  This work gives an overview of analytic tools for the design, analysis, and
modelling of communication systems which can be described by linear vector
channels such as y = Hx+z where the number of components in each vector is
large. Tools from probability theory, operator algebra, and statistical physics
are reviewed. The survey of analytical tools is complemented by examples of
applications in communications engineering. Asymptotic eigenvalue distributions
of many classes of random matrices are given. The treatment includes the
problem of moments and the introduction of the Stieltjes transform. Free
probability theory, which evolved from non-commutative operator algebras, is
explained from a probabilistic point of view in order to better fit the
engineering community. For that purpose freeness is defined without reference
to non-commutative algebras. The treatment includes additive and multiplicative
free convolution, the R-transform, the S-transform, and the free central limit
theorem. The replica method developed in statistical physics for the purpose of
analyzing spin glasses is reviewed from the viewpoint of its applications in
communications engineering. Correspondences between free energy and mutual
information as well as energy functions and detector metrics are established.
These analytic tools are applied to the design and the analysis of linear
multiuser detectors, the modelling of scattering in communication channels with
dual antennas arrays, and the analysis of optimal detection for communication
via code-division multiple-access and/or dual antenna array channels.
