We study random band matrices within the framework of traffic probability, an operadic non-commutative probability theory introduced by Male based on graph operations. As a starting point, we revisit the familiar case of the permutation invariant Wigner matrices and compare the situation to the general case in the absence of this invariance. Here, we find a departure from the usual free probabilistic universality of the joint distribution of independent Wigner matrices. We then show how the traffic space of Wigner matrices completely realizes the traffic central limit theorem. We further prove general Markov-type concentration inequalities for the joint traffic distribution of independent Wigner matrices. We then extend our analysis to random band matrices, as studied by Bogachev, Molchanov,

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We prove a sharp p N transition for the infinitesimal distribution of a periodically banded GUE matrix. For bandwidths b N h .p N /, we further prove that our model is infinitesimally free from the matrix units and the normalized all-1's matrix. Our results allow us to extend previous work of Shlyakhtenko on finite-rank perturbations of Wigner matrices in the infinitesimal framework. For finite-rank perturbations of our model, we find outliers at the classical positions from the deformed Wigner ensemble.

Let (σ (i) N ) 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) i∈I in terms of the geometry of the permutations. We also consider the analogous problem for the limiting joint distribution of (WIn 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.

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.