We discuss the continuous time percolation model in an ergodically defined
environment. Under minimal assumptions on the ergodic system, we show the
existence of sets of sampling functions with percolation or extinction
showing that the latter are dense open. We also discuss the related
spatially inhomogeneous continuous time random cluster model and the
topological properties of sets of sampling functions corresponding to
percolation and decay.

The Bethe strip is essentially the cartesian product of the Bethe lattice
with a finite set. Using supersymmetric integrals we show absolutely
continuous spectrum for random Schr\"odinger Operators with small random
matrix potential. The proof extends Abel Klein's original proof for a.c.
spectrum on the Bethe lattice. The considered models include the Anderson
moodel on the product of a finite graph with the Bethe lattice and the
Wegner m-orbital model on the Bethe lattice for a fixed number of
orbitals.

In the first part of the talk I will introduce the Becker-Doring
nucleation equation and describe its singular perturbation solution under
time-dependent conditions of a nucleation pulse. In the second part, I
will discuss a supersaturated lattice gas on a square lattice, where
steady-state and time-dependent nucleation can be described from first
principles. Comparison confirms qualitative (not quantitative) validity of
the Becker-Doring model at not too small temperatures T , but also reveals
its limitations due to neglect of "magic numbers", which become prominent
as T -> 0 .