  We study statistical properties of the Kolmogorov-Avrami-Johnson-Mehl
nucleation-and-growth model in one dimension. We obtain exact results for the
gap density as well as the island distribution. When all nucleation events
occur simultaneously, the island distribution has discontinuous derivatives on
the rays x_n(t)=nt, n=1,2,3... We introduce an accelerated growth mechanism
where the velocity increases linearly with the island size. We solve for the
inter-island gap density and show that the system reaches complete coverage in
a finite time and that the near-critical behavior of the system is robust,
i.e., it is insensitive to details such as the nucleation mechanism.
