Spectral Analysis of infinite grapgs with discrete group actions

本文从几何学的角度讨论了晶格上周期随机游动的一个长时间行为，特别是大偏差性质，并将其与有限图的第一同调群中的有理凸多面体联系起来，作为一个显著的组合特征，晶格具有与图距相关的度量结构。通过改变距离的尺度，得到了一类单参数度量空间。序列的Gromov-Hausdorff极限称为晶格无穷远处的渐近锥。当刻度归零时，由于晶格的周期性，渐近锥是存在的，我们用组合数据明确地确定了它的单位球.我们还在Amer.Math. Soc的Sugaku Expository上发表了一篇关于晶格的离散几何分析的综述文章.在那篇文章中，我们讨论了晶格上随机游动的谱性质和几何，如大数定律，中心极限定理，磁薛定谔算子与非对易几何大偏差和谱

We discuss ed a long time behavior of periodic random walks on a crystal lattice in view of geometry, a large deviation property in particular, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as remarkable combinatorial features,. A crystal lattice has a metric structure with the graph distance. By changing scale of the distance, we obtain a one-parameter family of metric spaces. The Gromov-Hausdorff limit of the sequence is called the asymptotic cone at the infinity of the crystal lattice. As the scale go to zero., because of the periodicity of the crystal lattice, the asymptotic cone exists and we determinded its unit ball explicitely in terms of combinatorial data.We also published a survey article on discrete geometric analysis of crystal lattice from Sugaku Expository, Amer.Math.Soc. In there, we discussed spectral properties and geometry of random walks on a crystal lattice, such as the law of large number, the central limit theorem, large deviation and spectrum of magnetic Schroedinger operators from non-commutative geometry.