Jamming and percolation of parallel squares in single-cluster growth model

This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size k x k squares (E-problem) or a mixture of k x k and m x m (m <= k) squares (M-problem). The larger k x k squares were assumed to be active (conductive) and the smaller m x m squares were assumed to be blocked (non-conductive). For equal size k x k squares (E-problem) the value of p(j) = 0.638 +/- 0.001 was obtained for the jamming concentration in the limit of k -> infinity. This value was noticeably larger than that previously reported for a random sequential adsorption model, p(j) = 0.564 +/- 0.002. It was observed that the value of percolation threshold p(c) (i.e., the ratio of the area of active k x k squares and the total area of k x k squares in the percolation point) increased with an increase of k. For mixture of k x k and m x m squares CM-problem), the value of p(c) noticeably increased with an increase of k at a fixed value of m and approached 1 at k >= 10m. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.