## Random matrices for wireless communications : when Wigner meets Shannon

### Merouane Debbah, Alcatel-Lucent Chair on Flexible Radio, Supélec

The asymptotic behaviour of the eigenvalues of large random matrices has been extensively studied since the fifties. One of the first related result was the work of Eugène Wigner in 1955 who remarked that the eigenvalue distribution of a standard Gaussian hermitian matrix converges to a deterministic probability distribution called the semi-circular law when the dimensions of the matrix converge to infinity. Since that time, the study of the eigenvalue distribution of random matrices has triggered numerous works, in the theoretical physics as well as probability theory communities. However, as far as communications systems are concerned, until the mid 90’s, intensive simulations were thought to be the only technique to get some insight on how communications behave with many parameters. All this changed in 1997 when large system analysis based on random matrix theory was discovered as an appropriate tool to gain intuitive insight into communication systems. In particular, the self-averaging effect of random matrices was shown to be able to capture the parameters of interest of communication schemes. Since the year 2000, the results led to very active research in many fields such as MIMO systems currently used in 4G technologies. This talk is intended to give a comprehensive overview of random matrices and their application to the analysis and design of wireless communication systems (4G and 5G systems).