Integrable Geometry, Random Matrices and Matrix Integrals

AbstractAward：DMS-0406287项目负责人：Mark Adler该项目旨在研究随机矩阵理论、各种统计过程、可积力学和Virasoro代数之间的各种联系。具体而言，一个项目是研究非碰撞布朗运动，包括Dyson椭圆布朗运动;与链中的矩阵模型的连接特别与找到运动的联合概率的PDE有关。使用在随机矩阵和排列的上下文中产生的缩放限制，上面的戴森运动倾向于新的极限随机过程。最外面的粒子的运动倾向于所谓的艾里过程。找到这些过程的随机微分方程取决于随机矩阵理论中出现的关于普适性定律的复杂统计问题。另一个项目是找到大时间渐近性，如渐近协方差，研究者们认为，随机矩阵理论中的每一个普适性定律都与一个可积系统和Virasoro约束的代数有关，问题是找到这些系统并提取关于分布函数及其微分方程的有趣信息。最后，戴森圆周运动在“随机Loewner方程”中有一个有趣的实现，提供了这个运动的共形映射实现。它的Ito随机微分方程以一种神秘的方式与Virasoro代数有关，Virasoro代数也自然地出现在非碰撞随机行走和Fokker-Planck方程的问题中。这些联系将被研究。上面的数学物理在统计分析中有应用，这些统计分析出现在涉及少量源的许多实际问题中，每一个都产生大量的数据，就像天线接收信息，分析来自少量来源的生态数据，每一个都产生大量的数据，等等。要点是，在许多实际问题中，出现了大的数据矩形阵列，在一个方向上很大，但不是另一个。出现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，在随机矩阵理论中出现的普适性定律在量子力学中很自然地出现，在研究大原子中，因此可能被证明有助于理解物理化学中的化学反应，从而有朝一日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.