Semicircular families of general covariance from Wigner matrices with permuted entries

Abstract

Let $({\sigma }_N^{(i)}{)}_{i\in I}$ be a family of symmetric permutations of the entries of a Wigner matrix ${\mathbf{W}}_N$. We characterize the limiting traffic distribution of the corresponding family of dependent Wigner matrices $({\mathbf{W}}_N^{{\sigma }_N^{(i)}}{)}_{i\in I}$ in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of $({\mathbf{W}}_N^{{\sigma }_N^{(i)}}{)}_{i\in I}$. 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.

21 pages, 7 figures