  This paper adresses the behaviour of the mutual information of correlated
MIMO Rayleigh channels when the numbers of transmit and receive antennas
converge to infinity at the same rate. Using a new and simple approach based on
Poincar\'{e}-Nash inequality and on an integration by parts formula, it is
rigorously established that the mutual information converges to a Gaussian
random variable whose mean and variance are evaluated. These results confirm
previous evaluations based on the powerful but non rigorous replica method. It
is believed that the tools that are used in this paper are simple, robust, and
of interest for the communications engineering community.
