[論文レビュー] Bond percolation in distorted simple cubic and body-centered cubic lattices

Bishnu Bhowmik, Sayantan Mitra+2 more|arXiv (Cornell University)|Feb 16, 2026

Theoretical and Computational Physics被引用数 0

ひとこと要約

The paper studies how geometric distortion and distance-based bond occupation affect bond percolation thresholds in distorted SC and BCC lattices using extensive Monte Carlo simulations and finite-size scaling.

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.

研究の動機と目的

• Understand how random site distortions modify nearest-neighbor distances in SC and BCC lattices.
• Investigate bond percolation thresholds under a distance-dependent occupation rule with a connection threshold d.
• Analyze how distortion α and threshold d jointly influence spanning connectivity in finite and infinite lattices.
• Determine critical parameters (d_c, α_c) for global spanning when all allowed bonds are occupied.

提案手法

• Introduce controlled random displacements of lattice sites within a cube of side 2α to distort SC and BCC lattices.
• Define bond occupation eligibility by a distance criterion δ ≤ d, where δ is the bond length after distortion.
• Use Monte Carlo simulations on L-sized lattices (DSC: L=2^7, DBCC: L=2^6) with 1000 realizations per point to estimate p_b(α,d).
• Compute cluster properties and spanning status using the Newman–Ziff algorithm.
• Estimate p_b(α,d) for finite systems and extrapolate to the thermodynamic limit via Binder cumulant intersections over lattice sizes.
• Identify d_c and α_c by progressively adjusting d or α and checking for spanning, averaging over realizations.

実験結果

リサーチクエスチョン

• RQ1 How does a fixed distortion α affect the bond percolation threshold p_b(α,d) for various connection thresholds d?
• RQ2 How does the connection threshold d influence p_b(α,d) at fixed α, and how do these effects differ between DSC and DBCC lattices?
• RQ3 What are the critical values d_c(α) and α_c(d) required for spanning when all allowed bonds are occupied?
• RQ4 How robust are the observed trends with respect to lattice size and finite-size effects?
• RQ5 What is the qualitative mechanism linking average coordination number and percolation in distorted lattices?

主な発見

• For d above the undistorted nearest-neighbor distance, p_b(α,d) increases monotonically with distortion α for both lattices.
• For d below the undistorted distance, p_b(α,d) varies non-monotonically with α, first decreasing and then increasing as distortion grows.
• When d equals the undistorted distance, a sharp rise in p_b occurs with any infinitesimal distortion, followed by a linear-like increase.
• The average coordination number z_avg(α) tracks the behavior of p_b(α,d), decreasing with α for d>1 and showing nontrivial behavior for d<1.
• Binder cumulant analysis yields thermodynamic-limit p_b^∞(α,d) values consistent with finite-size trends and confirms small finite-size effects.
• There exist nontrivial, nonmonotonic dependencies of the critical connection threshold d_c(α) and the critical distortion α_c(d), with d_c showing a minimum at intermediate α for both lattices and α_c decreasing with increasing d.