One-dimensional long-range percolation: A numerical study

In this paper we study bond percolation on a one-dimensional chain with power-law bond probability C /r^{d +σ} , where r is the distance length between distinct sites and d =1 . We introduce and test an order-N Monte Carlo algorithm and we determine as a function of σ the critical value C_c at which percolation occurs. The critical exponents in the range 0 <σ <1 are reported. Our analysis is in agreement, up to a numerical precision ≈10^-3 , with the mean-field result for the anomalous dimension η =2 -σ , showing that there is no correction to η due to correlation effects. The obtained values for C_c are compared with a known exact bound, while the critical exponent ν is compared with results from mean-field theory, from an expansion around the point σ =1 and from the ɛ -expansion used with the introduction of a suitably defined effective dimension d_eff relating the long-range model with a short-range one in dimension d_eff. We finally present a formulation of our algorithm for bond percolation on general graphs, with order N efficiency on a large class of graphs including short-range percolation and translationally invariant long-range models in any spatial dimension d with σ >0 .