Persistence in cluster-cluster aggregation

Persistence is considered in one-dimensional diffusion-limited cluster-cluster aggregation when the diffusion coefficient of a cluster depends on its size s as $D\left(s\right)\mathrm{}\sim s^{\gamma }.$ The probabilities that a site has been either empty or covered by a cluster all the time define the empty and filled site persistences. The cluster persistence gives the probability of a cluster remaining intact. The empty site and cluster persistences are universal whereas the filled site depends on the initial concentration. For $\gamma >0\mathrm{}$ the universal persistences decay algebraically with the exponent $2/\left(2-\gamma \right).\mathrm{}$ For the empty site case the exponent remains the same for $\gamma <0\mathrm{}$ but the cluster persistence shows a stretched exponential behavior as it is related to the small s behavior of the cluster size distribution. The scaling of the intervals between persistent regions demonstrates the presence of two length scales: the one related to the distances between clusters and that between the persistent regions.