DIRECTED SPIRAL PERCOLATION HULL ON THE SQUARE AND TRIANGULAR LATTICES

Abstract

Critical properties of hulls of directed spiral percolation clusters are studied on the square and triangular lattices in two dimensions (2D). The hull fractal dimension (dH) and some of the critical exponents associated with different moments of the hull size distribution function of the anisotropic DSP clusters are reported here. The values of dH and other critical exponents are found the same within error bars on both the lattices. The universality of the hull's critical exponents then holds true between the square and triangular lattices in 2D unlike the cluster's critical exponents which exhibit a breakdown of universality. The hull fractal dimension (dH ≈ 1.46) is also found close to 4/3 and away from 7/4, that of ordinary percolation cluster hull. A new conjecture is proposed for dH in terms of two connectivity length exponents (ν‖ and ν⊥) of the anisotropic clusters generated here. The values of dH and other critical exponents obtained here are very close to that of the spiral percolation cluster hull. The hull properties of the DSP clusters are then mostly determined by the rotational constraint and almost independent of the directional constraint present in the model.