RUI: Asymptotics of Determinants of Perturbations of Convolution Operators

摘要本课题的重点是研究卷积算子摄动的行列式的渐近性。我们的目标是将经典的极限定理扩展到这些算子，包括标量和矩阵值符号，以及光滑和奇异符号。对于这些算子中的许多，常数项是渐近展开中最难描述的部分。对于矩阵值符号，只有少数情况下可以显式地描述常数。特别地，我们将研究Toeplitz行列式被可能有不同符号的Hankel算子扰动的渐近性。其他类型的算子是Wiener-Hopf + Hankel算子和Bessel算子。经典的算子方法将用于研究这些问题以及新的发展。例如，使用BorodinGeronimo-Geronimo-Case标识在平滑符号和奇异符号之间架起桥梁就非常成功。人们对寻找卷积型算子的行列式的渐近展开越来越感兴趣，因为它们与数学物理中的许多问题有联系，包括Ising模型（二维（或非常薄）磁铁的模型），经典二聚体模型，自旋链模型中的纠缠问题，随机增长模型，以及随机矩阵理论的一般领域。在这些物理问题中，人们常常对模型的复杂的、不可预测的行为感兴趣。通常，描述系统某些统计性质的量可以被重新表述为行列式近似问题。物理系统给出了近似的正确形式的预测，并表明许多答案应该是相当普遍的。通用性是特别重要的，因为它表明许多复杂的系统和模型实际上是非常相似的。因此，这个想法不是简单地证明定理，然后找到定理的应用，而是用数学物理的思想来给出数学的预测，反过来，用数学来告诉我们一些关于物理系统的东西。

AbstractBasor The focus of this project is to investigate the asymptotics of determinants of perturbations of convolution operators. Our goal will be to extend the classical limit theorems to these operators, both for scalar and matrix-valued symbols, and for both smooth and singular symbols. For many of these operators, the constant term is the most difficult piece of the asymptotic expansion to describe. For matrix-valued symbols there are only a few cases where the constants can be explicitly described. In particular, we will investigate the asymptotics in the case of a perturbation of a Toeplitz determinant by a Hankel operator with possibly different symbol. Other classes of operators of interest are Wiener-Hopf plus Hankel operators and Bessel operators. Classical operator methods will be used to study these problems as well as newer developments. For example, using the BorodinGeronimo-Geronimo-Case identity to bridge between smooth and singular symbols has been highly successful.There is increasing interest in finding asymptotic expansions of determinants of convolution type operators because they have connections to many problems in mathematical physics, including the Ising model (a model of a two-dimensional (or very thin) magnets), the classical dimer model, the entanglement problem in spin chain model, random growth models, and to the general area of random matrix theory. In these physical problems one is often interested in the complicated, unpredictable behavior of the models. Often a quantity that describes some statistical property of a system can be reformulated as a determinant approximation problem. The physical systems give predictions as to the right form of the approximation and show that many of the answers should be quite universal. The universality is especially important since it shows that many complicated systems and models are actually quite similar. Hence the idea is not simply to prove theorems and then find applications for the theorems, but to use the ideas of mathematical physics to give predictions of the mathematics and then conversely, to use the mathematics to tell us something about physical systems.