Freeness and The Partial Transposes of Wishart Random Matrices

TL;DR

This paper demonstrates that partial transposes of complex Wishart matrices become asymptotically free, explores regimes with fixed block numbers, and examines real Wishart matrices, revealing new insights into their operator-level freeness.

Contribution

It introduces the asymptotic freeness of partial transposes of Wishart matrices and analyzes different regimes, including real matrices, at the operator level.

Findings

Partial transposes of complex Wishart matrices are asymptotically free.

Freeness persists when the number of blocks is fixed and block size increases.

Real Wishart matrices also exhibit similar freeness properties.

Abstract

We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives a example where the partial transpose produces freeness at the operator level. Finally we investigate the case of real Wishart matrices.

Equations

$$A=\left(\begin{array}[]{c|c|c}A(1,1)&\ \cdots&A(1,d_{1})\\ \hline\cr\vdots&&\vdots\\ \hline\cr A(d_{1},1)&\ \cdots&A(d_{1},d_{1}).\\ \end{array}\right).$$

$$w=\frac{1}{2}\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right),X_1=\left(\begin{array}{cc}{w}_{11}& 0\\ 0& w_{22}\end{array}\right),X_1=\left(\begin{array}{cc}0& w_{21}\\ w_{12}& 0\end{array}\right),$$

$$W=\frac{1}{d_{1}d_{2}}\left(\begin{array}[]{c}G_{1}\\ \hline\cr\vdots\\ \hline\cr G_{d_{1}}\end{array}\right)\left(\begin{array}[]{c|c|c}G_{1}^{*}&\cdots&G_{d_{1}}^{*}\end{array}\right)=\frac{1}{d_{1}d_{2}}(G_{i}G_{j}^{*})_{ij}$$

$$w=\frac{1}{2}\begin{pmatrix}w_{11}&w_{12}\\ w_{21}&w_{22}\end{pmatrix}\mbox{\ and\ }w^{\reflectbox{\tiny$\Gamma$}}=\frac{1}{2}\begin{pmatrix}w_{11}&w_{21}\\ w_{12}&w_{22}\end{pmatrix}.$$

$$\mathrm{Tr}(A_1\cdots A_n)=\sum _{i_{\pm 1},\dots ,i_{\pm n}=1}^N{a}_{i_1,i_{-1}}^{(1)}{a}_{i_2,i_{-2}}^{(2)}\cdots a_{i_n{i}_{-n}}^{(n)}$$

$$E(\mathrm{Tr}\otimes \mathrm{Tr}(W^{({ϵ}_1,{\eta }_1)}\cdots W^{({ϵ}_n,{\eta }_n)}))$$

$$\displaystyle\operatorname{Tr}\otimes\operatorname{Tr}\big{(}W^{(\epsilon_{1},\eta_{1})}\cdots W^{(\epsilon_{n},\eta_{n})}\big{)}$$

$$\displaystyle\operatorname{Tr}\big{(}W(j_{1},j_{-1})^{(\eta_{1})}\cdots W(j_{n},j_{-n})^{(\eta_{n})}\big{)}$$

$$\{\begin{array}{ll}s(-k)=s(\sigma (k))& \text{for }k>0\\ j(-k)=j(\sigma (k))& \text{for }k>0\\ t=t(\sigma (k))& \text{for }k>0\end{array}$$

$$\sum _{j_{\pm 1},\dots ,j_{\pm n}}\sum _{s_{\pm 1},\dots ,s_{\pm n}}\sum _{t_1,\dots ,t_n}E(g_{s_1{t}_1}^{(j_1)}\cdots g_{s_n{t}_n}^{(j_n)}\overline{g_{s_{-1}{t}_1}^{(j_{-1})}}\cdots \overline{g_{s_{-n}{t}_n}^{(j_{-n})}})$$

$$\operatorname{E}(\operatorname{tr}\otimes\operatorname{tr}(W^{(\epsilon_{1},\eta_{1})}\cdots W^{(\epsilon_{n},\eta_{n})}))=\sum_{\sigma\in S_{n}}\bigg{(}\frac{p}{d_{1}d_{2}}\bigg{)}^{\#(\sigma)}d_{1}^{\ f_{\epsilon}(\sigma)}d_{2}^{\ f_{\eta}(\sigma)}$$

$$E(\mathrm{tr}\otimes \mathrm{tr}(W^{({ϵ}_1,{\eta }_1)}\cdots W^{({ϵ}_n,{\eta }_n)}))$$

$$\#(\pi )+\#({\pi }^{-1}\sigma )+\#(\sigma )=n+2(1-g)$$

$$ϵ\delta \sigma \delta {\sigma }^{-1}ϵ(k)=\{\begin{array}{ll}ϵ\delta \sigma \delta {\sigma }^{-1}(k)& {ϵ}_k=1\\ ϵ\delta \sigma (k)& {ϵ}_k=-1\end{array}=\{\begin{array}{ll}{ϵ}_{{\sigma }^{-1}(k)}{\sigma }^{-1}(k)& {ϵ}_k=1\\ -{ϵ}_{\sigma (k)}\sigma (k)& {ϵ}_k=-1\end{array}.$$

$$ϵ\delta \sigma \delta {\sigma }^{-1}ϵ(k)=\{\begin{array}{ll}{\sigma }^{-1}(k)& {ϵ}_k=1\\ \sigma (k)& {ϵ}_k=-1\end{array}\in [n].$$

$$ϵ\delta \sigma \delta {\sigma }^{-1}ϵ(k)=\{\begin{array}{ll}{ϵ}_{{\sigma }^{-1}(k)}{\sigma }^{-1}(k)& {ϵ}_k=1\\ -{ϵ}_{\sigma (k)}\sigma (k)& {ϵ}_k=-1\end{array}$$

$$2\#(ϵ\gamma \delta {\gamma }^{-1}ϵ\vee \sigma \delta {\sigma }^{-1})$$

$$\#(ϵ\delta {\sigma }^{-1}\delta \sigma ϵ)+\#((ϵ\delta {\sigma }^{-1}\delta \sigma ϵ{)}^{-1}\gamma \delta {\gamma }^{-1}\delta )+\#(\gamma \delta {\gamma }^{-1}\delta )=2n+2(1-g),$$

$$f_{ϵ}(\sigma )=\#(ϵ\gamma \delta {\gamma }^{-1}ϵ\vee \sigma \delta {\sigma }^{-1})+\#(\sigma )-(n+1)=-(g+1)\le -1.$$

$$ϵ\delta \sigma \delta {\sigma }^{-1}ϵ(k)=\{\begin{array}{ll}{\sigma }^{-1}(k)& {ϵ}_k=1\\ \sigma (k)& {ϵ}_k=-1\end{array}.$$

$$f_{ϵ}(\sigma )$$

$$\#(\sigma \tau )=1+e(\sigma )$$

$$\lim_{d_{2}\rightarrow\infty}\operatorname{E}(\operatorname{tr}\otimes\operatorname{tr}((d_{1}W^{\reflectbox{\tiny$\Gamma$}})^{n}))=\kern-7.5pt\sum_{\sigma\in NC(n)}c^{\#(\sigma)}d_{1}^{f_{-1}(\sigma)+n}=\kern-7.5pt\sum_{\sigma\in NC(n)}(d_{1}c)^{\#(\sigma)}d_{1}^{\,e(\sigma)}.$$

$$w=\frac{1}{2}\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right).$$

$$2w^{\reflectbox{\tiny$\Gamma$}}=\begin{pmatrix}w_{11}&w_{21}\\ w_{12}&w_{22}\\ \end{pmatrix}.$$

$$X_1=\left(\begin{array}{cc}{w}_{11}& 0\\ 0& w_{22}\end{array}\right)\text{mbox}and{X}_2=\left(\begin{array}{cc}0& w_{21}\\ w_{12}& 0\end{array}\right)$$

$$w=\frac{1}{2}\left(\begin{array}{cc}{w}_{11}& w_{12}\\ w_{21}& w_{22}\end{array}\right).$$

$${\kappa }_n^{(1)}(w_{11},\dots w_{11})$$

$$Y_1=X_1,Y_2=\left(\begin{array}{cc}{w}_{21},& 0\\ 0& w_{12}\end{array}\right)\text{mbox}and{Y}_3=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).$$

Full text

∗ Research supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada

∗∗ Research supported by the Simons Foundation grant No. 360242.

1. Introduction

Suppose we have a matrix $A$ in $M_{d_{1}}(\mathbb{C})\otimes M_{d_{2}}(\mathbb{C})$. We can write this as a block matrix.

[TABLE]

with each $A(i,j)\in M_{d_{2}}(\mathbb{C})$. We can form two partial transposes of this matrix.

\kern-15.00002ptA^{\reflectbox{\tiny\Gamma}}=\left(\begin{array}[]{c|c|c}A(1,1)&\ \cdots&A(d_{1},1)\\ \hline\cr\vdots&&\vdots\\ \hline\cr A(1,d_{1})&\ \cdots&A(d_{1},d_{1}).\\ \end{array}\right),A^{\Gamma}=\left(\begin{array}[]{c|c|c}A(1,1)^{\mathrm{T}}&\ \cdots&A(1,d_{1})^{\mathrm{T}}\\ \hline\cr\vdots&&\vdots\\ \hline\cr A(d_{1},1)^{\mathrm{T}}&\ \cdots&A(d_{1},d_{1})^{\mathrm{T}}.\\ \end{array}\right).

In quantum information theory the partial transpose has been used as an entanglement detector. Suppose that $A$ is a positive matrix in $M_{d_{1}}(\mathbb{C})\otimes M_{d_{2}}(\mathbb{C})$. Recall that $A$ is entangled if we cannot find positive matrices $B_{1},\dots,\allowbreak B_{k}\in M_{d_{1}}(\mathbb{C})$ and $C_{1},\dots,C_{k}\in M_{d_{2}}(\mathbb{C})$ such that $A=\sum_{i=1}^{k}A_{i}\otimes B_{i}$. If a positive matrix fails to have a positive partial transpose then the matrix must be entangled. It was shown by Aubrun [2] that in a particular regime of the Wishart distribution, the partial transpose of a positive matrix is typically entangled. He also showed that the limiting distribution of a partially transposed Wishart matrix is semi-circular. This was quite unexpected. We revisit his theorem and show that the conclusion can be explained by the requirement that the non-crossing partitions that survive in the limit, have to remain non-crossing when the order of elements is reversed.

In this paper we show that in addition to transforming a Marchenko-Pastur into a semi-circular distribution, the partial transpose also produces freeness — in two different regimes. The first is when both $d_{1}$ and $d_{2}$ tend to $\infty$; we show that when $W$ is a Wishart matrix, $W$, W^{\reflectbox{\tiny\Gamma}}, $W^{\Gamma}$ and $W^{\mathrm{T}}$ are asymptotically free in the complex case and that $W=W^{\mathrm{T}}$ and W^{\Gamma}=W^{\reflectbox{\tiny\Gamma}} are asymptotically free in the real case.

The second regime is the one considered by Banica and Nechita [3] where we fix $d_{1}$ and let $d_{2}$ tend to $\infty$. In this case we show that $W$ and $W^{\Gamma}$ are asymptotically free. As (W^{\reflectbox{\tiny\Gamma}})^{\mathrm{T}}=W^{\Gamma}, one can has that W^{\reflectbox{\tiny\Gamma}} and $W^{\Gamma}$ have the same limit distributions. Banica and Nechita showed that the limit distribution of d_{1}W^{\reflectbox{\tiny\Gamma}} could be written as the free difference of two Marchenko-Pastur distributions. We can show that in the same regime and when $d_{1}=2$, one can write the limit distribution of d_{1}W^{\reflectbox{\tiny\Gamma}} as the sum of two free operators $X_{1}$ and $X_{2}$ coming from the diagonal and off-diagonal parts of d_{1}W^{\reflectbox{\tiny\Gamma}}. More precisely if we write a Marchenko-Pastur random variable $w$ as a block matrix

[TABLE]

then d_{1}w^{\reflectbox{\tiny\Gamma}}=X_{1}+X_{2} and $X_{1}$ and $X_{2}$ are free. Moreover $X_{1}$ is a Marchenko-Pastur random variable with the same distribution as $2w$ and $X_{2}$ is an even operator whose even cumulants are the same as those of $X_{1}$. So by writing d_{1}w^{\reflectbox{\tiny\Gamma}} as a sum as opposed to a difference we get a natural representation for the two operators.

The connection of the transpose to freeness goes back to the work of Emily Redelmeier on real second order freeness [13]. She showed that the fluctuation moments of real Gaussian and Wishart matrices require the transpose to taken into account. Later in [9] the authors showed that the transpose can also appear at the first order level. Namely, that for many complex ensembles a random matrix could be asymptotically free from its transpose. Before this the known examples of asymptotic freeness required independence of the entries, see [10, Ch. 4] and [11, Lect. 23, 24] for examples.

Our main tool here is the explicit evaluation mixed moments of the four matrices $W$, W^{\reflectbox{\tiny\Gamma}}, $W^{\Gamma}$ and $W^{\mathrm{T}}$ and then to show that mixed cumulants must vanish, thus demonstrating freeness. In order to achieve this we use the technique of doubling of indices which already appeared in the work of Redelmeier [12, 13].

Besides the transpose one can consider the action of other positive linear maps on the blocks of matrices and the effect on the limit eigenvalue distribution. This was considered in considerable generality in the recent paper of Arizmendi, Nechita, and Vargas [1]. A third regime was considered by Fukuda and Śniady in [6] and a connection to meander polynomials was found.

The outline of the paper is as follows. In section 2 we establish the notation needed for our calculations. The main method for computing mixed moments is to expand the expression as a sum over the symmetric group. This part is presented in Theorem 9 of Section 3. In Section 4 we determine which permutations contribute to the limit. The main result in this section is Proposition 16. In Section 5 we consider the limit distributions of our partially transposed operators in the two regimes. In particular we shall show that the operators $X_{1}$ and $X_{2}$ mentioned above are free. In Section 7 we present our main results, the asymptotic freeness of our partially transposed Wishart matrices. In Section 8 we consider the situation for real Wishart matrices.

2. Notation and statement of results

Suppose $G_{1},\dots,G_{d_{1}}$ are $d_{2}\times p$ random matrices where $G_{i}=(g^{(i)}_{jk})_{jk}$ and $g^{(i)}_{jk}$ are complex Gaussian random variables with mean 0 and (complex) variance 1, i.e. $\operatorname{E}(|g^{(i)}_{jk}|^{2})=1$. Moreover suppose that the random variables $\{g^{(i)}_{jk}\}_{i,j,k}$ are independent.

[TABLE]

is a $d_{1}d_{2}\times d_{1}d_{2}$ complex Wishart matrix. We write $W=d_{1}^{-1}(W(i,j))_{ij}$ as $d_{1}\times d_{1}$ block matrix with each entry the $d_{2}\times d_{2}$ matrix $d_{2}^{-1}G_{i}G_{j}^{*}$. From this we get four matrices \{W,W^{\reflectbox{\tiny\Gamma}},W^{\Gamma},W^{\mathrm{T}}\} defined as follows:

• $\circ$

$W^{\mathrm{T}}=\frac{1}{d_{1}}(W(j,i)^{\mathrm{T}})_{ij}$ is the “full” transpose

• $\circ$

W^{\reflectbox{\tiny\Gamma}}=\frac{1}{d_{1}}(W(j,i))_{ij} is the “left” partial transpose

• $\circ$

$W^{\Gamma}=\frac{1}{d_{1}}(W(i,j)^{\mathrm{T}})_{ij}$ is the “right” partial transpose

Note that the $X\mapsto X^{\Gamma}$ notation conceals the dependence on $d_{1}$ and $d_{2}$. Thus as the size of the matrix grows these operators might be expected to behave differently, depending on the way $d_{1}$ and $d_{2}$ grow.

If the random variables $\{g{(i)}_{jk}\}_{i,j,k}$ are real Gaussian random variables with mean 0 and variance 1 then $W$ is a real Wishart matrix. For many eigenvalue results there is no distinction between the real and complex case. In [9] we showed that when it comes to freeness there is a difference, in particular with respect to the behaviour of the transpose. In this paper we show that with the partial transpose we continue to see a difference between the real and complex cases.

If we assume that $d_{1},d_{2}\rightarrow\infty$ and that $\displaystyle\frac{p}{d_{1}d_{2}}\rightarrow c$, $0<c<\infty$, then the eigenvalue distributions of $W$ and $W^{\mathrm{T}}$ converge to Marchenko-Pastur with parameter $c$. This is the distribution on $\mathbb{R}^{+}$ that has density $\displaystyle\frac{\sqrt{(b-t)(t-a)}}{2\pi t}$ on $[a,b]$ and an atom of weight $(1-c)$ at [math] if $c<1$; we set $b=(1+\sqrt{c})^{2}$ and $a=(1-\sqrt{c})^{2}$.

Note that we are using what one might call the free probabilist’s Marchenko-Pastur law. In our normalization all the cumulants are equal to $c$. For the relation between the two see [10, Ch. 2 Remark 12]. With this normalization we can restate Aubrun’s theorem.

*Suppose $\displaystyle\lim_{d_{1},d_{2}\rightarrow\infty}\frac{p}{d_{1}d_{2}}=c$, then the eigenvalue distributions of W^{\reflectbox{\tiny\Gamma}} and $W^{\Gamma}$ converge to a shifted semi-circular operator with mean $c$ and variance $c$. *

Our main results are the following.

Theorem 1**.**

Suppose $\displaystyle\lim_{d_{1},d_{2}\rightarrow\infty}\frac{p}{d_{1}d_{2}}=c$, then the family \{W,W^{\reflectbox{\tiny\Gamma}},W^{\Gamma},\allowbreak W^{\mathrm{T}}\} is asymptotically free in the complex case and the family $\{W,W^{\Gamma}\}$ is asymptotically free in the real case.

Theorem 2**.**

Suppose $d_{1}$ is fixed and$\displaystyle\lim_{d_{2}\rightarrow\infty}\frac{p}{d_{1}d_{2}}=c$, then the family $\{W,W^{\Gamma}\}$ is asymptotically free in the complex case.

In $M_{2}(\mathbb{C})$ let $\{E_{ij}\}_{ij=1}^{2}$ be the standard matrix units. For convenience of notation we shall write $E_{ij}$ for $E_{ij}\otimes I_{d_{2}}\in M_{2}(\mathbb{C})\otimes M_{d_{2}}(\mathbb{C})$.

Theorem 3**.**

Suppose $d_{1}=2$, $p,d_{2}\rightarrow\infty$ and $\displaystyle\lim_{d_{2}\rightarrow\infty}\frac{p}{d_{1}d_{2}}=c$. Then \{W,W^{\reflectbox{\tiny\Gamma}},E_{ij}\}_{i,=1}^{2} has a limit joint distribution, \{w,w^{\reflectbox{\tiny\Gamma}},e_{ij}\}_{i,j=1}^{2}, in a non-commutative $*$-probability space $(\mathcal{A},\varphi)$. Relative to the matrix units $\{e_{ij}\}_{i,j=1}^{2}$ we write

[TABLE]

Then $X_{1}=\begin{pmatrix}w_{11}&0\\ 0&w_{22}\end{pmatrix}$ and $X_{2}=\begin{pmatrix}0&w_{21}\\ w_{12}&0\end{pmatrix}$ are free. $X_{1}$ has a Marchenko-Pastur distribution with parameter $2c$ and $X_{2}$ is an even operator with all even cumulants equal to $2c$.

3. A general formula for mixed moments

Let $A_{1},\dots,A_{n}$ be $N\times N$ matrices, then

[TABLE]

where the sum runs over all $i:[\pm n]\rightarrow[N]$ such that $i(-1)=i(2)$, $i(-2)=i(3)$, …, $i(-n)=i(1)$.

We wish to use the symmetric group $S_{n}$ of permutations on $[n]=\{1,2,3,\dots,n\}$ to keep track of the partial transposes. So we shall introduce the following notation. Given a permutation $\sigma$ in $S_{n}$ we extend $\sigma$ to be a permutation on $[\pm n]=\{1,-1,2,-2,3,-3,\dots,n,-n\}$ be setting $\sigma(-k)=-k$ for $k>0$. We let $\delta$ be the permutation of $[\pm n]$ given by $\delta(k)=-k$ for all $k\in[\pm n]$ and $\gamma\in S_{n}$ be the permutation with one cycle: $\gamma=(1,2,3,\dots,n)$. With our conventions we have $\gamma\delta\gamma^{-1}=(1,-n)(2,-1)(3,-2)\cdots(n,-(n-1))$. The condition in (1) now becomes $i=i\circ\gamma\delta\gamma^{-1}$.

To show that the family \{W,W^{\reflectbox{\tiny\Gamma}},W^{\Gamma},W^{\mathrm{T}}\} is asymptotically free we shall have to compute the expectation of the trace of arbitrary words in \{W,W^{\reflectbox{\tiny\Gamma}},W^{\Gamma},W^{\mathrm{T}}\}. For this we use the following notation.

Let $(\epsilon,\eta)\in\{-1,1\}^{2}=\mathbb{Z}_{2}^{2}$.

We set W^{(\epsilon,\eta)}=\left\{\begin{array}[]{cl}W&\mbox{\ if\ }(\epsilon,\eta)=(1,1)\\ W^{\reflectbox{\tiny\Gamma}}&\mbox{\ if\ }(\epsilon,\eta)=(-1,1)\\ W^{\Gamma}&\mbox{\ if\ }(\epsilon,\eta)=(1,-1)\\ W^{\mathrm{T}}&\mbox{\ if\ }(\epsilon,\eta)=(-1,-1)\\ \end{array}\right.

Let $(\epsilon_{1},\eta_{1}),\dots,(\epsilon_{n},\eta_{n})\in\mathbb{Z}_{2}^{n}$, then an arbitrary word in \{W,W^{\reflectbox{\tiny\Gamma}},W^{\Gamma},\allowbreak W^{\mathrm{T}}\} is $W^{(\epsilon_{1},\eta_{1})}\cdots W^{(\epsilon_{n},\eta_{n})}$ and we seek to write $\displaystyle\lim_{d_{1},d_{2}\rightarrow\infty}\operatorname{E}(\operatorname{tr}(W^{(\epsilon_{1},\eta_{1})}\allowbreak\cdots\allowbreak W^{(\epsilon_{n},\eta_{n})}))$ as a sum of free cumulants.

To achieve this we need to introduce still more notation. We shall suppose that $n$, the length of the word, is fixed for the moment. Now given $(\epsilon_{1},\epsilon_{2},\dots,\epsilon_{n})$ we denote by $\epsilon$ the permutation of $[\pm n]$ given by $\epsilon(k)=\epsilon_{|k|}k$; here $k\in[\pm n]$, but $|k|>0$, so $\epsilon_{|k|}$ means the $k^{th}$ element of our vector $(\epsilon_{1},\dots,\epsilon_{n})$. Similarly given $(\eta_{1},\dots,\eta_{n})$ we get the permutation $\eta$ of $[\pm n]$. Note that $\delta$, $\epsilon$ and $\eta$ all commute with each other.

We shall think of $W$, W^{\reflectbox{\tiny\Gamma}}, $W^{\Gamma}$, and $W^{\mathrm{T}}$ as random elements of $M_{d_{1}}(\mathbb{C})\otimes M_{d_{2}}(\mathbb{C})$. On this algebra we have a trace $\operatorname{Tr}\otimes\operatorname{Tr}$; we also have the normalized trace $\operatorname{tr}\otimes\operatorname{tr}=\displaystyle\frac{1}{d_{1}d_{2}}\operatorname{Tr}\otimes\operatorname{Tr}$.

Remark 4*.*

With the notations above, we have that

[TABLE]

where the summation is subject to the conditions that $j=j\circ\epsilon\gamma\delta\gamma^{-1}\epsilon$, $s=s\circ\eta\gamma\delta\gamma^{-1}\eta$.

Proof.

[TABLE]

To achieve the last step we let $j=i\circ\epsilon$. Also we have adopted the convention that for $A$ a matrix in $M_{d_{2}}(\mathbb{C})$ we let $A^{(1)}=A$ and $A^{(-1)}=A^{\mathrm{T}}$.

Next we must expand \operatorname{Tr}\Big{(}W(j_{1},j_{-1})^{(\eta_{1})}\cdots W(j_{n},j_{-n})^{(\eta_{n})}\Big{)}.

[TABLE]

hence the conclusion. ∎

Next we need to compute $\operatorname{E}(g^{(j_{1})}_{s_{1}t_{1}}\cdots g^{(j_{n})}_{s_{n}t_{n}}\ \overline{g^{(j_{-1})}_{s_{-1}t_{1}}}\cdots\overline{g^{(j_{-n})}_{s_{-n}t_{n}}})$. We shall use the complex form of Wick’s rule that says that if $g_{1},\dots,g_{m}$ are independent Gaussian random variables and $\alpha_{1},\dots,\alpha_{n},\beta_{1},\dots,\beta_{n}\in[m]$ then $\operatorname{E}(g_{\alpha(1)}\cdots g_{\alpha(n)}\overline{g_{\beta(1)}}\cdots\overline{g_{\beta(n)}})$ is the number of permutations $\sigma\in S_{n}$ such that for all $k\in[n]$ we have $\beta(k)=\alpha(\sigma(k))$; see Janson [7, page 13].

Thus we let $g_{\alpha(k)}=g_{s_{k}t_{k}}^{(j_{k})}$ and $g_{\beta(k)}=g_{s_{-k}t_{k}}^{(j_{-k})}$. So if $\sigma\in S_{n}$ and $\beta(k)=\alpha(\sigma(k))$ we have

[TABLE]

We wish to write the first two conditions as an equation involving functions on $[\pm n]$.

Lemma 5**.**

Let $\sigma\in S_{n}$ and $j:[\pm n]\rightarrow[d]$. We have $j(-k)=j(\sigma(k))$ for all $k>0$ if and only if $j=j\circ\sigma\delta\sigma^{-1}$.

Proof.

Suppose that for $k>0$ we have $j(-k)=j(\sigma(k))$. Then for $k>0$ we have $j\circ\sigma\delta\sigma^{-1}(k)=j(-\sigma^{-1}(k))=j(-l)=j(\sigma(l))=j(k)$ where $l=\sigma^{-1}(k)>0$. Also $j\circ\sigma\delta\sigma^{-1}(-k)=j(\sigma(k))=j(-k)$. Thus $j=j\circ\sigma\gamma\sigma^{-1}$.

Now suppose that $j=j\circ\sigma\delta\sigma^{-1}$. For $k>0$ we have that $j(-k)=j\circ\sigma\delta\sigma^{-1}(-k)=j(\sigma(k))$ as claimed. ∎

Lemma 6**.**

$\operatorname{E}(g^{(j_{1})}_{s_{1}t_{1}}\cdots g^{(j_{n})}_{s_{n}t_{n}}\ \overline{g^{(j_{-1})}_{s_{-1}t_{1}}}\cdots\overline{g^{(j_{-n})}_{s_{-n}t_{n}}})=|\{\sigma\in S_{n}\mid j=j\circ\sigma\delta\sigma^{-1}$, $s=s\circ\sigma\delta\sigma^{-1}$, and $t=t\circ\sigma\}|$.

Proof.

By Lemma 5 and Equation (3) we have to count the number of permutations $\sigma$ such that $j=j\circ\sigma\delta\sigma^{-1}$ on the set $[d_{1}]$, $s=s\circ\sigma\delta\sigma^{-1}$ on the set $[d_{2}]$, and $t=t\circ\sigma$ on the set $[p]$. ∎

Remark 7*.*

In the proposition below we use the notation $\#(\sigma)$ to denote the number of cycles in the cycle decomposition of $\sigma$. If $\sigma$ and $\pi$ are permutations we let $\sigma\vee\pi$ be the partition obtained by regarding $\sigma$ and $\pi$ as partitions where the blocks of the partition are the cycles of the permutation. Now $\sigma\vee\pi$ denotes the supremum of the two partitions in the lattice of partitions. Recall that the function $\sigma\mapsto\#(\sigma)$ is a central function on $S_{n}$, so $\#(\pi^{-1}\sigma\pi)=\#(\sigma)$ for all $\pi$ and $\sigma$.

Proposition 8**.**

Subject to the conditions $j=j\circ\epsilon\gamma\delta\gamma^{-1}\epsilon$ and $s=s\circ\eta\gamma\delta\gamma^{-1}\eta$,

[TABLE]

Proof.

According to Lemma 6, subject to the conditions $j=j\circ\epsilon\gamma\delta\gamma^{-1}\epsilon$ and $s=s\circ\eta\gamma\delta\gamma^{-1}\eta$,

[TABLE]

To get the last equality we recall that the condition on $j$ is that it must simultaneously satisfy $j=j\circ\epsilon\gamma\delta\gamma^{-1}\epsilon$ and $j=j\circ\sigma\delta\sigma^{-1}$. So $j$ must be constant on the cycles of $\epsilon\gamma\delta\gamma^{-1}\epsilon$ and of $\sigma\delta\sigma^{-1}$; so $j$ must be constant on the blocks of $\epsilon\gamma\delta\gamma^{-1}\epsilon\vee\sigma\delta\sigma^{-1}$. The same argument applies to $s$. The only condition on $t$ is that $t=t\circ\sigma$. ∎

Theorem 9**.**

[TABLE]

where $f_{\epsilon}(\sigma)=\#(\epsilon\gamma\delta\gamma^{-1}\epsilon\vee\sigma\delta\sigma^{-1})+\#(\sigma)-(n+1)$ and $f_{\eta}(\sigma)=\#(\eta\gamma\delta\gamma^{-1}\eta\vee\sigma\delta\sigma^{-1})+\#(\sigma)-(n+1)$

Proof.

According to Equation (2) and Proposition 8 we have

[TABLE]

∎

4. Asymptotics of Permutations

Theorem 9 gave us an expansion of mixed moments of \{W,W^{\reflectbox{\tiny\Gamma}},W^{\Gamma},\allowbreak W^{\mathrm{T}}\} as a sum over the symmetric group. We now have to determine which permutations contribute to the limit. We shall show that for all $\epsilon$ and all $\sigma$, $f_{\epsilon}(\sigma)\leq 0$ and determine for which $\sigma$ equality is achieved. Our first goal is to show that $f_{\epsilon}(\sigma)<0$ unless $\epsilon$ is constant on the cycles of $\sigma$. Since $\epsilon$ is arbitrary, whatever we show for $\epsilon$ will apply to $\eta$.

There is a fundamental equation that we shall frequently use in what follows. Given a subgroup, $G$, of the group $S_{n}$ of permutations of $[n]$, we shall say that the subgroup acts transitively on $[n]$ if given $k,l\in[n]$ we can find $\rho\in G$ such that $\rho(k)=l$.

Given two permutations $\pi$ and $\sigma$ of $S_{n}$ such that the subgroup generated $\pi$ and $\sigma$ acts transitively there is a non-negative integer $g$ such that

[TABLE]

Recall that a pairing of $[n]$ is a partition $\pi$ of $[n]$ with all blocks of size 2; note this implies that $n$ is even. The set of all pairings of $[n]$ is denoted $\mathcal{P}_{2}(n)$. We shall also regard such a $\pi$ as the permutation whose cycles are the blocks of $\pi$. In this case $\pi$ has no fixed points and $\pi^{2}=id$. In [8, Lemma 2] we proved the following.

Lemma 10**.**

Let $\pi,\sigma\in\mathcal{P}_{2}(n)$ be pairings and $(i_{1},i_{2},\dots,i_{k})$ a cycle of $\pi\sigma$. Let $j_{r}=\sigma(i_{r})$. Then $(j_{k},j_{k-1},\dots,j_{1})$ is also a cycle of $\pi\sigma$, and these two cycles are distinct; $\{i_{,}\dots,i_{k},j_{1},\dots,j_{k}\}$ is a block of $\pi\vee\sigma$ and all are of this form; $2\#(\pi\vee\sigma)=\#(\pi\sigma)$. The cycle decomposition of $\pi\sigma$ can be written $c_{1}c^{\prime}_{1}\cdots c_{k}c^{\prime}_{k}$ where $c^{\prime}_{i}=\sigma c_{i}^{-1}\sigma$. With this notation the blocks of $\pi\vee\sigma$ are $c_{i}\cup c^{\prime}_{i}$.

Lemma 11**.**

Let $\sigma\in S_{n}$ and $\epsilon\in\mathbb{Z}_{2}^{n}$ be given, then $\epsilon$ is constant on the cycles of $\sigma$ if and only if $\epsilon\delta\sigma\delta\sigma^{-1}\epsilon[n]=[n]$.

Proof.

We begin by noting that for $k>0$

[TABLE]

Suppose $\epsilon$ is constant on the cycles of $\sigma$. Then

[TABLE]

Conversely suppose that $\epsilon\delta\sigma\delta\sigma^{-1}\epsilon[n]=[n]$. Then for $k>0$ by Equation (5)

[TABLE]

and thus $\epsilon_{\sigma(k)}$ and $\epsilon_{k}$ have the same sign. ∎

Lemma 12**.**

Let $\sigma\in S_{n}$ and $\epsilon\in\mathbb{Z}_{2}^{n}$ be given then $f_{\epsilon}(\sigma)<0$ unless $\epsilon$ is constant on the cycles of $\sigma$.

Proof.

Suppose $\epsilon$ is not constant on the cycles of $\sigma$, then by Lemma 11 we have that $\epsilon\delta\sigma\delta\sigma^{-1}\epsilon[n]$ meets $[-n]$. Both $\epsilon\gamma\delta\gamma^{-1}\epsilon$ and $\sigma\delta\sigma^{-1}$ are pairings and as permutations $\delta$ and $\epsilon$ commute. Thus by Lemma 10

[TABLE]

Now $\#(\gamma\delta\gamma^{-1}\delta)=2$ and $\#(\epsilon\delta\sigma\delta\sigma^{-1}\epsilon)=\#(\delta\sigma\delta\sigma^{-1})=\#(\delta\sigma\delta)+\#(\sigma^{-1})=2\#(\sigma)$. Hence by Equation (4) there is $g\geq 0$ such that

[TABLE]

and thus

[TABLE]

∎

Lemma 13**.**

Suppose that $\sigma\in S_{n}$ and $\epsilon\in\mathbb{Z}_{2}^{n}$ and $\epsilon$ is constant on the cycles of $\sigma$. Then there is a permutation $\sigma_{\epsilon}\in S_{n}$ such that $\epsilon\delta\sigma\delta\sigma^{-1}\epsilon=\delta\sigma_{\epsilon}\delta\sigma_{\epsilon}^{-1}$. Moreover if $\sigma=c_{1}\cdots c_{k}$ is the cycle decomposition of $\sigma$ then $\sigma_{\epsilon}=c_{1}^{\lambda_{1}}\cdots c_{k}^{\lambda_{k}}$ where $\lambda_{i}$ is the (constant) value of $\epsilon$ on the cycle $c_{i}$.

Proof.

In the proof of Lemma 11 we showed that when $\epsilon$ is constant on the cycles of $\sigma$ we have that for $k>0$

[TABLE]

Thus on a cycle of $\sigma$ on which $\epsilon=1$ we have $\sigma_{\epsilon}^{-1}$ and $\epsilon\delta\sigma\delta\sigma^{-1}\epsilon$ agree and on a cycle on which $\epsilon=-1$ we have $\sigma_{\epsilon}$ and $\epsilon\delta\sigma\delta\sigma^{-1}\epsilon$ agree. ∎

Definition 14**.**

Let $\pi\in S_{n}$ be a permutation of $[n]$ and $\gamma=(1,2,3,\dots,\allowbreak n)$. We say that $\pi$ is a non-crossing permutation if $\#(\pi)+\#(\pi^{-1}\gamma)=n+1$. We shall denote by $S_{NC}(n)$ the non-crossing permutations of $[n]$.

Remark 15*.*

We have already used the idea of taking a permutation of $[n]$ and regard it as a partition of $[n]$ by using the decomposition of the permutation into disjoint cycles and making these the blocks of a partition. Biane [4] showed that the permutations that satisfy $\#(\pi)+\#(\pi^{-1}\gamma)=n+1$ i.e. $g=0$ in Equation (4), are exactly those whose cycles form a non-crossing partition of $[n]$.

Proposition 16**.**

Let $\sigma\in S_{n}$ and $\epsilon\in\mathbb{Z}_{2}^{n}$. Suppose that $\epsilon$ is constant on the cycles of $\sigma$. Then $f_{\epsilon}(\sigma)\leq 0$ with equality only if $\sigma_{\epsilon}$ is a non-crossing permutation.

Proof.

Let $\epsilon\delta\sigma\delta\sigma^{-1}\epsilon=\delta\sigma_{\epsilon}\delta\sigma_{\epsilon}^{-1}$ as in Lemma 13. As in the proof of Lemma 12 we have that

[TABLE]

By Equation (4) we have $f_{\epsilon}(\sigma)\leq 0$; and, according to Definition 14, $\sigma_{\epsilon}$ is a non-crossing permutation if and only if $f_{\epsilon}(\sigma)=0$ ∎

Remark 17*.*

As an illustration let us consider two examples: $\epsilon\equiv 1$ and $\epsilon\equiv-1$. First suppose $\epsilon\equiv 1$, then $\#(\epsilon\gamma\delta\gamma^{-1}\epsilon\vee\sigma\delta\sigma^{-1})={\textstyle\frac{1}{2}}\#(\gamma\delta\gamma^{-1}\sigma\delta\sigma^{-1})=\#(\sigma^{-1}\gamma)$, so $\sigma_{1}=\sigma$ and $f_{1}(\sigma)=\#(\sigma)+\#(\sigma^{-1}\gamma)-(n+1)=0$ only if $\sigma$ is non-crossing. When $\epsilon\equiv-1$ we have that $\#(\epsilon\gamma\delta\gamma^{-1}\epsilon\vee\sigma\delta\sigma^{-1})={\textstyle\frac{1}{2}}\#(\delta\gamma\delta\gamma^{-1}\delta\sigma\delta\sigma^{-1})=\#(\sigma^{-1}\gamma)$, so $\sigma_{{}_{1}}=\sigma^{-1}$ and $f_{-1}(\sigma)=\#(\sigma^{-1})+\#(\sigma\gamma)-(n+1)=0$ only if $\sigma^{-1}$ is non-crossing.

5. Limit Distributions

We assume that $d_{1}d_{2}\rightarrow\infty$ and that $\displaystyle\frac{p}{d_{1}d_{2}}\rightarrow c$, for some $0<c<\infty$. Since $W$ and $W^{\mathrm{T}}$ are Wishart matrices, their eigenvalue distributions converge to the Marchenko-Pastur law with parameter $c$. Setting $b=(1+\sqrt{c})^{2}$ and $a=(1-\sqrt{c})^{2}$, this is the distribution on $\mathbb{R}^{+}$ that has density $\displaystyle\frac{\sqrt{(b-t)(t-a)}}{2\pi t}$ on $[a,b]$ and an atom of weight $(1-c)$ at [math] if $c<1$.

The asymptotic eigenvalue distributions of W^{\reflectbox{\tiny\Gamma}} and $W^{\Gamma}$ were described by G. Auburn (see [2]) for the case when $d_{1},d_{2}\rightarrow\infty$, respectively by T. Banica and I. Nechita for the case when $d_{1}\rightarrow\infty$ (see [3]). The calculations below give another proof of these results and give some more insight on the limit distributions.

Lemma 18**.**

Let $\sigma\in S_{n}$ and suppose that both $\sigma$ and $\sigma^{-1}$ are non-crossing in the sense of Definition 14. Then $\sigma$ can have only cycles of size 1 or 2.

Before proving Lemma 18 we need to recall some standard facts about permutations and pairings. We let $[\pm n]=\{1,-1,2,-2,\dots,n,\allowbreak-n\}$. If $\sigma\in S_{n}$ is a permutation of $[n]$ then $\sigma\delta\sigma^{-1}$ is a pairing of $[\pm n]$; moreover if $(r,s)$ is a pair in this pairing then $r$ and $s$ have opposite signs. We let $\mathcal{P}_{2}^{\delta}(\pm n)$ be the set of pairings of $[\pm n]$ that only pair a positive number to a negative number. There is a standard bijection from $S_{n}$ to $\mathcal{P}_{2}^{\delta}(\pm n)$ that we shall use. For $\sigma\in S_{n}$ we have $\sigma\delta\sigma^{-1}\in\mathcal{P}_{2}^{\delta}(\pm n)$. If $\pi\in\mathcal{P}_{2}^{\delta}(\pm n)$ then $\pi\delta$ leaves $[n]$ invariant and so $\pi\delta|_{[n]}\in S_{n}$. These two maps are inverses of each other.

For example consider $\gamma=(1,2,\dots,n)\in S_{n}$. Then $\gamma\delta\gamma^{-1}=(-n,1)\allowbreak(-1,2)(-2,3)\cdots(-(n-1),n)\in\mathcal{P}_{2}^{\delta}(\pm n)$ and $(\gamma\delta\gamma^{-1})\delta|_{[n]}=\gamma$. Also the permutation $\gamma\delta$ has the one cycle $(1,-1,2,-2,\dots,n,-n)$.

Inside $\mathcal{P}_{2}^{\delta}(\pm n)$ we have the non-crossing pairings of $[\pm n]$ which only connect a positive number to a negative number; we shall denote this subset by $NC_{2}^{\delta}(\pm n)$.

Lemma 19**.**

The map $\sigma\mapsto\sigma\delta\sigma^{-1}$ is a bijection from $S_{NC}(n)$ to $NC_{2}^{\delta}(\pm n)$.

Proof.

We have to check that $\sigma\in S_{NC}(n)$ if and only if $\sigma\delta\sigma^{-1}\in NC_{2}^{\delta}(\pm n)$. Note that $\sigma\delta\sigma^{-1}$ is a pairing so that $\#(\sigma\delta\sigma^{-1})=n$. Also $\#((\sigma\delta\sigma^{-1})^{-1}\gamma\delta)=\#(\delta\sigma\delta\sigma^{-1}\gamma)=\#(\delta\sigma\delta)+\#(\sigma^{-1}\gamma)=\#(\sigma)+\#(\sigma^{-1}\gamma)$ because $\delta\sigma\delta$ acts trivially on $[n]$ and $\sigma^{-1}\gamma$ acts trivially on $[-n]$. Thus $\#(\sigma\delta\sigma^{-1})+\#((\sigma\delta\sigma^{-1})^{-1}\gamma\delta)=n+\#(\sigma)+\#(\sigma^{-1}\gamma)$. By Remark 15 we have that $\sigma$ is non-crossing if and only if $\sigma\delta\sigma^{-1}$ is non-crossing. ∎

Proof of Lemma 18. Suppose that $\sigma\in S_{NC}(n)$ and $i_{1}<i_{2}<i_{3}$ are distinct with $\sigma(i_{1})=i_{2}$ and $\sigma(i_{2})=i_{3}$. Then $\gamma\delta$ visits $\{i_{1},-i_{1},i_{2},-i_{2},i_{3},\allowbreak-i_{3}\}$ and $(i_{1},-i_{2})$ and $(i_{2},-i_{3})$ are pairs of $\sigma^{-1}\delta\sigma$. Thus $\sigma^{-1}\delta\sigma$ is not in $NC_{2}^{\delta}(\pm n)$ and hence by Lemma 19 $\sigma^{-1}\not\in S_{NC}(n)$. Thus the only permutations $\sigma\in S_{NC}(n)$ for which $\sigma^{-1}\in S_{NC}(n)$ are those where $\sigma=\sigma^{-1}$, i.e. all cycles are singletons or pairs. ∎

Theorem 20**.**

[2, Thm. 1]** If $d_{1},d_{2}\rightarrow\infty$, then the limit distributions of W^{\reflectbox{\tiny\Gamma}} and $W^{\Gamma}$ are semi-circular with mean $c$ and variance $c$.

Proof.

We have just shown that the only non-vanishing cumulants of the limiting distribution are $\kappa_{1}=\kappa_{2}=c$. Thus the limiting distribution is semi-circular. ∎

Remark 21*.*

The measure on $\mathbb{R}$ whose free cumulants are $\kappa_{1}=\kappa_{2}=c$ and $\kappa_{n}=0$ for $n\geq 3$ is the shifted semi-circle law. It has density $\frac{1}{2\pi c}\sqrt{4c^{2}-(t-c)^{2}}$ on the interval $[c-2\sqrt{c},c+2\sqrt{c}]$. We have used a different normalization for $W$ than Aubrun, (we used $\frac{1}{d_{1}d_{2}}$ and he used $\frac{1}{p}$), the advantage of ours is that the free cumulants are very simple with this normalization.

Next, we shall discuss the case when only one of the parameters $d_{1}$, $d_{2}$ approaches infinity and the other one is fixed.

The following remarkable result is due to T. Banica and I. Nechita [3, Lemma 1.1].

Lemma 22**.**

Suppose that $\sigma$ is a non-crossing permutation and that $\tau$ is a cycle of length $n$ in $S_{n}$. Then

[TABLE]

where $e(\sigma)$ is the number of cycles of $\sigma$ of even length.

Let us recall the main result of [3, Theorem 3.1], which computes the free cumulants of the limit distribution of d_{1}W^{\reflectbox{\tiny\Gamma}} as $p/(d_{1}d_{2})\rightarrow c$ but keeping $d_{1}$ fixed.

Theorem 23**.**

Suppose that $d_{1}$ is a fixed positive integer, and $\frac{p}{d_{1}d_{2}}\longrightarrow c$ with $0<c<\infty$. The free cumulants of the limit distribution of d_{1}W^{\reflectbox{\tiny\Gamma}} are $\kappa_{n}=cd_{1}^{\,2}$ for $n$ even and $\kappa_{n}=cd_{1}$ for $n$ odd. This limit distribution is the free difference of two Marchenko-Pastur laws one with parameter $cd_{1}\frac{d_{1}+1}{2}$ and the other $cd_{1}\frac{d_{1}-1}{2}$.

Proof.

Let $\epsilon\equiv 1$ and $\eta\equiv-1$ in Theorem 9. By Remark 17, $f_{\epsilon}(\sigma)<0$ unless $\sigma\in NC(n)$. For $\sigma\in NC(n)$ and $\eta\equiv-1$ we have by Remark 17 and Lemma 22, $f_{\eta}(\sigma)-\#(\sigma)+n=e(\sigma)$. Hence Theorem 9 gives

[TABLE]

Note that if we set $\kappa_{n}=d_{1}^{2}c$ for $n$ even and $\kappa_{n}=d_{1}c$ for $n$ odd then $\kappa_{\sigma}=(d_{1}c)^{\#(\sigma)}d_{1}^{e(\sigma)}$. This shows that the limit distribution of d_{1}W^{\reflectbox{\tiny\Gamma}} has the claimed cumulants. Since $\kappa_{n}=(d_{1}c)\frac{d_{1}+1}{2}+(-1)^{n}(d_{1}c)\frac{d_{1}-1}{2}$, we have the claim about the distribution being a free difference of Marchenko-Pastur laws. ∎

Remark 24*.*

If in Theorem 9 we let $\epsilon\equiv-1$ and $\eta\equiv 1$, the coefficients $d_{1}$ and $d_{2}$ switch roles, hence the argument above also gives an analogous statement for holding $d_{2}$ fixed. More precisely, if $d_{2}$ is fixed and $p/(d_{1}d_{2})\rightarrow c$, then the free cumulants of the limit distribution of $d_{2}W^{\Gamma}$ are given by $\kappa_{n}=d_{2}^{2}c$ for $n$ even and $\kappa_{n}=d_{2}c$ for $n$ odd. This distribution is also the free difference of two Marchenko-Pastur distributions one of parameter $d_{2}c\frac{d_{2}+1}{2}$ and one of $d_{2}c\frac{d_{2}-1}{2}$.

Remark 25*.*

Since taking transposes preserves eigenvalue distributions Theorem 23 and Remark 24 also gives us the free cumulants of the limit distribution of $d_{1}W^{\Gamma}$ and d_{2}W^{\reflectbox{\tiny\Gamma}}.

6. A natural free decomposition of d_{1}W^{\reflectbox{\tiny\Gamma}} when $d_{1}=2$

In [3] it was shown that the limit distribution of d_{1}W^{\reflectbox{\tiny\Gamma}} can be written as the free difference of two Marchenko-Pastur laws. The operators so obtained are not related to the operator d_{1}W^{\reflectbox{\tiny\Gamma}} though. In this section we shall show that there is a natural decomposition of d_{1}W^{\reflectbox{\tiny\Gamma}} when $d_{1}=2$, namely the diagonal and off diagonal blocks, into free summands. More precisely, we let $w$ be the limit distribution of $W$, which we can write $w$ as a $2\times 2$ matrix

[TABLE]

Relative to this block decomposition 2W^{\reflectbox{\tiny\Gamma}} converges to

[TABLE]

We consider the two operators

[TABLE]

The diagonal summand $X_{1}$ has the Marchenko-Pastur distribution and the off diagonal summand $X_{2}$ is even and has the same even cumulants as the diagonal summand. Our main result in this section is that $X_{1}$ and $X_{2}$ are free.

Notation 26*.*

Let $d_{1}=2$, and suppose $p/(d_{1}d_{2})$ converges to $c$ with $0<c<\infty$. Let $\{E_{11},E_{12},E_{21},E_{22}\}$ be the standard matrix units in $M_{2}(\mathbb{C})$, but viewed as elements of $M_{2}(\mathbb{C})\otimes M_{d_{2}}(\mathbb{C})$.

Lemma 27**.**

There is a $*$-non-commutative probability space $(\mathcal{A},\varphi)$ with elements $w,e_{11},e_{12},e_{21},e_{22}\in\mathcal{A}$ such that $w$ has the Marchenko-Pastur distribution with parameter $c$ and $\{e_{11},e_{12},e_{21},e_{22}\}$ are matrix units in $\mathcal{A}$ free from $w$. Moreover the joint distribution of $\{W,E_{11},E_{12},\allowbreak E_{21},E_{22}\}$ converges to that of $\{w,e_{11},e_{12},e_{21},e_{22}\}$

Proof.

As $W$, our Wishart matrix, is unitarily invariant, it is asymptotically free from our matrix units (see [10, Theorem 4.9]). This is exactly the claim of the lemma. ∎

Notation 28*.*

Thus we may write the matrix of $w$ with respect to the matrix units $\{e_{11},e_{12},e_{21},e_{22}\}$ as

[TABLE]

We will let $\varphi_{1}$ be the state on $e_{11}\mathcal{A}e_{11}$ given by $\varphi_{1}(x)=2\varphi(x)$. The elements $\{w_{11},w_{12},w_{21},w_{22}\}$ are in $e_{11}\mathcal{A}e_{11}$ so their cumulants must be computed relative to the state $\varphi_{1}$. When necessary we will denote these relative cumulants by $\kappa_{n}^{(1)}$.

Lemma 29**.**

Each of $w_{11}$ and $w_{22}$ have the Marchenko-Pastur distribution with parameter $d_{1}c$.

Proof.

By construction $w_{11}=e_{11}2we_{11}$. By [11, Theorem 14.18]

[TABLE]

∎

Remark 30*.*

Elements $\{a_{ij}\}_{i,j=1}^{n}$ in a non-commutative probability space $(\mathcal{A},\varphi)$, they are called $R$-cyclic if whenever $i_{1},j_{1},\dots,i_{l},j_{l}\in[n]$ we have $\kappa_{l}(a_{i_{1}j_{1}},\dots,a_{i_{l}j_{l}})=0$ unless $j_{1}=i_{2},\dots,j_{n-1}=i_{n}$ and $j_{n}=i_{1}$. By [11, Example 20.4] the elements $\{w_{11},w_{12},w_{21},w_{22}\}$ are $R$-cyclic. Moreover $\kappa_{l}(2w,\dots,2w)=2^{l}\kappa_{l}(w,\dots,w)=2^{l}c$. So by [11, Example 20.4], we have $\kappa_{l}^{(1)}(w_{i_{1}j_{1}},w_{i_{2}j_{2}},\dots,w_{i_{l}j_{l}})=2^{-l+1}\kappa_{l}(2w,\dots,2w)=2c$, when $j_{1}=i_{2},\dots,j_{n-1}=i_{n}$ and $j_{n}=i_{1}$.

Let $X_{1}=\begin{pmatrix}w_{11}&0\\ 0&w_{22}\\ \end{pmatrix}$ and $X_{2}=\begin{pmatrix}0&w_{21}\\ w_{12}&0\\ \end{pmatrix}$. Then 2w^{\reflectbox{\tiny\Gamma}}=X_{1}+X_{2}.

Lemma 31**.**

$X_{1}$* and $X_{2}$ are self-adjoint. The cumulants of $X_{1}$ are all equal to $2c$, i.e. $X_{1}$ is a Marchenko-Pastur operator with parameter $2c$. $X_{2}$ is an even operator in that it is self-adjoint and all of its odd moments are [math]. The even cumulants of $X_{2}$ are all equal to $2c$.*

Proof.

We have $\varphi(X_{1}^{l})=\varphi^{(1)}(w_{11}^{l})$ so $X_{1}$ and $w_{11}$ have the same cumulants, which by Remark 30 are all $2c$. Because $X_{2}$ is off diagonal and self-adjoint, it is an even operator. By [11, Proposition 15.12] the cumulants of $X_{2}$ are the $*$-cumulants of $w_{21}$. In Remark 30 we observed that these are all $2c$. ∎

Our next goal is to show that $X_{1}$ and $X_{2}$ are free in $(\mathcal{A},\varphi)$. This is somewhat surprising in that $X_{1}$ and X_{2}^{\reflectbox{\tiny\Gamma}} are not free. By X_{2}^{\reflectbox{\tiny\Gamma}} we mean the matrix $\begin{pmatrix}0&w_{12}\\ w_{21}&0\end{pmatrix}$. To see this note that \varphi(X_{1}X_{2}^{\reflectbox{\tiny\Gamma}}X_{2}^{\reflectbox{\tiny\Gamma}}X_{1})=2c+3(2c)^{2}+(2c)^{3} whereas if $X_{1}$ and $X_{2}$ were free we would have \varphi(X_{1}X_{2}^{\reflectbox{\tiny\Gamma}}X_{2}^{\reflectbox{\tiny\Gamma}}X_{1})=(2c)^{2}+(2c)^{3}. This gives another unexpected instance where a partial transpose produces freeness, but this time at the level of operators.

Now let us turn to the freeness of $X_{1}$ and $X_{2}$. Let

[TABLE]

Then $X_{2}=Y_{2}Y_{3}$. Let $i_{1},\dots,i_{n}$ be such that $\ker(i)<1_{n}$. Let $r$ be the number of times $i=2$, and $q=n+r$. Then there are $j_{1},j_{2},\dots,j_{q}\in\{1,2,3\}$ such that

[TABLE]

We now apply the formula for cumulants with products as entries [11, Theorem 11.12]. Then

[TABLE]

where the sum runs over all non-crossing partitions in $NC(q)$ such that $\pi\vee\rho=1_{q}$ and $\rho$ is the non-crossing partition whose blocks are have either 1 or 2 elements, and the singletons are where $j_{l}=1$ appears in the string $Y_{j_{1}}\cdots Y_{j_{q}}$, and the pairs are $(l,l+1)$ where $j_{l}=2$ and $j_{l+1}=3$. Since $\ker(i)<1_{n}$ there must both singletons and pairs. Let us consider a $\pi\in NC(q)$ which is such that $\pi\vee\rho=1_{q}$ and we shall show that by the $R$-cyclicity of $w$ we have $\kappa_{\pi}(Y_{j_{1}},\dots,Y_{j_{q}})=0$. Summing over all such $\pi$ we get that $\kappa_{n}(X_{i_{1}},\dots,X_{i_{n}})=0$. This shows that all mixed cumulants vanish and hence that $X_{1}$ and $X_{2}$ are free as claimed.

Lemma 32**.**

Given $p_{1},\dots,p_{s}\in[3]$ we have $\varphi(Y_{p_{1}}\cdots Y_{p_{s}})=0$ unless $Y_{3}$ appears an even number of times.

Proof.

$Y_{1}$ and $Y_{2}$ are diagonal so $Y_{p_{1}}\cdots Y_{p_{s}}$ will be 0 on the diagonal unless $Y_{3}$ appears an even number of times. ∎

Lemma 33**.**

Given $p_{1},\dots,p_{s}\in[3]$ we have $\kappa_{s}(Y_{p_{1}},\dots,Y_{p_{s}})=0$ unless $Y_{3}$ appears an even number of times.

Proof.

We write

[TABLE]

Given $\pi$, we have by Lemma 32, that each block of $\pi$ must contain an even number of $Y_{3}$’s, or else $\varphi_{\pi}(Y_{p_{1}},\dots,Y_{p_{s}})=0$. Summing over all blocks of $\pi$ we get that the number of $Y_{3}$’s is even. ∎

Definition 34**.**

Let $i_{1},\dots,i_{s}\in[3]$ we say that the $s$-tuple has the property (nvc) if each non-zero entry of $Y_{i_{1}}\cdots Y_{i_{s}}$ is of the form

[TABLE]

where $v_{1}=u_{2},v_{2}=u_{3},\dots,v_{k-1}=v_{k}$. Note that we do not require $v_{k}=u_{1}$ as in $R$-cyclicity. We say that the string has property (vc) if it does not have property (ncv).

Remark 35*.*

We now describe the generic sequences with property (nvc). First we have any power of $Y_{1}$. The product of two or more $Y_{2}$’s does not have property (nvc). No power of $Y_{3}$ has property (nvc), because all its entries are either 0 or 1.

Now suppose we start with a $Y_{2}$. We can only follow with a $Y_{1}$ or a $Y_{3}$. So our basic reduced sequence is (with the possibility that $k=0$)

[TABLE]

We can enhance this by putting an even power of $Y_{3}$ between any two letters above. Note that there cannot be an odd number of $Y_{3}$’s between two $Y_{1}$’s as

[TABLE]

So the most general string starting and ending with a $Y_{2}$ is

[TABLE]

with $l_{1},\dots,l_{r}$ even and $l_{r+1}$ odd.

Lemma 36**.**

Let $i_{1},\dots,i_{k}$ be a string with property (nvc) which starts and ends with $Y_{2}$ and has no other $Y_{2}$’s. Then the number of $Y_{3}$’s is odd.

Proof.

We just observed that the number of $Y_{3}$’s is $l_{1}+\cdots+l_{r}+l_{r+1}$ which is odd. ∎

Lemma 37**.**

If $\pi\in NC(q)$ and $\pi\vee\rho=1_{q}$ and $\kappa_{\pi}(Y_{i_{1}},\dots,Y_{i_{q}})\not=0$ then each block of $\pi$ must contain the same number of $Y_{2}$’s as $Y_{3}$, and both numbers are even.

Proof.

We have just observed that the number of $Y_{3}$’s between $Y_{2}$’s is odd. Thus to go all the way round a cycle the number of $Y_{3}$’s is equal to the number of $Y_{2}$ plus an even number which might be 0. However in our whole string the number of $Y_{2}$’s and $Y_{3}$’s is the same. If one cycle had an excess of $Y_{3}$’s then another cycle would have a deficit. Thus all cycles must be balanced. Since we already know that each cycle has an even number of $Y_{3}$’s it also has an equal even number of $Y_{2}$’s. ∎

Lemma 38**.**

Let $i_{1}\,\dots,i_{q}\in[3]$ be such that $\ker(i)<1_{q}$ and $\pi\in NC(q)$ be such that $\pi\vee\rho=1_{q}$. Then $\kappa_{\pi}(Y_{i_{1}},\dots,Y_{i_{k}})=0$.

Proof.

Let $V$ be a block of $\pi$ that contains a $l$ such that $j_{l}=1$. Then $(l)$ is a block of $\rho$. Since we are assuming that $\pi\vee\rho=1_{q}$ there must be $l_{0}\in V$ with $j_{l_{0}}\in\{2,3\}$. If the contribution of this block to $\kappa_{\pi}(Y_{i_{1}},\dots,Y_{i_{k}})$ is not 0 then there must be a $Y_{1}$ followed by a $Y_{3}$. So we may assume that we have a $l$ and $l^{\prime}$ such that $j_{l}=1$, $j_{l^{\prime}}=3$ and $l^{\prime}$ follows $l$ in $V$. We have that $\pi$ restricts to a non-crossing partition of $[l+1,l^{\prime}-1]$. Each block in this restriction contains the same number of $Y_{2}$’s as $Y_{3}$’s by Lemma 37. However this impossible because in the original string $Y_{i_{1}},\dots,Y_{i_{k}}$ a $Y_{2}$ is always followed by a $Y_{3}$ and we have removed one $Y_{3}$. Thus $\kappa_{\pi}(Y_{i_{1}},\dots,Y_{i_{k}})=0$. ∎

Theorem 39**.**

$X_{1}$* and $X_{2}$ are free in $(\mathcal{A},\varphi)$.*

Proof.

We have just shown that by the formula for cumulants with products for entries we have that mixed cumulants vanish. Thus $X_{1}$ and $X_{2}$ are free. ∎

Remark 40*.*

The distribution of w^{\reflectbox{\tiny\Gamma}} in $(\mathcal{A},\varphi)$ is the limit distribution of W^{\reflectbox{\tiny\Gamma}} which is the same as $W^{\Gamma}$. Thus the distribution of $d_{1}w^{\Gamma}$ is the same as that of d_{1}w^{\reflectbox{\tiny\Gamma}}.

Theorem 41**.**

For $d_{1}=2$ and $p/(d_{1}d_{2})\rightarrow c$ the limit distribution of 2W^{\reflectbox{\tiny\Gamma}} is the free additive convolution of a Marchenko-Pastur law with parameter $2c$ and an even operator with all even cumulants equal to $2c$.

7. Asymptotic Freeness

Since $W$ is unitarily invariant, a consequence of the results from [9] is that $W$ and $W^{T}$ are asymptotically free if $d_{1}d_{2}\rightarrow\infty$. In this section we will present the main results of the paper, which, using the relation form Theorem 9, gives an improvement of the result mentioned above.

Theorem 42**.**

If $d_{1}\rightarrow\infty$ and $d_{2}\rightarrow\infty,$ then the family \{W,W^{T},W^{\Gamma},\allowbreak W^{\reflectbox{\tiny\Gamma}}\} is asymptotically free.

Proof.

By Theorem 9 we have that

[TABLE]

and by Lemma 12 and Proposition 16 we have that

• $\circ$

$f_{\epsilon}(\sigma),f_{\eta}(\sigma)\leq 0$ for all $\sigma,\epsilon$, and $\eta$;

• $\circ$

$f_{\epsilon}(\sigma),f_{\eta}(\sigma)<0$ unless $\epsilon$ and $\eta$ are constant on the cycles of $\sigma$;

• $\circ$

$f_{\epsilon}(\sigma)<0$ unless $\sigma_{\epsilon}$ is non-crossing.

Thus when $d_{1},d_{2}\rightarrow\infty$ and $\displaystyle\frac{p}{d_{1}d_{2}}\rightarrow c$ we need only consider $\sigma$’s for which

1. i)

$\epsilon$ and $\eta$ are constant on the cycles of $\sigma$; 2. ii)

both $\sigma_{\epsilon}$ and $\sigma_{\eta}$ are non-crossing.

Note that as partitions $\sigma$, $\sigma_{\epsilon}$, and $\sigma_{\eta}$ are the same, as the only possible difference between them is whether we reverse the order of elements in a cycle of $\sigma$. Thus we have shown that the limit when $d_{1},d_{2}\rightarrow\infty$ of an arbitrary mixed moment can be written as a sum over non-crossing partitions; that means that the terms that appear are the free cumulants of the mixed moment we are considering. However, by (i), the blocks of $\sigma$ only connect $W^{(\epsilon_{i},\eta_{i})}$ to $W^{(\epsilon_{j},\eta_{j})}$ if $(\epsilon_{i},\eta_{i})=(\epsilon_{j},\eta_{j})$. This means we have shown that mixed cumulants vanish and this implies the conclusion.

∎

Theorem 43**.**

(i)* If $d_{1}\rightarrow\infty$ and $d_{2}$ is fixed, then the family $\{W,W^{\Gamma}\}$ is asymptotically free from the family \{W^{T},W^{\reflectbox{\tiny\Gamma}}\}, but $W$ is not asymptotically free from $W^{\Gamma}$, nor is $W^{T}$ from W^{\reflectbox{\tiny\Gamma}}.*

(ii) If $d_{1}$ is fixed and $d_{2}\rightarrow\infty$, then the family \{W,W^{\reflectbox{\tiny\Gamma}}\} is asymptotically free from the family $\{W^{T},W^{\Gamma}\}$ but $W$ is not asymptotically free from $W^{\Gamma}$, nor is $W^{T}$ from $W^{\Gamma}$.

Proof.

Suppose first that $d_{1}\rightarrow\infty$ and $d_{2}$ is fixed. Hence in the summation from Theorem 9 only the terms where $f_{\varepsilon}(\sigma)=0$ will contribute to the limit. As in the proof of Theorem 42, the last condition is equivalent to $\sigma_{\varepsilon}$ is non-crossing and $\varepsilon$ is constant on the cycles of $\sigma$. Since the partitions $\sigma$ and $\sigma_{\varepsilon}$ are the same, it follows that the limit as $d_{1}\rightarrow\infty$ of an arbitrary mixed moment is written as a sum over non-crossing partitions, so the terms in the right-hand side are in fact free cumulants. The condition that $\varepsilon$ is constant on the cycles of $\sigma$ gives that only free cumulants in elements from only one of the families from part (i) do not vanish, hence the asymptotic freeness is proved.

For the second part of (i), we will use the expansion of $E\circ\operatorname{tr}\otimes\operatorname{tr}(W\cdot W^{\Gamma})$ from Theorem 9 in the case $\varepsilon=(1,1)$ and $\eta=(1,-1)$. Note that $S_{2}$ contains only two permutations, $\gamma=(1,2)$ and $\sigma=(1),(2)$, both non-crossing. Also, since $\varepsilon$ is constant, it is constant on the cycles of $\sigma$ and $\gamma$. It follows that $f_{\varepsilon}(\sigma)=f_{\varepsilon}(\gamma)=0.$ Moreover, $\eta$ is constant on the cycles of $\sigma$ and $\sigma_{\eta}=\sigma$ is non-crossing, hence $f_{\eta}(\sigma)=0$. Therefore, Theorem 8 gives that

[TABLE]

As $d_{1}\rightarrow\infty$, the first moment of $W$ approaches $c$, and from Theorem 23, so does the first moment of $W^{\Gamma}$, hence

[TABLE]

The same argument also shows that $W^{T}$ and W^{\reflectbox{\tiny\Gamma}} are not asymptotically free, since W^{\reflectbox{\tiny\Gamma}}=\big{(}W^{\Gamma}\big{)}^{T}.

Finally, part (ii) also follows from the argument for part (i), since the relation from Theorem 9 is symmetric in $(d_{1},\varepsilon)$ and $(d_{2},\eta)$.

∎

8. The Case of Real Wishart Matrices

In this section we examine the case of real Wishart matrices. More precisely, $W$ will denote now the symmetric $d_{1}d_{2}\times d_{1}d_{2}$ random matrix

[TABLE]

where $\{G_{i}:1\leq i\leq d_{1}\}$ is a family of $d_{2}\times p$ random matrices whose entries are independent Gaussian random variables of mean 0 and variance 1.

Since $W=W^{\mathrm{T}}$ and W^{\reflectbox{\tiny\Gamma}}=W^{\Gamma} we shall only work with $W$ and $W^{\Gamma}$. For this reason we shall use slightly different notation that in the previous sections. For $\epsilon\in\mathbb{Z}_{2}=\{-1,1\}$ we let

[TABLE]

Thus our goal will be to consider, $W^{(\epsilon_{1})}\cdots W^{(\epsilon_{n})}$, an arbitrary word in $W$ and $W^{\Gamma}$ and find its limiting expectation.

Theorem 44**.**

With the notations from above, we have that

[TABLE]

where

[TABLE]

and $\epsilon\in S(\pm n)$ is, as before, given by

[TABLE]

Proof.

[TABLE]

In line 3 we momentarily break with our previous convention about $W^{(\epsilon)}$ indicating whether or not we take a partial transpose; in this case $W(i_{u},i_{-u})^{(-1)}$ means take the transpose of the $d_{2}\times d_{2}$ matrix $W(i_{u},i_{-u})$. In passing from line 3 to line 4 above we let $j=k\circ\epsilon$.

Now

[TABLE]

On the right hand side in the expression above we are extending $k$ as a function from $[n]$ to $[p]$ to a function on $[\pm n]$ by requiring $k_{r}=k_{-r}$. This means $k=k\circ\delta$. Now $\operatorname{E}(g^{(i_{r})}_{j_{r}k_{r}}g^{(i_{s})}_{j_{s}k_{s}})=0$ unless $i_{r}=i_{s}$, $j_{r}=j_{s}$, and $k_{r}=k_{s}$, in which case it is 1. Thus

[TABLE]

Since we require $k=k\circ\pi$ and $k=k\circ\delta$ we must have $k=k\circ\pi\delta$. Now as noted in Lemma 10 the cycles of $\pi\delta$ appear in pairs where one part of a pair is the conjugate by $\delta$ of the other. Since $k$ is a function on $[n]$, $\#(\pi\delta)$ double counts the degrees of freedom. Hence the exponent of $p$ is $\#(\pi\delta)/2$. Thus

[TABLE]

Finally, note that

[TABLE]

hence the conclusion. ∎

Next we shall show that $g(\pi)\leq 0$ and $g(\epsilon\pi\epsilon)\leq 0$ for all $\epsilon$ and $\pi$, and to find for which pairings $\pi$ we have equality.

Lemma 45**.**

Let $\pi\in\mathcal{P}_{2}(\pm n)$ be a pairing such that there is $(r,s)\in\pi$ with the same sign. Then $g(\pi)<0$.

Proof.

Since $\pi$ connects two elements with the same sign, $\pi\delta$ connects two elements with opposite signs. Then the subgroup generated by $\gamma\delta\gamma^{-1}\delta$ and $\pi\delta$ acts transitively on $[\pm n]$. Thus

[TABLE]

We have

[TABLE]

Thus $g(\pi)=\#(\gamma\delta\gamma^{-1}\vee\pi)+\#(\pi\delta)/2-(n+1)\leq-1$. ∎

Lemma 46**.**

Suppose $\pi\in\mathcal{P}_{2}(\pm n)$ and $\pi$ only connects elements of opposite sign. Then $\pi\delta$ leaves $[n]$ invariant and $g(\pi)\leq 0$ with equality only if $\pi\delta|_{[n]}$ is a non-crossing permutation.

Proof.

Since both $\pi$ and $\delta$ switch signs, $\pi\delta$ preserves signs. Thus $\pi\delta$ leaves $[n]$ invariant. By Lemma 10 we have $\#(\pi\delta)=2\#(\pi\delta|_{[n]})$. Also $2\#(\gamma\delta\gamma^{-1}\vee\pi)=\#(\gamma\delta\gamma^{-1}\delta\delta\pi)=\#((\pi\delta)^{-1}\gamma\delta\gamma^{-1}\delta)=2\#((\pi\delta|_{[n]})^{-1}\gamma)$. Hence

[TABLE]

with equality only if $\pi\delta|_{[n]}$ is a non-crossing permutation. ∎

Lemma 47**.**

Let $\epsilon\in\mathbb{Z}_{2}^{n}$ and $\pi\in\mathcal{P}_{2}(\pm n)$. Then $g(\epsilon\pi\epsilon)<0$ unless $\epsilon\pi\delta\epsilon$ leaves $[n]$ invariant. If $\epsilon\pi\delta\epsilon$ leaves $[n]$ invariant then $g(\epsilon\pi\epsilon)\leq 0$ with equality only if $\epsilon\pi\delta\epsilon|_{[n]}$ is a non-crossing permutation.

Proof.

By Lemma 46 we have $g(\epsilon\pi\epsilon)<0$ unless $\epsilon\pi\epsilon\delta=\epsilon\pi\delta\epsilon$ leaves $[n]$ invariant. If $\epsilon\pi\delta\epsilon$ leaves $[n]$ invariant then again by Lemma 46 we have $g(\epsilon\pi\epsilon)\leq 0$ with equality only if $\epsilon\pi\delta\epsilon|_{[n]}$ is a non-crossing permutation. ∎

Lemma 48**.**

Let $\epsilon\in\mathbb{Z}_{2}^{n}$ and $\pi\in\mathcal{P}_{2}(\pm n)$. Suppose $\pi\delta$ leaves $[n]$ invariant. Then $\epsilon\pi\delta\epsilon$ leaves $[n]$ invariant if and only if $\epsilon$ is constant on the cycles of $\pi\delta$.

Proof.

Suppose $(i_{1},\dots,i_{k})$ is a cycle of $\pi\delta$. All these elements must have the same sign. The the corresponding cycle of $\epsilon\pi\delta\epsilon$ is $(\epsilon(i_{1}),\dots,\allowbreak\epsilon(i_{k}))$. The elements of $\epsilon\pi\delta\epsilon$ is $(\epsilon(i_{1}),\dots,\allowbreak\epsilon(i_{k}))$ have the same sign if and only if $\epsilon$ is constant on $\epsilon\pi\delta\epsilon$ is $(\epsilon(i_{1}),\dots,\allowbreak\epsilon(i_{k}))$. ∎

The following theorem is the main result of this section. Recall from Lemma 13 that if $\epsilon$ is constant on the cycles of $\sigma$, then we obtain $\sigma_{\epsilon}$ from $\sigma$ by reversing the cycles on which $\epsilon=-1$.

Theorem 49**.**

[TABLE]

where the sum runs over all non-crossing permutations $\sigma$ such that $\epsilon$ is constant on the cycles of $\sigma$ and $\sigma_{\epsilon}$ is also non-crossing.

Proof.

In the formula from Theorem 44, only the pairings $\pi$ such that $g(\pi)=g(\epsilon\pi\epsilon)=0$ will contribute to the summation when $d_{1},d_{2}\rightarrow\infty$.

Recall that $\mathcal{P}_{2}^{\delta}(\pm n)$ denotes the pairings $\pi$ of $[\pm n]$ such that $\pi\delta$ leaves $[n]$ invariant. For such a $\pi$ we let $\sigma=\pi\delta|_{[n]}$ be the corresponding permutation. We already noted that this is a bijection from $\mathcal{P}_{2}^{\delta}(\pm n)$ to $S_{n}$ and $\pi\delta=\delta\sigma^{-1}\delta\sigma$. From Lemma 46, the condition $g(\pi)=0$ implies that $\sigma=\delta\sigma^{-1}\delta\sigma_{\ [n]}$ is noncrossing.

According to Lemmas 47 and 48, the condition $g(\epsilon\pi\epsilon)=0$ implies that $\epsilon$ is constant on the cycles of $\sigma$. As in Lemma 13, $\epsilon\pi\delta\epsilon=\delta\sigma_{\epsilon}^{-1}\delta\sigma_{\epsilon}$. Therefore

[TABLE]

which gives

[TABLE]

hence the formula (4) gives that $g(\epsilon\pi\epsilon)\leq 0$ with equality if and only if $\sigma_{\epsilon}$ is non-crossing.

∎

An immediate consequence of Theorem 49 is part (1) of the following result.

Theorem 50**.**

Suppose $p/(d_{1}d_{2})\rightarrow c$.

(1)* If $d_{1},d_{2}\rightarrow\infty$, then $W^{\Gamma}$ is asymptotically a shifted semi-circular operator with $\kappa_{1}=\kappa_{2}=c$.*

(2)* If $d_{1}\rightarrow\infty$ and $d_{2}\geq 2$ is fixed then the asymptotic distribution of $d_{2}W^{\Gamma}$, equals the distribution of the difference of two free variables with Marchenko-Pastur laws, the first of parameter $\displaystyle cd_{2}\frac{d_{2}+1}{2}$ and the second of parameter $\displaystyle cd_{2}\frac{d_{2}-1}{2}$.*

(3)* If $d_{1}$ is fixed and $d_{2}\rightarrow\infty$ then the asymptotic distribution of $d_{1}W^{\Gamma}$, equals the distribution of the difference of two free variables with Marchenko-Pastur laws, the first of parameter $\displaystyle cd_{1}\frac{d_{1}+1}{2}$ and the second of parameter $\displaystyle cd_{1}\frac{d_{1}-1}{2}$.*

Proof.

Letting $\epsilon_{j}=-1$ for all $j=1,\dots,n$ in Theorem 44, we obtain that

[TABLE]

Suppose first that $d_{1},d_{2}\rightarrow\infty$. Then, in the summation from (6), only terms with $g(\pi)=g(\delta\pi\delta)=0$ will contribute to the limit. From Theorem 49, this is equivalent to both $\sigma$ and $\sigma_{\delta}$ be noncrossing. But $\sigma_{\delta}=\sigma^{-1}$ so Lemma 18 implies that $\sigma$ has only cycles of length 1 or 2, hence part (1) is proved.

Suppose now that $d_{1}\rightarrow\infty$ and $d_{2}$ is fixed. Then only $\pi$ such that $g(\pi)=0$ will contribute to the limit in the summation (6). Applying again Lemma 46, this is equivalent to $\pi=\sigma\delta\sigma^{-1}$, for $\sigma$ a non-crossing permutation on $[n]$. In this case, we have that $g(\delta\pi\delta)=g(\delta\sigma\delta\sigma^{-1}\delta).$

Also, $\#(\gamma\delta\gamma^{-1}\vee\delta\pi\delta)=\frac{1}{2}\#(\gamma\delta\gamma^{-1}\delta\sigma\delta\sigma^{-1}\delta),$ and, if $k\in[n]$, we have that

[TABLE]

Moreover,

[TABLE]

hence, Lemma 22 gives that

[TABLE]

so equation (6) becomes

[TABLE]

Thus

[TABLE]

where $\kappa_{n}=cd_{2}$ for $n$ odd and $\kappa_{n}=cd_{2}^{2}$ for $n$ even. The conclusion follows because $\kappa_{n}=(cd_{2})\frac{d_{2}+1}{2}+(-1)^{n}(cd_{2})\frac{d_{2}-1}{2}$ (see the proof of Theorem 23). The case $d_{1}$ fixed and $d_{2}\rightarrow\infty$ is similar.

∎

Theorem 51**.**

If both $d_{1},d_{2}\rightarrow\infty$, then $\{W,W^{\Gamma}\}$ is an asymptotically free family.

Proof.

The result is a consequence of Theorems 49 and 50.

∎

Remark 52*.*

For $W$ a real Wishart random matrix, $W^{\Gamma}$ is not asymptotically free from $W$ if $d_{1}$ is fixed or if $d_{2}$ is fixed.

Indeed, for $n=2$ and $\epsilon_{1}=1$ and $\epsilon_{2}=-1$, the formula from Theorem 44 gives

[TABLE]

There are only 3 pairings in $P_{2}(\pm 2)$: $\pi_{1}=(1,-1),(2,-2)$, $\pi_{2}=(1,2),(-1,-2)$ and $\pi_{3}=(1,-2),(-1,2)$. Direct calculations give that $\pi_{1}\delta=\textrm{id}$, $\pi_{2}\delta=(1,-2),(-1,2)$ and $\pi_{2}\delta=(1,2),(-1,-2)$.

Moreover, $\epsilon\pi_{1}\epsilon=\pi_{1}$, while $\epsilon\pi_{2}\epsilon=\pi_{3}$ and $\epsilon\pi_{3}\epsilon=\pi_{2}$; also, for $n=2$, we have that $\gamma\delta\gamma^{-1}=(1,-2),(-1,2).$ Therefore $g(\pi_{1})=g(\pi_{2})=0$ and $g(\pi_{3})=1$, so

[TABLE]

and the second term in the equation above does not cancel asymptotically unless both $d_{1},d_{2}\rightarrow\infty$.