Random matrices with independent entries: Beyond non-crossing partitions

Abstract

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

Noncrossing partition flow and random matrix models

“…(2) 2 (l) with E blocks, studied in [4] and in Section IV. This result has been obtained recently in [17], where P (2) 2 (l) is called the set of special symmetric partitions SS(2l).…”

“…This property of the last block is taken as part of the definition of the special symmetric partition set SS(2l) in [17], which is…”

“…2 (l) is 1 or 2, i.e. the element is odd or even; as remarked above, consecutive elements in a block i and j in a block have j = i + 1 mod k, so that even and odd numbers alternate in a block in the case k = 2, which is also taken as part of the definition of SS(l) in [17].…”

“…In [17] it has recently been shown that the set of closed walks contributing to the moments of the Adjacency random matrix model on ER graphs in the large-N limit is isomorphic to a set of 2-divisible partitions, whose properties are investigated.…”

“…Define a k-gon tree as a directed connected graph formed by oriented k-gons joined at vertices and with no edge in common, such that no other loops apart from the k-gons are present; k-gon trees are a generalization of trees, the k = 2 case. There is an isomorphism between the set of length-kl closed walks covering rooted k-gon trees and the set P (k) k (l) of k-divisible partitions with kl elements, in which for any two blocks B i and B j of a partition, the number of elements of B j between two elements of B i is multiple of k. In the case k = 2 this set of partitions has been studied in [17], where it is shown its relation with the moments of the Adjacency random matrix on the ER graphs. From this isomorphism it follows that there is an isomorphism between the set of closed walks covering rooted k-gon trees, with noncrossing sequence of steps, and the set N C (k) (l) of noncrossing k-divisible partitions with kl elements.…”