2,097 research outputs found

    Pair Connectedness and Shortest Path Scaling in Critical Percolation

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    We present high statistics data on the distribution of shortest path lengths between two near-by points on the same cluster at the percolation threshold. Our data are based on a new and very efficient algorithm. For d=2d=2 they clearly disprove a recent conjecture by M. Porto et al., Phys. Rev. {\bf E 58}, R5205 (1998). Our data also provide upper bounds on the probability that two near-by points are on different infinite clusters.Comment: 7 pages, including 4 postscript figure

    Universality of critically pinned interfaces in 2-dimensional isotropic random media

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    Based on extensive simulations, we conjecture that critically pinned interfaces in 2-dimensional isotropic random media with short range correlations are always in the universality class of ordinary percolation. Thus, in contrast to interfaces in >2>2 dimensions, there is no distinction between fractal (i.e., percolative) and rough but non-fractal interfaces. Our claim includes interfaces in zero-temperature random field Ising models (both with and without spontaneous nucleation), in heterogeneous bootstrap percolation, and in susceptible-weakened-infected-removed (SWIR) epidemics. It does not include models with long range correlations in the randomness, and models where overhangs are explicitly forbidden (which would imply non-isotropy of the medium).Comment: 5 pages (including 8 figures) of main text + 5 pages (including 7 figures) supplemental materia

    Critical Behaviour of the Drossel-Schwabl Forest Fire Model

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    We present high statistics Monte Carlo results for the Drossel-Schwabl forest fire model in 2 dimensions. They extend to much larger lattices (up to 65536×6553665536\times 65536) than previous simulations and reach much closer to the critical point (up to θp/f=256000\theta \equiv p/f = 256000). They are incompatible with all previous conjectures for the (extrapolated) critical behaviour, although they in general agree well with previous simulations wherever they can be directly compared. Instead, they suggest that scaling laws observed in previous simulations are spurious, and that the density ρ\rho of trees in the critical state was grossly underestimated. While previous simulations gave ρ0.408\rho\approx 0.408, we conjecture that ρ\rho actually is equal to the critical threshold pc=0.592...p_c = 0.592... for site percolation in d=2d=2. This is however still far from the densities reachable with present day computers, and we estimate that we would need many orders of magnitude higher CPU times and storage capacities to reach the true critical behaviour -- which might or might not be that of ordinary percolation.Comment: 8 pages, including 9 figures, RevTe

    Free energy and extension of a semiflexible polymer in cylindrical confining geometries

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    We consider a long, semiflexible polymer, with persistence length PP and contour length LL, fluctuating in a narrow cylindrical channel of diameter DD. In the regime DPLD\ll P\ll L the free energy of confinement ΔF\Delta F and the length of the channel RR_\parallel occupied by the polymer are given by Odijk's relations ΔF/R=AkBTP1/3D2/3\Delta F/R_\parallel=A_\circ k_BTP^{-1/3}D^{-2/3} and R=L[1α(D/P)2/3]R_\parallel=L[1-\alpha_\circ(D/P)^{2/3}], where AA_\circ and α\alpha_\circ are dimensionless amplitudes. Using a simulation algorithm inspired by PERM (Pruned Enriched Rosenbluth Method), which yields results for very long polymers, we determine AA_\circ and α\alpha_\circ and the analogous amplitudes for a channel with a rectangular cross section. For a semiflexible polymer confined to the surface of a cylinder, the corresponding amplitudes are derived with an exact analytic approach. The results are relevant for interpreting experiments on biopolymers in microchannels or microfluidic devices.Comment: 15 pages without figures, 5 figure

    Self-organized criticality and directed percolation

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    A sandpile model with stochastic toppling rule is studied. The control parameters and the phase diagram are determined through a MF approach, the subcritical and critical regions are analyzed. The model is found to have some similarities with directed percolation, but the existence of different boundary conditions and conservation law leads to a different universality class, where the critical state is extended to a line segment due to self-organization. These results are supported with numerical simulations in one dimension. The present model constitute a simple model which capture the essential difference between ordinary nonequilibrium critical phenomena, like DP, and self-organized criticality.Comment: 9 pages, 10 eps figs, revtex, submitted to J. Phys.

    Epidemic analysis of the second-order transition in the Ziff-Gulari-Barshad surface-reaction model

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    We study the dynamic behavior of the Ziff-Gulari-Barshad (ZGB) irreversible surface-reaction model around its kinetic second-order phase transition, using both epidemic and poisoning-time analyses. We find that the critical point is given by p_1 = 0.3873682 \pm 0.0000015, which is lower than the previous value. We also obtain precise values of the dynamical critical exponents z, \delta, and \eta which provide further numerical evidence that this transition is in the same universality class as directed percolation.Comment: REVTEX, 4 pages, 5 figures, Submitted to Physical Review

    Scaling of loop-erased walks in 2 to 4 dimensions

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    We simulate loop-erased random walks on simple (hyper-)cubic lattices of dimensions 2,3, and 4. These simulations were mainly motivated to test recent two loop renormalization group predictions for logarithmic corrections in d=4d=4, simulations in lower dimensions were done for completeness and in order to test the algorithm. In d=2d=2, we verify with high precision the prediction D=5/4D=5/4, where the number of steps nn after erasure scales with the number NN of steps before erasure as nND/2n\sim N^{D/2}. In d=3d=3 we again find a power law, but with an exponent different from the one found in the most precise previous simulations: D=1.6236±0.0004D = 1.6236\pm 0.0004. Finally, we see clear deviations from the naive scaling nNn\sim N in d=4d=4. While they agree only qualitatively with the leading logarithmic corrections predicted by several authors, their agreement with the two-loop prediction is nearly perfect.Comment: 3 pages, including 3 figure