Research in Random Matrices and Integrable Systems

提议：DMS-9732687首席研究员：哈罗德·维多姆摘要：随机矩阵理论有非常广泛的适用性：在过去的几年里，随机矩阵理论中的间隔分布被证明在数论中有很深的应用；在数值分析和计算复杂性中，随机矩阵的条件数是重要的；随机矩阵理论促进了黎曼-希尔伯特方法的发展，这反过来又在可积系统和逆散射中找到了应用。在物理学中的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输运。明确了四个研究领域。第一个问题与这样一个事实有关，即在某些随机矩阵系综中，描述本征值分布的度量是通过逆温度β等于1、2或4(分别对应于正交、酉系综和辛系综)的势相互作用的电荷的吉布斯度量。这些系综的极限间距分布现在已经很清楚了，但这些方法只适用于这些Beta值。一般贝塔系数的问题虽然相当困难，但在数学上很有趣，在统计物理学中也很重要。一种新的方法看起来很有希望，我们打算继续下去。第二个研究领域是矩阵系综中最大特征值的极限分布的普适性问题。这将类似于独立随机变量和的高斯分布的普适性，即中心极限定理。第三，我们建议研究特征值之间的间距的顺序统计量(这与前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？已有关于独立随机变量的已知结果，但显然还没有关于随机矩阵的结果，因为其特征值远不是独立的。最后，我们希望通过确定所谓的临界曲线上的渐近性来完成早期关于周期Toda方程解的渐近性的工作，其中的渐近性将采取非常不同的形式。Wiener-Hopf算子和算子行列式理论应该在这一研究中起决定性作用。20世纪50年代，尤金·维格纳在他对大原子核的高激发态的经典研究中，引入了一个数学模型来描述这些态之间的间距。这个模型被冠以随机矩阵理论的名义。自从Wigner的开创性工作以来，随机矩阵的数学已经被证明在凝聚态物理、原子物理和量子混沌的新领域有着深远的应用。在数学本身，随机矩阵理论已经在许多不同的背景下浮出水面。人们很自然地会问，为什么这个主题具有如此广泛的适用性。在概率论中，钟形曲线是无处不在的，因为有一个定理粗略地说，当一个人把随机和独立的量相加时，其和遵循钟形曲线，而不管这些量本身的分布。随机矩阵理论的分布函数对于一类问题似乎具有类似的普适性，在这类问题中，潜在过程具有高度的相关性。在本项目中，随机矩阵理论的数学将进一步发展，着眼于可能的应用。

Proposal: DMS-9732687 Principal Investigator: Harold Widom Abstract: Random matrix theory has had remarkably wide applicability: the spacing distributions arising in random matrix theory have over the past few years been shown 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 developments in 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 are specified. 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 research is 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 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 probability distribution for the largest or smallest spacing? There are known results for independent random variabl es but, apparently, none yet for for random matrices, whose eigenvalues are far from independent. Finally, we hope to complete earlier work 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 operators and operator determinants should play a decisive role in this investigation. In the 1950s Eugene Wigner, in his now classic study of highly excited states of large nuclei of atoms, introduced a mathematical model to describe the spacing between these states. This model goes under the name of random matrix theory. Since Wigner's pioneering work, it has been shown that the mathematics of random matrices has far-reaching applications to condensed matter physics, atomic physics and the new area of quantum chaos. In mathematics itself, random matrix theory has surfaced in a multitude of different contexts. It is natural to ask why the subject has such wide applicability. In probability theory the bell-shaped curve is pervasive 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 quantities themselves. The distribution functions of random matrix theory appear to have a similar universality for a class of problems in which 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 toward possible applications.