Block-modified Wishart matrices and free Poisson laws

Abstract

We study the random matrices of type $\tilde{W}=(id\otimes \phi )W$, where $W$ is a complex Wishart matrix of parameters $(dn,dm)$, and $\phi :M_n(\mathbb{C})\to M_n(\mathbb{C})$ is a self-adjoint linear map. We prove that, under suitable assumptions, we have the $d\to \infty$ eigenvalue distribution formula $\delta m\tilde{W}\sim {\pi }_{mn\rho }⊠\nu$, where $\rho$ is the law of $\phi$, viewed as a square matrix, $\pi$ is the free Poisson law, $\nu$ is the law of $D=\phi (1)$, and $\delta =tr(D)$.

18 pages

References in corpus (1)

• Limit distributions of random matrices

Cited by in corpus (4)

• Realigning random states
• Partial transpose of random quantum states: exact formulas and meanders
• On the asymptotic distribution of block-modified random matrices
• On the reduction criterion for random quantum states