隨機矩陣

提示：此條目的主題不是轉移矩陣。

在概率論和數學物理中，隨機矩陣（英語：Random matrix）是一個矩陣值的隨機變量，也就是說，一個矩陣中的所有元素都是隨機變量。^[1]

應用

物理

• 原子核物理學^[2][3]，量子場論
• 量子混沌（quantum chaos）Bohigas–Giannoni–Schmit（BGS）猜想^[4]
• 量子光學^[5][6]
• 楊-米爾斯理論（量子色動力學）^[7]
• 兩維的量子引力，AdS/CFT對偶，^[8]
• 介觀物理學,^[9]
• 自旋轉移矩,^[10]
• 小數量子霍爾效果,^[11]
• 安德森的本地化（Anderson localization）^[12]
• 量子點,^[13]
• 超導現象^[14]

其他（AI、數學、統計）

• 數論，黎曼ζ函數和其他L函數的零分布，希爾伯特–波利亞猜想，黎曼猜想^[15]
• 多元變量統計^[16][17]
• 數值分析^[18][19]
• 最優控制^[20][21][22]
• 神經科學理論，混沌理論^{[22][23][24][25][26]}
• 人工智能，機器學習，深度學習，深度神經網絡^[27][28][29]

隨機矩陣模型

設${\displaystyle H}$是${\displaystyle N\times N}$的矩陣，有下面的概率測度：

${\displaystyle P(H)={\frac {1}{Z}}e^{-Ntr(V(H))}}$

例子，高斯模型：${\displaystyle V(H)=H^{2}/2}$。

• GUE (Gaussian Unitary Ensemble)：H是埃爾米特矩陣。通過1/N展開，維格納半圓分布描述H的大N特徵值的機率密度函數${\displaystyle \rho (E)}$。^[1]
• GOE (Orthogonal)：H是對稱矩陣
• GSE (Symplectic)：H是四元數的矩陣（Quaternion matrix）

參見

• 維格納半圓分布
• 弗里曼·戴森氣體模型（Dyson gas model）
• 1/N展開
• 普遍性 (物理學)（Universality）
• Spectral Theory
• 非古典機率（Free probability）

閱讀

• 陶哲軒的Topics in random matrix theory (https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf （頁面存檔備份，存於網際網路檔案館）)
• 其他書：^[30][31][32]
• 文章：^{[33][34][35][36]}
• 原始文章：^[37][38][39]
• Voiculescu, Free Probability Theory and Operator Algebras
• Speicher, Free Probability Theory (https://arxiv.org/pdf/0911.0087.pdf （頁面存檔備份，存於網際網路檔案館）)
• 徐一鴻的https://en.wikipedia.org/wiki/Quantum_Field_Theory_in_a_Nutshell （頁面存檔備份，存於網際網路檔案館） (Large N expansion)

參考文獻

1. Terence Tao 陶哲軒. Topics in random matrix theory (PDF). （原始內容 (PDF)存檔於2021-05-06） （英語）.
2. Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079.
3. Mehta, M.L. Random Matrices. Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7.
4. Bohigas, O.; Giannoni, M.J.; Schmit, Schmit. Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws. Phys. Rev. Lett. 1984, 52 (1): 1–4. Bibcode:1984PhRvL..52....1B. doi:10.1103/PhysRevLett.52.1.
5. Aaronson, Scott; Arkhipov, Alex. The computational complexity of linear optics. Theory of Computing. 2013, 9: 143–252. doi:10.4086/toc.2013.v009a004.
6. Russell, Nicholas; Chakhmakhchyan, Levon; O'Brien, Jeremy; Laing, Anthony. Direct dialling of Haar random unitary matrices. New J. Phys. 2017, 19 (3): 033007. Bibcode:2017NJPh...19c3007R. arXiv:1506.06220 . doi:10.1088/1367-2630/aa60ed.
7. Random Matrix Theory and Chiral Symmetry in QCD. Annu. Rev. Nucl. Part. Sci. 2000, 50: 343–410. Bibcode:2000ARNPS..50..343V. arXiv:hep-ph/0003017 . doi:10.1146/annurev.nucl.50.1.343.
8. Horizon in random matrix theory, the Hawking radiation, and flow of cold atoms. Phys. Rev. Lett. October 2009, 103 (16): 166401. Bibcode:2009PhRvL.103p6401F. PMID 19905710. arXiv:0905.3533 . doi:10.1103/PhysRevLett.103.166401.
9. Magnetic-field asymmetry of nonlinear mesoscopic transport. Phys. Rev. Lett. September 2004, 93 (10): 106802. Bibcode:2004PhRvL..93j6802S. PMID 15447435. arXiv:cond-mat/0404387 . doi:10.1103/PhysRevLett.93.106802.
10. Spin torque and waviness in magnetic multilayers: a bridge between Valet-Fert theory and quantum approaches. Phys. Rev. Lett. August 2009, 103 (6): 066602. Bibcode:2009PhRvL.103f6602R. PMID 19792592. arXiv:0902.4360 . doi:10.1103/PhysRevLett.103.066602.
11. Callaway DJE. Random matrices, fractional statistics, and the quantum Hall effect. Phys. Rev. B. April 1991, 43 (10): 8641–8643. Bibcode:1991PhRvB..43.8641C. PMID 9996505. doi:10.1103/PhysRevB.43.8641.
12. Correlated random band matrices: localization-delocalization transitions. Phys. Rev. E. June 2000, 61 (6 Pt A): 6278–86. Bibcode:2000PhRvE..61.6278J. PMID 11088301. arXiv:cond-mat/9911467 . doi:10.1103/PhysRevE.61.6278.
13. Spin-orbit coupling, antilocalization, and parallel magnetic fields in quantum dots. Phys. Rev. Lett. December 2002, 89 (27): 276803. Bibcode:2002PhRvL..89A6803Z. PMID 12513231. arXiv:cond-mat/0208436 . doi:10.1103/PhysRevLett.89.276803.
14. Bahcall SR. Random Matrix Model for Superconductors in a Magnetic Field. Phys. Rev. Lett. December 1996, 77 (26): 5276–5279. Bibcode:1996PhRvL..77.5276B. PMID 10062760. arXiv:cond-mat/9611136 . doi:10.1103/PhysRevLett.77.5276.
15. Keating, Jon. The Riemann zeta-function and quantum chaology. Proc. Internat. School of Phys. Enrico Fermi. 1993, CXIX: 145–185. ISBN 9780444815880. doi:10.1016/b978-0-444-81588-0.50008-0.
16. Wishart, J. Generalized product moment distribution in samples. Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32.
17. Tropp, J. User-Friendly Tail Bounds for Sums of Random Matrices. Foundations of Computational Mathematics. 2011, 12 (4): 389–434. arXiv:1004.4389 . doi:10.1007/s10208-011-9099-z.
18. von Neumann, J.; Goldstine, H.H. Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6.
19. Edelman, A.; Rao, N.R. Random matrix theory. Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236.
20. Chow, Gregory P. Analysis and Control of Dynamic Economic Systems. New York: Wiley. 1976. ISBN 0-471-15616-7.
21. Turnovsky, Stephen. Optimal stabilization policies for stochastic linear systems: The case of correlated multiplicative and additive disturbances. Review of Economic Studies. 1976, 43 (1): 191–194. JSTOR 2296741. doi:10.2307/2296614.
22. Turnovsky, Stephen. The stability properties of optimal economic policies. American Economic Review. 1974, 64 (1): 136–148. JSTOR 1814888.
23. García del Molino, Luis Carlos; Pakdaman, Khashayar; Touboul, Jonathan; Wainrib, Gilles. Synchronization in random balanced networks. Physical Review E. October 2013, 88 (4): 042824. Bibcode:2013PhRvE..88d2824G. arXiv:1306.2576 . doi:10.1103/PhysRevE.88.042824.
24. Rajan, Kanaka; Abbott, L. Eigenvalue Spectra of Random Matrices for Neural Networks. Physical Review Letters. November 2006, 97 (18): 188104. Bibcode:2006PhRvL..97r8104R. PMID 17155583. doi:10.1103/PhysRevLett.97.188104.
25. Wainrib, Gilles; Touboul, Jonathan. Topological and Dynamical Complexity of Random Neural Networks. Physical Review Letters. March 2013, 110 (11): 118101. Bibcode:2013PhRvL.110k8101W. PMID 25166580. arXiv:1210.5082 . doi:10.1103/PhysRevLett.110.118101.
26. Muir, Dylan; Mrsic-Flogel, Thomas. Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks (PDF). Phys. Rev. E. 2015, 91 (4): 042808 [2020-01-13]. Bibcode:2015PhRvE..91d2808M. PMID 25974548. doi:10.1103/PhysRevE.91.042808. （原始內容 (PDF)存檔於2018-07-21）.
27. Cosme Louart, Zhenyu Liao, and Romain Couillet. A RANDOM MATRIX APPROACH TO NEURAL NETWORKS (PDF). （原始內容 (PDF)存檔於2020-01-13）.
28. Zhenyu Liao, Romain Couillet. The Dynamics of Learning: A Random Matrix Approach (PDF). （原始內容 (PDF)存檔於2020-11-12）.
29. Jeffrey Pennington, Pratik Worah. Nonlinear random matrix theory for deep learning (PDF). （原始內容 (PDF)存檔於2020-11-03）.
30. Mehta, M.L. Random Matrices. Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7.
31. Anderson, G.W.; Guionnet, A.; Zeitouni, O. An introduction to random matrices.. Cambridge: Cambridge University Press. 2010. ISBN 978-0-521-19452-5.
32. Akemann, G.; Baik, J.; Di Francesco, P. The Oxford Handbook of Random Matrix Theory.. Oxford: Oxford University Press. 2011. ISBN 978-0-19-957400-1.
33. Edelman, A.; Rao, N.R. Random matrix theory. Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236.
34. Pastur, L.A. Spectra of random self-adjoint operators. Russ. Math. Surv. 1973, 28 (1): 1–67. Bibcode:1973RuMaS..28....1P. doi:10.1070/RM1973v028n01ABEH001396.
35. Diaconis, Persi. Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture. American Mathematical Society. Bulletin. New Series. 2003, 40 (2): 155–178. MR 1962294. doi:10.1090/S0273-0979-03-00975-3.
36. Diaconis, Persi. What is ... a random matrix?. Notices of the American Mathematical Society. 2005, 52 (11): 1348–1349 [2020-01-13]. ISSN 0002-9920. MR 2183871. （原始內容存檔於2019-03-28）.
37. Wigner, E. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079.
38. Wishart, J. Generalized product moment distribution in samples. Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32.
39. von Neumann, J.; Goldstine, H.H. Numerical inverting of matrices of high order. Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6.