Standard and inverse site percolation of straight rigid rods on triangular lattices: Isotropic and nematic deposition/removal

TL;DR

This study uses numerical simulations to analyze standard and inverse percolation of straight rigid rods on triangular lattices, revealing symmetric threshold behaviors and convergence patterns for various rod lengths under isotropic and nematic schemes.

Contribution

It introduces a detailed analysis of percolation thresholds for rods on triangular lattices, uncovering a symmetric property and convergence behaviors not previously observed in other lattices.

Findings

Percolation thresholds exhibit non-monotonous dependence on rod length, converging to specific values.

Standard and inverse thresholds are symmetric around 0.5, satisfying a complementary relation.

Percolation transition occurs across all rod lengths studied.

Abstract

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of straight rigid rods on triangular lattices. In the case of standard (inverse) percolation, the lattice is initially empty(occupied) and linear $k$-mers ($k$ linear consecutive sites) are randomly and sequentially deposited on(removed from) the lattice, considering an isotropic and nematic scheme. The study is conducted by following the behavior of four critical concentrations with the size $k$, determined for a wide range of $k$ : $(i)$[$(ii)$] standard isotropic[nematic] percolation threshold $\theta_{c,k}$[$\vartheta_{c,k}$], and $(iii)$[$(iv)$] inverse isotropic[nematic] percolation threshold $\theta^i_{c,k}$[$\vartheta^i_{c,k}$]. The obtained results indicate that: $(1)$ $\theta_{c,k}$[$\theta^i_{c,k}$] exhibits a non-monotonous dependence with $k$. It decreases[increases], goes through a minimum[maximum] around $k = 11$, then increases and asymptotically converges towards a definite value for large $k$ $\theta_{c,k \rightarrow \infty}=0.500(2)$[$\theta^i_{c,k \rightarrow \infty}=0.500(1)$]; $(2)$ $\vartheta_{c,k}$[$\vartheta^i_{c,k}$] rapidly increases[decreases] and asymptotically converges towards a definite value for infinitely long $k$-mers $\vartheta_{c,k \rightarrow \infty}=0.5334(6)$[$\vartheta^i_{c,k \rightarrow \infty}=0.4666(6)$]; $(3)$ for both models, the curves of standard and inverse percolation thresholds are symmetric with respect to $\theta = 0.5$. Thus, a complementary property is found $\theta_{c,k} + \theta^i_{c,k} = 1$ (and $\vartheta_{c,k} + \vartheta^i_{c,k} = 1$), which has not been observed in other regular lattices. This condition is analytically validated by using exact enumeration of configurations for small systems; and $(4)$ in all cases, the model presents percolation transition for the whole range of $k$.