The Gaussian central limit theorem says that for a wide class of stochastic systems, the bell curve (Gaussian distribution) describes the statistics for random fluctuations of important observables. In this talk I will look beyond this class of systems to a collection of probabilistic models which include random growth models, polymers, particle systems, matrices and stochastic PDEs, as well as certain asymptotic problems in combinatorics and representation theory. I will explain in what ways these different examples all fall into a single new universality class with a much richer mathematical structure than that of the Gaussian.

The KPZ equation is a stochastic PDE describing the motion of an interface
between a stable and an unstable phase. We will discuss solutions of the one-dimensional
equation with sharp wedge initial conditions. For long times the Tracy-Widom distribution
of GUE random matrices is recovered.
The talk is based on recent joint work with Tomohiro Sasamoto.

The limit shape of Young diagrams under the Plancherel
measure was found by Vershik \& Kerov (1977) and Logan \& Shepp
(1977). We obtain a central limit theorem for fluctuations of Young
diagrams in the bulk of the partition '`spectrum''. More
specifically, under a suitable (logarithmic) normalization, the
corresponding random process converges (in the FDD sense) to a
Gaussian process with independent values. We also discuss a link
with an earlier result by Kerov (1993) on the convergence to a
generalized Gaussian process. The proof is based on poissonization
of the Plancherel measure and an application of a general central
limit theorem for determinantal point processes. (Joint work with
Zhonggen Su.)

Stochastic averaging goes back to Khasminskii in the 1960's. The
standard result is that, given a separation of scales, one can find effective dynamics
for slow components. We investigate the motion of two particles in such a system, in
particular in a randomly-perturbed twist map. The nub of the issue
is how two points escape from a 1-1 resonance zone. Results of Pinsky
and Wihstutz indicate that there is a third scale at work, which we can use to study
the escape from resonance.

We study the stochastic heat equation ∂tu = u+σ(u)w in (1+1) dimensions, where w is space-time white noise, σ:R→R is Lipschitz continuous, and is the generator of a Lvy process. We assume that the underlying Lvy process has finite exponential moments in a neighborhood of the origin and u0 has exponential decay at ∞. Then we prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.
This is joint work with Daniel Conus (University of Utah)