Site and bond percolation in linearly distorted triangular and square lattices

TL;DR

This study explores how linear distortion affects site and bond percolation thresholds in triangular and square lattices, revealing anisotropic effects in triangular lattices and isotropic behavior in square lattices through extensive simulations.

Contribution

It introduces a novel analysis of linear distortion effects on percolation, highlighting directional dependence in triangular lattices and isotropy in square lattices, using Monte Carlo simulations.

Findings

Triangular lattices show directional dependence in percolation thresholds.

Square lattices remain isotropic under linear distortion.

Percolation thresholds converge reliably in large finite lattices.

Abstract

We investigate site and bond percolation in triangular and square lattices subjected to linear distortion. In contrast to previously studied distortion schemes that preserve lattice geometry, linear distortion dislocates regular lattice sites along a fixed direction. Nearest-neighbors of a regular lattice need to satisfy a distance-based connection criterion to remain neighbors in the linearly distorted lattice. Using extensive Monte Carlo simulations and finite-size scaling analyses, we examine how site and bond percolation thresholds vary with the distortion parameter and the connection threshold. For triangular lattices, we observe pronounced directional dependence of both site and bond percolation thresholds, as well as of the critical connection threshold. This arises from the distortion-induced anisotropic modification of nearest-neighbor separations. In particular, bond percolation exhibits nontrivial behavior that cannot be explained solely in terms of changes in the average coordination number. In contrast, square lattices remain effectively isotropic under linear distortion, resulting in identical percolation thresholds for distortions applied along different directions. Percolation thresholds in the thermodynamic limit, evaluated for a selected set of values of distortion parameter and connection threshold, confirm that the results for large finite lattices provide reliable estimates of the infinite-system behavior.