Standard and inverse bond percolation of straight rigid rods on square lattices

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

PHYSICAL REVIEW E

AMER PHYSICAL SOC

Lugar: New York; Año: 2018 vol. 97 p. 42113 - 42122

1539-3755

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse bondpercolation of straight rigid rods on square lattices. In the case of standard percolation, the lattice is initially empty.Then, linear bond k-mers (sets of k linear nearest-neighbor bonds) are randomly and sequentially deposited on thelattice. Jamming coverage pj,k and percolation threshold pc,k are determined for a wide range of k (1 k 120).pj,k and pc,k exhibit a decreasing behavior with increasing k, pj,k→∞ = 0.7476(1) and pc,k→∞ = 0.0033(9) beingthe limit values for large k-mer sizes. pj,k is always greater than pc,k , and consequently, the percolation phasetransition occurs for all values of k. In the case of inverse percolation, the process starts with an initial configurationwhere all lattice bonds are occupied and, given that periodic boundary conditions are used, the opposite sides ofthe lattice are connected by nearest-neighbor occupied bonds. Then, the system is diluted by randomly removinglinear bond k-mers from the lattice. The central idea here is based on finding the maximum concentration ofoccupied bonds (minimum concentration of empty bonds) for which connectivity disappears. This particular valueof concentration is called the inverse percolation threshold pic,k , and determines a geometrical phase transitionin the system. On the other hand, the inverse jamming coverage pij,k is the coverage of the limit state, in whichno more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbor bondsof appropriate size. It is easy to understand that pij,k = 1 − pj,k . The obtained results for pic,k show that theinverse percolation threshold is a decreasing function of k in the range 1 k 18. For k > 18, all jammedconfigurations are percolating states, and consequently, there is no nonpercolating phase. In other words, thelattice remains connected even when the highest allowed concentration of removed bonds pij,k is reached. Interms of network attacks, this striking behavior indicates that random attacks on single nodes (k = 1) are muchmore effective than correlated attacks on groups of close nodes (large k?s). Finally, the accurate determinationof critical exponents reveals that standard and inverse bond percolation models on square lattices belong to thesame universality class as the random percolation, regardless of the size k considered.