Random matrix

In probability theory and statistics (with main applications in theoretical physics ) a random matrix is a matrix-valued random variable ( English Random Matrix ). To distinguish it from the multivariate distributions, the distribution of a random matrix is called a matrix-variable probability distribution .

Many important properties of physical systems can be formulated mathematically with matrices and random matrices appear in problems of statistical mechanics. For example, the thermal conductivity of a crystalline solid can be calculated directly from the so-called dynamic matrix of the particle-particle interaction of the crystal lattice.

To motivate: Disordered systems

In the case of a disordered physical system (e.g. with so-called amorphous material) the relevant matrix elements are random variables. The physics of these systems can essentially be determined by the parameters of the respective matrices, e.g. B. by mean value and fluctuation of the respective size. The eigenvectors and the eigenvalues ​​of the random matrices are of special interest .

Spectral theory of random matrices

Mathematicians and physicists have developed many remarkable theoretical relationships and empirical evidence on the theory of random matrices. One of the main results is the so-called Wigner'sche law (see Eugene Wigner ): It states that the spectral measure of the eigenvalues of a symmetric random matrix, in physics known as the so-called density , a characteristic semicircle distribution is sufficient. It is about N  ×  N matrices with Gaussian elements in the Limes . Wigner's law applies not only to symmetric matrices (so-called orthogonal ensemble), but with slight modifications also to unitary or symplectic matrices (so-called unitary or symplectic ensemble). ${\displaystyle N\to \infty }$

Applications

• Applications concern, among other things, the L-functions of Dirichlet and others in mathematics . There are also numerous applications in number theory , operator algebra , and so-called free probability theory .
• In physics there are applications u. a. in magnetic systems, e.g. B. in multilayer systems of magnetic thin-film systems , the quantum Hall effect , so-called quantum dots and in superconductors .
• Applications in nuclear physics relate to a. the above-mentioned Gaussian orthogonal, the unitary and the symplectic ensemble: the energy spectrum and cross-section of an atomic nucleus are extremely complex, but precisely for this reason they are accessible to the theory of so-called chaotic behavior.
• Other applications concern signal processing and wireless networks ,
• as well as the so-called quantum chaos and mesoscopic physics .
• There are also applications in so-called quantum gravity in two-dimensional systems.
• Current studies suggest that the theory of random matrices could lead to improvements in search engines on the web.

literature

• Greg Anderson, Alice Guionnet , Ofer Zeitouni: An Introduction to Random Matrices . Cambridge University Press, 2010.
• Alice Guionnet : Large Random Matrices: Lectures on Macroscopic Asymptotics . Springer Verlag, 2009.
• Persi Diaconis : Patterns in Eigenvalues: The 70th Josiah Willard Gibbs Lecture . In: American Mathematical Society. Bulletin. New Series, 2003, pp. 155-178, doi: 10.1090 / S0273-0979-03-00975-3 .
• Persi Diaconis: What Is… a Random Matrix? In: Notices of the American Mathematical Society , 2005, pp. 1348-1349, ISSN  0002-9920 .
• ML Mehta : Random matrices . 3. Edition. In: Pure and Applied Mathematics , 142. Elsevier / Academic Press, Amsterdam 2004. xviii + 688 pp.
• Guhr, Müller-Groening, Weidenmüller: Random Matrix Theories in Quantum Physics: Common Concepts . In: Physics Reports , Volume 299, 1998, pp. 189-425, arxiv : cond-mat / 9707301 .
• Alan Edelman, N. Raj Rao: Random Matrix Theory . (PDF) In: Acta Numerica , Volume 14, 2005, pp. 233-297.
• Terence Tao : Topics in Random Matrix Theory . American Mathematical Society, 2012

Web links

• Random Matrix at MathWorld
• RMTool A MATLAB based Random Matrix Calculator
• Lecture by Terence Tao
• Kriecherbauer: A short and selective introduction to the theory of random matrices (PDF), University of Bochum 2008.

References and comments

1. ↑ arxiv : math.OA / 0412545
2. ↑ arxiv : math.OA / 0501238
3. ↑ VS Rychkov, S Borlenghi, H Jaffres, A Fert, X Waintal: Spin Torque and Waviness in Magnetic Multilayers: A Bridge Between Valet-Fert Theory and Quantum Approaches . In: Phys. Rev. Lett. . 103, No. 6, August 2009, p. 066602. doi : 10.1103 / PhysRevLett.103.066602 . PMID 19792592 .
4. ↑ DJE Callaway: Random Matrices, Fractional Statistics, and the Quantum Hall Effect . In: Phys. Rev., B Condens. Matter . 43, No. 10, April 1991, pp. 8641-8643. doi : 10.1103 / PhysRevB.43.8641 . PMID 9996505 .
5. ↑ M Janssen, K Pracz: Correlated random band matrices: Localization-delocalization transitions . In: Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics . 61, No. 6 Pt A, June 2000, pp. 6278-86. doi : 10.1103 / PhysRevE.61.6278 . PMID 11088301 .
6. ↑ DM Zumbühl, JB Miller, CM Marcus, K Campman, AC Gossard: Spin-orbit Coupling, Antilocalization, and Parallel Magnetic Fields in Quantum Dots . In: Phys. Rev. Lett. . 89, No. 27, December 2002, p. 276803. doi : 10.1103 / PhysRevLett.89.276803 . PMID 12513231 .
7. ^ SR Bahcall: Random Matrix Model for Superconductors in a Magnetic Field . In: Phys. Rev. Lett. . 77, No. 26, December 1996, pp. 5276-5279. doi : 10.1103 / PhysRevLett.77.5276 . PMID 10062760 .
8. ↑ Antonia M. Tulino, Sergio Verdu: Random Matrix Theory and Wireless Communications . Now , 2004.
9. ^ D Sánchez, M Büttiker: Magnetic-field Asymmetry of Nonlinear Mesoscopic Transport . In: Phys. Rev. Lett. . 93, No. 10, September 2004, p. 106802. doi : 10.1103 / PhysRevLett.93.106802 . PMID 15447435 .
10. ↑ F Franchini, VE Kravtsov: Horizon in Random Matrix Theory, the Hawking radiation, and Flow of Cold Atoms . In: Phys. Rev. Lett. . 103, No. 16, October 2009, p. 166401. doi : 10.1103 / PhysRevLett.103.166401 . PMID 19905710 .
11. ↑ newscientist.com