Space-filling Percolation

A region of two-dimensional space has been filled randomly with large number of growing circular discs allowing only a slight' overlapping among them just before their growth stop. More specifically, each disc grows from a nucleation center that is selected at a random location within the uncovered region. The growth rate $\delta$ is a continuously tunable parameter of the problem which assumes a specific value while a particular pattern of discs is generated. When a growing disc overlaps for the first time with at least another disc, it's growth is stopped and is said to be frozen’. In this paper we study the percolation properties of the set of frozen discs. Using numerical simulations we present evidence for the following: (i) The Order Parameter appears to jump discontinuously at a certain critical value of the area coverage; (ii) the width of the window of the area coverage needed to observe a macroscopic jump in the Order Parameter tends to vanish as $\delta \to 0$ and on the contrary (iii) the cluster size distribution has a power law decaying functional form. While the first two results are the signatures of a discontinuous transition, the third result is indicative of a continuous transition. Therefore we refer this transition as a sharp but continuous transition similar to what has been observed in the recently introduced Achlioptas process of Explosive Percolation. It is also observed that in the limit of $\delta \to 0$, the critical area coverage at the transition point tends to unity, implying the limiting pattern is space-filling. In this limit, the fractal dimension of the pore space at the percolation point has been estimated to be $1.42(10)$ and the contact network of the disc assembly is found to be a scale-free network.