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

摘要gioevpi将研究四个问题，所有这些问题都涉及随机矩阵理论（RMT）中的渐近技术。问题1和问题2分别讨论了在直线上和在半直线上具有多项式权值的随机矩阵的正交系和辛系的普适性证明。问题三是对具有不同多项式权值的正交和辛系综的配分函数（自由能）的渐近展开的严格证明，它在二维量子引力和弦理论中的图枚举问题中有应用。最后，问题IV涉及证明费米子系统在晶格和连续体上的纠缠熵的各种渐近公式（该问题与RMT有关）。随机矩阵理论（RMT）最初是由维格纳（Wigner）在20世纪50年代作为中子从大原子核散射的模型引入理论物理界的，在过去的50年里，人们已经清楚地认识到，它不仅在物理学中，而且在纯数学和应用数学以及其他科学（如心脏病学）中有着广泛的应用。更准确地说，事实证明，各种看似无关的对象（统计属性），如弹性板的振动频率，司机试图优化交通流量时的公共汽车到达时间，黎曼ζ函数的零点，大型电话簿中电话号码的分布，以及心跳峰值，例如，都可以很好地用大型随机矩阵（RM）的特征值来描述。最根本的问题是：为什么RMT建模如此广泛的现象？答案是，大RM的特征值应该有一定的极限分布规律，这些规律与矩阵集合的精确分布无关。这个松散的陈述被称为普适性猜想，它可以被视为概率论中中心极限定理的类比，但现在适用于某些类相关随机变量（即RM的特征值）。普适性问题在RMT中很早就出现了，普适性被广泛认为是正确的。然而，所谓的酉系综的普适性的数学证明直到20世纪90年代末才被发现。PI和Deift最近在剩下的两种情况，即正交系综和辛系综中，以极大的普遍性证明了其通用性。问题1和问题2，处理这些结果的延续和推广，将极大地扩展我们对普适性的理解到潜在的均衡测度由多个区间组成的重要情况。三种类型RM系综配分函数的渐近展开式在数学以及二维量子引力和弦理论中都有重要的应用。最近，在单一情况下，厄尔科拉尼和麦克劳克林严格地建立了扩张。在问题III中，PI计划严格证明一般正交和辛综的展开式。这将对二维量子引力模型产生影响，而二维量子引力模型不包括在单一情况下。在新兴的量子计算领域中，一个目前备受关注的课题，研究的基本量之一是所谓的纠缠熵，它测量从给定量子态中可以提取的最大纠缠对的数量。在问题四中，PI计划证明关于费米子的纠缠熵随子系统尺寸变大而渐近扩展的各种结果。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。

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.