Statistical Mechanics and Random Matrix Theory

Pi：Haru T.Pinson，亚利桑那大学DMS-0301830首席研究员对大自由度系统中普适性概念的问题感兴趣，特别是在两种情况下-二维临界伊辛模型和随机矩阵模型。我们想要严格地研究至少三个方向：1.二维临界伊辛模型各种关联函数的旋转不变性。我们想更全面地研究二维临界伊辛模型中普适性的某些方面。我们希望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某种渐近展开式。简而言之，我们感兴趣的是获得精确的宏观信息，而我们只给出了一些微观描述。这种情况在生活中无处不在，从生物学到物理学，我们只知道一些当地的法律，但我们想知道更大范围内发生的事情。除了这些一般性的考虑外，我们希望从这项研究中得到的结果和方法可以被证明是对其他未解决的猜想有用的，例如建立一般的二维临界Potts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.