权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Characteristic polynomials in Gaussian beta-ensembles and Calogero-Moser operators

高斯 beta 系综和 Calogero-Moser 算子中的特征多项式

基本信息

批准号：
EP/K010123/1
负责人：
Martin Hallnas
金额：
$ 12.6万
依托单位：
Loughborough University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK010123%2F1
关键词：
Characteristic polynomials Gaussian beta ensembles

项目摘要

Random matrices are square (or, more generally, rectangular) arrays of numbers that are drawn at random according to some probability distribution. Ever since the seminal work of Wigner in the 1950s, random matrix theory has attracted wide attention in both mathematics and physics. An important reason is the truly remarkable fact that statistical properties of large random matrices -- such as, e.g., the mean spacing of eigenvalues -- can be used as an effective model for a wide range of phenomena. Striking examples include highly excited states of heavy nuclei, the limiting distribution of the non-trivial zeros of the Riemann zeta function, and the enumeration of maps or graphs 'drawn' on surfaces.Depending on according to which distribution the elements of the matrices are chosen, one obtains different so-called ensembles of random matrices. Some of the most important and widely used are the Gaussian Orthogonal (GOE), Unitary (GUE) and Symplectic (GSE) ensembles: they are not only directly relevant to numerous applications but can also be understood in great detail. These ensembles are all contained in a one-parameter family known as the Gaussian beta-ensembles, where beta is any positive real number. This family of ensembles is not only interesting and important in its own right, but also provides a unifying point of view on the classical GOE, GUE and GSE ensembles. Although recent years has seen a surge in both interest and striking new results, the Gaussian beta-ensembles remain far from as well understood as these classical cases.The main aim of the proposed research is to bridge an important gap in the literature on Gaussian beta-ensembles: to determine the behaviour of averages of products and ratios of characteristic polynomials of random matrices drawn from such ensembles for large matrices. These averages are important in numerous applications, and they are also fundamental to random matrix theory itself. In addition, it would provide a unifying point of view on corresponding recent results for the GOE, GUE and GSE ensembles. In order to achieve this aim we will exploit a direct connection to integrable partial differential operators of so-called (deformed) Calogero-Moser type. This connection provides powerful techniques and results from the theory of partial differential operators in particular and Calogero-Moser operators in particular. Moreover, the proposed research will thus lead to important new results also on (deformed) Calogero-Moser operators.

随机矩阵是根据某种概率分布随机绘制的数字的正方形（或更一般地，矩形）阵列。自从20世纪50年代维格纳的开创性工作以来，随机矩阵理论在数学和物理学中都引起了广泛的关注。一个重要的原因是一个真正值得注意的事实，即大型随机矩阵的统计特性-例如，本征值的平均间距--可以作为一个有效的模型，用于广泛的现象。引人注目的例子包括重原子核的高激发态、黎曼zeta函数非平凡零点的极限分布以及在表面上“画”出的映射或图形的计数。根据矩阵元素的分布选择，可以得到不同的所谓随机矩阵系综。其中最重要和最广泛使用的是高斯正交（GOE），酉（GUE）和辛（GSE）系综：它们不仅与许多应用直接相关，而且可以非常详细地理解。这些系综都包含在一个单参数族中，称为高斯β系综，其中β是任何正的真实的数。这个合奏家庭不仅是有趣的和重要的，在其本身的权利，但也提供了一个统一的观点，对经典的GOE，GUE和GSE合奏。虽然近年来已经看到了激增的兴趣和惊人的新成果，高斯β-合奏仍然远远没有以及了解这些经典cases.The拟议的研究的主要目的是弥合一个重要的差距，在文献中的高斯β-合奏：以确定的行为的平均值的产品和比率的随机矩阵的特征多项式从这样的合奏大矩阵。这些平均值在许多应用中很重要，它们也是随机矩阵理论本身的基础。此外，它还将为GOE、GUE和GSE集合的相应最新结果提供统一的观点。为了实现这一目标，我们将利用一个直接连接到所谓的（变形）Calogero-Moser型可积偏微分算子。这种联系提供了强大的技术和结果，从理论的偏微分算子，特别是Calogero-Moser算子。此外，拟议的研究将导致重要的新成果也（变形）Calogero-Moser算子。