Abstract
A random cluster growth model is developed in terms of two parameters, the initial seed concen- tration ρ and the growth probability g of individual clusters. The model is studied on a two dimensional square lattice. For every ρ value, a critical value of g = gc is determined at which a percolation transition is observed. A scaling theory for this model is developed and numerically verified. The scaling functions are found to scale with ρ, g as well as the system size L with appropriate critical exponents. The values of the critical exponents are found to belong to the same universality class of percolation. Finally a phase dia- gram is developed in the ρ − g parameter space for percolating and non-percolating regions separated by a line of second order phase transition points. (doi: 10.5562/cca2313)
Highlights
Percolation is a model of disordered systems and has extensive applications in different branches of science
Percolation refers to the formation of long-range connectedness in a system and is known to exhibit a continuous phase transition from a disconnected to a fully connected phase at a sharply defined percolation threshold value.[9]
The interest in studying percolation problem in recent time has been triggered by the introduction of a controversial phenomenon called “explosive percolation” (EP).[10]
Summary
Percolation is a model of disordered systems and has extensive applications in different branches of science. In the present problem one starts with an initial seed concentration ρ and the empty sites around the clusters formed by the initial seeds are grown with probability g, the area fraction p at the end of the growth process is expected to be p ρ g 1 ρ (2). The scaling form of the cluster size distribution and that of all other related geometrical quantities can be obtained in terms of g and ρ. The scaling form of different geometrical quantities in terms of ρ and g can be derived from the above cluster size distribution ns(ρ, g) in terms of ρ and g as per their definitions in terms of the distribution function. Model are performed for several values of ρ varying the growth probability g on square lattices of size L.
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