Abstract
Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse site percolation by the removal of k×k square tiles (k^{2}-mers) from square lattices. The process starts with an initial configuration, where all lattice sites are occupied and, obviously, the opposite sides of the lattice are connected by occupied sites. Then the system is diluted by removing k^{2}-mers of occupied sites from the lattice following a random sequential adsorption mechanism. The process finishes when the jamming state is reached and no more objects can be removed due to the absence of occupied sites clusters of appropriate size and shape. The central idea of this paper is based on finding the maximum concentration of occupied sites, p_{c,k}, for which the connectivity disappears. This particular value of the concentration is called the inverse percolation threshold and determines a well-defined geometrical phase transition in the system. The results obtained for p_{c,k} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤4. For k≥5, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed sites is reached. The jamming exponent ν_{j} was measured, being ν_{j}=1 regardless of the size k considered. In addition, the accurate determination of the critical exponents ν, β, and γ reveals that the percolation phase transition involved in the system, which occurs for k varying between one and four, has the same universality class as the standard percolation problem.
Highlights
The percolation theory is one of the simplest models in probability theory, which has been applied to a wide range of phenomena in physics, chemistry, biology, and materials science where connectivity and clustering play an important role: flow in porous materials [1,2,3], network theory [4,5,6,7,8,9], thermal phase transitions [10, 11], spread of the computer virus [12], transport in disordered media [13, 14], electrical conductivity in alloys [15,16,17,18], simulated spread fire in multi-compartmented structures [19] and the spread of epidemics [20]
Two important observations can be drawn from the figure: (1) while the phase transition disappears for k2-mers with k > 4, percolating and non-percolating phases extend to infinity in the space of the parameter k for rigid kmers; and (2) the values of pc,k corresponding to k2-mers remain below the curve obtained by removing k-mers
In order to completely analyze the universality of the problem, the critical exponents β and γ can be obtained from the scaling behavior of the percolation order parameter P = SL /L2 and the corresponding percolation susceptibility χ = SL2 − SL 2 /L2, respectively [1]. . . . means an average over simulation runs
Summary
The percolation theory is one of the simplest models in probability theory, which has been applied to a wide range of phenomena in physics, chemistry, biology, and materials science where connectivity and clustering play an important role: flow in porous materials [1,2,3], network theory [4,5,6,7,8,9], thermal phase transitions [10, 11], spread of the computer virus [12], transport in disordered media [13, 14], electrical conductivity in alloys [15,16,17,18], simulated spread fire in multi-compartmented structures [19] and the spread of epidemics [20]. The study complements previous work in homogeneous lattices [24,25,26], revealing that the construction of networks with low local connectivity (or low clustering capacity), as occurs in the model for increasing values of the fraction of impurities, is an effective strategy against correlated attacks on groups of close nodes (large k’s). The aim of the present work is to extend previous studies to the removal of more compact objects such as k × k square tiles (or k2-mers) For this purpose, extensive numerical simulations supplemented by analysis using finite-size scaling theory have been carried out to study the problem of inverse percolation by removing k2mers from square lattices.
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