Site and bond percolation in linearly distorted triangular and square lattices

TL;DR

This work analyzes how linear distortion of lattice positions along a fixed axis alters site and bond percolation on triangular and square lattices. Using Monte Carlo simulations with finite-size scaling and Binder-cumulant analysis, it reveals pronounced anisotropy in the triangular lattice—affecting thresholds p_c, p_b, and the critical connection distance d_c—while the square lattice remains effectively isotropic. The results show that geometric distortion shifts connectivity beyond simple changes in average coordination, with site and bond thresholds exhibiting distinct directional trends, and that thermodynamic-limit estimates closely match large finite-size data. Overall, the study highlights how linear geometric constraints fundamentally modify percolation behavior in planar lattices and provides robust benchmarks for distorted lattice models.

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.

Figures (13)

• Figure 1: Mechanism of linear distortion in a triangular lattice: (a) distortion along the $x$ direction and (b) distortion along the $y$ direction. The corresponding maximum and minimum nearest-neighbor distances are shown.
• Figure 2: Mechanism of linear distortion in a square lattice: (a) distortion along the $x$ direction and (b) distortion along the $y$ direction. The corresponding maximum and minimum nearest-neighbor distances are shown.
• Figure 3: Variation of (a) $p_\mathrm{c}^{(\triangle)}$ and (b) $p_\mathrm{b}^{(\triangle)}$ for a triangular lattice of size $L=2048$ when linear distortion is applied along the $x$ and $y$ directions. The connection threshold is fixed at $d=1.1$. The obtained data points (not shown) are close enough so that the curves appear smooth. In panel (a), the four curves correspond to the site-percolation thresholds $p_\mathrm{c,xx}^{(\triangle)}(\alpha)$ (solid red), $p_\mathrm{c,xy}^{(\triangle)}(\alpha)$ (broken black), $p_\mathrm{c,yx}^{(\triangle)}(\alpha)$ (solid magenta), and $p_\mathrm{c,yy}^{(\triangle)}(\alpha)$ (broken blue). The corresponding four curves in panel (b) represent the bond-percolation thresholds $p_\mathrm{b,xx}^{(\triangle)}(\alpha)$, $p_\mathrm{b,xy}^{(\triangle)}(\alpha)$, $p_\mathrm{b,yx}^{(\triangle)}(\alpha)$, and $p_\mathrm{b,yy}^{(\triangle)}(\alpha)$, shown with the same line styles and colors, as defined in the text.
• Figure 4: Variation of (a) $p_\mathrm{c,x}^{(\triangle)}$ and (b) $p_\mathrm{b,x}^{(\triangle)}$ for a triangular lattice of size $L=2048$. Each data point represents an average over $1000$ independent realizations. The associated error bars are of the order of $10^{-5}$ and are therefore hidden by the plot markers. The data points are joined by lines as a guide to the eye. Each curve corresponds to a distinct value of $d$ and is indicated by a different color and plot marker. The corresponding percolation thresholds in the thermodynamic limit for some selected values of $d$ and $\alpha$ are shown by the symbol $\odot$ in the same colors.
• Figure 5: Plots of the average coordination number $z_\mathrm{avg,x}^{(\triangle)}$ for the same set of values of $d$ as in Fig. \ref{['fig:tri_x']}. Curves corresponding to identical values of $d$ in the two figures are indicated using the same color and plot symbol. Each data point represents an average over $100$ independent realizations. The associated error bars are of the order of $10^{-5}$ and are therefore hidden by the plot markers. The data points are joined by lines as a guide to the eye.