论文标题
具有统一局部结构的大（网格）图的渐近光谱
Asymptotic spectra of large (grid) graphs with a uniform local structure
论文作者
论文摘要
我们关注具有网格几何形状的图序列，在有限域$ω\ subset {\ Mathbb r}^d $，$ d \ ge 1 $中具有均匀的本地结构。我们假设可以通过常规边界来测量$ω$，并为方便起见，并将其包含在Cube $ [0,1]^d $中。当$ω= [0,1] $时，此类图包括标准的toeplitz图，对于$ω= [0,1]^d $，考虑的类包括$ d $ level-level toeplitz图。在一般情况下，邻接矩阵的基础序列在Weyl Sense中具有规范的特征值分布，我们表明我们可以将其关联为符号$ f $。符号及其基本分析特征的知识为特征值结构，定位，光谱差距，聚类和分布类型提供了许多信息。与众所周知的局部toeplitz序列的概念相关的概念也很少考虑，并讨论了应用，例如从通过数值方案对差分运算符的近似。
We are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain $Ω\subset {\mathbb R}^d$, $d\ge 1$. We assume $Ω$ to be Lebesgue measurable with regular boundary and contained, for convenience, in the cube $[0,1]^d$. When $Ω=[0,1]$, such graphs include the standard Toeplitz graphs and, for $Ω=[0,1]^d$, the considered class includes $d$-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and we show that we can associate to it a symbol $f$. The knowledge of the symbol and of its basic analytical features provide many informations on the eigenvalue structure, of localization, spectral gap, clustering, and distribution type. Few generalizations are also considered in connection with the notion of generalized locally Toeplitz sequences and applications are discussed, stemming e.g. from the approximation of differential operators via numerical schemes.
