Directed Spiral Site Percolation on the Square Lattice

Abstract

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold $p_c\approx 0.655$ is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension $d_f\approx 1.733$. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as $p\to p_c$ with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution $P_s(p)$ show power law behavior with $|p-p_c|$ in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of $P_s(p)$. The results obtained are in good agreement with other model calculations.