Percolation thresholds for discorectangles: Numerical estimation for a range of aspect ratios

Tarasevich, Yuri Yu.; Eserkepov, Andrei V.

doi:10.1103/physreve.101.022108

Cited by 16 publications

(13 citation statements)

References 31 publications

(63 reference statements)

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“…7for the square and cubic lattices with z = (2k) d , in the limit that η c /k d is small or in other words the continuum limit where k is large and η c (k) for a discrete system is replaced by η c of the continuum. For circular neighborhoods, where η c of a disk equals 1.128087 [31,[45][46][47][48], one should thus expect from Eq. 9p c = 4.512348 z ,…”

Section: B Asymptotic Behaviormentioning

confidence: 99%

Bond percolation on simple cubic lattices with extended neighborhoods

Xun

Ziff

2020

Phys. Rev. E

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By means of Monte Carlo simulations, we study long-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors, up to the eighth nearest neighbors for the square lattice and the ninth nearest neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods can be mapped to problems of lattice percolation of extended shapes, such as disks and spheres, and the thresholds can be related to the continuum thresholds ηc for objects of those shapes. This mapping implies zpc ∼ 4ηc = 4.51235 in 2D and zpc ∼ 8ηc = 2.73512 in 3D for large z for circular and spherical neighborhoods respectively, where z is the coordination number. Fitting our data to the form pc = c/(z + b) we find good agreement with c = 2 d ηc, where the constant b represents a finite-z correction term. We also study power-law fits of the thresholds.

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Section: B Asymptotic Behaviormentioning

confidence: 99%

Bond percolation on simple cubic lattices with extended neighborhoods

Xun

Ziff

2020

Phys. Rev. E

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“…We will choose neighborhoods with sites that fall within a radius r. As z → ∞, the shape of the neighborhood becomes more circular (or spherical), so one can naturally suppose that the asymptotic behavior of zp c should be universal, with η c for a disk or spheres depending upon the dimension of the system. For circular neighborhoods in two dimensions, where η c for disks equals 1.128087 [9][10][11][12]54], one should thus expect…”

Section: Theoretical Analysismentioning

confidence: 99%

“…The most common one is to occupy sites or bonds on a regular lattice with statistically independent probability p. Site and bond percolation can be distinguished depending on the method of obtaining the cluster. One can also consider continuum percolation systems [9][10][11][12], such as overlapping disks and spheres placed randomly.…”

Section: Introductionmentioning

confidence: 99%

Universal behavior of site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in 2 and 3 dimensions

Xun,

Hao,

Ziff

2021

Preprint

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Extended-range percolation on various regular lattices, including all eleven Archimedean lattices in two dimensions, and the simple cubic (sc), body-centered cubic (bcc), and face-centered cubic (fcc) lattices in three dimensions, is investigated. In two dimensions, correlations between coordination number z and site thresholds pc for Archimedean lattices up to 10th nearest neighbors (NN) are seen by plotting z versus 1/pc and z versus −1/ ln(1 − pc), using the data of d'Iribarne et al. [J. Phys. A 32:2611[J. Phys. A 32: , 1999] and others. The results show that all the plots overlap on a line with a slope consistent with the theoretically predicted asymptotic value of zpc ∼ 4ηc = 4.51235, where ηc is the continuum threshold for disks. In three dimensions, precise site and bond thresholds for bcc and fcc lattices with 2nd and 3rd NN, and bond thresholds for the sc lattice with up to the 13th NN, are obtained by Monte-Carlo simulations, using an efficient single-cluster growth method. For site percolation, the values of thresholds for different types of lattices with compact neighborhoods also collapse together, and linear fitting is consistent with the predicted value of zpc ∼ 8ηc = 2.7351, where ηc is the continuum threshold for spheres. For bond percolation, Bethe-lattice behavior pc = 1/(z − 1) is expected to hold for large z, and the finite-z correction is confirmed to satisfy zpc − 1 ∼ a1z −x , with x = 2/3 for three dimensions as predicted by Frei and Perkins [Electron. J. Probab. 21:56, 2016] and by Xu et al. [Phys. Rev. E, 103:022127, 2021]. Our analysis indicates that for compact neighborhoods, the asymptotic behavior of zpc is universal, depending only upon the dimension of the system and whether site or bond percolation, but not upon the type of lattice.

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“…Since the introduction of this model, a host of papers on the subject has appeared in the physics literature (see for example [8] and [10] and the references therein). Many of these deal with a three-dimensional variant where the two-dimensional stick is replaced by other percolation objects, such as nanotubes or nanowires, which are suspended in some other material.…”

Section: Introductionmentioning

confidence: 99%

Higher-Dimensional Stick Percolation

Broman

2021

J Stat Phys

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We consider two cases of the so-called stick percolation model with sticks of length L. In the first case, the orientation is chosen independently and uniformly, while in the second all sticks are oriented along the same direction. We study their respective critical values $$\lambda _c(L)$$ λ c ( L ) of the percolation phase transition, and in particular we investigate the asymptotic behavior of $$\lambda _c(L)$$ λ c ( L ) as $$L\rightarrow \infty $$ L → ∞ for both of these cases. In the first case we prove that $$\lambda _c(L)\sim L^{-2}$$ λ c ( L ) ∼ L - 2 for any $$d\ge 2,$$ d ≥ 2 , while in the second we prove that $$\lambda _c(L)\sim L^{-1}$$ λ c ( L ) ∼ L - 1 for any $$d\ge 2.$$ d ≥ 2 .

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Percolation thresholds for discorectangles: Numerical estimation for a range of aspect ratios

Cited by 16 publications

References 31 publications

Bond percolation on simple cubic lattices with extended neighborhoods

Bond percolation on simple cubic lattices with extended neighborhoods

Universal behavior of site and bond percolation thresholds on regular lattices with compact extended-range neighborhoods in 2 and 3 dimensions

Higher-Dimensional Stick Percolation

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