Multi-range percolation model on the square lattice

Abstract

In this paper, we present a numerical study to describe the behavior of the critical point in the multi-range percolation model on Z2, for bonds and sites, and a new approach for obtaining the critical point of the ordinary Bernoulli percolation model on Zd when d is even. It is known that, under certain conditions for the ranges sizes, the critical point of the multi-range percolation model on Z2, with n different ranges of sizes larger then 1, converges to that of the ordinary Bernoulli percolation model on Z2(n+1) when the ranges sizes diverge. We observe that this convergence is monotonous and follows a power law for n=1 and n=2. On the contrary, in the case that the conditions for the ranges sizes do not hold, we show that the behavior of the critical point is irregular and propose a heuristic to explain this peculiar behavior. We also present algorithms for simulating the model with two ranges of different size considering independent probabilities for the bond of each size to be open or closed and construct the phase diagram for this model. These results are a numerical evidence regarding the conjecture that the critical point of the multi-range percolation model on Z2 converges monotonically to the critical point of the ordinary Bernoulli percolation model on Z2(n+1).