Semicircular families of general covariance from Wigner matrices with permuted entries

Abstract

Let $$(\sigma _N^{(i)})_{i \in I}$$ ( σ N ( i ) ) i ∈ I be a family of symmetric permutations of the entries of a Wigner matrix $${\mathbf {W}}_N$$ W N . We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices $$({\mathbf {W}}_N^{\sigma _N^{(i)}})_{i \in I}$$ ( 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 $$({\mathbf {W}}_N^{\sigma _N^{(i)}})_{i \in I}$$ ( 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.