  The model of heavy Wigner matrices generalizes the classical ensemble of
Wigner matrices: the sub-diagonal entries are independent, identically
distributed along to and out of the diagonal, and the moments its entries are
of order 1/N, where N is the size of the matrices. Adjacency matrices of
Erd\"os-Renyi sparse graphs and matrices with properly truncated heavy tailed
entries are examples of heavy Wigner matrices. We consider a family X_N of
independent heavy Wigner matrices and a family Y_N of arbitrary random
matrices, independent of X_N, with a technical condition (e.g. the matrices of
Y_N are deterministic and uniformly bounded in operator norm, or are
deterministic diagonal). We characterize the possible limiting joint
*-distributions of (X_N,Y_N) in the sense of free probability. We find that
they depend on more than the *-distribution of Y_N. We use the notion of
distributions of traffics and their free product to quantify the information
needed on Y_N and to infer the limiting distribution of (X_N,Y_N). We give an
explicit combinatorial formula for joint moments of heavy Wigner and
independent random matrices. When the matrices of Y_N are diagonal, we give
recursion formulas for these moments. We deduce a new characterization of the
limiting eigenvalues distribution of a single heavy Wigner.
