随机矩阵

在概率論和數學物理中，隨機矩陣（英語：）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。[1]

應用

物理

原子核物理学[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对偶，[8]
介观物理学,[9]
自旋转移矩,[10]
小数量子霍爾效果,[11]
安德森的本地化（Anderson localization）[12]
量子点,[13]
超导現象[14]

其他（AI、数学、统计）

随机矩阵模特

设 $H$ 是 $N\times N$ 的矩阵，有下面的概率测度：

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

例子，高斯模特： $V(H)=H^{2}/2$ 。

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 $\rho (E)$ 。[1]
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·戴森气体模特（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)
其他书：[31][32][33]
文章：[34][35][36][37]
原始文章：[38][39][40]
Voiculescu, Free Probability Theory and Operator Algebras
Speicher, Free Probability Theory (https://arxiv.org/pdf/0911.0087.pdf)
徐一鴻的https://en.wikipedia.org/wiki/Quantum_Field_Theory_in_a_Nutshell (Large N expansion)

參考文獻

Terence Tao 陶哲轩. (PDF) （英语）.
Wigner, E. . Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079.
Mehta, M.L. . Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7.
Bohigas, O.; Giannoni, M.J.; Schmit, Schmit. . Phys. Rev. Lett. 1984, 52 (1): 1–4. Bibcode:1984PhRvL..52....1B. doi:10.1103/PhysRevLett.52.1.
Aaronson, Scott; Arkhipov, Alex. . Theory of Computing. 2013, 9: 143–252. doi:10.4086/toc.2013.v009a004.
Russell, Nicholas; Chakhmakhchyan, Levon; O'Brien, Jeremy; Laing, Anthony. . New J. Phys. 2017, 19 (3): 033007. Bibcode:2017NJPh...19c3007R. arXiv:1506.06220. doi:10.1088/1367-2630/aa60ed.
. 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.
. Phys. Rev. Lett. October 2009, 103 (16): 166401. Bibcode:2009PhRvL.103p6401F. PMID 19905710. arXiv:0905.3533. doi:10.1103/PhysRevLett.103.166401.
. Phys. Rev. Lett. September 2004, 93 (10): 106802. Bibcode:2004PhRvL..93j6802S. PMID 15447435. arXiv:cond-mat/0404387. doi:10.1103/PhysRevLett.93.106802.
. Phys. Rev. Lett. August 2009, 103 (6): 066602. Bibcode:2009PhRvL.103f6602R. PMID 19792592. arXiv:0902.4360. doi:10.1103/PhysRevLett.103.066602.
Callaway DJE. . Phys. Rev. B. April 1991, 43 (10): 8641–8643. Bibcode:1991PhRvB..43.8641C. PMID 9996505. doi:10.1103/PhysRevB.43.8641.
. 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.
. Phys. Rev. Lett. December 2002, 89 (27): 276803. Bibcode:2002PhRvL..89A6803Z. PMID 12513231. arXiv:cond-mat/0208436. doi:10.1103/PhysRevLett.89.276803.
Bahcall SR. . 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.
Keating, Jon. . Proc. Internat. School of Phys. Enrico Fermi. 1993, CXIX: 145–185. ISBN 9780444815880. doi:10.1016/b978-0-444-81588-0.50008-0.
Wishart, J. . Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32.
Tropp, J. . Foundations of Computational Mathematics. 2011, 12 (4): 389–434. arXiv:1004.4389. doi:10.1007/s10208-011-9099-z.
von Neumann, J.; Goldstine, H.H. . Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6.
Edelman, A.; Rao, N.R. . Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236.
Chow, Gregory P. . New York: Wiley. 1976. ISBN 0-471-15616-7.
Turnovsky, Stephen. . Review of Economic Studies. 1976, 43 (1): 191–194. JSTOR 2296741. doi:10.2307/2296614.
Turnovsky, Stephen. . American Economic Review. 1974, 64 (1): 136–148. JSTOR 1814888.
Turnovsky, Stephen. . American Economic Review. 1974, 64 (1): 136–148. JSTOR 1814888.
García del Molino, Luis Carlos; Pakdaman, Khashayar; Touboul, Jonathan; Wainrib, Gilles. . Physical Review E. October 2013, 88 (4): 042824. Bibcode:2013PhRvE..88d2824G. arXiv:1306.2576. doi:10.1103/PhysRevE.88.042824.
Rajan, Kanaka; Abbott, L. . Physical Review Letters. November 2006, 97 (18): 188104. Bibcode:2006PhRvL..97r8104R. PMID 17155583. doi:10.1103/PhysRevLett.97.188104.
Wainrib, Gilles; Touboul, Jonathan. . Physical Review Letters. March 2013, 110 (11): 118101. Bibcode:2013PhRvL.110k8101W. PMID 25166580. arXiv:1210.5082. doi:10.1103/PhysRevLett.110.118101.
Muir, Dylan; Mrsic-Flogel, Thomas. (PDF). Phys. Rev. E. 2015, 91 (4): 042808. Bibcode:2015PhRvE..91d2808M. PMID 25974548. doi:10.1103/PhysRevE.91.042808.
Cosme Louart, Zhenyu Liao, and Romain Couillet. (PDF).
Zhenyu Liao, Romain Couillet. (PDF).
Jeffrey Pennington, Pratik Worah. (PDF).
Mehta, M.L. . Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7.
Anderson, G.W.; Guionnet, A.; Zeitouni, O. . Cambridge: Cambridge University Press. 2010. ISBN 978-0-521-19452-5.
Akemann, G.; Baik, J.; Di Francesco, P. . Oxford: Oxford University Press. 2011. ISBN 978-0-19-957400-1.
Edelman, A.; Rao, N.R. . Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236.
Pastur, L.A. . Russ. Math. Surv. 1973, 28 (1): 1–67. Bibcode:1973RuMaS..28....1P. doi:10.1070/RM1973v028n01ABEH001396.
Diaconis, Persi. . American Mathematical Society. Bulletin. New Series. 2003, 40 (2): 155–178. MR 1962294. doi:10.1090/S0273-0979-03-00975-3.
Diaconis, Persi. . Notices of the American Mathematical Society. 2005, 52 (11): 1348–1349. ISSN 0002-9920. MR 2183871.
Wigner, E. . Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079.
Wishart, J. . Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32.
von Neumann, J.; Goldstine, H.H. . Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6.

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[:0-1] Terence Tao 陶哲轩. (PDF) （英语）.

[wigner-2] Wigner, E. . Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079.

[mehta-3] Mehta, M.L. . Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7.

[4] Bohigas, O.; Giannoni, M.J.; Schmit, Schmit. . Phys. Rev. Lett. 1984, 52 (1): 1–4. Bibcode:1984PhRvL..52....1B. doi:10.1103/PhysRevLett.52.1.

[5] Aaronson, Scott; Arkhipov, Alex. . Theory of Computing. 2013, 9: 143–252. doi:10.4086/toc.2013.v009a004.

[6] Russell, Nicholas; Chakhmakhchyan, Levon; O'Brien, Jeremy; Laing, Anthony. . New J. Phys. 2017, 19 (3): 033007. Bibcode:2017NJPh...19c3007R. arXiv:1506.06220. doi:10.1088/1367-2630/aa60ed.

[7] . 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] . Phys. Rev. Lett. October 2009, 103 (16): 166401. Bibcode:2009PhRvL.103p6401F. PMID 19905710. arXiv:0905.3533. doi:10.1103/PhysRevLett.103.166401.

[9] . Phys. Rev. Lett. September 2004, 93 (10): 106802. Bibcode:2004PhRvL..93j6802S. PMID 15447435. arXiv:cond-mat/0404387. doi:10.1103/PhysRevLett.93.106802.

[10] . 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. . Phys. Rev. B. April 1991, 43 (10): 8641–8643. Bibcode:1991PhRvB..43.8641C. PMID 9996505. doi:10.1103/PhysRevB.43.8641.

[12] . 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] . 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. . 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. . Proc. Internat. School of Phys. Enrico Fermi. 1993, CXIX: 145–185. ISBN 9780444815880. doi:10.1016/b978-0-444-81588-0.50008-0.

[wishart-16] Wishart, J. . Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32.

[17] Tropp, J. . Foundations of Computational Mathematics. 2011, 12 (4): 389–434. arXiv:1004.4389. doi:10.1007/s10208-011-9099-z.

[vng-18] von Neumann, J.; Goldstine, H.H. . Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6.

[er-19] Edelman, A.; Rao, N.R. . Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236.

[20] Chow, Gregory P. . New York: Wiley. 1976. ISBN 0-471-15616-7.

[21] Turnovsky, Stephen. . Review of Economic Studies. 1976, 43 (1): 191–194. JSTOR 2296741. doi:10.2307/2296614.

[22] Turnovsky, Stephen. . American Economic Review. 1974, 64 (1): 136–148. JSTOR 1814888.

[23] Turnovsky, Stephen. . American Economic Review. 1974, 64 (1): 136–148. JSTOR 1814888.

[24] García del Molino, Luis Carlos; Pakdaman, Khashayar; Touboul, Jonathan; Wainrib, Gilles. . Physical Review E. October 2013, 88 (4): 042824. Bibcode:2013PhRvE..88d2824G. arXiv:1306.2576. doi:10.1103/PhysRevE.88.042824.

[25] Rajan, Kanaka; Abbott, L. . Physical Review Letters. November 2006, 97 (18): 188104. Bibcode:2006PhRvL..97r8104R. PMID 17155583. doi:10.1103/PhysRevLett.97.188104.

[26] Wainrib, Gilles; Touboul, Jonathan. . Physical Review Letters. March 2013, 110 (11): 118101. Bibcode:2013PhRvL.110k8101W. PMID 25166580. arXiv:1210.5082. doi:10.1103/PhysRevLett.110.118101.

[27] Muir, Dylan; Mrsic-Flogel, Thomas. (PDF). Phys. Rev. E. 2015, 91 (4): 042808. Bibcode:2015PhRvE..91d2808M. PMID 25974548. doi:10.1103/PhysRevE.91.042808.

[28] Cosme Louart, Zhenyu Liao, and Romain Couillet. (PDF).

[29] Zhenyu Liao, Romain Couillet. (PDF).

[30] Jeffrey Pennington, Pratik Worah. (PDF).

[mehta2-31] Mehta, M.L. . Amsterdam: Elsevier/Academic Press. 2004. ISBN 0-12-088409-7.

[32] Anderson, G.W.; Guionnet, A.; Zeitouni, O. . Cambridge: Cambridge University Press. 2010. ISBN 978-0-521-19452-5.

[33] Akemann, G.; Baik, J.; Di Francesco, P. . Oxford: Oxford University Press. 2011. ISBN 978-0-19-957400-1.

[er2-34] Edelman, A.; Rao, N.R. . Acta Numerica. 2005, 14: 233–297. Bibcode:2005AcNum..14..233E. doi:10.1017/S0962492904000236.

[pastur72-35] Pastur, L.A. . Russ. Math. Surv. 1973, 28 (1): 1–67. Bibcode:1973RuMaS..28....1P. doi:10.1070/RM1973v028n01ABEH001396.

[36] Diaconis, Persi. . American Mathematical Society. Bulletin. New Series. 2003, 40 (2): 155–178. MR 1962294. doi:10.1090/S0273-0979-03-00975-3.

[37] Diaconis, Persi. . Notices of the American Mathematical Society. 2005, 52 (11): 1348–1349. ISSN 0002-9920. MR 2183871.

[wigner2-38] Wigner, E. . Annals of Mathematics. 1955, 62 (3): 548–564. JSTOR 1970079. doi:10.2307/1970079.

[wishart2-39] Wishart, J. . Biometrika. 1928, 20A (1–2): 32–52. doi:10.1093/biomet/20a.1-2.32.

[vng2-40] von Neumann, J.; Goldstine, H.H. . Bull. Amer. Math. Soc. 1947, 53 (11): 1021–1099. doi:10.1090/S0002-9904-1947-08909-6.