Free Deterministic Equivalents, Rectangular Random Matrix Models, and Operator-Valued Free Probability Theory

Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators satisfying certain freeness relations. We comment on the relation between our free deterministic equivalent and deterministic equivalents considered in the engineering literature. We do not only consider the case of square matrices, but also show how rectangular matrices can be treated. Furthermore, we emphasize how operator-valued free probability techniques can be used to solve our free deterministic equivalents. As an illustration of our methods we consider a random matrix model studied first by R. Couillet, J. Hoydis, and M. Debbah. We show how its free deterministic equivalent can be treated and we thus recover in a conceptual way their result. On a technical level, we generalize a result from scalar valued free probability, by showing that randomly rotated deterministic matrices of different sizes are asymptotically free from deterministic rectangular matrices, with amalgamation over a certain algebra of projections. In the Appendix, we show how estimates for differences between Cauchy transforms can be extended from a neighborhood of infinity to a region close to the real axis. This is of some relevance if one wants to compare the original random matrix problem with its free deterministic equivalent.