Growth model induced by a random walk of particles

Abstract

In this paper, we present the numerical study of a model of the growth of geometrical structures. Particles are randomly placed on a two-dimensional (2D) square lattice and move as random walkers which annihilate when encountering an occupied site. The model is studied for two cases. In case A, we study the critical properties as a function of the initial particle concentration, CI ,a fter the annihilation of all particles. We have found a critical behaviour characterized by the emergence of a percolative cluster for C ∗ = 0.098 82 ± 2 × 10 −4 .I ncase B, we do a kinetic study of the systems where the fraction of occupied sites, q ,i s am easure of time. For C ∗ I we obtain q ∗ = 0.4679 ± 0.0005. This kinetic study is also done for CI = 0.2, 0.3, 0.45 and 0.5927. The critical exponent ν and the exponent ratios β ν and γ are measured for all cases. We compare the results obtained with the known 2D percolation values. The results obtained suggest the existence, for an infinite system, of a critical line in the phase diagram CI − q (CI is the particle concentration, q is the fraction of occupied sites), q ∼ (CI ) 1−d � f /2 (where d � f = 1.74), connecting the points (C