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Register a new userSimple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors
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K. Malarz
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In the paper random-site percolation thresholds for simple cubic lattice with sites neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation [Bastas et al., arXiv:1411.5834] is implemented for the studies of the top-bottom wrapping probability. The obtained percolation thresholds are $p_C(text{4NN})=0.31160(12)$, $p_C(text{4NN+NN})=0.15040(12)$, $p_C(text{4NN+2NN})=0.15950(12)$, $p_C(text{4NN+3NN})=0.20490(12)$, $p_C(text{4NN+2NN+NN})=0.11440(12)$, $p_C(text{4NN+3NN+NN})=0.11920(12)$, $p_C(text{4NN+3NN+2NN})=0.11330(12)$, $p_C(text{4NN+3NN+2NN+NN})=0.10000(12)$, where 3NN, 2NN, NN stands for next-next-nearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with two times larger lattice constant the percolation threshold $p_C$(4NN) is exactly equal to $p_C$(NN). The simplified Bastas et al. method allows for reaching uncertainty of the percolation threshold value $p_C$ similar to those obtained with classical method but ten times faster.
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We determine thresholds $p_c$ for random site percolation on a triangular lattice for neighbourhoods containing nearest (NN), next-nearest (2NN), next-next-nearest (3NN), next-next-next-nearest (4NN) and next-next-next-next-nearest (5NN) neighbours, and their combinations forming regular hexagons (3NN+2NN+NN, 5NN+4NN+NN, 5NN+4NN+3NN+2NN, 5NN+4NN+3NN+2NN+NN). We use a fast Monte Carlo algorithm, by Newman and Ziff [M. E. J. Newman and R. M. Ziff, Physical Review E 64, 016706 (2001)], for obtaining the dependence of the largest cluster size on occupation probability. The method is combined with a method, by Bastas et al. [N. Bastas, K. Kosmidis, P. Giazitzidis, and M. Maragakis, Physical Review E 90, 062101 (2014)], of estimating thresholds from low statistics data. The estimated values of percolation thresholds are $p_c(text{4NN})=0.192410(43)$, $p_c(text{3NN+2NN})=0.232008(38)$, $p_c(text{5NN+4NN})=0.140286(5)$, $p_c(text{3NN+2NN+NN})=0.215484(19)$, $p_c(text{5NN+4NN+NN})=0.131792(58)$, $p_c(text{5NN+4NN+3NN+2NN})=0.117579(41)$, $p_c(text{5NN+4NN+3NN+2NN+NN})=0.115847(21)$. The method is tested on the standard case of site percolation on triangular lattice, where $p_c(text{NN})=p_c(text{2NN})=p_c(text{3NN})=p_c(text{5NN})=frac{1}{2}$ is recovered with five digits accuracy $p_c(text{NN})=0.500029(46)$ by averaging over one thousand lattice realisations only.
By means of Monte Carlo simulations, we study long-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors, up to the eighth neighbors for the square lattice and the ninth neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods can be mapped to problems of lattice percolation of extended shapes, such as disks and spheres, and the thresholds can be related to the continuum thresholds $eta_c$ for objects of those shapes. This mapping implies $zp_{c} sim 4 eta_c = 4.51235$ in 2D and $zp_{c} sim 8 eta_c = 2.73512$ in 3D for large $z$ for circular and spherical neighborhoods respectively, where $z$ is the coordination number. Fitting our data to the form $p_c = c/(z+b)$ we find good agreement with $c = 2^d eta_c$; the constant $b$ represents a finite-$z$ correction term. We also study power-law fits of the thresholds.
We study bond percolation on the simple cubic (SC) lattice with various combinations of first, second, third, and fourth nearest-neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds. Correlations between percolation thresholds and lattice properties are discussed, and our results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number $z$ quite accurately according to a power law $p_{c} sim z^{-a}$, with exponent $a = 1.111$. However, for large $z$, the threshold must approach the Bethe lattice result $p_c = 1/(z-1)$. Fitting our data and data for lattices with additional nearest neighbors, we find $p_c(z-1)=1+1.224 z^{-1/2}$.
The critical curves of the q-state Potts model can be determined exactly for regular two-dimensional lattices G that are of the three-terminal type. Jacobsen and Scullard have defined a graph polynomial P_B(q,v) that gives access to the critical manifold for general lattices. It depends on a finite repeating part of the lattice, called the basis B, and its real roots in the temperature variable v = e^K - 1 provide increasingly accurate approximations to the critical manifolds upon increasing the size of B. These authors computed P_B(q,v) for large bases (up to 243 edges), obtaining determinations of the ferromagnetic critical point v_c > 0 for the (4,8^2), kagome, and (3,12^2) lattices to a precision (of the order 10^{-8}) slightly superior to that of the best available Monte Carlo simulations. In this paper we describe a more efficient transfer matrix approach to the computation of P_B(q,v) that relies on a formulation within the periodic Temperley-Lieb algebra. This makes possible computations for substantially larger bases (up to 882 edges), and the precision on v_c is hence taken to the range 10^{-13}. We further show that a large variety of regular lattices can be cast in a form suitable for this approach. This includes all Archimedean lattices, their duals and their medials. For all these lattices we tabulate high-precision estimates of the bond percolation thresholds p_c and Potts critical points v_c. We also trace and discuss the full Potts critical manifold in the (q,v) plane, paying special attention to the antiferromagnetic region v < 0. Finally, we adapt the technique to site percolation as well, and compute the polynomials P_B(p) for certain Archimedean and dual lattices (those having only cubic and quartic vertices), using very large bases (up to 243 vertices). This produces the site percolation thresholds p_c to a precision of the order 10^{-9}.
We determine thresholds $p_c$ for random-site percolation on a triangular lattice for all available neighborhoods containing sites from the first to the fifth coordination zones, including their complex combinations. There are 31 distinct neighbourhoods. The dependence of the value of the percolation thresholds $p_c$ on the coordination number $z$ are tested against various theoretical predictions. The newly proposed single scalar index $xi=sum_i z_ir_i^2/i$ (depending on the coordination zone number $i$, the neighbourhood coordination number $z$ and the square-distance $r^2$ to sites in $i$-th coordination zone from the central site) allows to differentiate among various neighbourhoods and relate $p_c$ to $xi$. The thresholds roughly follow a power law $p_cproptoxi^{-gamma}$ with $gammaapprox 0.710(19)$.
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