AI 中文总结
本文提出随机撒布图模型，通过安德森-渗流表示刻画其谱，给出态密度的支撑集与定量界，并分析稀疏撒布下原子附近的态密度行为。
AI 中文摘要
在一个大型核心图 $\mathbb G$ 的每个顶点 $x$ 处，我们独立地绘制一个随机图，并将 $x$ 连接到其顶点的一个子集。这种随机撒布图模型捕捉了常见观察到的图的定性特征；它也可以被视为量子无序模型，其中无序源于图几何的局部扰动。我们通过推导其谱的安德森-渗流表示，表明它自然地与安德森模型和 $\mathbb G$ 上的位点渗流相关联。我们刻画了谱的支撑集，并导出了积分态密度的定量界。我们还详细分析了稀疏撒布的情形，特别是研究了原子附近的态密度行为。
英文摘要
Independently at every vertex $x$ of a large core graph $\mathbb G$, we draw a random graph and connect $x$ to a subset of its vertices. This model of a randomly sprinkled graph captures qualitative features of commonly observed graphs; it can also be regarded as a model of quantum disorder, where disorder arises from local perturbations to the graph geometry. We show that it is naturally connected both to the Anderson model and to site percolation on $\mathbb G$ by deriving an Anderson-percolation representation for its spectrum. We characterize the support of the spectrum and derive quantitative bounds on the integrated density of states. We also analyse in detail the regime of sparse sprinkling, in particular investigating the behaviour of the density of states in the vicinity of atoms.
