Integrable Geometry, Random Matrices and Matrix Integrals

项目主要研究随机矩阵理论、各种统计过程、可积力学和Virasoro代数之间的各种联系，其中一个项目是研究直线上和圆上的非碰撞布朗运动，包括戴森椭圆布朗运动；在链中与矩阵模型的联系与寻找运动联合概率的PDE特别相关。利用在随机矩阵和排列的背景下产生的尺度限制，上述戴森运动倾向于新的限制随机过程。最外层粒子的运动趋向于所谓的艾里过程。寻找这些过程的随机微分方程取决于随机矩阵理论中出现的普适性规律的复杂统计问题。另一个课题是寻找Dyson过程和极限过程的大时间渐近性，如渐近协方差。研究者认为随机矩阵理论中的每一个通用性定律都与不可积系统和Virasoro约束代数相联系。问题是找到这些系统，并提取有关分布函数及其微分方程的有趣信息。最后，Dyson圆周运动在“随机Loewnerequation”中有一个有趣的实现，提供了该运动的保角映射实现。它的伊托随机微分方程以一种神秘的方式与维拉索罗代数相关，后者也自然地出现在非碰撞随机漫步和福克-普朗克方程的问题中。这些联系将被调查。上面的数学物理在统计分析中有应用，它出现在许多实际问题中，涉及少量来源，每个来源产生大量数据，如天线接收信息，分析来自少量来源的生态数据，每个来源产生大量数据，等等。关键在于，在许多实际问题中，会出现较大的矩形数据数组，它们在一个方向上很大，但在另一个方向上很小。随之而来的统计过程似乎也出现在许多应该与工业过程相关的增长模型中。此外，随机矩阵理论中出现的普遍规律在量子力学中很自然地出现，在研究大原子时，因此可能被证明对理解物理化学中的化学反应有用，从而在某一天通过模拟实验制造药物。

AbstractAward: DMS-0406287Principal Investigator: Mark AdlerThe project aims at investigating various connections betweenrandom matrix theory, various statistical processes,integrablemechanics and the Virasoro algebra Specifically,one project is tostudy non-colliding Brownian motion a la Dyson,on the line and onthe circle,including the Dyson elliptic Brownian motion;theconnection with matrix models in a chain is particularly relevantto finding PDE's for the joint probabilities of the motion.Usingscaling limits arising in the context of random matrices andpermutations,the Dyson motions above tend to novel limitingrandom processes.The motion of the outmost particle tends to theso-called Airy process.Finding stochastic differential equationsfor these processes hinges on intricate statistical questionsabout the universality laws appearing in random matrixtheory.Another project is to find large time asymptotics,likeasymptotic covariances,for the Dyson processes and the limitingprocesses as well.The investigators believe that each of theuniversality laws in random matrix theory connects with anintegrable system and the algebra of Virasoro constraints.Theproblem is to find these sysems and to extract interestinginformation about the distribution functions and theirdifferential equations.Finally,the Dyson circular motion has aninteresting realization in terms of the "Stochastic Loewnerequation",providing a conformal map realization of thismotion.Its Ito stochastic differential equation is related-in amysterious way-to the Virasoro algebra,which also naturally comesup in questions of non-colliding random walks and theFokker-Planck equations.These connections will be investigated.The mathematical physics above has applications in thestatistiical analysis that comes up in many practical problemsinvolving a small number of sources ,each generating lots ofdata,like antennas receiving information,analyzing ecologicaldata from a small number of sources ,each generating lots ofdata,etc.The point being that in many practical problems largerectangular arrays of data come up,which are big in onedirection,but not the other.The statistical processes that comeup also seem to come up in lots of growth models that should berelevant in industrial processes.In addition the universalitylaws that arise in the random matrix theory arise quite naturallyin quantum mechanics,in studying large atoms and so may proveuseful in understanding chemical reactions in physical chemistryand hence in manufacturing drugs through simulation experimentsone day.