Non-perturbative effects in complex systems: A study through the theory of random matrices and orthogonal polynomials

复杂系统的范例是，一个极其简单的模型(例如，大量相同的粒子的集合，称为费米子，它们只通过一个极其简单的规则进行通信：不允许两个粒子占据空间中的同一个点)可以表现出高度有组织的行为，因为n(成分)的数量变大。随机矩阵是以n列x n行网格排列的数字阵列，数字由掷骰子确定，这引入了不可预测性元素。结果表明，对于一个特殊但普遍存在的随机矩阵族，其描述基础的某些实数(本征值)与上面提到的n个费米子一一对应。为了了解这个系统的集体行为，我们使用一个小的外部探测器并观察系统的响应。结果表明，如果探头很小，并且在某种意义上是光滑的扰动，系统将倾向于保持原始的无扰状态。然而，即使是一个微弱但非光滑的外部探头也可以产生与正常情况下预期的定性不同的响应。在这种情况下，通常的小参数展开方法-微扰处理--完全崩溃。在拟议的研究中，将开发处理非微扰效应的新方法来解释这种行为。

The paradigm of complex systems is that an extermely simple model (for example, a collection of a large number of identical particles, called Fermions, which only communicate through an extremely simple rule: no two particles are allowed to occupy the same point in space) can exhibit highly organised behaviour as n, the number of constituents, becomes large.A random matrix is an array of numbers arranged in an n columns by n rows grid with the numbers determined from say, the throw of a die, which introducesthe element of unpredictability. It turns out that for a particular, but ubiqiutous familyof random matrices, certain real numbers (eigenvalues) that are fundamental to its description are in one-to-one correspondence with the n Fermions mentioned above.To understand the collective behaviour of this system, we apply a small external probeand observe the response of the system. It turns out that if the probe is asmall and in some sense smooth perturbation, thesystem will tend to remain in the original unperturbed state.However, even a weak but non-smooth external probe can produce responses qualitatively distinct from what can be normally expected. In such situations, the usual approach of expansion in terms of small parameters --- the perturbative treatment --- breaks down completely. New methods to deal with non-perturbative effects will be developed in the proposed research to explain such behaviour.