Asymptotic Analysis of Random Matrices and Exactly Soluble Models

随机矩阵和精确可溶模型的渐近分析

• 批准号: EP/E022928/1
• 负责人: Igor Krasovsky
• 金额: $ 26.87万
• 依托单位: Brunel University London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2006
• 资助国家: 英国
• 起止时间: 2006 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FE022928%2F1
• 关键词: Asymptotic; Analysis; Random; Matrices; Exactly

Some magnets can be imagined as a collection of small elementary magnets ('compas needles' called spins). If an outside magnetic field is applied they tend to aline themselves in the direction of the field. If the magnetic field is removed,they tend to become disoriented again. One of the important questions is to know how quickly the alignment is destroyed. It depends on the type of a magnet and the temperature.This information is encoded in the so called correlation functions of the spins.Correlation functions tell us how well the spins 'feel' one another.If we pick one spin in the magnet,one can ask the probability that another spin points in the same direction. This probability is one of the examples of the correlation functions. It is in general impossible to calculate correlation functions exactly. However, in some simple magnets, it is possible to calculate them asymptotically. This last wordmeans, in the example above, that we may be able to calculate the correlation function of two spins when the distance between them becomes large.Such asymptotic calculations are the subject of this proposal.

一些磁铁可以想象成一个小的基本磁铁的集合（“罗盘针”称为自旋）。如果施加外部磁场，它们倾向于沿着磁场的方向排列。如果磁场被移除，它们往往会再次迷失方向。其中一个重要的问题是要知道多快的路线被破坏。这取决于磁体的类型和温度。这些信息被编码在所谓的自旋相关函数中。相关函数告诉我们自旋之间的“感觉”有多好。如果我们在磁体中选择一个自旋，我们可以问另一个自旋指向同一方向的概率。这个概率是相关函数的一个例子。通常不可能精确地计算相关函数。然而，在一些简单的磁体中，可以渐近地计算它们。在上面的例子中，最后一个词意味着我们可以计算两个自旋之间的距离变大时的相关函数，这种渐近计算就是这个建议的主题。