Anomalous Dimension and Spatial Correlations in a Point-Island Model

We examine the island size distribution function and spatial correlation function of a model for island growth in the submonolayer regime in both 1 and 2 dimensions. In our model the islands do not grow in shape, and a fixed number of adatoms are added, nucleate, and are trapped at islands as they diffuse. We study the cases of various critical island sizes $i$ for nucleation as a function of initial coverage. We found anomalous scaling of the island size distribution for large $i$ . Using scaling, random walk theory, a version of mean-field theory we obtain a closed form for the spatial correlation function. Our analytic results are verified by Monte Carlo simulations.