Jamming limiting the percolation of square tiles on square lattices*

Square tiles of k×k sites (k²-mers) are deposited irreversibly on L×L square lattices of exactly the same inter-site distance; no overlapping is allowed. Coverage is defined as q=Nk²/L², where N represents the number of deposited tiles. Percolation thresholds q_p(k) are reported with high precision for k=1,2, and 3. For k≥4 jamming suppresses percolation. The coverages at which jamming appears q_j(k) are also reported accurately [1]. It is observed that q_p(2)<q_j(2) and q_p(3)<q_j(3) while this inequality is reversed for k≥4, namely, q_p(k)>q_j(k), which explains the suppression of percolation for k≥4. Monte Carlo techniques are used to simulate these depositions for k²-mers from k=2 to k=100, and lattice sizes with sides much larger than k. Calculations based on exact enumeration were done for k≤6 and several L values to show that this property is inherent to these systems. Finite size scaling is used to estimate the thresholds in the thermodynamic limit. The universality class of this deposition corresponds to random percolation; the corresponding critical exponents nu, gamma, and beta are reported with good accuracy.
[1] P.M. Centres, A.-J. Ramirez-Pastor, E.E. Vogel, J.F. Valdés, submitted to Phys. Rev. E (2018).