Random matrices, free probability and the replica method

This survey gives an overview of analytic tools to 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. Asymptotic eigenvalue distributions of some classes of random matrices are given in terms of densities, moments and/or Stieltjes transforms. 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, and the S-transform. The replica method developed in statistical physics for the purpose of analyzing spin glasses is reviewed from the view point of its applications in communications engineering. Correspondences between free energy and mutual information as well as energy functions and detector metrics are established.