Extended-range percolation for neighborhoods of rectangular shape on square lattice

In this paper, extended-range percolation for neighborhoods of rectangular shape on a square lattice is investigated by means of Monte Carlo simulations. We choose a rectangle with an aspect ratio of 2 : 1, with the long side indicating its orientation. Three types of orientation are considered, namely horizontal (model I), horizontal or vertical randomly with equal probability (model II), and both horizontal and vertical simultaneously (thus in the shape of a cross) (model III). Simulations of both bond and site percolation are performed on the basis of an effective single-cluster growth algorithm, and precise percolation thresholds are found. The asymptotic and finite-size behaviors between the thresholds p_{c} and the coordination number z for each type are also discussed. For bond percolation, the asymptotic value zp_{c} of all three situations tends to Bethe-lattice behavior (zp_{c}∼1), and the finite-z correction is found to be consistent with zp_{c}∼1+a_{1}z^{-1/2}+a_{2}z^{-1}+a_{3}z^{-3/2} given by Deng et al. [J. Phys.: Conf. Ser. 1163, 012001, (2019)10.1088/1742-6596/1163/1/012001]. Regarding site percolation, for model I, data fitting gives the value of zp_{c}∼4.3951(1) for large z, compared to zp_{c}=2^{d}η_{c}∼4.3954, where η_{c}=1.09884280(9) [Phys. Rev. E 86, 061109 (2012)1539-375510.1103/PhysRevE.86.061109] is the continuum percolation threshold of aligned squares. For model II with site percolation, the nonlinear behavior of zp_{c} versus 1/z shows that the orientation of the rectangle has an effect on the asymptotic behavior of zp_{c}. A variation of model II in which reciprocal neighborhoods are taken into account however shows linear behavior on such a plot. For model III, the fitting of the data leads to zp_{c}∼4.414(2), which is in principle not related to the continuum threshold of overlapping crosses but close to the value zp_{c}≈4.51235 of overlapping disks. Some power laws of the simulation results are also discussed. The results show that the orientation of the rectangle has an impact on the thresholds of both bond percolation and site percolation, and a deeper analysis indicates that this effect gradually weakens as the coordination number z increases, following a power law of z^{-w} with w=1.49(1) for bond percolation and 0.91(1) for site percolation for the difference between models I and II.