The Semicircular Law and Its Scalings

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $n\in\mathbb{N}$.

1. (Semicircular law)¶ The semicircular law of $n$ variables is the law $\mathrm{sc}_{n}\in\Sigma_{n}$ that satisfies the Schwinger-Dyson equation: such a law exists by The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §vacuum-law and The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §equation, and there is only one by The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation §unique.

2. (Variance $t$)¶ For real $t\ge0$ let $\sqrt{t}$ be its nonnegative square root, and let $T_{t}=(A,0)$ be the affine datum from $n$ to $n$ variables with $A_{ij}=\sqrt{t}$ if $i=j$ and $A_{ij}=0$ if $i\neq j$. The semicircular law of variance $t$ is $\mathrm{sc}_{n,t}=\mathrm{sc}_{n}\circ\sigma_{T_{t}}$, with $\sigma_{T_{t}}$ the affine substitution; it belongs to $\Sigma_{n}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint.