Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m,n\in\mathbb{N}$, let $\iota^{1}:\mathcal{P}_{m}\to\mathcal{P}_{m+n}$ be as in Freeness of Two Groups of Variables under a Noncommutative Law §free, let $\star$ be the free product, and for $k\in\mathbb{N}$ let $\mathrm{sc}_{k}$ be the semicircular law of $k$ variables.

1. (Characterisation)¶ Let $\alpha\in\Sigma_{m}$ and let $\gamma\in\Sigma_{m+n}$ with $\gamma\circ\iota^{1}=\alpha$. Then $\gamma=\alpha\star\mathrm{sc}_{n}$ if and only if $\gamma$ satisfies the Schwinger-Dyson equation in the last $n$ variables.

2. (Semicircular systems)¶ $\mathrm{sc}_{m}\star\mathrm{sc}_{n}=\mathrm{sc}_{m+n}$.

3. (Rotations)¶ Let $k\in\mathbb{N}$ and let $O:[k]\times[n]\to\mathbb{R}$ satisfy $\sum_{l=1}^{n}O_{il}O_{jl}=1$ if $i=j$ and $=0$ if $i\neq j$, for all $i,j\in[k]$. Then $\mathrm{sc}_{n}\circ\sigma_{(O,0)}=\mathrm{sc}_{k}$, where $\sigma_{(O,0)}$ is the affine substitution of the affine datum $(O,0)$ from $n$ to $k$ variables.