Semicircular families of general covariance from Wigner matrices with permuted entries

Let (σ _N^(i))_i ∈ I be a family of symmetric permutations of the entries of a Wigner matrix 𝐖_N . We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices (𝐖_N^σ _N^(i))_i ∈ I in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of (𝐖_N^σ _N^(i))_i ∈ I . In particular, we obtain a description in terms of semicircular families with general covariance structures. As a special case, we derive necessary and sufficient conditions for traffic independence as well as sufficient conditions for free independence.