Gap size and capture zone distributions in one-dimensional point island nucleation and growth simulations: asymptotics and models

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.