Restoring site percolation on damaged square lattices

Abstract

Restoring site percolation on a damaged square lattice with the nearest neighbor (N2) is investigated using two different strategies. In the first one, a density y of new sites are created on the empty sites with longer range links, either next-nearest neighbor (N3) or next-next-nearest neighbor (N4), but without N2. In the second one, new longer range links N3 or N4 are added to N2 but only for a fraction v of the remaining non-destroyed sites. Starting at p(c)(N2), with a density x of randomly destroyed sites, the values of y(c) and v(c), which restore site percolation, are calculated for both strategies with, respectively, N3 and N4 using Monte Carlo simulations. Results are obtained for the whole range 0 < or = x < or = p(c)(N2).

Simple Cubic Random-Site Percolation Thresholds for Complex Neighbourhoods

“…The calculated thresholds p c are 0.1372(1), 0.1420(1), 0.0976(1), 0.1991(1), 0.1036(1), 0.2455(1) for (NN + 2NN), (NN + 3NN), (NN + 2NN + 3NN), 2NN, (2NN + 3NN), 3NN neighbourhoods, respectively. In contrast to the results obtained for a square lattice [13][14][15] the calculated percolation thresholds decrease monotonically with the site coordination number z, at least for our inspected neighbourhoods. Fig.…”

Percolation thresholds on triangular lattice for neighbourhoods containing sites up-to the fifth coordination zone

“…The quantity is based on the coordination zone ζ, the sites number z and the sites distances r to the central site in the neighbourhood. The dependency of p c on this newly proposed index ξ follows roughly a power law with an exponent close to −0.705 (23).…”

“…The percolation thresholds were initially estimated for nearest neighbour interactions [17,18] but later also complex (or extended) neighbourhoods were studied for 2D (square [19][20][21][22][23][24][25], triangular [19,20,26,27], honeycomb [19]), 3D (simple cubic [25,28,29]) and 4D (simple hypercubic [30]) lattices.…”

Universal behavior of site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in 2 and 3 dimensions

“…It has been suggested that these results may be applicable to a model of constrained percolation [38]. Malarz and coworkers [39][40][41][42][43][44][45][46] carried out extensive numerical simulations on lattices with combinations of "complex neighborhoods" in two, three, and four dimensions. Koza and collaborators [25,26] studied percolation of overlapping shapes on a lattice, which can be mapped to extended-range site percolation.…”