Orthogonal and Symplectic Random Matrix Ensembles: Universality and Asymptotics of the Partition Function

正交和辛随机矩阵系综：配分函数的普遍性和渐近性

• 批准号: 0556049
• 负责人: Dimitri Gioev
• 金额: $ 9.02万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0556049&HistoricalAwards=false
• 关键词: Orthogonal; Symplectic; Random; Matrix; Ensembles

AbstractGioevThe PI will work on four problems, all of which involve asymptotic techniques developed in Random Matrix Theory (RMT). Problems I and II are concerned with the proof of universality for orthogonal and symplectic ensembles of random matrices with varying polynomial weights on the line, and with polynomial weights on the half-line, respectively. Problem III deals with the rigorous proof of the asymptotic expansion for the partition function (free energy) for orthogonal and symplectic ensembles with varying polynomial weights on the line which has applications in graph enumeration problems arising in 2D quantum gravity and string theory. Finally, Problem IV concerns proving various asymptotic formulae for the entanglement entropy of fermion systems on the lattice and in the continuum (this problem has connections with RMT).Over the past fifty years it has become clear that Random Matrix Theory (RMT), initially introduced into the theoretical physics community by Wigner in 1950's as a model for the scattering of neutrons off large nuclei, has a long and extraordinarily varied list of applications not only in physics, but also in pure and applied mathematics, and in other sciences, such as cardiology. More precisely, it turns out that (the statistical properties of) various seemingly unrelated objects, such as vibration frequencies of elastic plates, bus arrival times when the drivers try to optimize the traffic flow, zeros of the Riemann zeta function, the distribution of phone numbers in a large phone directory, and heartbeat peaks, for example, are all remarkably well-described by eigenvalues of a large random matrix (RM). The fundamental question is: Why does RMT model such a broad variety of phenomena? The answer is that there should be certain limiting distribution laws for the eigenvalues of large RM's that are independent of the precise distribution of the matrix ensembles. This loose statement is known as the Universality Conjecture, and it can be viewed as an analog of the Central Limit Theorem in probability theory, but now for certain classes of correlated random variables (which are the eigenvalues of a RM). Universality questions arose very early on in RMT, and universality is widely believed to be true. However, a mathematical proof of universality for the so-called unitary ensembles was found only in the late 1990's. The PI and Deift have very recently proved universality in the remaining two cases, viz., the orthogonal and symplectic ensembles, in great generality. Problems I and II, which deal with continuation and extension of these results, will significantly extend our understanding of universality to the important case where the underlying equilibrium measure consists of more than one interval. The asymptotic expansion of the partition function for the three types of RM ensembles has important applications in mathematics and also in 2D quantum gravity and string theory. Very recently, in the unitary case, the expansion was rigorously established by Ercolani and McLaughlin. In Problem III, the PI plans to prove rigorously the expansion for general orthogonal and symplectic ensembles. This will have implications in 2D quantum gravity models which are not covered by the unitary case. In the emerging field of quantum computation, a subject currently of great interest, one of the basic quantities of study is the so-called entanglement entropy which measures the number of maximally entangled pairs that can be extracted from a given quantum state. In Problem IV, the PI plans to prove various results concerning asymptotic expansions of the entanglement entropy for fermions as the size of a subsystem becomes large. These results will considerably extend the understanding of the entanglement entropy for the lattice and also continuous fermion systems in higher dimensions.

摘要：PI将解决四个问题，所有这些问题都涉及随机矩阵理论(RMT)中发展的渐近技术。问题I和问题II分别涉及直线上具有变化多项式权的随机矩阵的正交和辛系综及在半直线上具有多项式权的随机矩阵集合的普适性的证明。问题三严格证明了直线上具有不同多项式权的正交系综和辛系综的配分函数(自由能)的渐近展开式，它在二维量子引力和弦理论中的图计数问题中有应用。最后，问题四涉及证明晶格上和连续介质中费米子系统纠缠熵的各种渐近公式(这个问题与RMT有关)。在过去的五十年里，随机矩阵理论(RMT)，作为一个中子从大核散射的模型，最初由Wigner在1950年引入理论物理界，不仅在物理学中，而且在理论数学和应用数学中，以及在其他科学，如心脏病学中，都有很长且非常不同的应用清单。更准确地说，事实证明，各种看似不相关的对象的(统计特性)，例如弹性板的振动频率、司机试图优化交通流量时的公交车到达时间、Riemann Zeta函数的零点、大型电话簿中电话号码的分布以及心跳峰值，都非常好地由大型随机矩阵(RM)的特征值描述。根本的问题是：为什么RMT模拟了如此广泛的各种现象？答案是，对于大Rm的本征值，应该有一定的极限分布规律，这些规律与矩阵系综的精确分布无关。这种松散的说法被称为普适性猜想，它可以被视为概率论中中心极限定理的类似，但现在是针对某些类别的相关随机变量(它们是Rm的特征值)。普遍性问题在RMT中很早就出现了，人们普遍认为普遍性是正确的。然而，所谓酉系综普适性的数学证明直到上世纪90年代末才被S发现。最近，PI和Deift证明了其余两种情形的普适性，即极具一般性的正交系综和辛系综。问题一和问题二涉及这些结果的延续和扩展，它们将极大地扩展我们对普遍性的理解，使之适用于潜在的均衡度量由一个以上的区间组成的重要情况。这三类RM系综配分函数的渐近展开式在数学、二维量子引力和弦理论中都有重要的应用。最近，在么正情形下，Ercolani和McLaughlin严格地建立了展开式。在问题三中，PI计划严格地证明一般的正交系综和辛系综展式。这将对不在么正情形下覆盖的2D量子引力模型产生影响。在新兴的量子计算领域，目前人们非常感兴趣的一个主题是所谓的纠缠熵，它衡量了从给定的量子态中可以提取的最大纠缠对的数量。在问题四中，PI计划证明随着子系统的大小，费米子纠缠熵的渐近展开式的各种结果。这些结果将极大地扩展对晶格和高维连续费米子系统纠缠熵的理解。