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

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 (particles occupying k consecutive sites along one of the lattice directions) 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 sets of k consecutive monomers (linear k-mers) from the lattice. Two schemes are used for the depositing/removing process: an isotropic scheme, where the deposition (removal) of the linear objects occurs with the same probability in any lattice direction, and an anisotropic (perfectly oriented) 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 size k: (i) [(ii)] standard isotropic[oriented] percolation threshold θc,k[ϑc,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. θc,k[ϑc,k] is reached by isotropic[oriented] deposition of straight rigid k-mers on an initially empty lattice; and (iii) [(iv)] inverse isotropic[oriented] percolation threshold θic,k[ϑic,k], which corresponds to the maximum concentration of occupied sites for which connectivity disappears. θic,k[ϑic,k] is reached after removing isotropic [completely aligned] straight rigid k-mers from an initially fully occupied lattice. θc,k, ϑc,k, θic,k, and ϑic,k are determined for a wide range of k (2≤k≤512). The obtained results indicate that (1)θc,k[θic,k] exhibits a nonmonotonous dependence on the size k. It decreases[increases] for small particle sizes, goes through a minimum[maximum] at around k=11, and finally increases and asymptotically converges towards a definite value for large segments θc,k→∞=0.500(2) [θic,k→∞=0.500(1)]; (2)ϑc,k[ϑic,k] depicts a monotonous behavior in terms of k. It rapidly increases[decreases] for small particle sizes and asymptotically converges towards a definite value for infinitely long k-mers ϑc,k→∞=0.5334(6) [ϑic,k→∞=0.4666(6)]; (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(ϑ)=0.5. Thus a complementary property is found θc,k+θic,k=1 (and ϑc,k+ϑic,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 nonpercolating region. These phases extend to infinity in the space of the parameter k so that the model presents percolation transition for the whole range of k.This work was supported in part by CONICET (Argentina) under Project No. PIP 112-201701-00673CO and Universidad Nacional de San Luis (Argentina) under Project No. 03-0816. W.L. is thankful for support from Dirección de Investigación Universidad de La Frontera (Chile) under Project DIUFRO, Grant No. DI20-0007. L.S.R. acknowledges the support through the Maria de Maeztu Program for units of Excellence in R&D (MDM-2017-0711)