随机矩阵

在概率論和數學物理中，隨機矩陣（英語：Random matrix）是一個矩陣值的随机变量，也就是说，一个矩阵中的所有元素都是随机变量。 原子核物理学，量子場論 量子混沌（quantum chaos）Bohigas–Giannoni–Schmit（BGS）猜想 量子光學 楊-米爾斯理論（量子色動力學） 兩維的量子引力，AdS/CFT对偶， 介观物理学, 自旋转移矩, 小数量子霍爾效果, 安德森的本地化（Anderson localization） 量子点, 超导現象 数论，黎曼ζ函數和其他L函数的零分布，希尔伯特–波利亚猜想，黎曼猜想 多元变量统计 數值分析 最优控制 神经科学理论，混沌理论 人工智能，机器学习，深度学习，深度神经网络 设 H {\displaystyle H} 是 N × N {\displaystyle N\times N} 的矩阵，有下面的概率测度： P ( H ) = 1 Z e − N t r ( V ( H ) ) {\displaystyle P(H)={\frac {1}{Z}}e^{-Ntr(V(H))}} 例子，高斯模型： V ( H ) = H 2 / 2 {\displaystyle V(H)=H^{2}/2} 。 GUE (Gaussian Unitary Ensemble)：H是埃尔米特矩阵。通过1/N展开，維格納半圓分布描述H的大N特征值的機率密度函數 ρ ( E ) {\displaystyle \rho (E)} 。 GOE (Orthogonal)：H是对称矩阵 GSE (Symplectic)：H是四元数的矩阵（Quaternion matrix） 維格納半圓分布 弗里曼·戴森气体模型（Dyson gas model） 1/N展开 普遍性 (物理学)（Universality） Spectral Theory 非古典机率（Free probability） 陶哲轩的Topics in random matrix theory (https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf （页面存档备份，存于互联网档案馆）) 其他书： 文章： 原始文章： Voiculescu, Free Probability Theory and Operator Algebras Speicher, Free Probability Theory (https://arxiv.org/pdf/0911.0087.pdf （页面存档备份，存于互联网档案馆）) 徐一鴻的https://en.wikipedia.org/wiki/Quantum_Field_Theory_in_a_Nutshell （页面存档备份，存于互联网档案馆） (Large N expansion) Terence Tao 陶哲轩. 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