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We study the very long-range bond-percolation problem on a linear chain with both sites and bonds dilution. Very long range means that the probability $p_{ij}$ for a connection between two occupied sites $i,j$ at a distance $r_{ij}$ decays as a power law, i.e. $p_{ij} = rho/[r_{ij}^alpha N^{1-alpha}]$ when $ 0 le alpha < 1$, and $p_{ij} = rho/[r_{ij} ln(N)]$ when $alpha = 1$. Site dilution means that the occupancy probability of a site is $0 < p_s le 1$. The behavior of this model results from the competition between long-range connectivity, which enhances the percolation, and site dilution, which weakens percolation. The case $alpha=0$ with $p_s =1 $ is well-known, being the exactly solvable mean-field model. The percolation order parameter $P_infty$ is investigated numerically for different values of $alpha$, $p_s$ and $rho$. We show that in the ranges $ 0 le alpha le 1$ and $0 < p_s le 1$ the percolation order parameter $P_infty$ depends only on the average connectivity $gamma$ of sites, which can be explicitly computed in terms of the three parameters $alpha$, $p_s$ and $rho$.
We compare the critical behavior of the short-range Ising spin glass with a spin glass with long-range interactions which fall off as a power sigma of the distance. We show that there is a value of sigma of the long-range model for which the critical
Long-range interacting systems such as nitrogen vacancy centers in diamond and trapped ions serve as useful experimental setups to probe a range of nonequilibrium many-body phenomena. In particular, via driving, various effective Hamiltonians with ph
We study critical behavior of the diluted 2D Ising model in the presence of disorder correlations which decay algebraically with distance as $sim r^{-a}$. Mapping the problem onto 2D Dirac fermions with correlated disorder we calculate the critical p
The dynamics and the stationary states of an exactly solvable three-state layered feed-forward neural network model with asymmetric synaptic connections, finite dilution and low pattern activity are studied in extension of a recent work on a recurren
We perform Monte Carlo simulations to determine the average excluded area $<A_{ex}>$ of randomly oriented squares, randomly oriented widthless sticks and aligned squares in two dimensions. We find significant differences between our results for rando