Bond percolation in distorted square and triangular lattices

TL;DR

This paper addresses how geometric disorder from random site distortions, controlled by $α$, and bond-length constraints via the connection threshold $d$ affect bond percolation in distorted square and triangular lattices. Using Monte Carlo simulations and the Newman–Ziff method, it computes $p_b(α,d)$ and identifies precise thresholds via Binder cumulants, demonstrating that the system remains in the same universality class as regular 2D percolation with $ν=4/3$, $γ=43/18$, and $β=5/36$. A new quantity, the critical connection threshold $d_c(α)$, captures the minimum bond length required for spanning and displays lattice-dependent evolution: monotonic growth in the square lattice and a non-monotonic behavior in the triangular lattice. The work highlights how distortion alters local connectivity ($z_{avg}$) and global spanning, with potential implications for disordered materials, forest-fire models, and porous media, while providing precise, universally consistent thresholds across lattice types.

Abstract

This article presents a Monte Carlo study on bond percolation in distorted square and triangular lattices. The distorted lattices are generated by dislocating the sites from their regular positions. The amount and direction of the dislocations are random, but can be tuned by the distortion parameter $α$. Once the sites are dislocated, the bond lengths $δ$ between the nearest neighbors change. A bond can only be occupied if its bond length is less than a threshold value called the connection threshold $d$. It is observed that when the connection threshold is greater than the lattice constant (assumed to be $1$), the bond percolation threshold $p_\mathrm{b}$ always increases with distortion. For $d\le 1$, no spanning configuration is found for the square lattice when the lattice is distorted, even very slightly. On the other hand, the triangular lattice not only spans for $d\le 1$, it also shows a decreasing trend for $p_\mathrm{b}$ in the low-$α$ range. These variation patterns have been linked with the average coordination numbers of the distorted lattices. A critical value $d_\mathrm{c}$ for the connection threshold has been defined as the value of $d$ below which no spanning configuration can be found even after occupying all the bonds satisfying the connection criterion $δ\le d$. The behavior of $d_\mathrm{c}(α)$ is markedly different for the two lattices.

Figures (9)

• Figure 1: Mechanism of distortion in (a) square and (b) triangular lattices. Green squares and triangles indicate possible regions of the dislocated sites.
• Figure 2: (a) Variation of bond percolation threshold with distortion parameter for distorted square lattice of linear size $L=2^{10}$. Six different curves correspond to different values of $d$. (b) Corresponding variation of the average coordination number $z_\mathrm{avg}(\alpha)$ satisfying $\delta\le d$. Curves for same values of $d$ are represented by same colors in (a) and (b). The black curve for $d=1.0$ shows a sudden jump from $4$ to $2$ for any small $\alpha$. Error bars of the obtained data points are of order $10^{-5}$ and are hidden by the plot markers.
• Figure 3: (a) Variation of bond percolation threshold with connection threshold for distorted square lattice of length $L=2^{10}$. Different curves correspond to different values of $\alpha$. Error bars of the obtained data points are of order $10^{-5}$ and are hidden by the plot markers. (b) Variation of bond percolation threshold with distortion and connection threshold for a distorted square lattice. Magnitude of $p_\mathrm{b}(\alpha , d)$ is represented by color variation. White portions indicate regions of no spanning. The dashed line indicates $\delta_\mathrm{Max}^\mathrm{Sq}(\alpha)$ from Eq. \ref{['eq:minmax']}.
• Figure 4: (a) Variation of bond percolation threshold with distortion parameter for distorted triangular lattice of length $L=2^{10}$. Nine different curves correspond to different values of $d$. (b) Corresponding variation of the average coordination number $z_\mathrm{avg}(\alpha)$ satisfying $\delta\le d$. Curves for same values of $d$ are represented by same colors in (a) and (b). Error bars of the obtained data points are of order $10^{-5}$ and are hidden by the plot markers.
• Figure 5: (a) Variation of bond percolation threshold with connection threshold for distorted triangular lattice of length $L=2^{10}$. Error bars of the obtained data points are of order $10^{-5}$ and are hidden by the plot markers. (b) Variation of bond percolation threshold with distortion and connection threshold for a distorted triangular lattice. Magnitude of $p_\mathrm{b}(\alpha , d)$ is represented by color variation. White portions indicate regions of no spanning. The dashed line indicates $\delta_\mathrm{Max}^\mathrm{Tr}(\alpha)$ from Eq. \ref{['eq:minmax']}.