Speaker: **Camille Male** (Univ. Paris VII)

Title: The spectrum of large random matrices, the non commutative random variables and the distribution of traffics

Time/Date: 4:30-6:00pm, Wednesday, November 20, 2013

Room: 118 Math. Sci. Building

Abstract: The purpose of this talk is to give an introductory presentation of a notion of distributions, which generalizes the notion of law of complex random variables, and is used to understand the spectrum of large random matrices.

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. This has yield 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. For instance, in the problem of computing the
asymptotic spectrum of H_{N} = A_{N} +
1/\sqrt N
X_{N} as above when the entries of X_{N}
can grow with N, one need more information on A_{N} that its
non commutative distribution. To answer this question, we mimic
Voiculescu's approach introducing 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. The notion
of freeness for traffics is an intriguing mixing between the classical
independence and Voiculescu's notion of freeness. Thanks to these concepts,
we prove an asymptotic freeness theorem for independent random matrices
invariant in law by conjugation by permutation matrices.

These notions are very related to the notion of convergence of large graphs introduced by Benjamini and Schramm. The free product of traffics reduces in this context to a natural notion of free product of random graphs and random groups.