A proposal for the visit of Dr. Vladimir Al. Osipov: From Random Matrices to Random Landscapes

随机矩阵理论（Random Matrix Theory，RMT）是近十年来发展起来的一种数学描述现象系统的普遍范式，其中混沌或/和潜在的无序起着重要作用。类似地，但有些独立地，能量守恒的概念贯穿于对玻璃、无序系统、蛋白质等的理论描述中，and recently最近reemed再现in stringtheory弦and cosmology宇宙学.在最简单的版本中，主要目标是通过一个在随机势中运动的单个点状粒子来描述整个系统或其中一个子系统的静态和动态，随机势编码了原始系统的复杂性。希望能够对随机势的可能类别进行分类，并建立通用的普适性质，与RMT中出现的那些性质没有什么不同。在一个更高级的（现实主义的）版本中，人们可能会对与聚合物构象相关的景观特性感兴趣，甚至对更高维的物体如膜感兴趣。目前的小规模的三个月的项目的目标是执行两个主要问题的试点调查：（i）计数的弹性物体的平衡配置，如聚合物或膜受到随机潜在景观的影响和（ii）调查统计力学的一个单一的点状粒子在一种新的类型ofrandom potential.In一系列的工作由首席研究员和其他研究人员它被证明，RMT方法和思想经常被证明是直接相关的景观上下文.到目前为止，它们只适用于最简单的点状粒子在最简单的可能类型的随机性（高斯）。我们希望将这些方法的实用性扩展到更高维的对象，并包括新类型的混乱。

Random Matrix Theory (RMT) acquired in the last decade the statusof a universal paradigm in mathematical description of phenomenain systems where chaos or/and underlying disorder play anessential role. Similarly but somewhat independently the idea of energy landscapespervades the theoretical description of glasses, disorderedsystems, proteins, etc., and recently re-emerged in stringtheory and cosmology. In the simplest version the main goalis to describe the statics and dynamics of the whole system, orone of its subparts, by a single point-like particle moving in a randompotential, which encodes the complexity of the original system.The hope then is to be able to classify the possible classes ofrandom potential and to establish generic, universal properties,not unlike those emerging in the RMT. In a more advanced ( andrealistic) version one may be interested in properties oflandscapes related to conformations of polymers, or evenhigher-dimensional objects like membranes. The goal of the present small-scale three months project is toperform pilot investigations of two main problems:(i) Counting equilibrium configurations of elastic objects like polymers or membranes subject to the influence of a random potential landscape and (ii) Investigating Statistical Mechanics of a single point-like particle in a new type ofrandom potential.In the series of works by the Principal Investigator and other researchers it was demonstrated that RMT methods and ideas frequentlyprove to be of direct relevance in the landscape context. So far they were applied only to the simplest point-like particles in the simplest possible typeof randomness (Gaussian). We hope to extend utility of those methods tohigher-dimensional objects and to include new types of disorder.