Q-state Bond-correlated Percolation Model Research Articles

Recent developments in the Monte Carlo approach to percolation problems

Using a Monte Carlo method, Hu, Lin and Chen found that bond and site percolation models on planar lattices have universal finite-size scaling functions for the probability W m for the appearance of m percolating clusters, which implies that the average number of percolating clusters, C, is a universal quantity. Hu and Lin found that C increases linearly with the aspect ratio, R, of the lattice for large R, i.e. in this case C= aR with a constant a. Hu and Lin also found that a is apparently independent of the boundary conditions. For the periodic boundary conditions in both horizontal and vertical directions, Ziff et al. found that the number of clusters per lattice site, n, for percolation on planar lattices of linear dimensions L can be written as n= n c + b/ N, where n c is n in the limit L→∞, b is a constant, and N is the number of lattice sites. Ziff et al. found that b is universal and argued that b is the number of percolating clusters so that its universality may be related to the universality of C. Hu found that for large R, b= b c R, but b c ≠ a. In this paper, we use a cluster Monte Carlo simulation method to calculate the number of clusters per site, n, of a q-state bond-correlated percolation model (QBCPM) which is equivalent to the q-state Potts on L× L square lattices. We find that for q≥2 the slopes of n as a function of 1/ L 2 are negative. For q=2, i.e. the Ising model, we find that our data can be well represented by n=n c−c/L+b/L 2+⋯ , where b can be calculated exactly from conformal field theory, c>0 and can be calculated exactly from the critical internal energy of the Ising model.

Geometry, thermodynamics, and finite-size corrections in the critical Potts model

We establish an intriguing connection between geometry and thermodynamics in the critical q-state Potts model on two-dimensional lattices, using the q-state bond-correlated percolation model (QBCPM) representation. We find that the number of clusters <N(c)> of the QBCPM has an energy-like singularity for q not equal to 1, which is reached and supported by exact results, numerical simulation, and scaling arguments. We also establish that the finite-size correction to the number of bonds, <N(b)>, has no constant term and explains the divergence of related quantities as q-->4, the multicritical point. Similar analyses are applicable to a variety of other systems.

Universal scaling functions and quantities in percolation models

We briefly review recent work on universal finite-size scaling functions (UFSSFs) and quantities in percolation models. The topics under discussion include: (a) UFSSFs for the existence probability (also called crossing probability) E p , the percolation probability P, and the probability W n of the appearance of n percolating clusters, (b) universal slope for average number of percolating clusters, (c) UFSSFs for a q-state bond-correlated percolation model corresponding to the q-state Potts model. We also briefly mention some very recent related developments and discuss implications of our results.

Histogram Monte Carlo approach to thermal properties of the Potts model on planar lattices

Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), we use the histogram Monte Carlo simulation method to calculate the free energy F e , the internal energy U, and the specific heat C h of the QPM on the honeycomb, the square, the Kagome, and the plane triangular lattices and for q being 2–5, where q = 2 corresponds to the Ising model. Our results for the Ising model agree very well with exact results for thermodynamic systems when the temperature is away from the critical temperature.

Histogram-importance-sampling Monte Carlo method for the q-state Potts model.

Based on Hu's histogram Monte Carlo method and the Swendsen-Wang algorithm, we calculate the geometrical factor of the q-state bond-correlated percolation model corresponding to the q-state Potts model (QPM). The free energy f, the internal energy U, and the specific heat ${\mathit{C}}_{\mathit{h}}$ of the Ising model, i.e., the QPM with q=2, calculated from this geometrical factor agree very well with exact results. This histogram-important-sampling Monte Carlo method is very simple and has many potential applications.

Scaling functions of the q-state Potts model on planar lattices

Based on the subgraph expansion of the q-state Potts model (QPM) in an external field, it has been shown that the QPM is corresponding to a q-state bond-correlated percolation model (QBCPM). The histogram Monte Carlo simulation method proposed by Hu is used to calculate the existence probability E P( G, p, q and the percolation probability P(G, p, q) of the QBCPM on the honeycomb, the Kagome, and the plane triangular lattices with various linear dimensions. From E p( G, p, q) and P(G, p, q) we obtain scaling functions of the QPM and QBCPM. We find that as q or the coordination number of the lattices increases, the widths of the scaling functions also increase. The implication of this study is discussed.

Percolation, fractals and the finite-size scaling of existence probability

Abstract Based on the connection between phase transitions of thermodynamic systems and percolation transitions, we obtain the geometrical meaning of critical exponents using finite-size scaling of the percolation probability P and the existence probability E p for percolating clusters and percolating subgraphs, respectively, where E p is the ratio of the total probability weight for percolating subgraphs to the total probability weight for all subgraphs. We use a histogram Monte Carlo simulation method to calculate E p for a q -state bond-correlated percolation model (QBCPM), which is corresponding to the q -state Potts model, on the square lattice with various linear dimensions. We find that for a given q , E p has very good scaling behavior. But E p for different q has different scaling functions.

Cluster Monte Carlo study of the q-state potts model on hypercubic lattices

The q-state Potts model (QPM) may be related to many interesting mathematical problems and physical systems. Previous studies of the QPM have been restricted mainly to small q values. Using the connection between the QPM and the q-state bond-correlated percolation model (QBCPM) and Swendsen and Wang's cluster Monte Carlo simulation method, in this paper we study the QPM on d-dimensional hypercubic lattices with d being 3, 4, 5 and 6, and q being 2, 3, 4, 8, 16, 32, 64, 128, 256, 512, 1024 and 2048. For q=2 where phase transitions are second order, a percolation Monte Carlo renormalization group method is used to determine more accurate critical points. We find that in the studied space dimensions, the critical points increase slowly with q. As q increases, the critical points for different d approach to each other such that when q=2048 the absolute values of the differences between such critical points are less than 1.1%. We also discuss some related theoretical problems, including implications of our results to the percolation theory of supercooled water.

FORTRAN code for the cluster Monte Carlo study of the q-state Potts model on D-dimensional hypercubic lattices

Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM) proposed by Hu and a fast cluster Monte Carlo simulation method proposed by Swendsen and Wang, we have written a FORTRAN program to calculate the spontaneous magnetization, the magnetic susceptibility, the internal energy, and the specific heat of the QPM on the D-dimensional hypercubic lattice with the periodic boundary condition.

Fast algorithm to calculate exact geometrical factors for the q-state Potts model.

Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model, Hu has defined the percolating geometrical factor ${\mathit{N}}_{\mathit{p}}$ and the nonpercolating geometrical factors ${\mathit{N}}_{\mathit{f}}$. From ${\mathit{N}}_{\mathit{p}}$ nd ${\mathit{N}}_{\mathit{f}}$, we may obtain the thermal properties of the QPM and construct a percolation renormalization-group method to study the QPM. In this paper we present a fast algorithm to calculate exact ${\mathit{N}}_{\mathit{p}}$ and ${\mathit{N}}_{\mathit{f}}$ for the QPM on finite cells.

Percolation renormalization-group calculations of equations of state for the Potts model.

The connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM) has been used to formulate a percolation renormalization group method (PRGM) for calculating the free energy, the critical point ${p}_{c}$, and critical exponents for the QPM. In this paper the PRGM is extended to calculate the percolation probability P, the backbone mass density ${P}_{B}$, and the backbone fractal dimensions ${D}_{B}$ for the QBCPM and the QPM on the square lattice.

Percolation renormalization-group approach to the q-state Potts model.

Based on the connection between the q-state Potts model (QPM) and the q-state bond-correlated percolation model (QBCPM), we have proposed percolation renormalization-group (PRG) methods to calculate the free energy, the critical point, and critical exponents for the QPM. Our methods are free from inconsistency in Larsson's method. We have carried out the ${\ensuremath{\lambda}}_{1}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{1}$ to ${\ensuremath{\lambda}}_{2}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{2}$ renormalization-group (RG) transformations in our methods for several values of original cell sizes ${\ensuremath{\lambda}}_{1}$ and final cell sizes ${\ensuremath{\lambda}}_{2}$ with various boundary conditions. We find that such RG transformations with large ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ and periodic boundary condition usually give accurate physical quantities for the QPM. The advantages and generalization of our approach are also discussed.

Exact cluster size distributions and mean cluster sizes for the q-state bond-correlated percolation model

The author shows that the q-state bond-correlated percolation model (QBCPM), which is the percolation representation of the q-state Potts model (QPM), on the lattice without closed loops is equivalent to the bond random percolation model (BRPM) on the same lattice. Using such results and exact results for the BRPM on the linear and Bethe lattices, the author obtains exact cluster size distributions and the mean cluster size S for the QBCPM on the linear and Bethe lattices. The mean cluster sizes obtained from this method are the same as those obtained by more tedious exact calculations. Near the critical point, the average number of m site clusters per site, nm, for the QBCPM on the linear and Bethe lattices may be written in the scaling form for large values of m, which is the geometrical basis for the scaling laws of critical exponents.

Cluster-size distribution and the magnetic property of a Potts model

Based on the connection between a q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), we propose that in the calculation of the free energy and a physical quantity, e.g., the magnetic susceptibility, we need only to retain the terms of the most probable cluster-size distribution (MPCSD) defined in the text. For the one-dimensional model, the MPCSD may be calculated exactly. The free energy and the magnetic susceptibility determined by such MPCSD are the same as those calculated exactly by the transfer-matrix method. For the QPM on the lattice with dimensions d\ensuremath{\ge}2, the above assumption about the MPCSD implies that the magnetic susceptibility of the QPM is proportional to the mean cluster sizes of the QBCPM for both T&gt;${T}_{c}$ and T${T}_{c}$. The hypothesis on the MPCSD may be extended to other spin models. The implication of this work on the theory of the scaling laws of the critical exponents is also discussed.

Geometrical factor and thermal properties of the Potts model

Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), we define a non-percolating geometrical factor g, and percolating geometrical factor gp which depend only on q and geometrical properties of the system. The thermal properties of the QPM could be derived simply from g, and g,. The idea is confirmed by an exact calculation for one-dimensional QPM and may be extended to other Ising-like spin models. Our formulation also provides a geometrical meaning for the finite-size scaling and broadening at first-order phase transitions of the spin model.