Percolation of dimers on square lattices

Lebrecht, W.; Valdés, J.F.; Vogel, E.E.; Nieto, F.; Ramírez-Pastor, A. J.

doi:10.1016/j.physa.2012.08.014

Cited by 17 publications

(25 citation statements)

References 37 publications

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“…An exact counting of configurations on finite cells was performed in order to back up the simulation predictions. This type of approach has been successfully applied to a variety of percolation problems [48][49][50]. Specifically, we will explore the relationship between standard and inverse thresholds from an analytical approach.…”

Section: B Exact Counting Of Configurations On Finite Cellsmentioning

confidence: 99%

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

et al. 2021

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Section: B Exact Counting Of Configurations On Finite Cellsmentioning

confidence: 99%

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

et al. 2021

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“…An exact counting of configurations on finite cells was performed in order to back up the simulation predictions. This type of approach has been successfully applied to a variety of percolation problems [43][44][45]. Specifically, we will explore the relationship between standard and inverse thresholds from an analytical approach.…”

Section: B Exact Counting Of Configurations On Finite Cellsmentioning

confidence: 99%

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

Ramirez,

Pasinetti,

Lebrecht

et al. 2020

Preprint

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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 percolation, the lattice is initially empty. Then, linear k-mers (sets of k linear consecutive sites) are randomly and sequentially deposited on the lattice. In the case of inverse percolation, the process starts with an initial configuration where all lattice sites are occupied and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then, the system is diluted by randomly removing linear k-mers from the lattice. Two schemes are used for the depositing/removing process: isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction; and nematic scheme, where one lattice direction is privileged for depositing (removing) the particles. The study is conducted by following the behavior of four critical concentrations with the size k: (i)[(ii)] standard isotropic[nematic] percolation threshold θ c,k [ϑ c,k ], which represents the minimum concentration of occupied sites at which an infinite cluster of occupied nearest-neighbor sites extends from one side of the system to the other. θ c,k [ϑ c,k ] is reached by isotropic[nematic] deposition of straight rigid k-mers on an initially empty lattice; and (iii)[(iv)] inverse isotropic[nematic] percolation threshold θ i c,k [ϑ i c,k ], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ i c,k [ϑ i c,k ] is reached after isotropically[nematically] removing straight rigid k-mers from an initially fully occupied lattice. θ c,k , ϑ c,k , θ i c,k and ϑ i c,k are determined for a wide range of k (2 ≤ k ≤ 512). The obtained results indicate that: (1) θ c,k [θ i c,k ] exhibits a non-monotonous dependence on the size k. It decreases[increases] for small particles sizes, goes through a minimum[maximum] around k = 11, and finally increases and asymptotically converges towards a definite value for large segments θ c,k→∞ = 0.500(2)[θ i c,k→∞ = 0.500(1)]; (2) ϑ c,k [ϑ i c,k ] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particles sizes and asymptotically converges towards a definite value for infinitely long k-mers ϑ c,k→∞ = 0.5334(6)[ϑ i c,k→∞ = 0.4666(6)];(3) for both isotropic and nematic models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ = 0.5. Thus, a complementary property is found θ c,k + θ i c,k = 1 (and ϑ c,k + ϑ 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 critical concentration curves divide the θ-space in a percolating region and a non-percolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for th...

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“…However, we should bear in mind that the effective conductivity and percolation itself could be influenced by the size of mono-grains [6] as well as by the grain size distribution in a random material [7] or polydispersity [8]. The usage of the finite size objects can also be extended to percolation of dimers on square lattices [9] and further, to the impact of defects on percolation in random sequential adsorption of linear k-mers on square lattices [10]. Along this line, but for a three-dimensional (3D) system of straight rigid rods of length k, the ratio between percolation threshold and jamming coverage (p /j) shows a non-universal behaviour, monotonically decreasing to zero, with increasing k [11].…”

Section: Introductionmentioning

confidence: 99%

On multi-scale percolation behaviour of the effective conductivity for the lattice model with interacting particles

Wiśniowski

Olchawa

Frączek

et al. 2016

Physica A: Statistical Mechanics and its Applications

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 We study the multi-scale site percolation in a system with interacting objects.  The interaction energy modifies the value of the percolation threshold in a limited range. When a repulsive interaction dominates thermal energy, the exact c at scales k = 2, 3 is found.  In turn, when k >> 1, the evaluated formula for c(k) implies the limit threshold value 0.75. A B S T R A C TRecently, the effective medium approach using 2  2 basic cluster of model lattice sites to predict the conductivity of interacting droplets has been presented by Hattori et al. To make a step aside from pure applications, we have studied earlier a multi-scale percolation, employing any k  k basic cluster for non-interacting particles. Here, with interactions included, we examine in what way they alter the percolation threshold for any cluster case. We found that at a fixed length scale k the interaction reduces the range of shifts of the percolation threshold. To determine the critical concentrations, the simplified model is used. It diminishes the number of local conductivities into two main ones. In the presence of a dominance of the repulsive interaction over the thermal energy, the exact percolation thresholds at scales k = 2 and 3 can be obtained from analytical formulas. Furthermore, by a simple reasoning, we obtain the limiting threshold formula for odd k. When k  1, the odd-even difference becomes negligible. Hence, the 0.75 is the highest possible value of the threshold.

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Percolation of dimers on square lattices

Cited by 17 publications

References 37 publications

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

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

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

On multi-scale percolation behaviour of the effective conductivity for the lattice model with interacting particles

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