发表机构
University of Bath（巴斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本讲义旨在推导随机矩阵理论经典结果并展示其应用，还为从业者提供分析技术参考指南，用多种形式体系重新推导经典结果，涵盖群体动力学方法等创新内容，讨论各方法优点及适用背景。
AI 中文摘要
在过去的一个世纪里，随机矩阵理论已成为数学科学的理论基石。它在核物理、金融、生态和无序系统等诸多研究领域有着不可否认的效用。本讲义有两个目的。其一，以教学方式推导最著名且广泛使用的经典结果，主要使用相对基础且透明的腔方法，并在特定应用背景下展示每个结果的意义，各部分结尾还有精选练习。其二，为随机矩阵/无序系统从业者提供分析技术参考指南，从第一原理引入图解、复制、路径积分和超对称形式体系，重新推导一些经典结果，尤其关注最简单的半圆定律，还包括群体动力学方法和自由概率论工具等创新内容，讨论每种分析方法的优点及特别有用的背景。
英文摘要
Random matrix theory has established itself as a theoretical cornerstone of the mathematical sciences over the past century. It has undeniable utility in areas of research as diverse as nuclear physics, finance, ecology and disordered systems. The purpose of these notes is twofold. First, the most famous and widely used classic results are derived in a pedagogical manner, mostly using the comparatively elementary and transparent cavity method. The significance of each result is then demonstrated in the context of a particular application. There are also some select exercises at the end of each section. In the second part of these notes, a reference guide of analytical techniques for the random-matrix/disordered-systems practitioner is provided. Introducing the diagrammatic, replica, path-integral, and supersymmetric formalisms from first principles, we rederive some of the aforementioned classic results, particularly focussing on the simplest one -- the semicircle law. Innovations such as the population dynamics method and the tools of free probability theory are also included. We discuss the merits of each analytical approach, and we highlight the contexts in which each becomes particularly useful.
Comments227 pages, 41 figures
