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Block-modified Wishart matrices and free Poisson laws

We study the random matrices of type $\tilde{W}=(id\otimesφ)W$, where $W$ is a complex Wishart matrix of parameters $(dn,dm)$, and $φ:M_n(\mathbb C)\to M_n(\mathbb C)$ is a self-adjoint linear map. We prove that, under suitable assumptions, we have the $d\to\infty$ eigenvalue distribution formula $δm\tilde{W}\simπ_{mnρ}\boxtimesν$, where $ρ$ is the law of $φ$, viewed as a square matrix, $π$ is the free Poisson law, $ν$ is the law of $D=φ(1)$, and $δ=tr(D)$.