Percolation theory

Essam, J W

doi:10.1088/0034-4885/43/7/001

Cited by 993 publications

(439 citation statements)

References 138 publications

(93 reference statements)

Supporting

Mentioning

408

Contrasting

Unclassified

Order By: Relevance

“…The second series method directly uses low-and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value Γ − /Γ + = 162.5 ± 2, a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.Percolation is one of the fundamental problems in statistical mechanics [1,2] and is perhaps the simplest system exhibiting critical behavior. Through its mapping to the q-state Potts model (for q → 1) many theoretical predictions follow, such as exact critical exponents in two dimensions.…”

supporting

confidence: 52%

Universal amplitude ratioΓ−∕Γ+for two-dimensional percolation

Jensen

Ziff

2006

Phys. Rev. E

View full text Add to dashboard Cite

The amplitude ratio of the susceptibility (or second size-moment) for two-dimensional percolation is calculated by two series methods and also by Monte-Carlo simulation. The first series method is a new approach based upon integrating approximations to the scaling function. The second series method directly uses low-and high-density series expansions of the susceptibility, going to unprecedented orders for both bond and site percolation on the square lattice. Putting all methods together we find a consistent value Γ − /Γ + = 162.5 ± 2, a significant improvement over previous results that placed the value of this ratio variously in the range of 14 to 220.Percolation is one of the fundamental problems in statistical mechanics [1,2] and is perhaps the simplest system exhibiting critical behavior. Through its mapping to the q-state Potts model (for q → 1) many theoretical predictions follow, such as exact critical exponents in two dimensions. Yet many unanswered questions remain. One of these is the value of amplitude ratios, which represent universal ratios of quantities related to integrals of the scaling function. A great amount of work has been done investigating amplitude ratios of various systems, both to demonstrate that systems expected to be in a given universality class have the same ratios, and to determine their values accurately [3]. The study of amplitude ratios remains an active area of research (e.g., [4,5,6,7,8,9,10,11]).In this paper we study specifically the universal amplitude ratio Γ − /Γ + for percolation, where Γ − and Γ + are the amplitudes of the second size-moment (also called the susceptibility) in the low-and high-density phases, respectively. This ratio has been especially difficult to estimate accurately and the status of the estimates is very controversial. In [3] values are quoted in a wide range from 14 to 220 based on numerical estimates from MonteCarlo simulations and series expansions for various percolation models. Furthermore, there are no reliable fieldtheoretical estimate for this quantity. Several years ago Delfino and Cardy [12] studied the q-state Potts model using methods from quantum field theory and predicted a value of 74.2 for percolation, by extrapolating the results for q = 2, 3, and 4 to q = 1. This was consistent with the numerical work of Corsten et al. [13], who gave the value 75 (+40, −26), but inconsistent with many other measurements [3]. In this work, we study bond and site percolation on the square lattice and use extensive exact enumerations to obtain estimates for Γ − /Γ + , using two different approaches: one a novel approach based upon directly integrating approximations to the scaling function, and the second a more conventional analysis of the high-and low-density series for the susceptibility. The estimates for the amplitude ratio are consistent with the value Γ − /Γ + = 162 ± 3. We also carried out a MonteCarlo calculation, which gave an almost identical value of 163 ± 2. These results are a significant improvement on previous published numerical est...

show abstract

supporting

confidence: 52%

Universal amplitude ratioΓ−∕Γ+for two-dimensional percolation

Jensen

Ziff

2006

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…This critical point appears to always be in the Ising universality class. Nonzero field values have been considered in scalar percolation studies previously [27,28,29,30,31,32,33]. A field may be introduced in percolation by allowing for the existence of "ghost bonds" which are present with probability h and connect sites directly to a solid background (or to "infinity").…”

Section: Introductionmentioning

confidence: 99%

Rigidity percolation in a field

Moukarzel

2003

Phys. Rev. E

View full text Add to dashboard Cite

Rigidity Percolation with g degrees of freedom per site is analyzed on randomly diluted Erdős-Renyi graphs with average connectivity γ, in the presence of a field h. In the (γ, h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities nF of uncanceled degrees of freedom and γr of redundant bonds. Upon crossing the coexistence line, γr and nF are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely that the density of uncanceled degrees of freedom is a "free energy" for Rigidity Percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g + 1)-core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point γc, Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g + 1)-rigid-core, i.e. the average coordination of these subgraphs is exactly 2g, although γc, the average coordination of the whole system, is smaller than 2g. γc is found to converge to 2g for large g, i.e. in this limit Maxwell counting is exact globally as well.

show abstract

“…Random breakdown in networks can be seen as a case of infinite-dimensional percolation. Two cases that have been solved exactly are Cayley trees [12] and Erdős-Rényi (ER) random graphs [13], where the networks collapse at known thresholds p c . Percolation on small-world networks (i.e., networks where every node is connected to its neighbors, plus some random long-range connections [9]) has also been studied by Moore and Newman [11].…”

mentioning

confidence: 99%

Resilience of the Internet to Random Breakdowns

Cohen¹,

Erez²,

Havlinl³

2011

The Structure and Dynamics of Networks

285

415

View full text Add to dashboard Cite

A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P (k) = ck −α . We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, pc, that need to be removed before the network disintegrates. We show analytically and numerically that for α ≤ 3 the transition never takes place, unless the network is finite. In the special case of the physical structure of the Internet (α ≈ 2.5), we find that it is impressively robust, with pc > 0. 99. 02.50.Cw, 05.40.a, 05.50.+q, 64.60.Ak

show abstract

Percolation theory

Cited by 993 publications

References 138 publications

Universal amplitude ratioΓ−∕Γ+for two-dimensional percolation

Universal amplitude ratioΓ−∕Γ+for two-dimensional percolation

Rigidity percolation in a field

Resilience of the Internet to Random Breakdowns

Contact Info

Product

Resources

About