Directed Spiral Percolation Hull on the Square and Triangular Lattices

TL;DR

This study investigates the critical properties and fractal dimensions of hulls in directed spiral percolation clusters on square and triangular lattices, revealing universality of hull exponents and proposing a new conjecture for hull fractal dimension.

Contribution

It provides the first detailed analysis of hull properties in DSP clusters, demonstrating universality across lattices and introducing a new conjecture relating hull fractal dimension to connectivity length exponents.

Findings

Hull fractal dimension approximately 1.46, close to 4/3.

Hull exponents are universal across square and triangular lattices.

Hull properties are mainly influenced by rotational constraints.

Abstract

Critical properties of hulls of directed spiral percolation (DSP) clusters are studied on the square and triangular lattices in two dimensions (2D). The hull fractal dimension ($d_H$) 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 $d_H$ 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 $(d_H \approx 1.46)$ is also found close to 4/3 and away from 7/4, that of ordinary percolation cluster hull. A new conjecture is proposed for the hull fractal dimension ($d_H$) in terms of two connectivity length exponents ($\nu_\|$ $&$ $\nu_\perp$) of the anisotropic clusters generated here. The values of $d_H$ 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 on the directional constraint present in the model.