Standard and inverse site percolation of triangular tiles 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 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 $k$: $\left(i\right) \left[\right(ii\left)\right]$ standard isotropic (oriented) percolation threshold ${\theta }_{c,k}$ (${\vartheta }_{c,k}$), 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. ${\theta }_{c,k}$ (${\vartheta }_{c,k}$) is reached by isotropic (oriented) deposition of $k$-tiles on an initially empty lattice; and $\left(iii\right) \left[\right(iv\left)\right]$ inverse isotropic (oriented) percolation threshold ${\theta }_{c,k}^i$ (${\vartheta }_{c,k}^i$), which corresponds to the maximum concentration of occupied sites for which connectivity disappears. ${\theta }_{c,k}^i$ (${\vartheta }_{c,k}^i$) is reached after removing isotropic (completely aligned) $k$-tiles from an initially fully occupied lattice. The obtained results indicate that $\left(1\right){\theta }_{c,k}$ (${\theta }_{c,k}^i$) is an increasing (decreasing) function of $k$ in the range $1\le k\le 6$. For $k\ge 7$, all jammed configurations are nonpercolating (percolating) states and, consequently, the percolation phase transition disappears. $\left(2\right){\vartheta }_{c,k}$ (${\vartheta }_{c,k}^i$) 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 $\theta \left(\vartheta \right)=0.5$. Thus, a complementary property is found ${\theta }_{c,k}+{\theta }_{c,k}^i=1$ (and ${\vartheta }_{c,k}+{\vartheta }_{c,k}^i=1$), which has not been observed in other regular lattices. (4) Finally, in all cases, the jamming exponent ${\nu }_j$ was measured, being ${\nu }_j=1$ regardless of the orientation (isotropic or nematic) or the size $k$ considered. In addition, the accurate determination of the critical exponents $\nu , \beta$, and $\gamma$ 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.