Equivalent-neighbor percolation models in two dimensions: Crossover between mean-field and short-range behavior

We investigate the influence of the range of interactions in the two-dimensional bond percolation model by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number z of equivalent neighbors. We also consider the z -> infinity limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite z are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the short-range and long-range behavior, such as is known to occur for q = 3 and q = 4 Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as y(r) approximate to 2/3. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.