Percolation phase transition by removal of $k^{2}$-mers from fully occupied lattices

Abstract

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse site percolation by the removal of $k \times k$ 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 \textit{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 $1 \leq k \leq 4$. For $k \geq 5$, all jammed configurations are percolating states, and consequently, there is no non-percolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed sites is reached. The jamming exponent $\nu_j$ was measured, being $\nu_j = 1$ regardless of the size $k$ considered. In addition, the accurate determination of the critical exponents $\nu$, $\beta$ and $\gamma$ reveals that the percolation phase transition involved in the system, which occurs for $k$ varying between 1 and 4, has the same universality class as the standard percolation problem.