Inverse percolation by removing extended objects from two-dimensional lattices

CENTRES, P. M.; RAMIREZ PASTOR, A. J.; RAMIREZ L. S.

Conferencia; STATPHYS 27. The XXIV International Conference on Statistical Physics of the International Union for Pure and Applied Physics (IUPAP); 2019

Percolation is a well-known problem that is interesting because it displays a threshold phenomenon and contributes to the understanding of the behavior of many systems when considering geometric connectivity. The percolation model can be represented as a site/bond lattice, where each element is occupied with probability p in the interval [0, 1] or empty with probability 1-p. In this work, we used the percolation theory to describe the response of the system to the removal of components, phenomena of primary interest in robustness. We studied the response of an initially fully occupied lattice when it is diluted by removing groups of components to find the minimum concentration at which connectivity is lost. We called this percolation scheme as inverse percolation. By using numerical simulations and finite-size analysis, five different systems were treated[1-4]: (1) removing rigid rods of k size (k-mers) from square lattices and (2) from triangular lattices; (3) removing k-mers of bonds from square lattices; (4) removing k-mers from square lattices in the presence of impurities; and (5) removing k×k tiles from square lattices. The behavior of the inverse threshold and the jamming properties strongly differ from the standard direct problem. For cases (1) and (2), 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 range of k. For (3),(4), and (5), the interplay between the percolation and jamming is responsible for the existence of a maximum value of k from which percolation phase transition no longer occurs. This behavior had not been observed previously for k-mer site percolation and has strong implications since it means that from a certain value, the system cannot be disconnected. The last is discussed in the work in terms of network attacks. Finally, an exhaustive study of critical exponents and universality was carried out, which revealed that the problem belongs to the same universality class as 2D random percolation model.