Asymptotics of correlation functions for random matrices

随机矩阵是在Wigner和Dyson的开创性工作中发现的一个分析和概率领域。最初被认为是通过用随机变量来建模许多具有未知相互作用的粒子系统的行为，此后随机矩阵在物理、数学和工程中得到了广泛的应用。当矩阵的规模较大时，研究随机矩阵的特征值之间的相关性是主要的课题，但仍有许多未解决的问题。典型的这种对象是实线的给定区间不包含随机矩阵的特征值的概率。它简称为缺口概率。对于许多有趣的随机矩阵集合，这个概率被表示为Hankel或Toeplitz行列式。问题是，对于大间隙的情况，要找到这个对象的显式展开。这个问题很容易表述，但获得解决方案是一项非常重要的任务，需要开发适当的方法。非常有效的方法依赖于所谓的Riemann-Hilbert方法，该方法将问题归结为复变量函数的问题，对于一些基本的随机矩阵集合，这个问题已经得到解决，但有一些非常重要的问题仍然是开放的。这个项目的目的是在一两个这样的公开案例中解决这个问题。这个项目中使用的Riemann-Hilbert最陡下降技术是由Deift和周在20年前提出的，并从那时起一直在发展。事实证明，它们可能是最强大的渐近分析工具，并帮助解决了许多重要问题。用这种方法解决新问题也是对方法本身的发展，因为必须设计新的技巧来克服新的障碍。因此，除了解决新的问题外，我们还有助于开发一种重要的方法。

Random matrices is an area of analysis and probability which was discovered in pioneering works of Wigner and Dyson.Initially considered to describe the behaviour of many particle systems with unknown interactions via modelling these interactions by random variables, random matrices have since found a multitude of applicationsin physics, mathematics, and engineering. The main subject, still containing many unresolved problems, is the study of correlations between eigenvalues of random matrices when the size of the matrix is large. A typical such object is a probability that a given interval of the real line contains no eigenvalues of the random matrices. It is briefly called gap probability. This probability is expressed for many interesting ensembles ofrandom matrices as a Hankel or Toeplitz determinant. The question is to find explicit expansions of this object for the case of a large gap. This question is very easy to formulate, but obtaining the solution is a highly nontrivial task involvingdevelopment of appropriate methods. Very efficient ones rely on a so called Riemann-Hilbert approach, where the question is reduced to a problem for functions of complex variable.For some basic ensembles of random matrices, this question has been solved, but there are very important ones wherethe problem remains open. The purpose of the project is to address this question in one or two such open cases.Riemann-Hilbert steepest descent techniques used in this project have been initiated by Deift and Zhou 20 years ago, and have been developing ever since. They proved to be probably the most powerful tool of asymptotic analysis and helped to solve many important problems. Solution of a new problem using this approach also provides a development of the method itself, as new tricks have to be designed to overcome new obstacles.Thus, in addition to solving a new problem, we also contribute to the development of an important method.