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，它们只能通过一个极其简单的规则进行交流：不允许两个粒子占据空间中的同一点），当n（组成部分的数量）变大时，可以表现出高度有组织的行为。随机矩阵是一组数字，排列在n列乘n行的网格中，这些数字是由掷骰子决定的，这就引入了不可预测性。事实证明，对于一个特定的，但无处不在的随机矩阵家族，某些实数（特征值）是其描述的基础，与上面提到的n个费米子是一一对应的。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来观察系统的响应。结果表明，如果探针很小并且在某种意义上是平滑的摄动，系统将倾向于保持在原始的非摄动状态。然而，即使是一个微弱但不光滑的外部探针也能产生与通常预期的性质不同的响应。在这种情况下，通常用小参数展开的方法——微扰处理——完全失效了。新的方法来处理非扰动效应将在拟议的研究中发展，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.