Equivalent-neighbor Percolation Models In Two Dimensions

In statistical physics and mathematics,percolation theory describes the behavior of connected clusters in a random graph.The application of percolation theory to material science and other domains are discussed.Assume that some liquid is poured on top of some porus material.This physical question is modelled mathematically as a three-dimensional network of n x n x n vertices,usually called "sites",in which the edge or"bonds" between each two neighbors may be open(allowing the liquid through)with probability p,or closed with probability 1-p,and they are assumed to be independent.Therefore,for a given p,what is the probablity that an open path(each of whose links is an "open" bond)exists from the top to the bottom?The behavior for large n is of primary interest.This problem,called now bond percolation,was introduced in the mathematics literature by Broadbent&Hammersley(1957),and has been studied intensively by mathematicians and physicists since then.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 complete graph case,where percolation bonds are al-lowed between each pair of sites and the model becomes mean-field-like.For finite z,all investigated models belong to the short-range universality class.There is no ev-idence of a tricritical point seperating the short-range and long-range behavior,such as is known to occur for q = 3 and q = 4 Potts models.Instead,we find continu-ous crossover between mean-field percolation univerality and short-range percolation universality.The crossover exponent at the mean-field fixed point and approximate re-lations between the coordination numbers z and amplitudes of the correction terms for finite interaction ranges are determined by finite-size scaling methods.We determine th renormalization exponent describing a finite-range perturbation at the mean-field limit as yr? 2/3.Its relevance confirms the continuous crossover from mean-field percola-tion universality to short-range percolation universality.For finite interaction ranges,e find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size analysis.