Integrable Geometry, Random Matrices and Matrix Integrals

摘要奖：DMS-0406287首席研究员：Mark Adler该项目旨在研究随机矩阵理论、各种统计过程、可积力学与Virasoro代数之间的各种联系。具体地说，一个项目是研究非碰撞布朗运动，如Dyson运动、线上运动和圆上运动，包括Dyson椭圆布朗运动；与链中的矩阵模型的联系对于寻找运动的联合概率的偏微分方程组是特别相关的。利用在随机矩阵和排列的背景下产生的标度极限，上面的Dyson运动倾向于新的极限随机过程。最外面的粒子的运动趋向于所谓的Ary过程。寻找这些过程的随机微分方程取决于关于随机矩阵理论中出现的普适性规律的复杂的统计问题。另一个项目是寻找大的时间渐近，如渐近协方差，对于Dyson过程和极限过程，研究人员认为随机矩阵理论中的每个普适性定律都与一个可积系统和Virasoro约束代数有关。问题是找到这些系统，并提取关于分布函数及其微分方程组的有趣信息。最后，Dyson圆周运动有一个有趣的实现，它提供了这种运动的保角映射实现。它与随机微分方程以神秘的方式联系在一起，这自然也会出现在无碰撞随机游动和福克-普朗克方程的问题中。这些联系将被研究。上面的数学物理在统计分析中有应用，这些统计分析在涉及少量源的许多实际问题中出现，每个源产生大量数据，如天线接收信息，分析来自少数源的生态数据，每个源产生大量数据，等等。要点是在许多实际问题中出现大的矩形数据阵列，这些数据在一个方向上是大的，但另一个不是这样。统计过程似乎也出现在许多增长模型中，这些模型应该在工业过程中起作用。此外，随机矩阵理论中出现的普适性定律在研究大原子时非常自然地出现在量子力学中，因此可能有助于理解物理化学中的化学反应，因此有朝一日通过模拟实验来制造药物。

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.