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 2kth moment of the limit equals the number of non-crossing pair-partitions of the set {1, 2,…, 2k}. There are several extensions of this result in the literature. In this paper, we consider a unifying extension which also yields additional results. Suppose Wn is an n × n 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.

中文翻译：

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具有独立条目的随机矩阵：超越非交叉分区

尺度化的标准 Wigner 矩阵（均值为 0 对称，方差为 1 独立同分布）及其极限特征值分布，即半圆分布，备受关注。这2ķ极限的时刻等于集合的非交叉对分区的数量{1, 2,…, 2ķ}. 文献中对该结果有几个扩展。在本文中，我们考虑了一个统一的扩展，它也产生了额外的结果。认为Wn是一个n × n对称矩阵，其中条目独立分布。我们表明，在对条目的适当假设下，极限光谱分布以概率或几乎肯定存在。极限矩可以通过一组分区来描述，该分区通常大于一组非交叉对分区。这个集合引起了有趣的枚举组合问题。我们的结果得出了几个现有的限制光谱分布结果。这些包括标准 Wigner 矩阵、稀疏齐次 Erdős–Rényi 图的邻接矩阵、重尾 Wigner 矩阵、一些带状 Wigner 矩阵和具有方差分布的 Wigner 矩阵的结果。这些模型及其扩展的一些新结果也来自我们的主要结果。