Takashi Kumagai

This book develops limit theorems for a natural class of long range random walks on finitely generated torsion free nilpotent groups. The limits in these limit theorems are Lévy processes on some simply connected nilpotent Lie groups. Both the limit Lévy process and the limit Lie group carrying this process are determined by and depend on the law of the original random walk. The book offers the first systematic study of such limit theorems involving stable-like random walks and stable limit Lévy processes in the context of (non-commutative) nilpotent groups.

The hydrogen bond represents an important interaction between molecules, and the dynamics of hydrogen bonds in water create an ever-present question associated with the process of chemical and biological reactions. In spite of numerous studies, the process remains poorly understood at the microscopic level because hydrogen-bond dynamics, such as bond rearrangements and hydrogen/proton transfer reactions, are extremely difficult to probe. Those studies have been carried out by means of spectroscopic methods where the signal stems from the ensemble of a system and the hydrogen-bond dynamics were inferred indirectly. This book addresses the direct imaging of hydrogen-bond dynamics within water-based model systems assembled on a metal surface, using a scanning tunneling microscope (STM). The dynamics of individual hydrogen bonds in water clusters, hydroxyl clusters, and water-hydroxyl complexes are investigated in conjunction with density functional theory. In these model systems, quantum dynamics of hydrogen bonds, such as tunneling and zero-point nuclear motion, are observed in real space. Most notably, hydrogen atom relay reactions, which are frequently invoked across many fields of chemistry, are visualized and controlled by STM. This work presents a means of studying hydrogen-bond dynamics at the single-molecule level, providing an important contribution to wide fields beyond surface chemistry.

In these lecture notes, we will analyze the behavior of random walk on disordered media by means of both probabilistic and analytic methods, and will study the scaling limits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media. The first few chapters of the notes can be used as an introduction to discrete potential theory. Recently, there has been significant progress on the theory of random walk on disordered media such as fractals and random media. Random walk on a percolation cluster(‘the ant in the labyrinth’)is one of the typical examples. In 1986, H. Kesten showed the anomalous behavior of a random walk on a percolation cluster at critical probability. Partly motivated by this work, analysis and diffusion processes on fractals have been developed since the late eighties. As a result, various new methods have been produced to estimate heat kernels on disordered media. These developments are summarized in the notes.

The hydrogen bond represents an important interaction between molecules, and the dynamics of hydrogen bonds in water create an ever-present question associated with the process of chemical and biological reactions. In spite of numerous studies, the process remains poorly understood at the microscopic level because hydrogen-bond dynamics, such as bond rearrangements and hydrogen/proton transfer reactions, are extremely difficult to probe. Those studies have been carried out by means of spectroscopic methods where the signal stems from the ensemble of a system and the hydrogen-bond dynamics were inferred indirectly. This book addresses the direct imaging of hydrogen-bond dynamics within water-based model systems assembled on a metal surface, using a scanning tunneling microscope (STM). The dynamics of individual hydrogen bonds in water clusters, hydroxyl clusters, and water-hydroxyl complexes are investigated in conjunction with density functional theory. In these model systems, quantum dynamics of hydrogen bonds, such as tunneling and zero-point nuclear motion, are observed in real space. Most notably, hydrogen atom relay reactions, which are frequently invoked across many fields of chemistry, are visualized and controlled by STM. This work presents a means of studying hydrogen-bond dynamics at the single-molecule level, providing an important contribution to wide fields beyond surface chemistry.