Studies in Interacting Particle Systems and Statistical Mechanics

03006772 Schonmann Schonmann将在相互作用粒子系统、统计力学、渗透和概率论中的其他相关学科领域从事各种项目。本研究的主要目标是几种相互作用粒子系统的遍历行为和弛豫模式，包括Ising模型、接触过程和其他晶格自旋系统。渗透理论也是本研究的一个主要组成部分，因为它与上述过程的密切关系以及它的简单性，使其成为研究具有许多组成部分的概率模型的范例。研究的一部分涉及对任意图上上述过程的研究——这是该领域的一个重要趋势。一些问题的动机来自于自然界中观察到的现象，例如相变、临界行为、相变区附近系统的亚稳行为以及畴的形成和生长。在非正式和广泛的层面上，我们可以看到本研究中感兴趣的对象如下。一个系统包含大量相似的组件，每个组件以一种简单的方式与其他几个在某种意义上接近它的组件进行交互。例如，人们可以想象晶体中的原子在距离足够近时相互作用，或者一个大群体中的个体可能会受到或不会受到邻居之间发生的污染的感染。局部交互可能包含大量的随机性，但由于系统整体很大，人们期望并确实观察到全局层面上更可预测的行为。对这类系统的极大兴趣源于这样一个事实，即整个系统的定性行为可能以非平凡的方式依赖于内置于局部相互作用中的参数，但这些参数只以平滑的方式影响那些参数（例如，在上面第二个例子中相邻个体之间的感染率）。我们感兴趣的一般问题是理解这种类型的随机动力学可以达到哪种平衡，如何达到平衡状态，以及影响局部规则的参数如何影响这些问题的答案。

0300672Schonmann Schonmann will work on various projects in the areas of interacting particle systems, statistical mechanics, percolation, and other related subjects in probability theory. The main targets in this research are the ergodic behavior and the relaxation patterns of several interacting particle systems, including Ising models, contact processes and other lattice spin systems. Percolation theory is also a major component of this research, because of its close relations to the processes indicated above and its simplicity, which makes it a paradigm in the study of probabilistic models with many components. Part of the research addresses the study of the above mentioned processes on arbitrary graphs -- an important current trend in the area. Several of the problems have their motivation in phenomena observed in nature as, for example, phase transitions, critical behavior, metastable behavior of systems in the vicinity of phase transition regions and domain formation and growth. On an informal and broad level one can see the objects of interest in this research in the following fashion. A system contains a large number of similar components and each component interacts in a simple way with a few other ones, which are near to it in some sense. For instance one can think of the atoms in a crystal interacting when they are close enough, or individuals in a large population which may suffer or not from some infection with contamination occurring between neighbors. The local interaction may involve a substantial amount of randomness, but since the system as a whole is large, one expects and indeed observes a much more predictable behavior at the global level. A great deal of interest in such systems stems from the fact that the qualitative behavior of the whole system may depend in non-trivial ways on parameters which are built into the local interactions, but which only affect those in a smooth way (e.g., the rate of infection between neighboring individuals in our second example above). The general issues of interest are then the understanding of which sort of equilibria can be reached by such types of stochastic dynamics, how an equilibrium state is reached, and how the parameters which affect the local rules may affect the answers to these questions.