Disorder Effects and Critical Behavior in Statistical Mechanics and in Quantum Systems

9512729艾森曼在一个尺度上发现的物理结构是一种普遍的经验，它是一个更精细的复杂系统的集体模式的表现，具有更多的自由度。微观上由粒子、布朗运动等组成的流体。值得注意的是，较粗尺度的物理通常通过简单的自治定律来表示，而在较精细的描述中找不到对称性。这项拟议的研究将在统计力学和场论的一些模型的背景下，从数学上分析这种“比例极限”的出现。将特别注意对关键系统中所见的大尺度特征的数学分析。另一个研究主题将涉及猝灭无序的影响，它经常改变振动模式的特征、导电性质和相变的性质。这项研究旨在达成物理上的理解和数学上令人满意的方法来分析这些领域中的重要现象。***

9512729 Aizenman It is common experience that physical structure found on one scale, is a manifestation of collective modes of a much finer complex system with many more degrees of freedom. Fluids which microscopically consist of particles, Brownian motion, etc. Remarkably, often the physics of the coarser scale is expressed through simple autonomous laws, and symmetries not found in the finer description. The proposed research will analyze mathematically the emergence of such "scaling limits", in the context of a number of models of statistical mechanics, and field theory. Particular attention will be devoted to the mathematical analysis of the large scale features seen in critical systems. Another topic of study will concern the effects of quenched disorder, which quite often modifies the characteristics of vibration modes, electrical conduction properties, and the nature of phase transitions. The study aims at reaching both physical understanding and mathematically satisfactory means of analyzing significant phenomenon these fields. ***