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Journal of Physics A Mathematical and General

Study of droplets for correlated site-bond percolation in two dimensions

Naeem Jan, Antonio Coniglio A, Dietrich Stauffer

Published Dec 1, 1982• 16 citations
Publisher

Abstract

The authors study the droplet size distribution of the correlated site-bond percolation model introduced by Coniglio and Klein (1980), and also the usual clusters of two-dimensional Ising models near the critical point. Equilibrium configurations of the Ising model with nearest-neighbour interaction and also one with nearest- and next-nearest-neighbour interactions are generated through a Monte Carlo simulation, and then a cluster analysis is performed. The exponents beta and gamma for the Coniglio and Klein droplet distribution are found to agree, for both the nearest-neighbour and the next-nearest-neighbour model, with the corresponding exponents of the Ising model. The usual Ising clusters diverge only at T c in the Ising model with nearest-neighbour interaction but not for the model with next-nearest-neighbour interaction. The Potts model formulation is used to predict the behaviour of the droplet for general further-neighbour interactions.

DOI
10.1088/0305-4470/15/12/008

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The authors study the droplet size distribution of the correlated site-bond percolation model introduced by Coniglio and Klein (1980), and also the usual clusters of two-dimensional Ising models near the critical point. Equilibrium configurations of the Ising model with nearest-neighbour interaction and also one with nearest- and next-nearest-neighbour interactions are generated through a Monte Carlo simulation, and then a cluster analysis is performed. The exponents beta and gamma for the Coniglio and Klein droplet distribution are found to agree, for both the nearest-neighbour and the next-nearest-neighbour model, with the corresponding exponents of the Ising model. The usual Ising clusters diverge only at T c in the Ising model with nearest-neighbour interaction but not for the model with next-nearest-neighbour interaction. The Potts model formulation is used to predict the behaviour of the droplet for general further-neighbour interactions.
In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point Tc{T}_{c}Tc​ of the thermal transition and the percolation exponents belong to a special universality class. By introducing a bond probability pB<1,{p}_{B}<1,pB​<1, the corresponding site-bond clusters keep on percolating at Tc{T}_{c}Tc​ and the exponents do not change, until ${p}_{B}{=p}_{\mathrm{CK}}=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}2J/kT):$ for this special expression of the bond weight the critical percolation exponents switch to the two-dimensional Ising universality class. We show here that the result is valid for a wide class of bidimensional models with a continuous magnetization transition: there is a critical bond probability pc{p}_{c}pc​ such that, for any pB> pc,{p}_{B}>~{p}_{c},pB​> pc​, the onset of percolation of the site-bond clusters coincides with the critical point of the thermal transition. The percolation exponents are the same for pc<pB< 1{p}_{c}<{p}_{B}<~1pc​<pB​< 1 but, for pB=pc,{p}_{B}{=p}_{c},pB​=pc​, they suddenly change to the thermal exponents, so that the corresponding clusters are critical droplets of the phase transition. Our result is based on Monte Carlo simulations of various systems near criticality.
The critical behaviour of several spin models can be simply described as percolation of some suitably defined clusters or droplets: the onset of the geometrical transition coincides with the critical point, and the percolation exponents are equal to the thermal exponents. It is still unknown whether, given a model, one can define at all the droplets. In the cases where this is possible, the droplet definition depends in general on the specific model at study and can be quite involved. We propose here a simple general definition for the droplets: they are clusters obtained by joining nearest-neighbour spins of the same sign with some bond probability p CK , which is the minimal probability that still allows the existence of a percolating cluster at the critical temperature T c . This definition has been recently introduced and indeed satisfies the conditions required for the droplets, for many classical spin models, discrete and continuous, in two dimensions. By means of lattice Monte Carlo simulations we show here that the definition works as well in three dimensions. In particular, our prescription allows us to describe exactly the confinement–deconfinement transition of SU ( N ) gauge theories as Polyakov loop percolation.
We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than the nearest neighbours. We show that the phase diagram for next-nearest neighbour percolation can be exactly obtained from the nearest-neighbour case, which implies that the percolation threshold, , is still equal to the Ising critical temperature . In addition, we present Monte Carlo calculations of the finite size behaviour of the correlated resistor network defined on the Ising model. The thermal exponent, t, of the conductivity that follows from it is found to be . We observe no corrections to scaling in its finite size behaviour.
Clusters in the Potts model are connected sets of nearest neighbour sites for which the spin variables are in the same state. Droplets can be obtained by adding bonds with probability p=1-exp(-K) between the sites in a cluster (where K is the Potts model's inverse temperature). When q to 4, renormalization group (RG) fixed points describing clusters and droplets will coalesce, leading to logarithmic corrections. We calculate the precise form of these corrections used in a differential RG method. Our predictions are then tested using extensive Monte Carlo calculations. Theory and the simulations are found to be in excellent agreement.
Using formal arguments based on conformal invariance and on the connection between correlated-site percolation and the q-state Potts model with vacancies, we show that the exponents describing Ising clusters at Onsager's critical point are those of the tricritical q=1 Potts model. This implies, in particular, a fractal dimension d\ifmmode\bar\else\textasciimacron\fi{}=(187/96 and a percolative susceptibility exponent \ensuremath{\gamma}=(91/48, in good agreement with existing numerical estimates. This d\ifmmode\bar\else\textasciimacron\fi{} is also clearly supported by a new very accurate Monte Carlo finite-size scaling determination. We also conjecture an exponent yJ{\mathrm{y}}_{\mathrm{J}}yJ​=(13/24 controlling the crossover between clusters and droplets.