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Zhipeng Xun,
Physical Review E, Volume 102; https://doi.org/10.1103/physreve.102.012102

Abstract:
We study bond percolation on the simple cubic 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 pc∼z−a with exponent a=1.111. However, for large z, the threshold must approach the Bethe lattice result pc=1/(z−1). Fitting our data and data for additional nearest neighbors, we find pc(z−1)=1+1.224z−1/2.
S. S. Manna,
Physical Review E, Volume 101; https://doi.org/10.1103/physreve.101.062143

Abstract:
We consider a percolation process in which k points separated by a distance proportional the system size L simultaneously connect together (k>1), or a single point at the center of a system connects to the boundary (k=1), through adjacent connected points of a single cluster. These processes yield new thresholds p¯ck defined as the average value of p at which the desired connections first occur. These thresholds not sharp, as the distribution of values of pck for individual samples remains broad in the limit of L→∞. We study p¯ck for bond percolation on the square lattice and find that p¯ck are above the normal percolation threshold pc=1/2 and represent specific supercritical states. The p¯ck can be related to integrals over powers of the function P∞(p) equal to the probability a point is connected to the infinite cluster; we find numerically from both direct simulations and from measurements of P∞(p) on L×L systems that for L→∞, p¯c1=0.51755(5), p¯c2=0.53219(5), p¯c3=0.54456(5), and p¯c4=0.55527(5). The percolation thresholds p¯ck remain the same, even when the k points are randomly selected within the lattice. We show that the finite-size corrections scale as L−1/νk where νk=ν/(kβ+1), with β=5/36 and ν=4/3 being the ordinary percolation critical exponents, so that ν1=48/41, ν2=24/23, ν3=16/17, ν4=6/7, etc. We also study three-point correlations in the system and show how for p>pc, the correlation ratio goes to 1 (no net correlation) as L→∞, while at pc it reaches the known value of 1.022.
Physical Review E, Volume 101; https://doi.org/10.1103/physreve.101.052138

Abstract:
We study the Ising model on the square lattice (Z2) and show, via numerical simulation, that allowing interactions between spins separated by distances 1 and m (two ranges), the critical temperature, Tc(m), converges monotonically to the critical temperature of the Ising model on Z4 as m→∞. Only interactions between spins located in directions parallel to each coordinate axis are considered. We also simulated the model with interactions between spins at distances of 1, m, and u (three ranges), with u a multiple of m; in this case our results indicate that Tc(m,u) converges to the critical temperature of the model on Z6. For percolation, analogous results were proven for the critical probability pc [B. N. B. de Lima, R. P. Sanchis, and R. W. C. Silva, Stochast. Process. Appl. 121, 2043 (2011)].
Zhipeng Xun,
Physical Review Research, Volume 2; https://doi.org/10.1103/physrevresearch.2.013067

Abstract:
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 pc=0.1601312(2), which confirms previous results (based on other methods), and find a new value pc=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 pc=0.074212(1) and 0.049517(1), which are substantially more precise than previous values. We also find critical exponents τ=2.3135(5) and Ω=0.40(3), consistent with previous numerical results and the recent four-loop series result of Gracey [Phys. Rev. D 92, 025012 (2015)].
, Mehran Kardar
Physical Review E, Volume 100; https://doi.org/10.1103/PhysRevE.100.022125

Abstract:
How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of N=uLd steps on a d-dimensional hypercubic lattice of size Ld (with periodic boundaries). We systematically explore dependence of the probability Πd(L,u) of percolation (existence of a spanning cluster) of sites not removed by the RW on L and u. The concentration of unvisited sites decays exponentially with increasing u, while the visited sites are highly correlated—their correlations decaying with the distance r as 1/rd−2 (in d>2). On increasing L, the percolation probability Πd(L,u) approaches a step function, jumping from 1 to 0 when u crosses a percolation threshold uc that is close to 3 for all 3≤d≤6. Within numerical accuracy, the correlation length associated with percolation diverges with exponents consistent with ν=2/(d−2). There is no percolation threshold at the lower critical dimension of d=2, with the percolation probability approaching a smooth function Π2(∞,u)>0.
Giulio Biroli, ,
Published: 12 February 2019
Physical Review E, Volume 99; https://doi.org/10.1103/physreve.99.022118

Abstract:
Recent advances on the glass problem motivate reexamining classical models of percolation. Here we consider the displacement of an ant in a labyrinth near the percolation threshold on cubic lattices both below and above the upper critical dimension of simple percolation, du=6. Using theory and simulations, we consider the scaling regime and obtain that both caging and subdiffusion scale logarithmically for d≥du. The theoretical derivation, which considers Bethe lattices with generalized connectivity and a random graph model, confirms that logarithmic scalings should persist in the limit d→∞. The computational validation employs accelerated random walk simulations with a transfer-matrix description of diffusion to evaluate directly the dynamical critical exponents below du as well as their logarithmic scaling above du. Our numerical results improve various earlier estimates and are fully consistent with our theoretical predictions.
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