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Damron M, Hanson J, Harper D, Lam WK. Transitions for exceptional times in dynamical first-passage percolation. Probab Theory Relat Fields 2022. [DOI: 10.1007/s00440-022-01178-1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
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2
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Hutchcroft T. Power-law bounds for critical long-range percolation below the upper-critical dimension. Probab Theory Relat Fields 2021. [DOI: 10.1007/s00440-021-01043-7] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
Abstract
AbstractWe study long-range Bernoulli percolation on $${\mathbb {Z}}^d$$
Z
d
in which each two vertices x and y are connected by an edge with probability $$1-\exp (-\beta \Vert x-y\Vert ^{-d-\alpha })$$
1
-
exp
(
-
β
‖
x
-
y
‖
-
d
-
α
)
. It is a theorem of Noam Berger (Commun. Math. Phys., 2002) that if $$0<\alpha <d$$
0
<
α
<
d
then there is no infinite cluster at the critical parameter $$\beta _c$$
β
c
. We give a new, quantitative proof of this theorem establishing the power-law upper bound $$\begin{aligned} {\mathbf {P}}_{\beta _c}\bigl (|K|\ge n\bigr ) \le C n^{-(d-\alpha )/(2d+\alpha )} \end{aligned}$$
P
β
c
(
|
K
|
≥
n
)
≤
C
n
-
(
d
-
α
)
/
(
2
d
+
α
)
for every $$n\ge 1$$
n
≥
1
, where K is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $$(2-\eta )(\delta +1)\le d(\delta -1)$$
(
2
-
η
)
(
δ
+
1
)
≤
d
(
δ
-
1
)
relating the cluster-volume exponent $$\delta $$
δ
and two-point function exponent $$\eta $$
η
.
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van den Berg J, Don H. A lower bound for point-to-point connection probabilities in critical percolation. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2020. [DOI: 10.1214/20-ecp326] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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5
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Koza Z. Critical p=1/2 in percolation on semi-infinite strips. Phys Rev E 2019; 100:042115. [PMID: 31770978 DOI: 10.1103/physreve.100.042115] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2019] [Indexed: 06/10/2023]
Abstract
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 a continuous limit of 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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Affiliation(s)
- Zbigniew Koza
- Faculty of Physics and Astronomy, University of Wrocław, 50-204 Wrocław, Poland
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6
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7
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Garban C, Pete G, Schramm O. The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane. ANN PROBAB 2018. [DOI: 10.1214/17-aop1252] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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8
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9
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10
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van den Berg J, Nolin P. Two-dimensional volume-frozen percolation: Exceptional scales. ANN APPL PROBAB 2017. [DOI: 10.1214/16-aap1198] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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11
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van den Berg J, Nolin P. Boundary rules and breaking of self-organized criticality in 2D frozen percolation. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2017. [DOI: 10.1214/17-ecp98] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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12
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Chelkak D, Duminil-Copin H, Hongler C. Crossing probabilities in topological rectangles for the critical planar FK-Ising model. ELECTRON J PROBAB 2016. [DOI: 10.1214/16-ejp3452] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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13
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Hammond A, Pete G, Schramm O. Local time on the exceptional set of dynamical percolation and the incipient infinite cluster. ANN PROBAB 2015. [DOI: 10.1214/14-aop950] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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14
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Cames van Batenburg W. The dimension of the incipient infinite cluster. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2015. [DOI: 10.1214/ecp.v20-3570] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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15
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16
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Posé N, Schrenk KJ, Araújo NAM, Herrmann HJ. Shortest path and Schramm-Loewner evolution. Sci Rep 2014; 4:5495. [PMID: 24975019 PMCID: PMC4074827 DOI: 10.1038/srep05495] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2014] [Accepted: 06/11/2014] [Indexed: 11/08/2022] Open
Abstract
We numerically show that the statistical properties of the shortest path on critical percolation clusters are consistent with the ones predicted for Schramm-Loewner evolution (SLE) curves for κ = 1.04 ± 0.02. The shortest path results from a global optimization process. To identify it, one needs to explore an entire area. Establishing a relation with SLE permits to generate curves statistically equivalent to the shortest path from a Brownian motion. We numerically analyze the winding angle, the left passage probability, and the driving function of the shortest path and compare them to the distributions predicted for SLE curves with the same fractal dimension. The consistency with SLE opens the possibility of using a solid theoretical framework to describe the shortest path and it raises relevant questions regarding conformal invariance and domain Markov properties, which we also discuss.
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Affiliation(s)
- N. Posé
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - K. J. Schrenk
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - N. A. M. Araújo
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - H. J. Herrmann
- Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
- Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil
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17
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Yao CL. Law of large numbers for critical first-passage percolation on the triangular lattice. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2014. [DOI: 10.1214/ecp.v19-3268] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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18
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Xu X, Wang J, Zhou Z, Garoni TM, Deng Y. Geometric structure of percolation clusters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012120. [PMID: 24580185 DOI: 10.1103/physreve.89.012120] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2013] [Indexed: 06/03/2023]
Abstract
We investigate the geometric properties of percolation clusters by studying square-lattice bond percolation on the torus. We show that the density of bridges and nonbridges both tend to 1/4 for large system sizes. Using Monte Carlo simulations, we study the probability that a given edge is not a bridge but has both its loop arcs in the same loop and find that it is governed by the two-arm exponent. We then classify bridges into two types: branches and junctions. A bridge is a branch iff at least one of the two clusters produced by its deletion is a tree. Starting from a percolation configuration and deleting the branches results in a leaf-free configuration, whereas, deleting all bridges produces a bridge-free configuration. Although branches account for ≈43% of all occupied bonds, we find that the fractal dimensions of the cluster size and hull length of leaf-free configurations are consistent with those for standard percolation configurations. By contrast, we find that the fractal dimensions of the cluster size and hull length of bridge-free configurations are given by the backbone and external perimeter dimensions, respectively. We estimate the backbone fractal dimension to be 1.643 36(10).
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Affiliation(s)
- Xiao Xu
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Junfeng Wang
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
| | - Zongzheng Zhou
- School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia
| | - Timothy M Garoni
- School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia
| | - Youjin Deng
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
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19
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Schrenk KJ, Posé N, Kranz JJ, van Kessenich LVM, Araújo NAM, Herrmann HJ. Percolation with long-range correlated disorder. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:052102. [PMID: 24329209 DOI: 10.1103/physreve.88.052102] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/11/2013] [Indexed: 06/03/2023]
Abstract
Long-range power-law correlated percolation is investigated using Monte Carlo simulations. We obtain several static and dynamic critical exponents as functions of the Hurst exponent H, which characterizes the degree of spatial correlation among the occupation of sites. In particular, we study the fractal dimension of the largest cluster and the scaling behavior of the second moment of the cluster size distribution, as well as the complete and accessible perimeters of the largest cluster. Concerning the inner structure and transport properties of the largest cluster, we analyze its shortest path, backbone, red sites, and conductivity. Finally, bridge site growth is also considered. We propose expressions for the functional dependence of the critical exponents on H.
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Affiliation(s)
- K J Schrenk
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - N Posé
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - J J Kranz
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - L V M van Kessenich
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - N A M Araújo
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
| | - H J Herrmann
- Computational Physics for Engineering Materials, Institute for Building Materials, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland and Departamento de Física, Universidade Federal do Ceará, 60451-970 Fortaleza, Ceará, Brazil
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20
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Beffara V, Duminil-Copin H. Planar percolation with a glimpse of Schramm–Loewner evolution. PROBABILITY SURVEYS 2013. [DOI: 10.1214/11-ps186] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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21
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Zhou Z, Yang J, Ziff RM, Deng Y. Crossover from isotropic to directed percolation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:021102. [PMID: 23005718 DOI: 10.1103/physreve.86.021102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2012] [Revised: 06/07/2012] [Indexed: 06/01/2023]
Abstract
We generalize the directed percolation (DP) model by relaxing the strict directionality of DP such that propagation can occur in either direction but with anisotropic probabilities. We denote the probabilities as p(↓) = pp(d) and p(↑) = p(1-p(d)), with p representing the average occupation probability and p(d) controlling the anisotropy. The Leath-Alexandrowicz method is used to grow a cluster from an active seed site. We call this model with two main growth directions biased directed percolation (BDP). Standard isotropic percolation (IP) and DP are the two limiting cases of the BDP model, corresponding to p(d) =1/2 and p(d) = 0,1 respectively. In this work, besides IP and DP, we also consider the 1/2 < p(d) <1 region. Extensive Monte Carlo simulations are carried out on the square and the simple-cubic lattices, and the numerical data are analyzed by finite-size scaling. We locate the percolation thresholds of the BDP model for p(d) = 0.6 and 0.8, and determine various critical exponents. These exponents are found to be consistent with those for standard DP. We also determine the renormalization exponent associated with the asymmetric perturbation due to p(d)-1/2 ≠ 0 near IP, and confirm that such an asymmetric scaling field is relevant at IP.
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Affiliation(s)
- Zongzheng Zhou
- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230027, People's Republic of China
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26
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Sapozhnikov A. The incipient infinite cluster does not stochastically dominate the invasion percolation cluster in two dimensions. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2011. [DOI: 10.1214/ecp.v16-1684] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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27
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Sun N. Conformally invariant scaling limits in planar critical percolation. PROBABILITY SURVEYS 2011. [DOI: 10.1214/11-ps180] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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29
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30
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31
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32
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Camia F. Scaling limits of two-dimensional percolation: an overview. STAT NEERL 2008. [DOI: 10.1111/j.1467-9574.2008.00400.x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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33
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Affiliation(s)
- Pierre Nolin
- Ecole Normale Supérieure and Université Paris-Sud
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34
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Riordan O, Walters M. Rigorous confidence intervals for critical probabilities. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:011110. [PMID: 17677413 DOI: 10.1103/physreve.76.011110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/09/2007] [Indexed: 05/16/2023]
Abstract
We use the method of Balister, Bollobás, and Walters [Random Struct. Algorithms 26, 392 (2005)] to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases.
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Affiliation(s)
- Oliver Riordan
- Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, United Kingdom
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35
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Camia F, Newman CM. Critical percolation exploration path and SLE
6: a proof of convergence. Probab Theory Relat Fields 2007. [DOI: 10.1007/s00440-006-0049-7] [Citation(s) in RCA: 37] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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36
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van den Berg J, Jarai A, Vagvolgyi B. The size of a pond in 2D invasion percolation. ELECTRONIC COMMUNICATIONS IN PROBABILITY 2007. [DOI: 10.1214/ecp.v12-1327] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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37
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Course 3 Conformal random geometry. ACTA ACUST UNITED AC 2006. [DOI: 10.1016/s0924-8099(06)80040-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register]
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38
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Lawler GF, Schramm O, Werner W. Conformal invariance of planar loop-erased random walks and uniform spanning trees. ANN PROBAB 2004. [DOI: 10.1214/aop/1079021469] [Citation(s) in RCA: 243] [Impact Index Per Article: 12.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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39
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40
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41
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Hara T, Slade G, van der Hofstad R. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. ANN PROBAB 2003. [DOI: 10.1214/aop/1046294314] [Citation(s) in RCA: 57] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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