The limiting distributions of large heavy Wigner and arbitrary random matrices

A heavy Wigner matrix XNXN is defined similarly to a classical Wigner one. It is Hermitian, with independent sub-diagonal entries. The diagonal entries and the non-diagonal entries are identically distributed. Nevertheless, the moments of the entries of NXN tend to infinity with N  , as for matrices with truncated heavy tailed entries or adjacency matrices of sparse Erdös–Rényi graphs. Consider a family XNXN of independent heavy Wigner matrices and an independent family YNYN of arbitrary random matrices with a bound condition and converging in ⁎-distribution in the sense of free probability. We characterize the possible limiting joint ⁎-distributions of (XN,YN)(XN,YN), giving explicit formulas for joint ⁎-moments. We find that they depend on more than the ⁎-distribution of YNYN and that in general XNXN and YNYN are not asymptotically ⁎-free. We use the traffic distributions and the associated notion of independence [21] to encode the information on YNYN and describe the limiting ⁎-distribution of (XN,YN)(XN,YN). We develop this approach for related models and give recurrence relations for the limiting ⁎-distribution of heavy Wigner and independent diagonal matrices.