Correction-to-scaling exponent for two-dimensional percolation

Abstract. We study percolation on self-dual hypergraphs that contain hyperedges with four bounding vertices, or "four-edges", using three different generators, each containing bonds or sites with three distinct probabilities p, r, and t connecting the four vertices. We find explicit values of these probabilities that satisfy the selfduality conditions discussed by Bollobás and Riordan. This demonstrates that explicit solutions of the self-duality conditions can be found using generators containing bonds and sites with independent probabilities. These solutions also provide new examples of lattices where exact percolation critical points are known. One of the generators exhibits three distinct criticality solutions (p, r, t). We carry out Monte-Carlo simulations of two of the generators on two different hypergraphs to confirm the critical values. For the case of the hypergraph and uniform generator studied by Wierman et al., we also determine the threshold p = 0.441374 ± 0.000001, which falls within the tight bounds that they derived. Furthermore, we consider a generator in which all or none of the vertices can connect, and find a soluble inhomogeneous percolation system that interpolates between site percolation on the union-jack lattice and bond percolation on the square lattice.arXiv:1506.06125v2 [cond-mat.dis-nn]

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“…9. Here we show an alternate way of analyzing the data, where we plot s τ −2 P ≥s vs. s −Ω where Ω = 72/91 [21,22] …”

Section: Generator II On Hypergraph Amentioning

confidence: 99%

Percolation on hypergraphs with four-edges

Damavandi

Ziff

2015

J. Phys. A: Math. Theor.

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“…The curves become more and more sharp as L is increased. All these curves intersect approximately at the same point [p vc , Π(p vc )] with p vc ≈ 0.5577 and Π(p vc ) ≈ 0.61, which is slightly lower than the value 0.636454001 [38] obtained using Cardy's formula for cylindrical geometry [39]. plying a finite-size scaling form…”

Section: Percolation Through the Sites Occupied By Similarly Orienmentioning

confidence: 67%

Jamming and percolation properties of random sequential adsorption with relaxation

2018

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The random sequential adsorption (RSA) model is a classical model in Statistical Physics for adsorption on two-dimensional surfaces. Objects are deposited sequentially at random and adsorb irreversibly on the landing site, provided that they do not overlap any previously adsorbed object. The kinetics of adsorption ceases when no more objects can be adsorbed (jamming state). Here, we investigate the role of post-relaxation on the jamming state and percolation properties of RSA of dimers on a two-dimensional lattice. We consider that, if the deposited dimer partially overlaps with a previously adsorbed one, a sequence of dimer displacements may occur to accommodate the new dimer. The introduction of this simple relaxation dynamics leads to a more dense jamming state than the one obtained with RSA without relaxation. We also consider the anisotropic case, where one dimer orientation is favored over the other, finding a non-monotonic dependence of the jamming coverage on the strength of anisotropy. We find that the density of adsorbed dimers at which percolation occurs is reduced with relaxation, but the value depends on the strength of anisotropy.

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“…Similar universality is believed to hold also in higher dimensions, an important ingredient of advanced numerical methods [9][10][11][12][13][14]. Many nontrivial properties of critical percolation clusters were also derived using conformal field theory [15][16][17][18].…”

mentioning

confidence: 82%

Critical p=1/2 in percolation on semi-infinite strips

Koza

2019

Phys. Rev. E

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We study site percolation on lattices confined to a semi-infinite strip. For triangular and square lattices we find that the probability that a cluster touches the three sides of such a system at the percolation threshold has the continuous limit 1/2 and argue that this limit is universal for planar systems. This value is also expected to hold for finite systems for any self-matching lattice. We attribute this result to the asymptotic symmetry of the separation lines between alternating spanning clusters of occupied and unoccupied sites formed on the original and matching lattice, respectively.

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Correction-to-scaling exponent for two-dimensional percolation

Cited by 43 publications

References 47 publications

Percolation on hypergraphs with four-edges

Percolation on hypergraphs with four-edges

Jamming and percolation properties of random sequential adsorption with relaxation

Critical p=1/2 in percolation on semi-infinite strips

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