20 May 2013
Free probability theory has been introduced by Voiculescu in the 80's for the study of the von Neumann algebras of the free groups. It consists in an algebraic setting of non commutative probability, where one encodes "non commutative random variables" in abstract (non commutative) algebras endowed with linear forms (which satisfies properties in order to play the role of the expectation). In this context, Voiculescu introduce the notion of freeness which is the analogue of the classical independence. A decade later, he realized that a family of independent random matrices invariant in law by conjugation by unitary matrices are asymptotically free. This phenomenon is called asymptotic freeness. It had a deep impact in operator algebra and probability and has been generalized in many directions. A simple particular case of Voiculescu's theorem gives an estimate, for N large, of the spectrum of an N by N Hermitian matrix H_N = A_N + 1/\sqrt N X_N, where A_N is a given deterministic Hermitian matrix and X_N has independent gaussian standard sub-diagonal entries. Nevertheless, it turns out that asymptotic freeness does not hold in certain situations, e.g. when the entries of X_N as above have heavy-tails. To infer the spectrum of a larger class of matrices, we go further into Voiculescu's approach and introduce the distributions of traffics and their free product. This notion of distribution is richer than Voiculescu's notion of distribution of non commutative random variables and it generalizes the notion of law of a random graph. The notion of freeness for traffics is an intriguing mixing between the classical independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem in that context for independent random matrices invariant in law by conjugation by permutation matrices. The purpose of this talk is to give an introductory presentation of these notions.
- Stochastic Analysis Seminar