Inverse percolation by removing straight rigid rods from triangular lattices

The problem of inverse percolation by removing straight rigid rods from two-dimensional triangular lattices has been studied by using numerical simulations and finite-size scaling analysis. 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 straight rigid rods of a length k (k-mers) from the surface. Based on percolation theory, the maximum concentration of occupied sites (minimum concentration of holes) at which the connectivity disappears was obtained. This particular value of the concentration is named the inverse percolation threshold, and determines a well-defined geometrical (second order) phase transition in the system. The corresponding critical exponents were also calculated. The results, obtained for k ranging from 2 to 256, revealed that (i) the inverse percolation threshold exhibits nonmonotonic behavior as a function of the k-mer size: it grows from k = 1 to k = 10, goes through a maximum at k = 11, and finally decreases again and asymptotically converges towards a definite value for large values of k; (ii) the percolating and non-percolating phases extend to infinity in the space of the parameter k and, consequently, the model presents percolation transition in all the ranges of k; and (iii) the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered.