Monte Carlo calculation of the size distribution of supported clusters

In an effort to understand the growth of clusters from metal vapor deposited on amorphous carbon substrates, we have studied the random walk of ``atoms'' among randomly distributed sinks. We find that the number of atoms accumulated at a sink has a power-law relationship with the area associated with that sink. The value of the exponent depends only weakly on the density of sinks for sinks of fixed size, but strongly on the total coverage when the sinks (i.e., clusters) are allowed to grow. For cluster growth in three dimensions, the number of atoms accumulated likewise has a power-law relationship with the volume associated with the sink, but the exponent depends strongly on the density of sinks. For both two and three dimensions, we find that the exponent is determined by a single parameter related to the mean number of cells visited in each random walk.