Random matrices with independent entries: Beyond non-crossing partitions

The scaled standard Wigner matrix (symmetric with mean zero, variance one i.i.d. entries), and its limiting eigenvalue distribution, namely the semi-circular have attracted much attention. [Formula: see text]th moment of limit equals number non-crossing pair-partitions set text]. There are several extensions this result in literature. In paper, we consider a unifying extension which also yields additional results. Suppose text] is an symmetric where entries independently distributed. We show that under suitable assumptions on entries, spectral distribution exists probability or almost surely. moments can be described through partitions general larger than pair-partitions. This gives rise to interesting enumerative combinatorial problems. Several existing results follow from our These include matrix, adjacency sparse homogeneous Erdős–Rényi graph, heavy tailed some banded matrices, matrices profile. Some new these models their main