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Percolation of overlapping squares or cubes on a lattice

Porous media are often modelled as systems of overlapping obstacles, which leads to the problem of two percolation thresholds in such systems, one for the porous matrix and the other one for the void space. Here we investigate these percolation thresholds in the model of overlapping squares or cubes of linear size $k>1$ randomly distributed on a regular lattice. We find that the percolation threshold of obstacles is a nonmonotonic function of $k$, whereas the percolation threshold of the void space is well approximated by a function linear in $1/k$. We propose a generalization of the excluded volume approximation to discrete systems and use it to investigate the transition between continuous and discrete percolation, finding a remarkable agreement between the theory and numerical results. We argue that the continuous percolation threshold of aligned squares on a plane is the same for the solid and void phases and estimate the continuous percolation threshold of the void space around aligned cubes in a 3D space as 0.036(1). We also discuss the connection of the model to the standard site percolation with complex neighborhood.