查看“︁随机矩阵”︁的源代码

{{not|轉移矩陣}}
在[[概率論]]和[[數學物理]]中，'''隨機矩陣'''（{{lang-en|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。<ref name=":0">{{Cite web|title=Topics in random matrix theory|url=https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf|accessdate=|author=Terence Tao [[陶哲轩]]|date=|format=|publisher=|language=en|archive-date=2021-05-06|archive-url=https://web.archive.org/web/20210506230749/https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf}}</ref>

== 應用 ==

=== 物理 ===

* [[原子核物理学]]<ref name="wigner">{{cite journal|title=Characteristic vectors of bordered matrices with infinite dimensions|url=https://archive.org/details/sim_annals-of-mathematics_1955-11_62_3/page/548|last=Wigner|first=E.|journal=Annals of Mathematics|issue=3|doi=10.2307/1970079|year=1955|volume=62|pages=548–564|jstor=1970079}}</ref><ref name="mehta">{{cite book|last=Mehta|first=M.L.|title=Random Matrices|year=2004|publisher=Elsevier/Academic Press|location=Amsterdam|isbn=0-12-088409-7}}</ref>，[[量子場論]]
* [[量子混沌]]（quantum chaos）Bohigas–Giannoni–Schmit（BGS）猜想<ref>{{Cite journal|title=Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws|last=Bohigas|first=O.|last2=Giannoni|first2=M.J.|journal=Phys. Rev. Lett.|issue=1|doi=10.1103/PhysRevLett.52.1|year=1984|volume=52|pages=1–4|bibcode=1984PhRvL..52....1B|last3=Schmit|first3=Schmit}}</ref>
* [[量子光學]]<ref>{{Cite journal|title=The computational complexity of linear optics|last=Aaronson|first=Scott|last2=Arkhipov|first2=Alex|date=2013|journal=Theory of Computing|doi=10.4086/toc.2013.v009a004|volume=9|pages=143–252}}</ref><ref>{{Cite journal|title=Direct dialling of Haar random unitary matrices|last=Russell|first=Nicholas|last2=Chakhmakhchyan|first2=Levon|date=2017|journal=New J. Phys.|issue=3|doi=10.1088/1367-2630/aa60ed|volume=19|pages=033007|arxiv=1506.06220|bibcode=2017NJPh...19c3007R|last3=O'Brien|first3=Jeremy|last4=Laing|first4=Anthony}}</ref>
* [[楊-米爾斯理論]]（[[量子色動力學]]）<ref>{{Cite journal|title=Random Matrix Theory and Chiral Symmetry in QCD|journal=Annu. Rev. Nucl. Part. Sci.|doi=10.1146/annurev.nucl.50.1.343|year=2000|volume=50|pages=343–410|arxiv=hep-ph/0003017|bibcode=2000ARNPS..50..343V}}</ref>
* 兩維的[[量子引力]]，[[AdS/CFT对偶]]，<ref>{{Cite journal|title=Horizon in random matrix theory, the Hawking radiation, and flow of cold atoms|date=October 2009|journal=Phys. Rev. Lett.|issue=16|doi=10.1103/PhysRevLett.103.166401|volume=103|pages=166401|arxiv=0905.3533|bibcode=2009PhRvL.103p6401F|pmid=19905710}}</ref>
* [[介观物理学]],<ref>{{Cite journal|title=Magnetic-field asymmetry of nonlinear mesoscopic transport|date=September 2004|journal=Phys. Rev. Lett.|issue=10|doi=10.1103/PhysRevLett.93.106802|volume=93|pages=106802|arxiv=cond-mat/0404387|bibcode=2004PhRvL..93j6802S|pmid=15447435}}</ref>
* [[自旋转移矩]],<ref>{{Cite journal|title=Spin torque and waviness in magnetic multilayers: a bridge between Valet-Fert theory and quantum approaches|date=August 2009|journal=Phys. Rev. Lett.|issue=6|doi=10.1103/PhysRevLett.103.066602|volume=103|pages=066602|arxiv=0902.4360|bibcode=2009PhRvL.103f6602R|pmid=19792592}}</ref> 
* [[分數量子霍爾效應|小数量子霍爾效果]],<ref>{{Cite journal|title=Random matrices, fractional statistics, and the quantum Hall effect|last=Callaway DJE|authorlink=David J E Callaway|date=April 1991|journal=Phys. Rev. B|issue=10|doi=10.1103/PhysRevB.43.8641|volume=43|pages=8641–8643|bibcode=1991PhRvB..43.8641C|pmid=9996505}}</ref> 
* [[安德森的本地化]]（Anderson localization）<ref>{{Cite journal|title=Correlated random band matrices: localization-delocalization transitions|date=June 2000|journal=Phys. Rev. E|issue=6 Pt A|doi=10.1103/PhysRevE.61.6278|volume=61|pages=6278–86|arxiv=cond-mat/9911467|bibcode=2000PhRvE..61.6278J|pmid=11088301}}</ref>
* [[量子点]],<ref>{{Cite journal|title=Spin-orbit coupling, antilocalization, and parallel magnetic fields in quantum dots|date=December 2002|journal=Phys. Rev. Lett.|issue=27|doi=10.1103/PhysRevLett.89.276803|volume=89|pages=276803|arxiv=cond-mat/0208436|bibcode=2002PhRvL..89A6803Z|pmid=12513231}}</ref> 
* [[超导现象|超导現象]]<ref>{{Cite journal|title=Random Matrix Model for Superconductors in a Magnetic Field|last=Bahcall SR|date=December 1996|journal=Phys. Rev. Lett.|issue=26|doi=10.1103/PhysRevLett.77.5276|volume=77|pages=5276–5279|arxiv=cond-mat/9611136|bibcode=1996PhRvL..77.5276B|pmid=10062760}}</ref>

=== 其他（AI、数学、统计） ===

*[[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|希尔伯特–波利亚猜想]]，[[黎曼猜想]]<ref>{{Cite journal|title=The Riemann zeta-function and quantum chaology|last=Keating|first=Jon|journal=Proc. Internat. School of Phys. Enrico Fermi|doi=10.1016/b978-0-444-81588-0.50008-0|year=1993|volume=CXIX|pages=145–185|isbn=9780444815880}}</ref>
*[[多元变量统计]]<ref name="wishart">{{Cite journal|title=Generalized product moment distribution in samples|last=Wishart|first=J.|journal=Biometrika|issue=1–2|doi=10.1093/biomet/20a.1-2.32|year=1928|volume=20A|pages=32–52}}</ref><ref>{{cite journal|title=User-Friendly Tail Bounds for Sums of Random Matrices|last=Tropp|first=J.|journal=Foundations of Computational Mathematics|issue=4|doi=10.1007/s10208-011-9099-z|year=2011|volume=12|pages=389–434|arxiv=1004.4389}}</ref>
* [[數值分析]]<ref name="vng">{{cite journal|title=Numerical inverting of matrices of high order|last=von Neumann|first=J.|last2=Goldstine|first2=H.H.|journal=Bull. Amer. Math. Soc.|issue=11|doi=10.1090/S0002-9904-1947-08909-6|year=1947|volume=53|pages=1021–1099}}</ref><ref name="er">{{cite journal|title=Random matrix theory|last=Edelman|first=A.|last2=Rao|first2=N.R|journal=Acta Numerica|doi=10.1017/S0962492904000236|year=2005|volume=14|pages=233–297|bibcode=2005AcNum..14..233E}}</ref>
* [[最优控制]]<ref>{{Cite book|last=Chow|first=Gregory P.|year=1976|title=Analysis and Control of Dynamic Economic Systems|location=New York|publisher=Wiley|isbn=0-471-15616-7}}</ref><ref>{{Cite journal|title=Optimal stabilization policies for stochastic linear systems: The case of correlated multiplicative and additive disturbances|last=Turnovsky|first=Stephen|journal=[[Review of Economic Studies]]|issue=1|doi=10.2307/2296614|year=1976|volume=43|pages=191–194|jstor=2296741}}</ref><ref name="#1">{{Cite journal|title=The stability properties of optimal economic policies|url=https://archive.org/details/sim_american-economic-review_1974-03_64_1/page/136|last=Turnovsky|first=Stephen|journal=American Economic Review|issue=1|year=1974|volume=64|pages=136–148|jstor=1814888}}</ref>
* [[神经科学]]理论，[[混沌理论]]<ref name="#1"/><ref>{{cite journal|title=Synchronization in random balanced networks|last=García del Molino|first=Luis Carlos|date=October 2013|journal=Physical Review E|issue=4|doi=10.1103/PhysRevE.88.042824|volume=88|pages=042824|arxiv=1306.2576|bibcode=2013PhRvE..88d2824G|author3=Touboul, Jonathan|author4=Wainrib, Gilles|author2=Pakdaman, Khashayar}}</ref><ref>{{cite journal|title=Eigenvalue Spectra of Random Matrices for Neural Networks|last=Rajan|first=Kanaka|date=November 2006|journal=Physical Review Letters|issue=18|doi=10.1103/PhysRevLett.97.188104|volume=97|pages=188104|bibcode=2006PhRvL..97r8104R|pmid=17155583|author2=Abbott, L.}}</ref><ref>{{cite journal|title=Topological and Dynamical Complexity of Random Neural Networks|last=Wainrib|first=Gilles|date=March 2013|journal=Physical Review Letters|issue=11|doi=10.1103/PhysRevLett.110.118101|volume=110|page=118101|arxiv=1210.5082|bibcode=2013PhRvL.110k8101W|pmid=25166580|author2=Touboul, Jonathan}}</ref><ref>{{cite journal|title=Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks|url=http://edoc.unibas.ch/41441/1/20160120100936_569f4ed0ddeee.pdf|first1=Dylan|last2=Mrsic-Flogel|first2=Thomas|date=2015|journal=Phys. Rev. E|issue=4|doi=10.1103/PhysRevE.91.042808|volume=91|page=042808|bibcode=2015PhRvE..91d2808M|pmid=25974548|last1=Muir|access-date=2020-01-13|archive-date=2018-07-21|archive-url=https://web.archive.org/web/20180721141013/https://edoc.unibas.ch/41441/1/20160120100936_569f4ed0ddeee.pdf}}</ref>
*[[人工智能]]，[[机器学习]]，[[深度学习]]，[[深度神经网络]]<ref>{{Cite web|title=A RANDOM MATRIX APPROACH TO NEURAL NETWORKS|url=https://arxiv.org/pdf/1702.05419.pdf|accessdate=|author=Cosme Louart, Zhenyu Liao, and Romain Couillet|date=|format=|publisher=|language=|archive-date=2020-01-13|archive-url=https://web.archive.org/web/20200113085633/https://arxiv.org/pdf/1702.05419.pdf}}</ref><ref>{{Cite web|title=The Dynamics of Learning: A Random Matrix Approach|url=https://arxiv.org/pdf/1805.11917.pdf|accessdate=|author=Zhenyu Liao, Romain Couillet|date=|format=|publisher=|language=|archive-date=2020-11-12|archive-url=https://web.archive.org/web/20201112031548/https://arxiv.org/pdf/1805.11917.pdf}}</ref><ref>{{Cite web|title=Nonlinear random matrix theory for deep learning|url=https://papers.nips.cc/paper/6857-nonlinear-random-matrix-theory-for-deep-learning.pdf|accessdate=|author=Jeffrey Pennington, Pratik Worah|date=|format=|publisher=|language=|archive-date=2020-11-03|archive-url=https://web.archive.org/web/20201103090400/http://papers.nips.cc/paper/6857-nonlinear-random-matrix-theory-for-deep-learning.pdf}}</ref>

== 随机矩阵模型 ==
设<math>H</math>是<math>N\times N</math>的矩阵，有下面的[[概率测度]]：

<math>P(H) = \frac{1}{Z} e^{-N tr(V(H))}</math>

例子，'''高斯模型：'''<math>V(H) = H^2 / 2</math>。

* 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]]<math>\rho(E)</math>。<ref name=":0" />
* GOE (Orthogonal)：H是[[对称矩阵]]
* GSE (Symplectic)：H是[[四元数]]的矩阵（Quaternion matrix）

== 參見 ==

* [[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）

== 阅读 ==

* [[陶哲轩]]的Topics in random matrix theory (https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf {{Wayback|url=https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf |date=20210506230749 }})
* 其他书：<ref name="mehta2">{{Cite book|last=Mehta|first=M.L.|title=Random Matrices|year=2004|publisher=Elsevier/Academic Press|location=Amsterdam|isbn=0-12-088409-7}}</ref><ref>{{Cite book|last=Anderson|first=G.W.|last2=Guionnet|first2=A.|last3=Zeitouni|first3=O.|title=An introduction to random matrices.|year=2010|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-19452-5}}</ref><ref>{{Cite book|last=Akemann|first=G.|last2=Baik|first2=J.|last3=Di Francesco|first3=P.|title=The Oxford Handbook of Random Matrix Theory.|year=2011|publisher=Oxford University Press|location=Oxford|isbn=978-0-19-957400-1}}</ref>
* 文章：<ref name="er2">{{Cite journal|title=Random matrix theory|last=Edelman|first=A.|last2=Rao|first2=N.R|journal=Acta Numerica|doi=10.1017/S0962492904000236|year=2005|volume=14|pages=233–297|bibcode=2005AcNum..14..233E}}</ref><ref name="pastur72">{{Cite journal|title=Spectra of random self-adjoint operators|last=Pastur|first=L.A.|journal=Russ. Math. Surv.|issue=1|doi=10.1070/RM1973v028n01ABEH001396|year=1973|volume=28|pages=1–67|bibcode=1973RuMaS..28....1P}}</ref><ref>{{Cite journal|title=Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture|url=https://archive.org/details/sim_american-mathematical-society-bulletin-new-series_2003-04_40_2/page/155|last=Diaconis|first=Persi|journal=American Mathematical Society. Bulletin. New Series|issue=2|doi=10.1090/S0273-0979-03-00975-3|year=2003|volume=40|pages=155–178|mr=1962294}}</ref><ref>{{Cite journal|title=What is ... a random matrix?|url=http://www.ams.org/notices/200511/|last=Diaconis|first=Persi|journal=[[Notices of the American Mathematical Society]]|issue=11|year=2005|volume=52|pages=1348–1349|issn=0002-9920|mr=2183871|access-date=2020-01-13|archive-date=2019-03-28|archive-url=https://web.archive.org/web/20190328014942/http://www.ams.org/notices/200511/}}</ref>
* 原始文章：<ref name="wigner2">{{Cite journal|title=Characteristic vectors of bordered matrices with infinite dimensions|url=https://archive.org/details/sim_annals-of-mathematics_1955-11_62_3/page/548|last=Wigner|first=E.|journal=Annals of Mathematics|issue=3|doi=10.2307/1970079|year=1955|volume=62|pages=548–564|jstor=1970079}}</ref><ref name="wishart2">{{Cite journal|title=Generalized product moment distribution in samples|last=Wishart|first=J.|journal=Biometrika|issue=1–2|doi=10.1093/biomet/20a.1-2.32|year=1928|volume=20A|pages=32–52}}</ref><ref name="vng2">{{Cite journal|title=Numerical inverting of matrices of high order|last=von Neumann|first=J.|last2=Goldstine|first2=H.H.|journal=Bull. Amer. Math. Soc.|issue=11|doi=10.1090/S0002-9904-1947-08909-6|year=1947|volume=53|pages=1021–1099}}</ref>
*Voiculescu, Free Probability Theory and Operator Algebras
*Speicher, Free Probability Theory (https://arxiv.org/pdf/0911.0087.pdf {{Wayback|url=https://arxiv.org/pdf/0911.0087.pdf |date=20200113070312 }})
* [[徐一鴻]]的<nowiki/>https://en.wikipedia.org/wiki/Quantum_Field_Theory_in_a_Nutshell {{Wayback|url=https://en.wikipedia.org/wiki/Quantum_Field_Theory_in_a_Nutshell |date=20201023045736 }} (Large N expansion)

== 參考文獻 ==
{{reflist}}

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[[Category:数学物理]]
[[Category:随机矩阵]]
[[Category:概率论]]