Percolation in a distorted square lattice

TL;DR

This study uses Monte Carlo simulations to explore how lattice distortion affects percolation, revealing that the system's universality class remains consistent with ordinary percolation despite distortions.

Contribution

It introduces a novel percolation model on distorted lattices with variable site positions and connection criteria, expanding understanding of percolation in irregular systems.

Findings

Percolation threshold increases with lattice distortion parameter.

Higher connection thresholds facilitate percolation in distorted lattices.

The critical behavior aligns with the universality class of ordinary percolation.

Abstract

This paper presents a Monte-Carlo study of percolation in a distorted square lattice, in which, the adjacent sites are not equidistant. Starting with an undistorted lattice, the position of the lattice sites are shifted through a tunable parameter $\alpha$ to create a distorted empty lattice. In this model, two neighboring sites are considered to be connected to each other in order to belong to the same cluster, if both of them are occupied as per the criterion of usual percolation and the distance between them is less than or equal to a certain value, called connection threshold $d$. While spanning becomes difficult in distorted lattices as is manifested by the increment of the percolation threshold $p_c$ with $\alpha$, an increased connection threshold $d$ makes it easier for the system to percolate. The scaling behavior of the order parameter through relevant critical exponents and the fractal dimension $d_f$ of the percolating cluster at $p_c$ indicate that this new type of percolation may belong to the same universality class as ordinary percolation. This model can be very useful in various realistic applications since it is almost impossible to find a natural system that is perfectly ordered.