Studies in Interacting Particle Systems and Statistical Mechanics

0300672 Schonmann Schonmann将在相互作用的粒子系统，统计力学，渗流领域的各种项目，以及概率论的其他相关主题。本研究的主要目标是研究几种相互作用粒子系统的遍历行为和弛豫模式，包括伊辛模型、接触过程和其他晶格自旋系统。逾渗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和它的简单性，这使它成为研究具有许多组件的概率模型的范例。部分研究解决了任意图上的上述过程的研究-该领域的一个重要的当前趋势。这些问题中有几个是在自然界中观察到的现象中产生的，例如相变、临界行为、相变区域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，人们可以看到以下方式在这项研究中感兴趣的对象。一个系统包含大量相似的组件，每个组件都以简单的方式与其他一些组件进行交互，这些组件在某种意义上与它接近。例如，人们可以想象晶体中的原子在足够接近时相互作用，或者在大量人口中的个体可能遭受或不遭受邻居之间发生的污染的感染。局部相互作用可能涉及大量的随机性，但由于系统作为一个整体是大的，人们期望并确实观察到一个更可预测的行为在全球范围内。对这种系统的极大兴趣源于这样的事实，即整个系统的定性行为可能以非平凡的方式取决于内置于局部相互作用中的参数，但这些参数仅以平滑的方式影响那些参数（例如，在我们上面的第二个例子中，相邻个体之间的感染率）。感兴趣的一般问题，然后了解哪种平衡可以达到这种类型的随机动力学，如何达到平衡状态，以及如何影响局部规则的参数可能会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.