Statistical Mechanics and Random Matrix Theory

PI: Haru T. Pinson，亚利桑那大学adms -0301830首席研究员对大自由度系统中普遍性概念的问题感兴趣，特别是在两种情况下-二维临界Ising模型和随机矩阵模型。我们希望至少从三个方向进行严格的调查——1。二维临界Ising模型各相关函数的旋转不变性我们想在二维临界伊辛模型中更充分地研究普遍性的某些方面。利用超对称随机矩阵的形式来理解或推导一类随机带矩阵模型的渐近展开式。简而言之，我们感兴趣的是获得精确的宏观信息，而我们只给出一些微观的描述。这种情况在生活中无处不在，从生物学到物理学，我们只知道一些局部规律，但我们想知道在更大的范围内发生了什么。除了这些一般性的考虑之外，我们希望从这项研究中获得的结果和方法可以证明对其他杰出的猜想有用，例如建立一般二维临界波茨模型的普适性以及与高维随机薛定谔算子相关的各种形式的普适性现象。

PI: Haru T. Pinson, University of ArizonaDMS-0301830The principal investigator is interested in the issues surrounding the notion of universality in systems with large degrees of freedom, specifically in two contexts-2d critical Ising model and the random matrix models. We would like to investigate rigorously at least three directions - 1. Rotational invariance of various correlation functions of the 2d critical Ising model.2. We would like to investigate more fully certain aspects of universality in the 2d critical Ising model.3. We would like to understand or derive a certain asymptotic expansion for a class of the random band matrix models using the supersymmetric random matrix formalism. In a nutshell, we are interested in obtaining precise macroscopic information while we areonly give some microscopic descriptions. This situation is ubiquitous in life ranging from biologyto physics where we only know some local laws, yet we would like to know what happens at a larger scale. Aside from such general considerations, we hope that the results and methods obtained from this investigation may prove useful for other outstanding conjectures such as establishing universality for the general 2d critical Potts models and the various forms of universality phenomena associated with the higher dimensional random Schroedinger operator.