Percolation transitions in two dimensions

Abstract

We investigate bond-and site-percolation models on several two-dimensional lattices numerically, by means of transfer-matrix calculations and Monte Carlo simulations. The lattices include the square, triangular, honeycomb kagome, and diced lattices with nearest-neighbor bonds, and the square lattice with nearest-and next-nearest-neighbor bonds. Results are presented for the bond-percolation thresholds of the kagome and diced lattices, and the site-percolation thresholds of the square, honeycomb, and diced lattices. We also include the bond-and site-percolation thresholds for the square lattice with nearest-and next-nearest-neighbor bonds. We find that corrections to scaling behave according to the second temperature dimension X t2 = 4 predicted by the Coulomb gas theory and the theory of conformal invariance. In several cases there is evidence for an additional term with the same exponent, but modified by a logarithmic factor. Only for the site-percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent. The amplitude of the power-law correction associated with X t2 = 4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems.

Incoherent dynamics in the toric code subject to disorder

“…If and how the polarization at this threshold is related to the site and bond percolation thresholds of the square lattice, which are η ≈ 0.5927 and 0.5 (see, e.g., Ref. [28]), respectively, is not completely clear.…”

Nonmonotonic size dependence of the critical concentration in 2D percolation of straight rigid rods under equilibrium conditions

“…Note that the IN phase transition on a square lattice with two allowed orientations occurs only for k ≥ 7 [22,23,24,25,26]. In the case of k = 1, the problem of percolation of monomers on square lattices has been one of the most studied percolation models in the literature, and the percolation threshold has been measured multiple times as 0.592746xx, where the last two decimal places (and error) are 21(13) [40], 21(33) [41], 03(09) [42], and 06(05) [43].…”

Modelling planar polarity of epithelia: the role of signal relay in collective cell polarization

“…The threshold for proper actin filament alignment near 50 per cent fat2 mutant fraction is reminiscent of the critical percolation threshold at this value that has been established theoretically for the node percolation problem on the triangular lattice [16]. The percolation theory describes the behaviour of connected clusters on lattices with randomly occupied sites, and the percolation threshold refers to the critical occupation probability at which long-range connectivity first occurs.…”