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点）可以表现出高度有组织的行为，因为N，组分的数量，随机矩阵是一个排列成n列n行网格的数字数组，其中的数字由骰子的掷数决定，这引入了不可预测性。事实证明，对于一个特殊的，但普遍存在的随机矩阵家族，某些真实的数（本征值）是其描述的基础，是在一对一的对应与上述n费米子。为了理解这个系统的集体行为，我们应用一个小的外部探针，并观察系统的响应。事实证明，如果探针是小的，并且在某种意义上是光滑的扰动，那么微扰将倾向于保持在原始的未扰动状态。然而，即使是一个弱的但非光滑的外部探针也可以产生与通常预期的性质不同的响应。在这种情况下，通常的方法扩展的小参数-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，以解释这种行为。

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.