论文标题
稀疏随机块矩阵：普遍性
Sparse Random Block Matrices : universality
论文作者
论文摘要
我们研究了从Erdös-Renyi随机图产生的稀疏随机块矩阵的集合，该矩阵具有平均度$ z $的$ n $顶点，在每个非呈现条目中插入一个真实的对称$ d \ times d $随机块。 我们考虑一些随机块矩阵的合奏，级别$ r <d $并且最高等级，$ r = d $。稀疏随机块矩阵的光谱矩对$ n \ to \ infty $，$ d $有限或无限进行了评估，以及对块的几个概率分布（例如固定跟踪，有界的跟踪和高斯）。 由于概率度量的浓度在$ d \至\ infty $限制中，光谱矩量次数独立于块的概率度量（带有各向同性，平滑度和次高斯尾巴的轻度假设）。 有效的培养基近似是具有有限等级的稀疏随机块集合的限制光谱密度。 Laplacian稀疏块集合的类似类别的类似类别。使用随机常规图代替Erdös-Renyi图获得了相同的限制分布。
We study ensembles of sparse random block matrices generated from the adjacency matrix of a Erdös-Renyi random graph with $N$ vertices of average degree $Z$, inserting a real symmetric $d \times d$ random block at each non-vanishing entry. We consider some ensembles of random block matrices with rank $r < d$ and with maximal rank, $r=d$. The spectral moments of the sparse random block matrix are evaluated for $N \to \infty$, $d$ finite or infinite, and several probability distributions for the blocks (e.g. fixed trace, bounded trace and Gaussian). Because of the concentration of the probability measure in the $d \to \infty$ limit, the spectral moments are independent of the probability measure of the blocks (with mild assumptions of isotropy, smoothness and sub-gaussian tails). The Effective Medium Approximation is the limiting spectral density of the sparse random block ensembles with finite rank. Analogous classes of universality hold for the Laplacian sparse block ensemble. The same limiting distributions are obtained using random regular graphs instead of Erdös-Renyi graphs.
