Finite size scaling study of a two parameter percolation model: Constant and correlated growth

A new percolation model of enhanced parameter space with nucleation and growth is developed taking the initial seed concentration ρ and a growth parameter g as two tunable parameters. Percolation transition is determined by the final static configurations of spanning clusters once taking uniform growth probability for all the clusters and then taking a cluster size dependent dynamic growth probability. The uniform growth probability remains constant over time and leads to a constant growth model whereas the dynamically varying growth probability leads to a correlated growth model. In the first case, the growth of a cluster will encounter partial hindrance due to the presence of other clusters whereas in the second case the growth of a larger cluster will be further suppressed in comparison to the growth of smaller clusters. A finite size scaling theory for percolation transition is developed and numerically verified for both the models. The scaling functions are found to depend on both g and ρ. At the critical growth parameter gc, the values of the critical exponents are found to be same as that of the original percolation at all values of ρ for the constant growth model whereas in the case of correlated growth model the scaling behavior deviates from ordinary percolation in the dilute limit of ρ. The constant growth model then belongs to the same universality class of percolation for a wide range of ρ whereas the correlated growth model displays a continuously varying universality class as ρ decreases towards zero.