Phase transitions in random matrices and infinite dimensional diffusions

随机矩阵理论中的这个项目源于随机矩阵特征值的戴森扩散，它迫使特征值根据非相交布朗运动进化。当随机矩阵的大小变得任意大时，特征值就变成了一个由无限不相交的粒子组成的“马尔可夫云”，这些粒子按照一定的平衡度量分布。对于每个给定的时间，它的支持将集中在间隔上，其数量可能随时间而变化。因此，随着时间的发展，区间可以合并，可以消失，也可以产生，从而在时空中形成一个区域R，该区域的边界除了各种奇点之外都是规则的。在R的边界点附近，不相交的布朗运动在极限下趋向于马尔可夫云，在接近奇点时发生相变；因此，这些无限维扩散是关键现象，应该表现出普遍的性质。我们希望推导（非线性）PDE的转移概率和各种缩放极限，这将产生边界条件，适当地理解，为这些painleletype PDE。这也将是一个从一个关键现象传递到另一个关键现象的工具。沿着同样的方向，该项目的另一个目标是通过布朗运动使用随机变化的域，将保角映射、无色散2D-Toda和随机Lowner方程连接起来。随机矩阵理论与许多数学和物理学科有着不同的接口，一方面，Fredholm行列式，可积力学和Painleve方程，一方面，共形场论和统计力学，特别是。批判现象与普遍性。基本动机是使用Painleve型偏微分方程理论将这些主题联系在一起，以解释各种临界现象如何相互融合并相互产生，也许可以为各种临界现象创建一种家谱。所提到的各个领域的工具将在描述现象和推导现象如何演变的概率预测方程中发挥作用。

This project in random matrix theory stems from Dyson diffusion for the eigenvalues of a random matrix ,which forces the eigenvalues to evolve according to non-intersecting Brownian motion.Upon letting the size of the random matrices grow arbitrarily large, the eigenvalues turn into a "Markov cloud " of infinite non-intersecting particles,distributed according to a certain equilibrium measure. For each given time,its support will be concentrated on intervals, whose number may vary with time..Therefore, when time evolves, intervals may merge, may disappear and be created, leading to a region R in space- time,whose boundary will be regular,except for various singularities. Near the boundary points of R the non-intersecting Brownian motions will, in the limit, tend to a Markov cloud performing phase transitions when approaching a singularity; these infinte dimensional diffusion are thus critical phenomena and should exhibit universal properties.We wish to derive (nonlinear) PDE's for the transition probabilities and various scaling limits which will yield boundary conditions, appropriatedly understood ,for these Painleve type PDE"s.This will also be a tool to pass from one critical phenomena to another.Along the same vein, another goal of the project is to connect conformal maps, dispersionless 2D-Toda and the Stochastic Lowner equation, through using a stochastically changing domain, via Brownian motion. Random matrix theory has a diverse interface with numerous mathematical and physical disciplines, on the one hand, Fredholm determinants ,integrable mechanics and Painleve equations and on the one hand conformal field theory and statistical mechanics, and in particular. critical phenomena and universality.The basic motivation is to tie these topics together using a Painleve type theory of partial differential equations to explain how various critical phenomena merge into each other and emerge out of each other, perphaps creating a sort of familty-tree for various critical phenomena.The tools of the various fields alluded to will come into play in both describing the phenomena and deriving equations for the probabilistic prediction of how the phenomena evolves.