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

随机矩阵是根据某种概率分布随机绘制的正方形（或者更一般地说是矩形）数字数组。自20世纪50年代维格纳的开创性工作以来，随机矩阵理论引起了数学和物理学的广泛关注。一个重要的原因是，大型随机矩阵的统计特性——例如，特征值的平均间距——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核的高激发态，黎曼ζ函数的非平凡零的极限分布，以及“绘制”在表面上的地图或图形的枚举。根据选择矩阵元素的分布，可以得到不同的所谓随机矩阵的集合。其中一些最重要和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系综中，称为高斯系综，其中系综是任意正实数。这个合奏家族不仅在其自身的权利是有趣和重要的，而且还提供了一个统一的观点，对经典的GOE， GUE和GSE合奏。尽管近年来人们对高斯-系综的兴趣激增，并取得了惊人的新结果，但人们对高斯-系综的理解仍远不及这些经典情况。所提出的研究的主要目的是弥合高斯β系综文献中的一个重要空白：确定从大矩阵的这种系综中抽取的随机矩阵的乘积的平均值和特征多项式的比率的行为。这些平均值在许多应用中都很重要，它们也是随机矩阵理论本身的基础。此外，它将为GOE， GUE和GSE系统的相应最新结果提供统一的观点。为了达到这个目的，我们将利用与所谓的（变形的）Calogero-Moser型可积偏微分算子的直接联系。这种联系提供了强大的技术和结果，特别是偏微分算子的理论，特别是卡罗伽罗-莫泽算子。此外，所提出的研究也将在（变形）Calogero-Moser算子上产生重要的新结果。

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.