Lecture notes in physics

Finite-Dimensional Percolation

Anders Malthe‐Sørenssen

Published Jan 1, 2024• 0 citations

Abstract

Abstract In this chapter we apply the scaling theories developed in the one-dimensional system and the infinite-dimensional system to systems of finite dimensions. The lowest dimension with an interesting behavior is two dimensions. Here, we introduce effective ways to measure the cluster number density $$n(s,p)$$ n ( s , p ) in two dimension. We develop the scaling theory for $$n(s,p)$$ n ( s , p ) and demonstrate how to use data-collapse plots as an efficient method to measure the critical exponents. We also demonstrate how we can use the scaling theory for $$n(s,p)$$ n ( s , p ) to derive expressions for the density of the spanning cluster, P, and the average cluster size, S. Finally, we demonstrate how the scaling theory provides scaling relations, that is, relations between exponents, and bounds for the values of the critical exponents.

DOI: 10.1007/978-3-031-59900-2_4

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arXiv:2412.06228Dec 9, 20241 citation

Lecture notes in physics

Finite-Dimensional Percolation

Anders Malthe‐Sørenssen

Published Jan 1, 2024• 0 citations

Publisher

Abstract

n(s,p)

n ( s , p ) in two dimension. We develop the scaling theory for

n(s,p)

n ( s , p ) and demonstrate how to use data-collapse plots as an efficient method to measure the critical exponents. We also demonstrate how we can use the scaling theory for

n(s,p)

n ( s , p ) to derive expressions for the density of the spanning cluster, P, and the average cluster size, S. Finally, we demonstrate how the scaling theory provides scaling relations, that is, relations between exponents, and bounds for the values of the critical exponents.

DOI: 10.1007/978-3-031-59900-2_4

Related papers

Measures of Critical Exponents in the Four Dimensional Site Percolation

H. G. Ballesteros, L. A. Fernández, Víctor Martín-Mayor +3 more

Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-toscaling exponent and, with great accuracy, the critical density. Key words: Lattice. Monte Carlo. Percolation. Critical exponents. Finite size scaling. ffl-expansion. PACS: 12.40.Ee;64.60.A;75.40.Mg;75.40.Cx hector@lattice.fis.ucm.es y laf@lattice.fis.ucm.es z victor@lattice.fis.ucm.es x sudupe@lattice.fis.ucm.es -- parisi@roma1.infn.it k ruiz@chimera.roma1.infn.it 1 Introduction From the point of view of its definition, the simplest statistical system is perhaps the percolation. In the case of the site percolation, we fill the sites of a given lattice with probability p. Then we construct the clusters as sets of contiguous filled sites. The critical properties of the system can be described in terms of the...Using finite-size scaling methods we measure the thermal and magnetic exponents of the site percolation in four dimensions, obtaining a value for the anomalous dimension very different from the results found in the literature. We also obtain the leading corrections-toscaling exponent and, with great accuracy, the critical density. Key words: Lattice. Monte Carlo. Percolation. Critical exponents. Finite size scaling. ffl-expansion. PACS: 12.40.Ee;64.60.A;75.40.Mg;75.40.Cx hector@lattice.fis.ucm.es y laf@lattice.fis.ucm.es z victor@lattice.fis.ucm.es x sudupe@lattice.fis.ucm.es -- parisi@roma1.infn.it k ruiz@chimera.roma1.infn.it 1 Introduction From the point of view of its definition, the simplest statistical system is perhaps the percolation. In the case of the site percolation, we fill the sites of a given lattice with probability p. Then we construct the clusters as sets of contiguous filled sites. The critical properties of the system can be described in terms of the...

Jan 1, 199632 citations

Finite-size effects in one-dimensional percolation: a verification of scaling theory

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Canadian Journal of PhysicsApr 1, 19793 citations

Finite-size scaling of energylike quantities in percolation

Youjin Deng, Xianqing Yang

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Physical Review EJun 13, 20064 citations

Scaling behavior in very small percolation lattices

J. B. Elliott, M. L. Gilkes, J. A. Hauger +7 more

We examine the average cluster distribution as a function of lattice probability for a very small $(L=6)$ lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of $L=6$ and $L=63$ lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.We examine the average cluster distribution as a function of lattice probability for a very small $(L=6)$ lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of $L=6$ and $L=63$ lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.

Physical Review CMar 1, 199727 citations

Critical dynamics at the percolation threshold

Anthony C. Maggs, Robin B. Stinchcombe

Dilution-induced criticality is discussed using exact and scaling descriptions. A simple exact argument is used to obtain the scaling form for the density of states in dilute chains. For higher-dimensional systems it is shown that the condition for the existence of a 'fraction' edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster, d and various static exponents. In addition the result dd/d f is obtained for the fracton dimensionality of the finite clusters near the percolation threshold; d f is the fractal dimension of the infinite cluster.Dilution-induced criticality is discussed using exact and scaling descriptions. A simple exact argument is used to obtain the scaling form for the density of states in dilute chains. For higher-dimensional systems it is shown that the condition for the existence of a 'fraction' edge is equivalent to the violation of a scaling relation between the fracton dimension of the infinite cluster, d and various static exponents. In addition the result dd/d f is obtained for the fracton dimensionality of the finite clusters near the percolation threshold; d f is the fractal dimension of the infinite cluster.

Journal of Physics A Mathematical and GeneralFeb 1, 19863 citations

Crossover Finite-Size Scaling Theory and Its Applications in Percolation

Ming Li, Sheng Fang, Jingfang Fan +1 more

Finite-size scaling (FSS) for a critical phase transition ( $t=0$ ) states that within a window of size $|t|\sim L^{-1/ν}$ , the scaling behavior of any observable $Q$ in a system of linear size $L$ asymptotically follows a scaling form as $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})$ , where $ν$ is the correlation-length exponent, $Y_Q$ is an FSS exponent and ${\tilde Q}(x)$ is a function of the scaled distance-to-criticality $x \equiv tL^{1/ν}$ . We systematically study the asymptotic scaling behavior of ${\tilde Q}(|x|\to\infty)$ for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with $t \to 0$ and $|x|\to\infty$ . This crossover FSS theory predicts that when the criticality is approached at a slower speed as $|t|\sim L^{-λ}$ with $λ<1/ν$ , the FSS becomes $λ$ -dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality $t=0$ can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.Finite-size scaling (FSS) for a critical phase transition ($t=0$) states that within a window of size $|t|\sim L^{-1/ν}$, the scaling behavior of any observable $Q$ in a system of linear size $L$ asymptotically follows a scaling form as $Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})$, where $ν$ is the correlation-length exponent, $Y_Q$ is an FSS exponent and ${\tilde Q}(x)$ is a function of the scaled distance-to-criticality $x \equiv tL^{1/ν}$. We systematically study the asymptotic scaling behavior of ${\tilde Q}(|x|\to\infty)$ for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with $t \to 0$ and $|x|\to\infty$. This crossover FSS theory predicts that when the criticality is approached at a slower speed as $|t|\sim L^{-λ}$ with $λ<1/ν$, the FSS becomes $λ$-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality $t=0$ can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.

arXiv:2412.06228Dec 9, 20241 citation

n(s,p)

n ( s , p ) in two dimension. We develop the scaling theory for

n(s,p)

n ( s , p ) and demonstrate how to use data-collapse plots as an efficient method to measure the critical exponents. We also demonstrate how we can use the scaling theory for

n(s,p)

{N}_{c}

and

{N}_{b}

, respectively. These quantities include ${C}_{2c}=(⟨{N}_{c}^{2}⟩\ensuremath{-}{⟨{N}_{c}⟩}^{2})∕{L}^{d}$ and ${C}_{cb}=(⟨{N}_{c}{N}_{b}⟩\ensuremath{-}⟨{N}_{c}⟩⟨{N}_{b}⟩)∕{L}^{d}$, where

L

is the linear system size and

d

is the spatial dimensionality. In statistical models with thermal fluctuations,

{C}_{2c}

and

{C}_{cb}

are specific heatlike quantities. Despite the absence of thermal fluctuations in percolation, we find that the leading finite-size scaling of

{C}_{2c}

and

{C}_{cb}

is described by the thermal critical exponent ${y}_{t}\text{\ensuremath{-}}d$. We also measure quantity ${\ensuremath{\kappa}}_{b}=2⟨{N}_{b}{S}_{2}⟩∕⟨{S}_{2}⟩\ensuremath{-}⟨{N}_{b}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩∕⟨{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}⟩\ensuremath{-}⟨{N}_{b}⟩$ and an analogous quantity ${\ensuremath{\kappa}}_{c}$ for

{N}_{c}

, where

{S}_{2}

and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}}_{4}$ are quantities associated with the second and the fourth moments of cluster sizes, respectively. At criticality, we show that ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ diverge as

{L}^{{y}_{t}}

for $L\ensuremath{\rightarrow}\ensuremath{\infty}$. The analysis of the data of ${\ensuremath{\kappa}}_{b}$ and ${\ensuremath{\kappa}}_{c}$ yields

{y}_{t}=1.145(2)

for the three-dimensional percolation, in good agreement with existing results.

We examine the average cluster distribution as a function of lattice probability for a very small

(L=6)

lattice and determine the scaling function of three-dimensional percolation. The behavior of the second moment, calculated from the average cluster distribution of

L=6

and

L=63

lattices, is compared to power-law behavior predicted by the scaling function. We also examine the finite-size scaling of the critical point and the size of the largest cluster at the critical point. This analysis leads to estimates of the critical exponent $\ensuremath{\nu}$ and the ratio of critical exponents $\ensuremath{\beta}/\ensuremath{\nu}$.

Finite-size scaling (FSS) for a critical phase transition (

t=0

) states that within a window of size

|t|\sim L^{-1/ν}

, the scaling behavior of any observable

Q

in a system of linear size

L

asymptotically follows a scaling form as

Q(t,L)=L^{Y_Q}\tilde{Q}(tL^{1/ν})

, where

ν

is the correlation-length exponent,

Y_Q

is an FSS exponent and

{\tilde Q}(x)

is a function of the scaled distance-to-criticality

x \equiv tL^{1/ν}

. We systematically study the asymptotic scaling behavior of

{\tilde Q}(|x|\to\infty)

for a broad variety of observables by requiring that the FSS and infinite-system critical behaviors match with each other in the crossover critical regime with

t \to 0

and

|x|\to\infty

. This crossover FSS theory predicts that when the criticality is approached at a slower speed as

|t|\sim L^{-λ}

with

λ<1/ν

, the FSS becomes

λ

-dependent and the exponent can be derived. As applications, explosive percolation and high-dimensional percolation are considered. For the former, it is shown that the widely observed anomalous phenomena at the infinite-system criticality

t=0

can be attributed to the mixing effects of the standard FSS behaviors around the pseudocritical point in an event-based ensemble. For the latter, FSS exponents are found to be different at the infinite-system critical and the pseudocritical point if free boundary conditions are used, and they are related to each other by using the crossover FSS theory. From these observations, the FSS of percolation systems falls into three classifications. Extensive simulations are carried out to affirm these predictions.