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

Abstract

A random growth lattice filling model of percolation with a touch and stop growth rule is developed and studied numerically on a two dimensional square lattice. Nucleation centers are continuously added one at a time to the empty lattice sites and clusters are grown from these nucleation centers with a growth probability g. For a given g (), the system passes through a critical point during the growth process where the transition from a disconnected to a connected phase occurs. The model is found to exhibit second order continuous percolation transitions as ordinary percolation for whereas for it exhibits weak first order discontinuous percolation transitions. The continuous transitions are characterized by estimating the values of the critical exponents associated with the order parameter fluctuation and the fractal dimension of the spanning cluster over the whole range of g. The discontinuous transitions, however, are characterized by a compact spanning cluster, lattice size independent fluctuation of the order parameter per lattice, departure from power law scaling in the cluster size distribution and weak bimodal distribution of the order parameter. The nature of transitions are further confirmed by studying the Binder cumulant. Instead of a sharp tricritical point, a tricritical region is found to occur for 0.5 < g < 0.8 within which the values of the critical exponents change continuously until the crossover from continuous to discontinuous transition is completed.