Spectral Analysis of infinite grapgs with discrete group actions

从几何的角度讨论了晶格上周期随机游动的长时间行为，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，作为显著的组合特征，.晶格具有具有图形距离的度规结构。通过改变距离的尺度，我们得到了一个单参数度量空间族。序列的Gromov-Hausdorff极限称为晶格无穷远处的渐近锥体。当尺度趋于零时，由于晶格的周期性，渐近圆锥的存在，我们用组合数据明确地确定了它的单位球。我们还发表了Sugaku说明书，amer.Math.Soc的一篇关于晶格离散几何分析的综述文章。在这里，我们讨论了晶格上随机游动的谱性质和几何性质，如大数定律、中心极限定理、大偏差和磁薛定谔算子与非对易几何的谱。

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.