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

TL;DR

This study uses Monte Carlo simulations to explore how interaction range affects phase transitions in 2D bond percolation, revealing a continuous crossover from mean-field to short-range universality without evidence of a tricritical point.

Contribution

It provides the first detailed analysis of the crossover between mean-field and short-range universality in 2D percolation models with variable interaction ranges.

Findings

Finite-range models belong to the short-range universality class.

No tricritical point separating the two behaviors was observed.

The renormalization exponent for finite-range perturbations is approximately 2/3.

Abstract

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 $z \to \infty$ 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 $y_r \approx 2/3$. 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.