Lecture Notes on Free Probability Theory

Overview & Synopsis

to free probability theory, covering the basic combinatorial and analytic theory, as well as the relations to random matrices and operator algebras. This note contains below topics, Very Short Introduction into Subject and History, The Notion of Freeness Definition Example and Basic Properties, Emergence of the Combinatorics of Free Probability Theory Free Central Limit Theorem, The Combinatorics of Free Probability Theory Free Cumulants, Free Convolution of Compactly Supported Probability Measures and the RTransform, Free Convolution of Arbitrary Probability Measures and the Subordination Function, Gaussian Random Matrices and Asymptotic Freeness, Unitary Random Matrices and Asymptotic Freeness and a Remark on Wigner Matrices, Von Neumann Algebras The Free Group Factors and Relation to Freeness, Circular and RDiagonal Operators, Applications of Freeness to von Neumann Algebras Compression of Free Group Factors, Some More Operator Algebraic Applications of Free Probability.