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Introduction
Random matrix theory (RMT) is the study of matrices whose entries are random variables.[1]
The main goal of the Random Matrix Theory is to provide understanding of the diverse properties of matrices with entries drawn randomly from various probability distributions, traditionally referred to as the random matrix ensembles, and explore their applications.[1]
History of RMT
en.wikipedia .org
Although origins of RMT could be traced back to works by John Wishart and in the 1928 in the field of Statistics the real start of the field is usually attributed to highly influential papers by Eugene Wigner in 1950's motivated by applications in Nuclear physics[3].
What is Random matrix theory?
In probability theory a random matrix is a matrix whose all element are random variables. With the help of this technique many important properties of physical system can be modeled mathematically as a matrix.[2]
A general random matrix can be written as:
Based on symmetry properties one can introduce certain classes of random matrices-
The three types of classical Gaussian random matrix ensembles [1]
a) Gaussian Unitary Ensemble (GUE)
b) Gaussian orthogonal ensemble(GOE)
c) Gaussian Symplectic Ensemble (GSE).
Applications of RMT
In Nuclear Physics
Random matrix theory was introduced by Wigner in order to describe the statistics of energy level spectra in nuclei.[1]
In Multiple input multiple output (MIMO) Communications
RMT is used to characterized the fundamental limit of various channel in the wireless communication.[7]
In Finance
Financial covariance matrix are random so, RMT is used to find relationship between in sample risk and out sample risk . In this case random matrix is in form of covariance matrix.[3]
RMT has also widely used in solid state physics, mesoscopic physics, sensor array, numerical analysis and many more.
Conclusion
Random matrix theory can be both a valuable tool and source of insight into problems of highly chaotic or disorder system. While sensitive to the scales within a given problem, it is perhaps the best tool for understanding physics, where other tools fail precisely because of their dependence on structure [6].
In the context of disordered systems the method of free probability is quite powerful in calculating the density of states of large dimension matrices, that otherwise becomes difficult to deal with other methods.