arXiv:cond-mat/0402294 · Physical Review E

Tails of the crossing probability

Oleg A. Vasilyev

Published Sep 15, 2005• 1 citation

Abstract

The scaling of the tails of the probability of a system to percolate only in the horizontal direction ${\ensuremath{\pi}}_{hs}$ was investigated numerically for the correlated site-bond percolation model ($q$-state Potts model) for $q=1$, 2, 3, 4 (where $q$ is the number of spin states). We have to demonstrate that the crossing probability ${\ensuremath{\pi}}_{hs}(p)$ far from the critical point ${p}_{c}$ has the shape ${\ensuremath{\pi}}_{hs}(p)\ensuremath{\simeq}D\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[cL{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\nu}}]$ where $\ensuremath{\nu}$ is the correlation length index, and $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed. For the tail region the correlation length is smaller than the lattice size. At criticality the correlation length reaches the sample size and we observe crossover to another scaling ${\ensuremath{\pi}}_{hs}(p)\ensuremath{\simeq}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}{\ensuremath{-}b{[L{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\nu}}]}^{x}}$. Here $x$ is a scaling index describing the central part of the crossing probability.

DOI: 10.1103/physreve.72.036115

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Universality of the crossing probability for the Potts model for q=1,2,3,4

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arXiv:cond-mat/0402294 · Physical Review E

Tails of the crossing probability

Oleg A. Vasilyev

Published Sep 15, 2005• 1 citation

arXiv HTML PDF

Abstract

The scaling of the tails of the probability of a system to percolate only in the horizontal direction

{\ensuremath{\pi}}_{hs}

was investigated numerically for the correlated site-bond percolation model (

q

-state Potts model) for

q=1

, 2, 3, 4 (where

q

is the number of spin states). We have to demonstrate that the crossing probability

{\ensuremath{\pi}}_{hs}(p)

far from the critical point

{p}_{c}

has the shape

{\ensuremath{\pi}}_{hs}(p)\ensuremath{\simeq}D\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[cL{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\nu}}]

where

\ensuremath{\nu}

is the correlation length index, and

p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})

is the probability of a bond to be closed. For the tail region the correlation length is smaller than the lattice size. At criticality the correlation length reaches the sample size and we observe crossover to another scaling

{\ensuremath{\pi}}_{hs}(p)\ensuremath{\simeq}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}{\ensuremath{-}b{[L{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\nu}}]}^{x}}

. Here

x

is a scaling index describing the central part of the crossing probability.

DOI: 10.1103/physreve.72.036115

Related papers

Universality of the crossing probability for the Potts model for q=1,2,3,4

Oleg A. Vasilyev

The universality of the crossing probability ${\ensuremath{\pi}}_{\mathrm{hs}}$ of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the q-state Potts model for $q=2,3,4$ and for percolation $q=1.$ We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction ${\ensuremath{\pi}}_{\mathrm{hs}}$ has the universal form ${\ensuremath{\pi}}_{\mathrm{hs}}=A(q)Q(z)$ for $q=1,2,3,4$ as a function of the scaling variable $z={{b(q)L}^{1/\ensuremath{\nu}(q)}[p\ensuremath{-}{p}_{c}(q,L)]{}}^{\ensuremath{\zeta}(q)}.$ Here, $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed, $A(q)$ is the nonuniversal crossing amplitude, $b(q)$ is the nonuniversal metric factor, $\ensuremath{\nu}(q)$ is the correlation length index, and $\ensuremath{\zeta}(q)$ is the additional scaling index. The universal function $Q(x)\ensuremath{\simeq}\mathrm{exp}(\ensuremath{-}|z|).$ The nonuniversal scaling factors were found numerically.The universality of the crossing probability ${\ensuremath{\pi}}_{\mathrm{hs}}$ of a system to percolate only in the horizontal direction was investigated numerically by a cluster Monte Carlo algorithm for the q-state Potts model for $q=2,3,4$ and for percolation $q=1.$ We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction ${\ensuremath{\pi}}_{\mathrm{hs}}$ has the universal form ${\ensuremath{\pi}}_{\mathrm{hs}}=A(q)Q(z)$ for $q=1,2,3,4$ as a function of the scaling variable $z={{b(q)L}^{1/\ensuremath{\nu}(q)}[p\ensuremath{-}{p}_{c}(q,L)]{}}^{\ensuremath{\zeta}(q)}.$ Here, $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed, $A(q)$ is the nonuniversal crossing amplitude, $b(q)$ is the nonuniversal metric factor, $\ensuremath{\nu}(q)$ is the correlation length index, and $\ensuremath{\zeta}(q)$ is the additional scaling index. The universal function $Q(x)\ensuremath{\simeq}\mathrm{exp}(\ensuremath{-}|z|).$ The nonuniversal scaling factors were found numerically.

arXiv:cond-mat/0204398Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsAug 25, 20033 citations

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arXiv:cond-mat/0005452May 25, 20000 citations

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The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability $p$ depending on the NN coupling constant $J$ and the temperature $T$ , we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.The phase transition in the Ising model and the percolation transition in the lattice percolation model have many common characteristics which have motivated researchers to explore whether the former is a percolation transition of a correlated percolation model. Previous attempts to draw such a connection have been either unsuccessful or unsatisfactory. Considering each lattice site with an Ising spin occupied and the nearest-neighbor (NN) coupling between occupied sites as a bond with a bond probability $p$ depending on the NN coupling constant $J$ and the temperature $T$, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

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A Study of Cross-Over Effects For The 2D Random Bond Potts Model

Marco Picco

We present results of a numerical simulation of the $q$ -state random bond Potts model in two dimensions and for large $q$ . In particular, care is taken to study the crossover from the pure model to the random model, as well as the crossover from the percolation to the random model. We show how to determine precisely the random fixed point and measure critical exponents at this point.We present results of a numerical simulation of the $q$-state random bond Potts model in two dimensions and for large $q$. In particular, care is taken to study the crossover from the pure model to the random model, as well as the crossover from the percolation to the random model. We show how to determine precisely the random fixed point and measure critical exponents at this point.

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Bond and site percolation in three dimensions

Junfeng Wang, Zongzheng Zhou, Wei Zhang +2 more

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to $L=512$ , and estimate the percolation thresholds to be ${p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)$ and ${p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)$ . By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent ${d}_{\mathrm{min}}=1.3756(3)$ . Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to $L=512$, and estimate the percolation thresholds to be ${p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)$ and ${p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)$. By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent ${d}_{\mathrm{min}}=1.3756(3)$. Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.

arXiv:1302.0421Physical Review EMay 7, 2013170 citations

Long-range correlated percolation

Abel Weinrib

This paper is a study of the percolation problem with long-range correlations in the site or bond occupations. An extension of the Harris criterion for the relevance of the correlations is derived for the case that the correlations decay as ${x}^{\ensuremath{-}a}$ for large distances $x$ .. For $a<d$ the correlations are relevant if $a\ensuremath{\nu}\ensuremath{-}2<0$ (where $\ensuremath{\nu}$ is the percolation correlation-length exponent), while for $a>d$ the correlations are relevant if $d\ensuremath{\nu}\ensuremath{-}2<0$.. Applying this criterion to the behavior that results when the correlations are relevant, we argue that the new behavior will have ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$.. It is shown that the correlated bond percolation problem is equivalent to a $q$ -state Potts model with quenched disorder in the limit $q\ensuremath{\rightarrow}1$.. With the use of this result, a renormalization-group study of the problem is presented, expanding in $\ensuremath{\epsilon}=6\ensuremath{-}d$ and in $\ensuremath{\delta}=4\ensuremath{-}a$.. In addition to the normal percolation fixed point, we find a new long-range fixed point. The crossover to this new fixed point follows the extended Harris criterion, and the fixed point has exponents ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$ (as predicted) and ${\ensuremath{\eta}}_{\mathrm{long}}=\frac{1}{11}(\ensuremath{\delta}\ensuremath{-}\ensuremath{\epsilon})$. Finally, several results on the percolation properties of the Ising model at its critical point are shown to be in agreement with the predictions of this paper.This paper is a study of the percolation problem with long-range correlations in the site or bond occupations. An extension of the Harris criterion for the relevance of the correlations is derived for the case that the correlations decay as ${x}^{\ensuremath{-}a}$ for large distances $x$.. For $a<d$ the correlations are relevant if $a\ensuremath{\nu}\ensuremath{-}2<0$ (where $\ensuremath{\nu}$ is the percolation correlation-length exponent), while for $a>d$ the correlations are relevant if $d\ensuremath{\nu}\ensuremath{-}2<0$.. Applying this criterion to the behavior that results when the correlations are relevant, we argue that the new behavior will have ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$.. It is shown that the correlated bond percolation problem is equivalent to a $q$-state Potts model with quenched disorder in the limit $q\ensuremath{\rightarrow}1$.. With the use of this result, a renormalization-group study of the problem is presented, expanding in $\ensuremath{\epsilon}=6\ensuremath{-}d$ and in $\ensuremath{\delta}=4\ensuremath{-}a$.. In addition to the normal percolation fixed point, we find a new long-range fixed point. The crossover to this new fixed point follows the extended Harris criterion, and the fixed point has exponents ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$ (as predicted) and ${\ensuremath{\eta}}_{\mathrm{long}}=\frac{1}{11}(\ensuremath{\delta}\ensuremath{-}\ensuremath{\epsilon})$. Finally, several results on the percolation properties of the Ising model at its critical point are shown to be in agreement with the predictions of this paper.

Physical review. B, Condensed matterJan 1, 1984177 citations

The scaling of the tails of the probability of a system to percolate only in the horizontal direction

{\ensuremath{\pi}}_{hs}

was investigated numerically for the correlated site-bond percolation model (

q

-state Potts model) for

q=1

, 2, 3, 4 (where

q

is the number of spin states). We have to demonstrate that the crossing probability

{\ensuremath{\pi}}_{hs}(p)

far from the critical point

{p}_{c}

has the shape

{\ensuremath{\pi}}_{hs}(p)\ensuremath{\simeq}D\phantom{\rule{0.2em}{0ex}}\mathrm{exp}[cL{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\nu}}]

where

\ensuremath{\nu}

is the correlation length index, and

p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})

{\ensuremath{\pi}}_{hs}(p)\ensuremath{\simeq}A\phantom{\rule{0.2em}{0ex}}\mathrm{exp}{\ensuremath{-}b{[L{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\nu}}]}^{x}}

. Here

x

is a scaling index describing the central part of the crossing probability.

q=2,3,4

and for percolation

q=1.

We check the percolation through Fortuin-Kasteleyn clusters near the critical point on the square lattice by using representation of the Potts model as the correlated site-bond percolation model. It was shown that probability of a system to percolate only in the horizontal direction ${\ensuremath{\pi}}_{\mathrm{hs}}$ has the universal form ${\ensuremath{\pi}}_{\mathrm{hs}}=A(q)Q(z)$ for

q=1,2,3,4

as a function of the scaling variable $z={{b(q)L}^{1/\ensuremath{\nu}(q)}[p\ensuremath{-}{p}_{c}(q,L)]{}}^{\ensuremath{\zeta}(q)}.$ Here, $p=1\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\beta})$ is the probability of a bond to be closed,

A(q)

is the nonuniversal crossing amplitude,

b(q)

is the nonuniversal metric factor, $\ensuremath{\nu}(q)$ is the correlation length index, and $\ensuremath{\zeta}(q)$ is the additional scaling index. The universal function $Q(x)\ensuremath{\simeq}\mathrm{exp}(\ensuremath{-}|z|).$ The nonuniversal scaling factors were found numerically.

We present study of finite-size scaling and universality of crossing probabilities for the

q

-state Potts model. Crossing probabilities of the Potts model are similar ones in percolation problem. We numerically investigated scaling of

π_{s}

- the probability of a system to percolate only in one direction for two-dimensional site percolation, the Ising model, and the q-state Potts model for

q=3,4,5,6,8,10

. We found the thermal scaling index

y= \frac{1}ν

for

q<4

. In contrast,

y \ne \frac{1}ν

for

q=4

p

depending on the NN coupling constant

J

and the temperature

T

, we formally show that the partition function of the Ising model is the generating function of bond-correlated percolation model (BCPM) with a bond probability $p=1\ensuremath{-}\mathrm{exp}(\frac{\ensuremath{-}2J}{\mathrm{kT}})$. The BCPM has the Ising critical temperature and exponents, including $\ensuremath{\nu},{\ensuremath{\nu}}^{\ensuremath{'}},\ensuremath{\eta},\ensuremath{\beta},\ensuremath{\alpha},{\ensuremath{\alpha}}^{\ensuremath{'}}$, and $\ensuremath{\gamma}$ (perhaps also ${\ensuremath{\gamma}}^{\ensuremath{'}}$). From the connection between the Ising model and the BCPM, we also derive and hence give a geometrical meaning of the finite-size scaling and broadening at first-order phase transitions of the Ising model. Our approach may easily be extended to many spin models and give geometrical meaning to other properties of spin models.

We present results of a numerical simulation of the

q

-state random bond Potts model in two dimensions and for large

q

. In particular, care is taken to study the crossover from the pure model to the random model, as well as the crossover from the percolation to the random model. We show how to determine precisely the random fixed point and measure critical exponents at this point.

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to

L=512

, and estimate the percolation thresholds to be

{p}_{c}(\mathrm{bond})=0.248\phantom{\rule{0.16em}{0ex}}811\phantom{\rule{0.16em}{0ex}}82(10)

and

{p}_{c}(\mathrm{site})=0.311\phantom{\rule{0.16em}{0ex}}607\phantom{\rule{0.16em}{0ex}}7(2)

. By performing extensive simulations at these estimated critical points, we then estimate the critical exponents $1/\ensuremath{\nu}=1.1410(15)$, $\ensuremath{\beta}/\ensuremath{\nu}=0.477\phantom{\rule{0.16em}{0ex}}05(15)$, the leading correction exponent ${y}_{i}=\ensuremath{-}1.2(2)$, and the shortest-path exponent

{d}_{\mathrm{min}}=1.3756(3)

. Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent $\ensuremath{\approx}$$\ensuremath{-}2$, rather than ${y}_{i}\ensuremath{\approx}\ensuremath{-}1.2$.

x

.. For

a<d

the correlations are relevant if $a\ensuremath{\nu}\ensuremath{-}2<0$ (where $\ensuremath{\nu}$ is the percolation correlation-length exponent), while for

a>d

the correlations are relevant if $d\ensuremath{\nu}\ensuremath{-}2<0$.. Applying this criterion to the behavior that results when the correlations are relevant, we argue that the new behavior will have ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$.. It is shown that the correlated bond percolation problem is equivalent to a

q

-state Potts model with quenched disorder in the limit $q\ensuremath{\rightarrow}1$.. With the use of this result, a renormalization-group study of the problem is presented, expanding in $\ensuremath{\epsilon}=6\ensuremath{-}d$ and in $\ensuremath{\delta}=4\ensuremath{-}a$.. In addition to the normal percolation fixed point, we find a new long-range fixed point. The crossover to this new fixed point follows the extended Harris criterion, and the fixed point has exponents ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$ (as predicted) and ${\ensuremath{\eta}}_{\mathrm{long}}=\frac{1}{11}(\ensuremath{\delta}\ensuremath{-}\ensuremath{\epsilon})$. Finally, several results on the percolation properties of the Ising model at its critical point are shown to be in agreement with the predictions of this paper.