Number of spanning clusters at the high-dimensional percolation thresholds

Fortunato, Santo; Aharony, Amnon; Coniglio, Antonio; Stauffer, Dietrich

doi:10.1103/physreve.70.056116

Cited by 23 publications

(19 citation statements)

References 15 publications

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“…Now, if a site is active, it is either in the infinite percolating cluster (of which there is only one for p > p ∞ ) or in one of the finite clusters, so [2,3,34]…”

Section: Regime Of Validitymentioning

confidence: 99%

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Universal finite-size scaling for percolation theory in high dimensions

Kenna¹,

Berche²

2017

J. Phys. A: Math. Theor.

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We present a unifying, consistent, finite-size-scaling picture for percolation theory bringing it into the framework of a general, renormalization-group-based, scaling scheme for systems above their upper critical dimensions d c . Behaviour at the critical point is non-universal in d > d c = 6 dimensions. Proliferation of the largest clusters, with fractal dimension 4, is associated with the breakdown of hyperscaling there when free boundary conditions are used. But when the boundary conditions are periodic, the maximal clusters have dimension D = 2d/3, and obey random-graph asymptotics. Universality is instead manifest at the pseudocritical point, where the failure of hyperscaling in its traditional form is universally associated with random-graph-type asymptotics for critical cluster sizes, independent of boundary conditions.

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“…Now, if a site is active, it is either in the infinite percolating cluster (of which there is only one for p > p ∞ ) or in one of the finite clusters, so [2,3,34]…”

Section: Regime Of Validitymentioning

confidence: 99%

“…[20,21,32,33] and summarised in Ref. [34] as "incomplete, leaving an open challenge for future research." Here we seek to meet that challenge.…”

Section: Introductionmentioning

confidence: 99%

Universal finite-size scaling for percolation theory in high dimensions

Kenna¹,

Berche²

2017

J. Phys. A: Math. Theor.

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“…The introduction of critical point symmetries concept [1,2,3,4,5,6,7,8], describing nuclei at points of shape-phase transitions between different limiting symmetries, was originally suggested by Iachello [1,2,3]. It is still one of the hot topics in nuclear structure physics.…”

Section: Introductionmentioning

confidence: 99%

Collective motion in prolate γ-rigid nuclei within minimal length concept via a quantum perturbation method

et al. 2018

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Based on the minimal length concept, inspired by Heisenberg algebra, a closed analytical formula is derived for the energy spectrum of the prolate γ-rigid Bohr-Mottelson Hamiltonian of nuclei, within a quantum perturbation method (QPM), by considering a scaled Davidson potential in β shape variable. In the resulting solution, called X(3)-D-ML, the ground state and the first β-band are all studied as a function of the free parameters. The fact of introducing the minimal length concept with a QPM makes the model very flexible and a powerful approach to describe nuclear collective excitations of a variety of vibrational-like nuclei. The introduction of scaling parameters in the Davidson potential enables us to get a physical minimum of this latter in comparison with previous works. The analysis of the corrected wave function, as well as the probability density distribution, shows that the minimal length parameter has a physical upper bound limit.

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“…Moving on to D = 4 and D = 5 dimensions, nothing spectacular happens and in particular we do not observe any indication of logarithmic corrections to the critical behavior in D = 5, in accordance with renormalization group expectations and also with newer simulations, cf. the careful discussion of five-dimensional percolation in (Fortunato et al, 2004).…”

Section: A Up To the Upper Critical Dimensionmentioning

confidence: 99%

High-temperature series expansions for theq-state Potts model on a hypercubic lattice and critical properties of percolation

Hellmund

Janke

2006

Phys. Rev. E

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We present results for the high-temperature series expansions of the susceptibility and free energy of the q-state Potts model on a D-dimensional hypercubic lattice Z D for arbitrary values of q. The series are up to order 20 for dimension D ≤ 3, order 19 for D ≤ 5 and up to order 17 for arbitrary D. Using the q → 1 limit of these series, we estimate the percolation threshold pc and critical exponent γ for bond percolation in different dimensions. We also extend the 1/D expansion of the critical coupling for arbitrary values of q up to order D −9 .

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Number of spanning clusters at the high-dimensional percolation thresholds

Cited by 23 publications

References 15 publications

Universal finite-size scaling for percolation theory in high dimensions

Universal finite-size scaling for percolation theory in high dimensions

Collective motion in prolate γ-rigid nuclei within minimal length concept via a quantum perturbation method

High-temperature series expansions for theq-state Potts model on a hypercubic lattice and critical properties of percolation

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