The limiting distributions of large heavy Wigner and arbitrary random matrices

Abstract

A heavy Wigner matrix XN 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 XN of independent heavy Wigner matrices and an independent family YN 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), giving explicit formulas for joint ⁎-moments. We find that they depend on more than the ⁎-distribution of YN and that in general XN and YN are not asymptotically ⁎-free. We use the traffic distributions and the associated notion of independence [21] to encode the information on YN and describe the limiting ⁎-distribution of (XN,YN). We develop this approach for related models and give recurrence relations for the limiting ⁎-distribution of heavy Wigner and independent diagonal matrices.