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

在过去十年中，随机矩阵理论（RMT）在混沌或/和潜在无序起重要作用的现象系统的数学描述中获得了普遍范式的地位。类似的，但在某种程度上是独立的，能量景观的概念遍布于对玻璃、无序系统、蛋白质等的理论描述中，最近又重新出现在弦理论和宇宙学中。在最简单的版本中，主要目标是通过在随机势中移动的单个点状粒子来描述整个系统或其子部分的静力学和动力学，这编码了原始系统的复杂性。我们希望能够对随机势的可能类别进行分类，并建立一般的、通用的性质，就像RMT中出现的那些性质一样。在一个更高级（和现实）的版本中，人们可能会对与聚合物构象有关的景观特性感兴趣，甚至是像膜这样的高维物体。目前为期三个月的小规模项目的目标是对两个主要问题进行试点调查：(i)计算受随机势景观影响的聚合物或膜等弹性物体的平衡构型，以及（ii）研究新型随机势中的单个点状粒子的统计力学。在首席研究员和其他研究人员的一系列工作中，RMT的方法和想法经常被证明与景观环境直接相关。到目前为止，它们只应用于最简单的随机类型（高斯）中最简单的点状粒子。我们希望将这些方法的效用扩展到更高维度的对象，并包括新的无序类型。

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.