Research in Random Matrices and Integrable Systems

随机矩阵和可积系统研究

• 批准号: 9802122
• 负责人: Craig Tracy
• 金额: $ 23.26万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802122&HistoricalAwards=false
• 关键词: Research; Random; Matrices; Integrable; Systems

Random matrix theory has had remarkably wide applicability. The spacing distributions arising in random matrix theory have over the past few years beenshown to have deep applications in number theory; there are applications in numerical analysis and computational complexity where condition numbers of random matrices are important; random matrix theory has motivated developmentsin the Riemann-Hilbert method which in turn finds applications to a variety of problems in integrable systems and inverse scattering. In physics the applications range from many-body systems (both atomic and nuclear), to quantum chaos to quantum transport in mesoscopic systems. Four areas for research arespecified. The first is related to the fact that in certain random matrix ensembles the measure describing the eigenvalue distribution is the Gibbs measure for charges interacting via a potential at inverse temperature beta equal to one, two or four (corresponding to orthogonal, unitary and symplectic ensembles, respectively). The limiting spacing distributions for these ensembles are now quite well understood but the methods are applicable to these values of beta only. The question for general beta, while quite difficult, is mathematically interesting and quite important in statistical physics. A new approach looks promising and we intend to pursue it. The second area of researchis the question of universality of the limiting distribution of the largest eigenvalue in matrix ensembles. This would be analogous to the universality of the Gaussian distribution for sums of independent random variables, the famous Central Limit Theorem. Thirdly, we propose to study the order statistics of the spacings between eigenvalues (which is different from the spacing distributions between consecutive eigenvalues mentioned above). For example, what is the probabilitydistribution for the largest or smallest spacing? There are known results for independent random variables but none yet for for random matrices,whose eigenvalues are far from independent. Finally, we expect to complete earlierwork on the asymptotics of solutions to the periodic Toda equations by determining the asymptotics on the so-called critical curves, where the asymptotics will take a very different form. The theory of Wiener-Hopf operatorsand operator determinants should play a decisive role in this investigation. No doubt the pursuit of these four questions will lead to others.In the 1950s Eugene Wigner, in his now classic study of highly excited states of large nuclei of atoms, introduced a mathematical modelto describe the spacing between these states. This model goes under the name of random matrix theory.Since Wigner's work in nuclear physics, it has been shown that the mathematics of random matrix theory has far-reaching applications to condensed matter physics, atomic physics and the new area of quantum chaos. In mathematics itself, random matrix theory has begun to appear in such diverse areas as number theory, combinatorics and numerical analysis. It is natural to ask why there is such wide applicability of random matrix theory. In probability theory the bell-shaped curve is widely applicable because of a theorem which says roughly that when one adds quantities which are random and independent, the sum follows the bell shaped curve regardless of the distribution of the randomobjects being added. The distribution functions of random matrix theory appear to have a similar universality for a class of problems where there is a high degree of dependence in the underlying processes. In the present project the mathematics of random matrix theory will be further developed with an eye kept on possible applications. In earlier work a general mathematical framework was developed that related the distribution functions of random matrix theory with solutions to certain equations which are said to be integrable. This mathematical theory gives exact formulas for distribution functions in random matrix theory and provides efficient numerical methods for their computation. Computing these distribution functions will allow oneto compare them with experimental data.

随机矩阵理论具有非常广泛的适用性。随机矩阵理论中出现的间距分布在过去几年中已被证明在数论中具有深入的应用。在数值分析和计算复杂性方面有一些应用，其中随机矩阵的条件数很重要；随机矩阵理论推动了黎曼-希尔伯特方法的发展，该方法反过来又应用于可积系统和逆散射中的各种问题。在物理学中，应用范围从多体系统（原子和核）到量子混沌再到介观系统中的量子传输。 指定了四个研究领域。第一个与以下事实有关：在某些随机矩阵系综中，描述特征值分布的测度是通过等于一、二或四的反温度β下的电势相互作用的电荷的吉布斯测度（对应于正交、酉和辛系综，分别）。现在已经很好地理解了这些系综的极限间距分布，但这些方法仅适用于这些 beta 值。一般贝塔值的问题虽然相当困难，但在数学上很有趣，并且在统计物理学中非常重要。一种新方法看起来很有希望，我们打算继续采用。 第二个研究领域是矩阵系综中最大特征值极限分布的普遍性问题。 这类似于独立随机变量之和的高斯分布的普遍性，即著名的中心极限定理。 第三，我们建议研究特征值之间的间距的阶次统计（这与上面提到的连续特征值之间的间距分布不同）。 例如，最大或最小间距的概率分布是什么？ 独立随机变量已有已知结果，但随机矩阵尚无已知结果，其特征值远非独立。 最后，我们期望通过确定所谓的临界曲线上的渐近性来完成关于周期 Toda 方程解的渐近性的早期工作，其中渐近性将采取非常不同的形式。 维纳-霍普夫算子和算子行列式理论在这项研究中应发挥决定性作用。毫无疑问，对这四个问题的追求将会引出其他问题。 20世纪50年代，尤金·维格纳（Eugene Wigner）在他现在对大原子核高激发态的经典研究中，引入了一个数学模型来描述这些状态之间的间距。这个模型被称为随机矩阵理论。自从维格纳在核物理方面的工作以来，已经表明随机矩阵理论的数学在凝聚态物理、原子物理和量子混沌的新领域具有深远的应用。在数学本身中，随机矩阵理论已经开始出现在数论、组合学和数值分析等不同领域。人们很自然会问为什么随机矩阵理论有如此广泛的适用性。 在概率论中，钟形曲线广泛适用，因为有一个定理，粗略地说，当一个人将随机且独立的数量相加时，无论所添加的随机对象的分布如何，总和都遵循钟形曲线。随机矩阵理论的分布函数对于底层过程具有高度依赖性的一类问题似乎具有类似的普遍性。在本项目中，随机矩阵理论的数学将进一步发展，并着眼于可能的应用。在早期的工作中，开发了一种通用数学框架，它将随机矩阵理论的分布函数与某些据说可积的方程的解联系起来。该数学理论给出了随机矩阵理论中分布函数的精确公式，并为其计算提供了有效的数值方法。 计算这些分布函数将允许人们将它们与实验数据进行比较。