Bond percolation between k separated points on a square lattice

Q factor: A measure of competition between the topper and the average in percolation and in self-organized criticality

We define the Q factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability p is increased, the Q factor for the system size L grows systematically to its maximum value Q_{max}(L) at a specific value p_{max}(L) and then gradually decays. Our numerical study of site percolation problems on the square, triangular, and simple cubic lattices exhibits that the asymptotic values of p_{max}, though close, are distinct from the corresponding percolation thresholds of these lattices. We also show, using scaling analysis, that at p_{max} the value of Q_{max}(L) diverges as L^{d} (d denoting the dimension of the lattice) as the system size approaches its asymptotic limit. We further extend this idea to nonequilibrium systems such as the sandpile model of self-organized criticality. Here the Q(ρ,L) factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches, with ρ the drop density of the driving mechanism. This study was prompted by some observations in sociophysics.

Site percolation in distorted square and simple cubic lattices with flexible number of neighbors

This paper exhibits a Monte Carlo study on site percolation using the Newmann-Ziff algorithm in distorted square and simple cubic lattices where each site is allowed to be directly linked with any other site if the Euclidean separation between the pair is at most a certain distance d, called the connection threshold. Distorted lattices are formed from regular lattices by a random but controlled dislocation of the sites with the help of a parameter α, called the distortion parameter. The distinctive feature of this study is the relaxation of the restriction of forming bonds with only the nearest neighbors. Owing to this flexibility and the intricate interplay between the two parameters α and d, the site percolation threshold may either increase or decrease with distortion. The dependence of the percolation threshold on the average degree of a site has been explored to show that the obtained results are consistent with those on percolation in regular lattices with an extended neighborhood and continuum percolation.

Percolation in a simple cubic lattice with distortion

Site percolation in a distorted simple cubic lattice is characterized numerically employing the Newman-Ziff algorithm. Distortion is administered in the lattice by systematically and randomly dislocating its sites from their regular positions. The amount of distortion is tunable by a parameter called the distortion parameter. In this model, two occupied neighboring sites are considered connected only if the distance between them is less than a predefined value called the connection threshold. It is observed that the percolation threshold always increases with distortion if the connection threshold is equal to or greater than the lattice constant of the regular lattice. On the other hand, if the connection threshold is less than the lattice constant, the percolation threshold first decreases and then increases steadily as distortion is increased. It is shown that the variation of the percolation threshold can be well explained by the change in the fraction of occupied bonds with distortion. The values of the relevant critical exponents of the transition strongly indicate that percolation in regular and distorted simple cubic lattices belong to the same universality class. It is also demonstrated that this model is intrinsically distinct from the site-bond percolation model.