Large deviations for random matrices with varying potential and sum rules

在这个项目中，我们研究了大随机矩阵的谱测度。随机矩阵用于模拟复杂的物理系统，而谱测度是描述矩阵性质的一种方法。由于矩阵是随机的，所以谱测度也是一个随机对象。然而，如果矩阵很大，光谱测度通常接近确定性极限测度，偏离指数不大可能。我们研究大偏差原理，它精确地测量这种波动偏离这个极限的可能性有多大。这个项目的新方法是当产生随机矩阵的机制不是固定的，而是随着它的大小而变化时，显示这些浓度估计。在这种情况下，已知谱测度有几个极限定理，但在大偏差尺度上的极限定理很少。有两种方法来描述随机光谱测量：通过它的光谱信息或通过一系列系数。然而，在以谱形式给出的信息和以系数形式给出的信息之间建立联系并不容易。我们利用这两种编码来研究变化的电位如何影响光谱测量的浓度，这允许以两种不同的方式证明大偏差原理。在两种描述中，两个独立证明的最大好处是一个结果和规则。这些求和规则是重要的方程，它允许在光谱信息和系数之间进行转换。除了纯概率的大偏差估计外，发现这样的恒等式是另一个主要目标。通过改变产生随机谱测度的机制，我们可以推导出新的和规则形式。

In this project we investigate spectral measures of large random matrices. Random matrices are used to model complex physical systems and the spectral measure is a way to describe properties of the matrix. Since the matrix is random, the spectral measure is a random object as well. However, if the matrix is large, the spectral measure is typically close to a deterministic limit measure and deviations are exponentially unlikely. We study large deviation principles, which measure how likely such a fluctuation from this limit is precisely. The novel approach of this project is to show these concentration estimates when the mechanism generating the random matrix, determined by a potential, is not fixed, but changes with its size. In this case, several limit theorems are known for the spectral measure, but very few on the scale of large deviations. There are two ways to describe a random spectral measure: by its spectral information or by a series of coefficients. It is however not easy to relate between information given in spectral form and information given in terms of the coefficients. We utilize both encodings to study how a varying potential influences the concentration of the spectral measure, which allows to prove large deviation principles in two different ways. The great benefit of two independent proofs in both descriptions is a resulting sum rule. These sum rules are important equations, which allow to translate between spectral information and the coefficients. Besides the purely probabilistic large deviation estimate, the discovery of such identities is another main goal. By varying the mechanism generating the random spectral measure, we can derive new forms of sum rules.