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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 $p_c(z-1)=1+1.224 z^{-1/2}$.
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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 several four-dimensional (4D) lattices, including the simple (hyper) cubic (SC), the SC with combinations of nearest neighbors and second nearest neighbors (SC-NN+2NN), the body-centered cubic (BCC), and the face-centered cubic (FCC) lattices, using an efficient single-cluster growth algorithm. For the SC lattice, we find $p_c = 0.1601312(2)$, which confirms previous results (based on other methods), and find a new value $p_c=0.035827(1)$ for the SC-NN+2NN lattice, which was not studied previously for bond percolation. For the 4D BCC and FCC lattices, we obtain $p_c=0.074212(1)$ and 0.049517(1), which are substantially more precise than previous values. We also find critical exponents $tau = 2.3135(5)$ and $Omega = 0.40(3)$, consistent with previous numerical results and the recent four-loop series result of Gracey [Phys. Rev. D 92, 025012, (2015)].
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.
We investigate kinetically constrained models of glassy transitions, and determine which model characteristics are crucial in allowing a rigorous proof that such models have discontinuous transitions with faster than power law diverging length and time scales. The models we investigate have constraints similar to that of the knights model, introduced by Toninelli, Biroli, and Fisher (TBF), but differing neighbor relations. We find that such knights-like models, otherwise known as models of jamming percolation, need a ``No Parallel Crossing rule for the TBF proof of a glassy transition to be valid. Furthermore, most knight-like models fail a ``No Perpendicular Crossing requirement, and thus need modification to be made rigorous. We also show how the ``No Parallel Crossing requirement can be used to evaluate the provable glassiness of other correlated percolation models, by looking at models with more stable directions than the knights model. Finally, we show that the TBF proof does not generalize in any straightforward fashion for three-dimensiona
We study numerically the geometrical properties of minimally weighted paths that appear in the negative-weight percolation (NWP) model on two-dimensional lattices assuming a combination of periodic and free boundary conditions (BCs). Each realization of the disorder consists of a random fraction 1-rho of bonds with unit strength and a fraction rho of bond strengths drawn from a Gaussian distribution with zero mean and unit width. For each such sample, the path is forced to span the lattice along the direction with the free BCs. The path and a set of negatively weighted loops form a ground state (GS). A ground state on such a lattice can be determined performing a non-trivial transformation of the original graph and applying sophisticated matching algorithms. Here we examine whether the geometrical properties of the paths are in accordance with predictions of Schramm-Loewner evolution (SLE). Measuring the fractal dimension and reviewing Schramms left passage formula indicates that the paths cannot be described in terms of SLE.
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