Commutative and Non-commutative Bloch Theory

目前的研究是晶格的布洛赫理论，晶格是无限的图形，以抽象的方式描述晶体中的原子如何通过内力相互结合。晶体结构通过图上某些阿贝尔群作用的存在来体现。我们的兴趣是在几何和分析的离散拉普拉斯算子（和他们的磁性或弹性版本）的晶格作为一个特殊的类的差异运营商。特别地，我们研究了随机游动的渐近性态，从几何的角度建立了随机游动的大偏差性质。我们采用的方法是布洛赫理论的真实的版本。我们可以把大偏差性质与有限图的第一同调群中的有理凸多面体联系起来，它具有显著的组合特征，并且也表现在晶格的Gromov-Hausdorff极限中。

The present research is the Bloch theory for crystal lattices, which are infinite graphs to be introduced to describe, in an abstract way, how the atoms in a crystal are bound to each other by internal force. Crystalline structure is embodied by the presence of certain abelian group actions on graphs. Our interest is in geometry and analysis of the discrete Laplacians (and their magnetic or elastic versions) on crystal lattices as a special class of difference operators. Especially we have studied asymptotic behaviors of random walks, and established a large deviation property in view of geometry. The method we took up is a real version of Bloch theory. We could relate the large deviation property to a rational convex polyhedron in the first homology group of a finite graphs, which has remarkable combinatorial features, and show up also in the Gromov-Hausdorff limit of a crystal lattice.