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--- freeness [2023/03/09 14:10] – ngilliers
+++ freeness [2023/03/09 14:35] (current) – ngilliers
@@ Line 1: / Line 1: @@
+========= Freeness =========
 ----
 <typo font-size:medium;>**Definition**
@@ Line 15: / Line 15: @@
-Let $A_N$ and $B_N$ be two ensembles of matrices, bounded independently of $N$
-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 $R$