Results for a critical threshold, the correction-to-scaling exponent and susceptibility amplitude ratio for 2d percolation

We summarize several decades of work in finding values for the percolation threshold p_c for site percolation on the square lattice, the universal correction-to-scaling exponent Omega, and the susceptibility amplitude ratio C^+/C^-, in two dimensions. Recent studies have yielded the precise values p_c = 0.59274602(4), Omega = 72/91 = 0.791, and C^+/C^- = 161.5(1.5), resolving long-standing controversies about the last two quantities and verifying the widely used value p_c = 0.592746 for the first.Comment: Talk presented at 24th Annual Workshop:"Recent Developments in Computer Simulational Studies in Condensed Matter Physics," Center for Computational Physics, University of Georgia, Athens, Georgia Feb. 21-25, 201

On the critical behavior of the Susceptible-Infected-Recovered (SIR) model on a square lattice

By means of numerical simulations and epidemic analysis, the transition point of the stochastic, asynchronous Susceptible-Infected-Recovered (SIR) model on a square lattice is found to be c_0=0.1765005(10), where c is the probability a chosen infected site spontaneously recovers rather than tries to infect one neighbor. This point corresponds to an infection/recovery rate of lambda_c = (1-c_0)/c_0 = 4.66571(3) and a net transmissibility of (1-c_0)/(1 + 3 c_0) = 0.538410(2), which falls between the rigorous bounds of the site and bond thresholds. The critical behavior of the model is consistent with the 2-d percolation universality class, but local growth probabilities differ from those of dynamic percolation cluster growth, as is demonstrated explicitly.Comment: 9 pages, 5 figures. Accepted for publication, Physical Review

Boundary conditions in random sequential adsorption

The influence of different boundary conditions on the density of random packings of disks is studied. Packings are generated using the random sequential adsorption algorithm with three different types of boundary conditions: periodic, open, and wall. It is found that the finite size effects are smallest for periodic boundary conditions, as expected. On the other hand, in the case of open and wall boundaries it is possible to introduce an effective packing size and a constant correction term to significantly improve the packing densities.Comment: 9 pages, 7 figure

Re-examination of seven-dimensional site percolation thresholds

Monte Carlo simulations alone could not clarify the corrections to scaling for the size-dependent p_c(L) above the upper critical dimension. Including the previous series estimate for the bulk threshold $p_c(\infty)$ gives preference for the complicated corrections predicted by renormalization group and against the simple 1/L extrapolation. Additional Monte-Carlo simulations using the Leath method corroborate the series result for p_c.Comment: 3 pages, 3 figures, submitte