “Non-commutative distributions in free probability” (grant no. 339760). The second author would like to thank Roland Speicher for allowing his stay in Saarbrücken, Germany so the authors had a chance to collaborate.