Deterministic equivalents, polynomials in free variables, and analytic theory of operator-valued convolution

Abstract

The notion of a “deterministic equivalent” for random matrices, which can be found in the engineering literature, is a non-rigorous concept which amounts to replacing a random matrix model of finite size (which is usually unsolvable) by another problem which is solvable, in such a way that, for large N, the distributions of both problems are close to each other. Motivated by our example in the last chapter, we will in this chapter propose a rigorous definition for this concept, which relies on asymptotic freeness results. This “free deterministic equivalent” was introduced by Speicher and Vargas in [166].