### Talk by Todd Kemp (UCSD)

Title: Chaos and the Fourth Moment

Abstract:
In 2005, Nualart and Peccati proved a very surprising
central limit theorem for multiple Wiener-Itô integrals: if X_k
is a (variance-normalized) sequence of nth Wiener integrals (for
fixed n ≥ 2), then X_k --> N(0,1) in distribution if and only if E(X_k^4) --> 3.
That is: in a fixed order of Wiener chaos, convergence to a normal
distribution is equivalent to the a priori dramatically weaker
*convergence of the fourth moment alone*.

In this talk, I will discuss extensions of this theorem to
the world of random matrices. Matrix chaos is built from Wiener
integrals with respect to matrix-valued Brownian motion. In the
limit as matrix size goes to infinity, a coherent theory emerges
describing a chaos of limiting eigenvalue distributions. In this
context, Nualart and Peccati's theorem has an exact analogue, but the
condition for convergence is E(X_k^4) --> 2 (the fourth moment of
the semicircle law). The tools of Malliavin calculus are used to
give quantitative estimates on the distance to the limit
distribution.

This is joint work with Ivan Nourdin, Giovanni Peccati, and
Roland Speicher.