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--- freeness [2023/01/13 13:55] – [Models] ngilliers
+++ freeness [2023/03/09 14:35] (current) – ngilliers
@@ Line 1: / Line 1: @@
-======== Freeness ========
+========= Freeness =========
+----
-===== Definition =====
+<typo font-size:medium;>**Definition**
+</typo>[(:cite:freeness:voiculescu1987multiplication)]
-**Definition** [Freeness [(:cite:freeness:voiculescu1987multiplication)]
-]
 Let $\mathcal{A}$ be a unital complex algebra equipped with a linear functional $\psi:\mathcal{A}\to \mathbb{C}$ where $\psi$ is unital. Let $\mathcal{A}_1 ,\mathcal{A}_2 \subset \mathcal{A}$ be two unital subalgebras. The two subalgebras $\mathcal{A}_1$ and $\mathcal{A}_2$ are //free// with respect to $\psi$ if, for any sequence $(a_1,\ldots,a_n)$ of $\mathcal{A}_1\cup \mathcal{A}_2$ which is alternating and centered with respect to $\psi$, we have $$\label{eqn:freemoments}
     \psi(a_1\cdots a_n) = 0.$$
-===== Models ======
+===== - Models =====
-** Theorem **
-Let $R>0$ and, for each $N\geq 1$, let $U_N$ be a Haar distributed unitary random matrix. Then, for any $P\in \mathbb{C}\langle X_k,Y_k:k\in K\rangle$, we have $$\mathbb{E}\left[\frac{1}{N}{\rm Tr}(P({\bf A}_N,U_N{\bf B}_NU_N^{\star}))\right] = \psi_{{\bf A}_N} * \psi_{{\bf B}_N}(P)+O(N^{-2})$$ uniformly for the the choice of any sequences ${\bf A}_N = \{A_N^k: k \in K\}$ and ${\bf B}_N = \{B^k_N:~k \in K\}$ of $N\times N$ matrices ($N\geq 1$) bounded in operator norm by $R$
+Let $R > 0$ a real number. For each $N\geq 1$, let $U_N$ be a Haar distributed unitary random matrix. Then, for any $P\in \mathbb{C}\langle X_k,Y_k:k\in K\rangle$, we have $$\mathbb{E}\left[\frac{1}{N}{\rm Tr}(P({\bf A}_N,U_N{\bf B}_NU_N^{\star}))\right] = \psi_{{\bf A}_N} * \psi_{{\bf B}_N}(P)+O(N^{-2})$$ uniformly for the the choice of any sequences ${\bf A}_N = \{A_N^k: k \in K\}$ and ${\bf B}_N = \{B^k_N:~k \in K\}$ of $N\times N$ matrices ($N\geq 1$) bounded in operator norm by $R$