In this talk I would like to share with you what I have learned about random matrix theory(RMT) these days. The contents are mainly divided into two parts, one is the concept of Gaussian random matrix ensemble in mathematics, and another is its application in physics.

Specifically, I will introduce the concept of RMT and discuss detailedly three types of random matrix ensembles, namely, Gaussian orthogonal ensemble (GOE), Gaussian symplectic ensemble (GSE) and Gaussian unitary ensemble (GUE), which are distinguished by imposed symmetry. It is noteworthy that these three ensembles are closely related to the famed “standard class” pertaining to “ten fold way”. From physical considerations, the statistical properties of energy levels in these matrix ensembles will be explained, such as joint probability density function (j.p.d.f) for characteristic values of these matrices, and level density. It shows that adjacent energy levels mutually repulse each other in the presence of symmetry, which is different from Poisson distribution.

Reference:

- Dyson F J. Statistical theory of the energy levels of complex systems. I [J]. Journal of Mathematical Physics, 1962, 3(1): 140-156.
- Wigner E P. Random matrices in physics [J]. SIAM review, 1967, 9(1): 1-23.
- Mehta M L. Random matrices [M]. Elsevier, 2004.