Universal finite-size scaling for percolation theory in high dimensions

Kenna, Ralph; Berche, Bertrand

doi:10.1088/1751-8121/aa6bd5

Cited by 23 publications

(23 citation statements)

References 94 publications

(467 reference statements)

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“…with γ 3 = 0.790 (26) in 3D. Similar power-law relations for various systems were studied by Galam and Mauger [40], van der Marck [28], and others, usually in terms of (z − 1) −γ4 rather than vs. z −γ4 .…”

Section: Sc-nn+2nnsupporting

confidence: 61%

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Precise bond percolation thresholds on several four-dimensional lattices

Xun

Ziff

2020

Phys. Rev. Research

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We study bond percolation on several four-dimensional (4D) lattices, including the simple (hyper) cubic (SC), the SC with combinations of nearest neighbors and second nearest neighbors (SC-NN+2NN), the body-centered cubic (BCC), and the face-centered cubic (FCC) lattices, using an efficient single-cluster growth algorithm. For the SC lattice, we find pc = 0.1601312(8), which confirms previous results (based on other methods), and find a new value pc = 0.035827(2) for the SC-NN+2NN lattice, which was not studied previously for bond percolation. For the 4D BCC and FCC lattices, we obtain pc = 0.074212(2) and 0.049517(2), which are substantially more precise than previous values. We also find critical exponents τ = 2.3135(5) and Ω = 0.40(3), consistent with previous results, including the recent four-loop series result of Gracey [Phys. Rev. D 92, 025012, (2015)], Ω = 0.4003.

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Section: Sc-nn+2nnsupporting

confidence: 61%

“…By examining wrapping probabilities, Wang et al [16,17] simulated the bond and site percolation models on several threedimensional lattices, including simple cubic (SC), the diamond, body-centered cubic (BCC), and face-centered cubic (FCC) lattices. Other recent work on percolation includes [18][19][20][21][22][23][24][25][26][27].…”

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confidence: 99%

Precise bond percolation thresholds on several four-dimensional lattices

Xun

Ziff

2020

Phys. Rev. Research

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“…We summarize the derivation of the temperature and magnetic exponents. These exponents do also apply in high-dimensional systems [19] with shortrange couplings. In the MF case, one has z = L 2 , and the edges of the lattice thus form a complete graph.…”

Section: Mean-field Percolationmentioning

confidence: 90%

“…(12) leads to an additional correction term proportional to L yt−2d in Eq. (19). The relative scale of this correction is L yt−2y h .…”

Section: B Finite-size Scalingmentioning

confidence: 99%

Equivalent-neighbor percolation models in two dimensions: Crossover between mean-field and short-range behavior

2018

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We investigate the influence of the range of interactions in the two-dimensional bond percolation model, by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges, as expressed by the number z of equivalent neighbors. We also consider the z → ∞ limit, i.e., the complete graph case, where percolation bonds are allowed between each pair of sites, and the model becomes mean-field-like. All investigated models with finite z are found to belong to the short-range universality class. There is no evidence of a tricritical point separating the shortrange and long-range behavior, such as is known to occur for q = 3 and q = 4 Potts models. We determine the renormalization exponent describing a finite-range perturbation at the mean-field limit as yr ≈ 2/3. Its relevance confirms the continuous crossover from mean-field percolation universality to short-range percolation universality. For finite interaction ranges, we find approximate relations between the coordination numbers and the amplitudes of the leading correction terms as found in the finite-size scaling analysis.

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“…Exponent d f is the standard fractal dimension of the clusters. Although well established for d < d u , FSS for d > d u is surprisingly subtle and depends on boundary conditions [17]. For instance, at the percolation threshold p c , it is predicted that the fractal dimension is d f = 4 [18] and 2d/3 [19] for systems with free and periodic boundary conditions, respectively.…”

Section: Introductionmentioning

confidence: 99%

Critical percolation clusters in seven dimensions and on a complete graph

et al. 2018

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We study critical bond percolation on a seven-dimensional hypercubic lattice with periodic boundary conditions (7D) and on the complete graph (CG) of finite volume (number of vertices) V. We numerically confirm that for both cases, the critical number density n(s,V) of clusters of size s obeys a scaling form n(s,V)∼s^{-τ}n[over ̃](s/V^{d_{f}^{*}}) with identical volume fractal dimension d_{f}^{*}=2/3 and exponent τ=1+1/d_{f}^{*}=5/2. We then classify occupied bonds into bridge bonds, which includes branch and junction bonds, and nonbridge bonds; a bridge bond is a branch bond if and only if its deletion produces at least one tree. Deleting branch bonds from percolation configurations produces leaf-free configurations, whereas deleting all bridge bonds leads to bridge-free configurations composed of blobs. It is shown that the fraction of nonbridge (biconnected) bonds vanishes, ρ_{n,CG}→0, for large CGs, but converges to a finite value, ρ_{n,7D}=0.0061931(7), for the 7D hypercube. Further, we observe that while the bridge-free dimension d_{bf}^{*}=1/3 holds for both the CG and 7D cases, the volume fractal dimensions of the leaf-free clusters are different: d_{lf,7D}^{*}=0.669(9)≈2/3 and d_{lf,CG}^{*}=0.3337(17)≈1/3. On the CG and in 7D, the whole, leaf-free, and bridge-free clusters all have the shortest-path volume fractal dimension d_{min}^{*}≈1/3, characterizing their graph diameters. We also study the behavior of the number and the size distribution of leaf-free and bridge-free clusters. For the number of clusters, we numerically find the number of leaf-free and bridge-free clusters on the CG scale as ∼lnV, while for 7D they scale as ∼V. For the size distribution, we find the behavior on the CG is governed by a modified Fisher exponent τ^{'}=1, while for leaf-free clusters in 7D, it is governed by Fisher exponent τ=5/2. The size distribution of bridge-free clusters in 7D displays two-scaling behavior with exponents τ=4 and τ^{'}=1. The probability distribution P(C_{1},V)dC_{1} of the largest cluster of size C_{1} for whole percolation configurations is observed to follow a single-variable function P[over ¯](x)dx, with x≡C_{1}/V^{d_{f}^{*}} for both CG and 7D. Up to a rescaling factor for the variable x, the probability functions for CG and 7D collapse on top of each other within the entire range of x. The analytical expressions in the x→0 and x→∞ limits are further confirmed. Our work demonstrates that the geometric structure of high-dimensional percolation clusters cannot be fully accounted for by their complete-graph counterparts.

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Universal finite-size scaling for percolation theory in high dimensions

Cited by 23 publications

References 94 publications

Precise bond percolation thresholds on several four-dimensional lattices

Precise bond percolation thresholds on several four-dimensional lattices

Equivalent-neighbor percolation models in two dimensions: Crossover between mean-field and short-range behavior

Critical percolation clusters in seven dimensions and on a complete graph

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