We present the results of a study of random sequential adsorption of linear segments (needles) on sites of a square lattice. We show that the percolation threshold is a nonmonotonic function of the length of the adsorbed needle, showing a minimum for a certain length of the needles, while the jamming threshold decreases to a constant with a power law. The ratio of the two thresholds is also nonmonotonic and it remains constant only in a restricted range of the needles length. We determine the values of the correlation length exponent for percolation, jamming, and their ratio.
A one-dimensional Ising model is studied, via Monte Carlo simulations, on a small world network, where each site has, apart from couplings to its two nearest neighbors, a certain probability to be linked to one of its farther neighbors. It is demonstrated that even a small fraction of such links enables the system to order at finite temperatures. The critical exponent beta is smaller than the two-dimensional value, and seems to be independent of the concentration of the extra links. The dependence of the magnetization and the critical temperature on the concentration of the small world links is also presented.
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