Research in Random Matrices and Integrable Systems

随机矩阵理论具有非常广泛的适用性。在随机矩阵理论中产生的间距分布在过去的几年中被证明在数论中有很深的应用;在随机矩阵的条件数很重要的数值分析和计算复杂性中有应用;随机矩阵理论促进了Riemann-Hilbert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）到量子混沌到介观系统中的量子输运。 四个领域的研究被指定。第一个是有关的事实，即在某些随机矩阵系综的措施，描述的特征值分布是吉布斯措施的收费相互作用，通过一个潜在的反温度β等于一，二或四（相应的正交，酉和辛系综，分别）。这些集合的极限间距分布现在已经很好地理解了，但是这些方法只适用于这些β值。一般β的问题，虽然很难，但在数学上很有趣，在统计物理学中也很重要。一个新的方法看起来很有前途，我们打算追求it. The第二个研究领域是矩阵合奏的最大特征值的极限分布的普遍性问题。 这类似于高斯分布对于独立随机变量之和的普适性，即著名的中心极限定理。 第三，我们提出研究特征值间距的顺序统计量（不同于前面提到的连续特征值间距分布）。 例如，最大或最小间距的概率分布是什么？ 对于独立的随机变量，已有一些结果，但对于特征值远非独立的随机矩阵，还没有结果。 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户田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。 Wiener-Hopf算子和算子行列式理论在这一研究中起着决定性的作用。毫无疑问，对这四个问题的追求会引出其他问题。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。自魏格纳在核物理领域的工作以来，随机矩阵理论在凝聚态物理、原子物理和量子混沌等新领域有着深远的应用。在数学本身，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，钟形曲线被广泛应用，因为有一个定理粗略地说，当一个人添加随机和独立的数量时，无论被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似乎对一类在底层过程中具有高度依赖性的问题具有类似的普遍性。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，一个通用的数学框架被开发出来，它将随机矩阵理论的分布函数与某些方程的解联系起来，这些方程被认为是可积的。这一数学理论给出了随机矩阵理论中分布函数的精确公式，并为它们的计算提供了有效的数值方法。 计算这些分布函数将允许一个比较它们与实验数据。

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.