Discontinuous percolation in diffusion-limited cluster aggregation

Abstract

Recently, the diffusion-limited cluster aggregation (DLCA) model was restudied as a real-world example of showing discontinuous percolation transitions (PTs). Because a larger cluster is less mobile in Brownian motion, it comes into contact with other clusters less frequently. Thus, the formation of a giant cluster is suppressed in the DLCA process. All clusters grow continuously with respect to time, but the largest cluster grows drastically with respect to the number of cluster merging events. Here, we study the discontinuous PT occurring in the DLCA model in more general dimensions such as two, three, and four dimensions. PTs are also studied for a generalized velocity, which scales with cluster size $s$ as $v_{s} \propto s^η$. For Brownian motion of hard spheres in three dimensions, the mean relative speed scales as $s^{-1/2}$ and the collision rate $σv_s$ scales as $\sim s^{1/6}$. We find numerically that the PT type changes from discontinuous to continuous as $η$ crosses over a tricritical point $η_{c} \approx 1.2$ (in two dimensions), $η_{c} \approx 0.8$ (in three dimensions), and $η_{c} \approx 0.4$ (in four dimensions). We illustrate the root of this crossover behavior from the perspective of the heterogeneity of cluster-size distribution. Finally, we study the reaction-limited cluster aggregation (RLCA) model in the Brownian process, in which cluster merging takes place with finite probability $r$. We find that the PTs in two and three dimensions are discontinuous even for small $r$ such as $r=10^{-3}$, but are continuous in four dimensions.