Bond dimer percolation on square lattices

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

doi:10.1016/j.physa.2013.12.038

Cited by 8 publications

(14 citation statements)

References 20 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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“…This is particularly true for exponent ν, which is where we focus our attention now. We can make use of the expression [21] ln L = ν X ln λ X ∞ , (10) where λ X ∞ represents the maximum value of the first derivative of the percolation function at p = p c in the limit of a cell of infinite dimensions, for the different deposition cases. In the case of finite dimensions both ν and the maximum λ are size dependent, namely,…”

Section: B Exponent νmentioning

confidence: 99%

“…In fact, most of the studies have been based on computer simulations and the obtained results have not been corroborated yet by analytical methods. In recent works from our group [20,21], bond and site dimer percolation have been studied by using analytical approximations based on exacts results for small cells that can be extrapolated to larger sizes. In this context, the main objective of the present work is to extend such results to larger objects like the trimers considered here.…”

Section: Introductionmentioning

confidence: 99%

“…In this context, the main objective of the present work is to extend such results to larger objects like the trimers considered here. The study is a natural continuation of our previous work [20,21] and focuses on the percolation properties of site trimers with all possibleshapes and orientations. The study of trimer phases is not only of analytical interest, but also of considerable practical importance in surface physics [5,6,[22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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Site trimer percolation on square lattices

et al. 2015

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Percolation of site trimers (k-mers with k=3) is investigated in a detailed way making use of an analytical model based on renormalization techniques in this problem. Results are compared to those obtained here by means of extensive computer simulations. Five different deposition possibilities for site trimers are included according to shape and orientation of the depositing objects. Analytical results for the percolation threshold p(c) are all close to 0.55, while numerical results show a slight dispersion around this value. A comparison with p(c) values previously reported for monomers and dimers establishes the tendency of p(c) to decrease as k increases. Critical exponent ν was also obtained both by analytical and numerical methods. Results for the latter give values very close to the expected value 4/3 showing that this percolation case corresponds to the universality class of random percolation.

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Bond dimer percolation on square lattices

Cited by 8 publications

References 20 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

Site trimer percolation on square lattices

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