RESUMO
Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse percolation of triangular tiles of side k (k-tiles) on triangular lattices. In the case of standard percolation, the lattice is initially empty. Then, k-tiles 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 by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then, the system is diluted by randomly removing k-tiles [composed by k(k+1)/2 monomers] from the lattice. Two schemes are used for the depositing and removing process: the isotropic scheme, where the deposition (removal) of the objects occurs with the same probability in any lattice direction; and the anisotropic (perfectly oriented or nematic) scheme, where one lattice direction is privileged for depositing (removing) the tiles. The study is conducted by following the behavior of four critical concentrations with the size k: (i) [(ii)] standard isotropic (oriented) percolation threshold θ_{c,k} (Ï_{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} (Ï_{c,k}) is reached by isotropic (oriented) deposition of k-tiles on an initially empty lattice; and (iii) [(iv)] inverse isotropic (oriented) percolation threshold θ_{c,k}^{i} (Ï_{c,k}^{i}), which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θ_{c,k}^{i} (Ï_{c,k}^{i}) is reached after removing isotropic (completely aligned) k-tiles from an initially fully occupied lattice. The obtained results indicate that (1)θ_{c,k} (θ_{c,k}^{i}) is an increasing (decreasing) function of k in the range 1≤k≤6. For k≥7, all jammed configurations are nonpercolating (percolating) states and, consequently, the percolation phase transition disappears. (2)Ï_{c,k} (Ï_{c,k}^{i}) show a behavior qualitatively similar to that observed for isotropic deposition. In this case, the minimum value of k at which the phase transition disappears is k=5. (3) For both isotropic and perfectly oriented models, the curves of standard and inverse percolation thresholds are symmetric to each other with respect to the line θ(Ï)=0.5. Thus, a complementary property is found θ_{c,k}+θ_{c,k}^{i}=1 (and Ï_{c,k}+Ï_{c,k}^{i}=1), which has not been observed in other regular lattices. (4) Finally, in all cases, the jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the orientation (isotropic or nematic) or the size k considered. In addition, the accurate determination of the critical exponents ν, ß, and γ reveals that the percolation phase transition involved in the system, which occurs for k varying between one and five (three) for isotropic (nematic) deposition scheme, has the same universality class as the standard percolation problem.
RESUMO
Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing semirigid rods from a L×L square lattice that contains two layers (and M=L×L×2 sites). The process starts with an initial configuration where all lattice sites are occupied by single monomers (each monomer occupies one lattice site) and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then the system is diluted by removing groups of k consecutive monomers according to a generalized random sequential adsorption mechanism. The study is conducted by following the behavior of two critical concentrations with size k: (1) jamming coverage θ_{j,k}, which represents the concentration of occupied sites at which the jamming state is reached, and (2) inverse percolation threshold θ_{c,k}, which corresponds to the maximum concentration of occupied sites for which connectivity disappears. The obtained results indicate that (1) the jamming coverage exhibits an increasing dependence on the size k-it rapidly increases for small values of k and asymptotically converges towards a definite value for infinitely large k sizes θ_{j,kâ∞}≈0.2701-and (2) the inverse percolation threshold is a decreasing function of k in the range 1≤k≤17. For k≥18, all jammed configurations are percolating states (the lattice remains connected even when the highest allowed concentration of removed sites is reached) and, consequently, there is no nonpercolating phase. This finding contrasts with the results obtained in literature for a complementary problem, where straight rigid k-mers are randomly and irreversibly deposited on a square lattice forming two layers. In this case, percolating and nonpercolating phases extend to infinity in the space of the parameter k and the model presents percolation transition for the whole range of k. The results obtained in the present study were also compared with those reported for the case of inverse percolation by removal of rigid linear k-mers from a square monolayer. The differences observed between monolayer and bilayer problems were discussed in terms of vulnerability and network robustness. Finally, the accurate determination of the critical exponents ν, ß, and γ reveals that the percolation phase transition involved in the system has the same universality class as the standard percolation problem.