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Let $(\sigma_N^{(i)})_{i \in I}$ be a family of symmetric permutations of theentries of a Wigner matrix $\mathbf{W}_N$. We characterize the limiting trafficdistribution of the corresponding family of dependent Wigner matrices$(\mathbf{W}_N^{\sigma_N^{(i)}})_{i \in I}$ in terms of the geometry of thepermutations. We also consider the analogous problem for the limiting jointdistribution of $(\mathbf{W}_N^{\sigma_N^{(i)}})_{i \in I}$. In particular, weobtain a description in terms of semicircular families with general covariancestructures. As a special case, we derive necessary and sufficient conditionsfor traffic independence as well as sufficient conditions for freeindependence.

Semicircular families of general covariance from Wigner matrices with permuted entries