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freeness [2023/03/09 14:09] ngilliersfreeness [2023/03/09 14:35] (current) ngilliers
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-Freeness +========= Freeness =========
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 <typo font-size:medium;>**Definition**  <typo font-size:medium;>**Definition** 
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-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 $R$
  
  
freeness.1678370944.txt.gz · Last modified: 2023/03/09 14:09 by ngilliers