Percolation of straight slots on a square grid.

ABSTRACT

This work is devoted to the emergence of a connected network of slots (cracks) on a square grid. Accordingly, extensive Monte Carlo simulations and finite-size scaling analysis have been conducted to study the site percolation of straight slots with length l measured in the number of elementary cells of the grid with the edge size L. A special focus was made on the dependence of the percolation threshold p_{C}(l,L) on the slot length l varying in the range 1≤l≤L-2 for the square grids with edge size in the range 50≤L≤1000. In this way, we found that p_{C}(l,L) strongly decreases with increase of l, whereas the variations of p_{C}(l=const,L) with the variation of ratio l/L are very small. Consequently, we acquire the functional dependencies of the critical filling factor and percolation strength on the slot length. Furthermore, we established that the slot percolation model interpolates between the site percolation on square lattice (l=1) and the continuous percolation of widthless sticks (l→∞) aligned in two orthogonal directions. In this regard, we note that the critical number of widthless sticks per unit area is larger than in the case of randomly oriented sticks. Our estimates for the critical exponents indicate that the slot percolation belongs to the same universality class as standard Bernoulli percolation.