Criticality of trapping in a dynamic epidemic model

We study a simple model mimicking the two-dimensional growth of a solid interface B through a liquid L in presence of particles A which are pushed by the advancing front. The model considers a short-range repulsive interaction between the particles and the advancing front (the so-called Uhlmann, Chalmers and Jackson mechanism). As particles are pushed by the advancing front, this leads to the formation of aggregates which are hindrances to the growth and which can be trapped leading to the formation of internal patterns. A transition between indefinitely growing clusters and frozen ones takes place for a critical particle fraction x(c) = 0.560 +/- 0.005 which is larger than the critical fraction of the corresponding epidemic model with static particles. At that critical threshold x, both percolating clusters and internal patterns are numerically found to be fractal with the same dimension D-f = 1.87 +/- 0.03 close to the classical percolation exponent 91/48. The correlation length exponent v is found to be v = 1.34 +/- 0.08 close to the classical percolation exponent 4/3. The criticality of the internal patterns is unexpected.