Bond percolation in distorted simple cubic and body-centered cubic lattices

TL;DR

This work analyzes how geometric distortion, parameterized by $\alpha$, and a distance-dependent bond criterion, $\delta \le d$, influence bond percolation on distorted simple cubic and body-centered cubic lattices. Using large-scale Monte Carlo simulations with the Newman–Ziff algorithm and Binder cumulant finite-size scaling, the authors map how $p_b$ evolves with $\alpha$ and $d$, identify the thermodynamic-limit thresholds $p_b^{\infty}(\alpha,d)$, and determine critical thresholds $d_c(\alpha)$ and $\alpha_c(d)$ for global spanning. Key findings include monotonic increases of $p_b$ with distortion for $d\ge1$, nonmonotonic behavior for $d<1$, and nonmonotonic $d_c(\alpha)$ while $\alpha_c(d)$ decreases with increasing $d$. The results, robust across SC and BCC lattices, clarify the interplay between geometric disorder and connectivity in 3D crystalline networks.

Abstract

We investigate the effect of structural distortion on bond percolation in simple cubic and body-centered cubic lattices using extensive Monte Carlo simulations. Distortion is introduced through controlled random displacements of lattice sites, thereby modifying nearest-neighbor distances. Bond occupation is permitted only when the bond length is smaller than a prescribed connection threshold, directly coupling geometric disorder to connectivity. Finite-size scaling analysis is employed to determine percolation thresholds for finite systems and in the thermodynamic limit. We find that when the connection threshold exceeds the nearest-neighbor distance of the undistorted lattice, the percolation threshold increases monotonically with distortion strength, indicating a systematic suppression of spanning. In contrast, this monotonic behavior breaks down when the connection threshold is below the nearest-neighbor distance of the undistorted lattice, highlighting a nontrivial interplay between geometric distortion and connectivity. We further identify critical values of the connection threshold and the distortion amplitude required for global spanning when all the allowed bonds are occupied. All qualitative behaviors remain robust across both lattice geometries. These results clarify how geometric disorder reshapes percolation in three-dimensional crystalline networks.

Figures (5)

• Figure 1: Mechanism of distortion of simple cubic (SC) and body-centered cubic (BCC) lattices. Cubic regions around red lattice point indicate the possible regions of dislocated sites. The minimum and maximum nearest-neighbor distances of corresponding lattices are shown in the figure.
• Figure 2: (a) Variation of bond percolation threshold $(p_\mathrm{b})$ with distortion parameter $(\alpha)$ for distorted SC lattice of linear size $L=2^{7}$. Ten different curves correspond to different values of $d$. (b) Corresponding variation of the average coordination number $z_\mathrm{avg}(\alpha)$ satisfying $\delta\le d$ for DSC lattice. Curves for same values of $d$ are represented by same colors in (a) and (b). (c) Variation of bond percolation threshold $(p_\mathrm{b})$ with distortion parameter $(\alpha)$ for distorted BCC lattice of linear size $L=2^{6}$. Seven different curves correspond to different values of $d$. (d) Corresponding variation of the average coordination number $z_\mathrm{avg}(\alpha)$ satisfying $\delta\le d$ for DBCC lattice. Curves for same values of $d$ are represented by same colors in (c) and (d). Each data point of panels (a) and (c) represents an average over $1000$ independent realizations and of panels (b) and (d) represents an average over $100$ independent realizations. The data points are joined by lines as a guide to the eye. 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 4: Determination of bond percolation threshold for an infinite simple cubic lattice when $d=1.0$ and $\alpha=0.05$ by intersection of Binder cumulant curves for different lattice size. Each data point represents an average over $10^{4}$ independent realizations. The data points are joined by lines as a guide to the eye. The vertical line indicates the threshold value which is shown in Table \ref{['Tab:precise']}.
• Figure 5: (a) Variation of critical connection threshold $(d_\mathrm{c})$ with distortion parameter $(\alpha)$ for distorted SC lattice of linear size $L=2^{7}$. (b) Variation of critical connection threshold $(d_\mathrm{c})$ with distortion parameter $(\alpha)$ for distorted BCC lattice of linear size $L=2^{6}$. Each data point represents an average over $1000$ independent realizations. The data points are joined by lines as a guide to the eye.
• Figure 6: (a) Variation of critical distortion parameter ($\alpha_\mathrm{c}$) with connection threshold $(d)$ for distorted SC lattice of linear size $L=2^{7}$. (b) Variation of critical distortion parameter ($\alpha_\mathrm{c}$) with connection threshold $(d)$ for distorted BCC lattice of linear size $L=2^{6}$. Each data point represents an average over $1000$ independent realizations. The data points are joined by lines as a guide to the eye.