  The nucleation and growth of point islands during submonolayer deposition on
a one-dimensional substrate is simulated for critical island size $i=0,1,2,3$.
The small and large size asymptotics for the gap size and capture zone
distributions (GSD and CZD) are studied. Comparisons to theoretical predictions
from fragmentation equation analyses are made, along with those from the
recently proposed Generalised Wigner Surmise (GWS). We find that the simulation
data can be fully understood in the framework provided by the fragmentation
equations, whilst highlighting the theoretical areas that require further
development. The GWS works well for the small-size CZD behaviour, but
completely fails to describe the large-size CZD asymptotics of the
one-dimensional system.
