  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, and Pastur, and investigate the
extent to which the joint traffic distribution of independent copies of these
matrices deviates from the Wigner case.
