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.
This paper presents a Monte-Carlo study of percolation in a distorted square lattice, in which, the adjacent sites are not equidistant. Starting with an undistorted lattice, the position of the lattice sites are shifted through a tunable parameter $α$ to create a distorted empty lattice. In this model, two neighboring sites are considered to be connected to each other in order to belong to the same cluster, if both of them are occupied as per the criterion of usual percolation and the distance between them is less than or equal to a certain value, called connection threshold $d$. While spannin…
Monte Carlo simulations and finite-size scaling analysis have been performed to study the jamming and percolation behavior of linear $k$-mers (also known as rods or needles) on the two-dimensional triangular lattice, considering an isotropic RSA process on a lattice of linear dimension $L$ and periodic boundary conditions. Extensive numerical work has been done to extend previous studies to larger system sizes and longer $k$-mers, which enables the confirmation of a nonmonotonic size dependence of the percolation threshold and the estimation of a maximum value of $k$ from which percolation wou…
We determine thresholds $p_c$ for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighbourhoods. The dependence of the value of the percolation thresholds $p_c$ on the coordination number $z$ are tested against various theoretical predictions. The newly proposed single scalar index $ξ=\sum_i z_ir_i^2/i$ (depending on the coordination zone number $i$, the neighbourhood coordination number $z$ and the square-distance $r^2$ to sites in $i$-t…
The percolation behavior of aligned rigid rods of length $k$ ($k$-mers) on two-dimensional triangular lattices has been studied by numerical simulations and finite-size scaling analysis. The $k$-mers, containing $k$ identical units (each one occupying a lattice site), were irreversibly deposited along one of the directions of the lattice. The connectivity analysis was carried out by following the probability $R_{L,k}(p)$ that a lattice composed of $L \times L$ sites percolates at a concentration $p$ of sites occupied by particles of size $k$. The results, obtained for $k$ ranging from 2 to 80,…
We determine thresholds $p_c$ for random site percolation on a triangular lattice for neighbourhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN) and next-next-next-next-nearest (5NN) neighbours, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [M. E. J. Newman and R. M. Ziff, Physical Review E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with …
The ground state of the spin-$1/2$ Heisenberg antiferromagnet on a distorted triangular lattice is studied using a numerical-diagonalization method. The network of interactions is the $\sqrt{3}\times\sqrt{3}$ type; the interactions are continuously controlled between the undistorted triangular lattice and the dice lattice. We find new states between the nonmagnetic 120-degree-structured state of the undistorted triangular case and the up-up-down state of the dice case. The intermediate states show spontaneous magnetizations that are smaller than one third of the saturated magntization correspo…
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] p…
Site percolation in a distorted simple cubic lattice is characterized numerically employing the Newman-Ziff algorithm. Distortion is administered in the lattice by systematically and randomly dislocating its sites from their regular positions. The amount of distortion is tunable by a parameter called the distortion parameter. In this model, two occupied neighboring sites are considered connected only if the distance between them is less than a predefined value called the connection threshold. It is observed that the percolation threshold always increases with distortion if the connection thres…
We study the triangular-lattice Ising model with dipolar interactions, inspired by its realisation in artificial arrays of nanomagnets. We show that a classical spin-liquid forms at intermediate temperatures, and that its behaviour can be tuned by temperature and/or a small lattice distortion between a string Luttinger liquid and a domain-wall-network state. At low temperature there is a transition into a magnetically ordered phase, which can be first-order or continous with a crossover in the critical behaviour between Pokrovsky-Talapov and 2D-Ising universality. When the Pokrovsky-Talapov cr…
We study site percolation on lattices confined to a semi-infinite strip. For triangular and square lattices we find that the probability that a cluster touches the three sides of such a system at the percolation threshold has the continuous limit 1/2 and argue that this limit is universal for planar systems. This value is also expected to hold for finite systems for any self-matching lattice. We attribute this result to the asymptotic symmetry of the separation lines between alternating spanning clusters of occupied and unoccupied sites formed on the original and matching lattice, respectively…
Restricted-valence random sequential adsorption~(RSA) is studied in its pure and disordered versions, on the square and triangular lattices. For the simplest case~(pure on the square lattice) we prove the absence of percolation for maximum valence $V_{\rm max}=2$. In other cases, Monte Carlo simulations are used to investigate the percolation threshold, universality class, and jamming limit. Our results reveal a continuous transition for the majority of the cases studied. The percolation threshold is computed through finite-size scaling analysis of seven properties; its value increases with th…
Mitra et al. [Phys. Rev. E 99 (2019) 012117] proposed a new percolation model that includes distortion in the square lattice and concluded that it may belong to the same universality class as the ordinary percolation. But the conclusion is questionable since their results of critical exponents are not consistent. In this paper, we reexamined the new model with high precision in the square, triangular, and honeycomb lattices by using the Newman-Ziff algorithm. Through the finite-size scaling, we obtained the percolation threshold of the infinite-size lattice and critical exponents ($ν$ and $β$)…
Three-dimensional three-colour percolation on a lattice made of tetrahedra is a direct generalization of two-dimensional two-colour percolation on the triangular lattice. The interfaces between one-colour clusters are made of bicolour surfaces and tricolour non-intersecting and non-self-intersecting curves. Because of the three-dimensional space, these curves describe knots and links. The present paper presents a construction of such random knots using particular boundary conditions and a numerical study of some invariants of the knots. The results are sources of precise conjectures about the …
We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a "near-loop" when it comes within one lattice spacing of itself. We define a new curve by erasing these near-loops as we traverse the interface. Our Monte Carlo simulations of this model lead us to conclude that the scaling limit of this loop-erased percolation interface is conformally invariant and has fractal dimension 4/3. However, it is not SLE_8/3. We also consider the process in…
Percolation and critical O($n$) loop configurations
研究了蜂窝格子上O（$n$）圈模型的临界圈结构的渗流问题。我们将二元团簇定义为二元三角晶格上不被环分开的位点组，并研究了这些二元团簇的键渗透性质。在临界Potts模型中，逾渗阈值的普适性质与Kasteleyn-Fortuin随机团簇相匹配。通过对$1\leq n \leq 2$范围内的几个O（$n$）模型的团簇模拟，对这种关系进行了数值检验。模拟结果包括几个值的$n$的渗流阈值，以及与键稀释和在渗流阈值的稀释团簇的尺寸分布的普适指数。我们的数值计算结果的指数与现有的库仑气体结果的随机集团模型，这证实了两种模型之间的关系。我们讨论了重整化流的键稀释参数$p$作为$n$的函数，并提供了一个表达式，准确地描述了一行不稳定的固定点作为$n$的函数，对应的逾渗阈值。此外，重整化方案表明，在$p$与$n$图，另一条线的不动点在$p=1$，这是稳定的关于$p$。
We calculate exact analytic expressions for the average cluster numbers $\langle k \rangle_{Λ_s}$ on infinite-length strips $Λ_s$, with various widths, of several different lattices, as functions of the bond occupation probability, $p$. It is proved that these expressions are rational functions of $p$. As special cases of our results, we obtain exact values of $\langle k \rangle_{Λ_s}$ and derivatives of $\langle k \rangle_{Λ_s}$ with respect to $p$, evaluated at the critical percolation probabilities $p_{c,Λ}$ for the corresponding infinite two-dimensional lattices $Λ$. We compare these exact…
