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Deterministic Equivalents, Polynomials in Free Variables, and Analytic Theory of Operator-Valued Convolution

Chapter
Part of the Fields Institute Monographs book series (FIM, volume 35)

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].

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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada
  2. 2.FB MathematikUniversität des SaarlandesSaarbrückenGermany

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