Universality class of the two-dimensional randomly distributed growing-cluster percolation model

Abstract

We consider the "Touch and Stop" cluster growth percolation (CGP) model on the two-dimensional square lattice. A key parameter in the model is the fraction p of occupied "seed" sites that act as nucleation centers from which a particular cluster growth procedure is started. Here, we consider two growth styles: rhombic and disk-shaped cluster growth. For intermediate values of p the final state, attained by the growth procedure, exhibits a cluster of occupied sites that span the entire lattice. Using numerical simulations we investigate the percolation probability and the order parameter and perform a finite-size scaling analysis for lattices of side length up to L=1024 in order to carefully determine the critical exponents that govern the respective transition. In contrast to previous studies, reported in Tsakiris et al. [Phys. Rev. E 82, 041108 (2010)], we find strong numerical evidence that the CGP model is in the standard percolation universality class.

Random growth lattice filling model of percolation: a crossover from continuous to discontinuous transition

“…initial seed concentration [39], but for intermediate seed concentrations the transitions are found to be continuous which belong to OP universality class [40,47].…”

“…The cumulants when plotted against the area fraction for dierent system sizes L are expected to cross each other at a definite p(t) corresponding to the critical threshold of the system for a continuous transition, whereas no such crossing is expected to occur in the case of a discontinuous transition [20]. Though the cumulant has some unusual behavior [64,65], it is rarely used in the study of recent models of percolation except in a few references [47,66]. The values of B L (t) are plotted against p(t) for dierent system sizes L in figure 11(a) for g = 0.1 and in figure 11(b) for g = 0.8.…”