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Frontiers of Physics

ISSN 2095-0462

ISSN 2095-0470(Online)

CN 11-5994/O4

邮发代号 80-965

2019 Impact Factor: 2.502

Frontiers of Physics  2020, Vol. 15 Issue (4): 41501   https://doi.org/10.1007/s11467-020-0972-6
  本期目录
N-cluster correlations in four- and five-dimensional percolation
Xiao-Jun Tan1,2, You-Jin Deng1,2(), Jesper Lykke Jacobsen3,4,5()
1. Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China
2. CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China
3. Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, Paris, France
4. Sorbonne Université, École Normale Supérieure, CNRS, Laboratoire de Physique (LPENS), 75005 Paris, France
5. Institut de Physique Théorique, Université Paris Saclay, CEA, CNRS, 91191 Gif-sur-Yvette, France
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Abstract

We study N-cluster correlation functions in four- and five-dimensional (4D and 5D) bond percolation by extensive Monte Carlo simulation. We reformulate the transfer Monte Carlo algorithm for percolation [Phys. Rev. E72, 016126 (2005)] using the disjoint-set data structure, and simulate a cylindrical geometry Ld−1 × ∞, with the linear size up to L = 512 for 4D and 128 for 5D. We determine with a high precision all possible N-cluster exponents, for N =2 and 3, and the universal amplitude for a logarithmic correlation function. From the symmetric correlator with N=2, we obtain the correlationlength critical exponent as 1/ν=1.4610(12) for 4D and 1/ν=1.737(2) for 5D, significantly improving over the existing results. Estimates for the other exponents and the universal logarithmic amplitude have not been reported before to our knowledge. Our work demonstrates the validity of logarithmic conformal field theory and adds to the growing knowledge for high-dimensional percolation.

Key wordscritical exponents    percolation    logarithmic conformal field theory    Monte Carlo algorithm
收稿日期: 2020-05-12      出版日期: 2020-07-21
Corresponding Author(s): You-Jin Deng,Jesper Lykke Jacobsen   
 引用本文:   
. [J]. Frontiers of Physics, 2020, 15(4): 41501.
Xiao-Jun Tan, You-Jin Deng, Jesper Lykke Jacobsen. N-cluster correlations in four- and five-dimensional percolation. Front. Phys. , 2020, 15(4): 41501.
 链接本文:  
https://academic.hep.com.cn/fop/CN/10.1007/s11467-020-0972-6
https://academic.hep.com.cn/fop/CN/Y2020/V15/I4/41501
1 S. R. Broadbent and J. M. Hammersley, Percolation processes, Proc. Camb. Philos. Soc. 53(3), 629 (1957)
https://doi.org/10.1017/S0305004100032680
2 D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd Ed., London: Taylor Francis, 1994
3 G. R. Grimmett, Percolation, 2nd Ed., Berlin: Springer, 1999
https://doi.org/10.1007/978-3-662-03981-6
4 B. Bollobás and O. Riordan, Percolation, Cambridge: Cambridge University Press, 2006
https://doi.org/10.1017/CBO9781139167383
5 P. W. Kasteleyn and C. M. Fortuin, Phase transitions in lattice systems with random local properties, J. Phys. Soc. Jpn. 26(Suppl.), 11 (1969)
6 R. B. Potts, Some generalized order-disorder transformations, Proc. Camb. Philos. Soc. 48(1), 106 (1952)
https://doi.org/10.1017/S0305004100027419
7 F. Y. Wu, The Potts model, Rev. Mod. Phys. 54(1), 235 (1982)
https://doi.org/10.1103/RevModPhys.54.235
8 H. A. Kramers and G. H. Wannier, Statistics of the twodimensional ferromagnet (Part I), Phys. Rev. 60(3), 252 (1941)
https://doi.org/10.1103/PhysRev.60.252
9 E. H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Lett. 18(17), 692 (1967)
https://doi.org/10.1103/PhysRevLett.18.692
10 R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys. 70(1), 193 (1972)
https://doi.org/10.1016/0003-4916(72)90335-1
11 A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory,Nucl. Phys. B 241(2), 333 (1984)
https://doi.org/10.1016/0550-3213(84)90052-X
12 D. Friedan, Z. Qiu, and S. Shenker, Conformal invariance, unitarity, and critical exponents in two dimensions, Phys. Rev. Lett. 52(18), 1575 (1984)
https://doi.org/10.1103/PhysRevLett.52.1575
13 B. Nienhuis, in: Phase Transition and Critical Phenomena, edited by C. Domb, M. Green, and J. L. Lebowitz, Academic Press, London, 1987, Vol. 11
14 J. L. Cardy, in: Phase Transition and Critical Phenomena, edited by C. Domb, M. Green, and J. L. Lebowitz, Academic Press, London, 1987, Vol. 11
15 G. F. Lawler, O. Schramm, and W. Werner, The dimension of the planar Brownian frontier is 4/3, Math. Res. Lett. 8(4), 401 (2001)
https://doi.org/10.4310/MRL.2001.v8.n4.a1
16 S. Smirnov and W. Werner, Critical exponents for twodimensional percolation, Math. Res. Lett. 8(6), 729 (2001)
https://doi.org/10.4310/MRL.2001.v8.n6.a4
17 A. Aharony, Y. Gefen, and A. Kapitulnik, Scaling at the percolation threshold above six dimension, J. Phys. A 17, L197 (1984)
https://doi.org/10.1088/0305-4470/17/4/008
18 T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128(2), 333 (1990)
https://doi.org/10.1007/BF02108785
19 R. Fitzner and R. van der Hofstad, Mean-field behavior for nearest-neighbor percolation in d>10, Electron. J. Probab. 22(0), 22 (2017)
https://doi.org/10.1214/17-EJP56
20 J. A. Gracey, Four loop renormalization of ϕ3 theory in six dimensions, Phys. Rev. D 92(2), 025012 (2015)
https://doi.org/10.1103/PhysRevD.92.025012
21 J. F. Wang, Z. Z. Zhou, W. Zhang, T. M. Garoni, and Y. J. Deng, Bond and site percolation in three dimensions, Phys. Rev. E 87(5), 052107 (2013)
https://doi.org/10.1103/PhysRevE.87.052107
22 X. Xu, J. F. Wang, J. P. Lv, and Y. J. Deng, Simultaneous analysis of three-dimensional percolation models, Front. Phys. 9(1), 113 (2014)
https://doi.org/10.1007/s11467-013-0403-z
23 G. Paul, R. M. Ziff, and H. E. Stanley, Percolation threshold, Fisher exponent, and shortest path exponent for four and five dimensions, Phys. Rev. E 64(2), 026115 (2001)
https://doi.org/10.1103/PhysRevE.64.026115
24 H. E. Stanley, in: Percolation Theory and Ergodic Theory of Infinite Particle Systems, edited by H. Kesten, IMA Volumes in Mathematics and Its Applications Vol. 8, New York: Springer-Verlag, 1987
25 V. Beffara and P. Nolin, On monochromatic arm exponents for 2D critical percolation, Ann. Probab. 39(4), 1286 (2011)
https://doi.org/10.1214/10-AOP581
26 M. Aizenman, B. Duplantier, and A. Aharony, Pathcrossing exponents and the external perimeter in 2D percolation, Phys. Rev. Lett. 83(7), 1359 (1999)
https://doi.org/10.1103/PhysRevLett.83.1359
27 R. Vasseur, J. L. Jacobsen, and H. Saleur, Logarithmic observables in critical percolation, J. Stat. Mech.: Theory Exp. 07, L07001 (2012)
https://doi.org/10.1088/1742-5468/2012/07/L07001
28 R. Vasseur and J. L. Jacobsen, Operator content of the critical Potts model in ddimensions and logarithmic correlations, Nucl. Phys. B 880, 435 (2014)
https://doi.org/10.1016/j.nuclphysb.2014.01.013
29 R. Couvreur, J. Lykke Jacobsen, and R. Vasseur, Nonscalar operators for the Potts model in arbitrary dimension, J. Phys. A Math. Theor. 50(47), 474001 (2017)
https://doi.org/10.1088/1751-8121/aa7f32
30 X. J. Tan, R. Couvreur, Y. J. Deng, and J. L. Jacobsen, Observation of nonscalar and logarithmic correlations in two- and three-dimensional percolation, Phys. Rev. E 99(5), 050103 (2019)
https://doi.org/10.1103/PhysRevE.99.050103
31 V. Gurarie and A. W. W. Ludwig, Conformal field theory at central charge c= 0 and two-dimensional critical systems with quenched disorder, arXiv: hep-th/0409105 (2004)
https://doi.org/10.1142/9789812775344_0032
32 P. Mathieu and D. Ridout, From percolation to logarithmic conformal field theory, Phys. Lett. B 657(1–3), 120 (2007)
https://doi.org/10.1016/j.physletb.2007.10.007
33 R. Vasseur, J. L. Jacobsen, and H. Saleur, Indecomposability parameters in chiral logarithmic conformal field theory, Nucl. Phys. B 851(2), 314 (2011)
https://doi.org/10.1016/j.nuclphysb.2011.05.018
34 V. Gurarie and A. W. W. Ludwig, Conformal algebras of two-dimensional disordered systems, J. Phys. Math. Gen. 35(27), L377 (2002)
https://doi.org/10.1088/0305-4470/35/27/101
35 W. Huang, P. C. Hou, J. F. Wang, R. M. Ziff, and Y. J. Deng, Critical percolation clusters in seven dimensions and on a complete graph, Phys. Rev. E 97(2), 022107 (2018)
https://doi.org/10.1103/PhysRevE.97.022107
36 Y. J. Deng and H. W. J. Blöte, Monte Carlo study of the site-percolation model in two and three dimensions, Phys. Rev. E 72(1), 016126 (2005)
https://doi.org/10.1103/PhysRevE.72.016126
37 J. Hoshen and R. Kopelman, Percolation and cluster distribution (I): Cluster multiple labeling technique and critical concentration algorithm, Phys. Rev. B 14(8), 3438 (1976)
https://doi.org/10.1103/PhysRevB.14.3438
38 B. A. Galler and M. J. Fisher, An improved equivalence algorithm, Commun. ACM 7(5), 301 (1964)
https://doi.org/10.1145/364099.364331
39 R. E. Tarjan and J. Van Leeuwen, Worst-case analysis of set union algorithms, J. Assoc. Comput. Mach. 31(2), 245 (1984)
https://doi.org/10.1145/62.2160
40 R. E. Tarjan, A class of algorithms which require nonlinear time to maintain disjoint sets, J. Comput. Syst. Sci. 18(2), 110 (1979)
https://doi.org/10.1016/0022-0000(79)90042-4
41 M. E. J. Newman and R. M. Ziff, Efficient Monte Carlo algorithm and high-precision results for percolation, Phys. Rev. Lett. 85(19), 4104 (2000)
https://doi.org/10.1103/PhysRevLett.85.4104
42 M. E. J. Newman and R. M. Ziff, Fast Monte Carlo algorithm for site or bond percolation, Phys. Rev. E 64(1), 016706 (2001)
https://doi.org/10.1103/PhysRevE.64.016706
43 M. M. Danziger, B. Gross, and S. V. Buldyrev, Faster calculation of the percolation correlation length on spatial networks, Phys. Rev. E 101(1), 013306 (2020)
https://doi.org/10.1103/PhysRevE.101.013306
44 H. W. J. Blöte and M. P. Nightingale, Critical behaviour of the two-dimensional Potts model with a continuous number of states; A finite size scaling analysis, Physica A 112(3), 405 (1982)
https://doi.org/10.1016/0378-4371(82)90187-X
45 Z. Koza and J. Poła, From discrete to continuous percolation in dimensions 3 to 7, J. Stat. Mech.: Theory Exp 2016(10), 103206 (2016)
https://doi.org/10.1088/1742-5468/2016/10/103206
46 M. Borinsky, J. A. Gracey, M. Kompaniets, and O. Schnetz (in preparation) (2020)
47 S. Mertens and C. Moore, Percolation thresholds and Fisher exponents in hypercubic lattices, Phys. Rev. E 98(2), 022120 (2018)
https://doi.org/10.1103/PhysRevE.98.022120
48 Z. P. Xun and R. M. Ziff, Precise bond percolation thresholds on several four-dimensional lattices, Phys. Rev. Research 2(1), 013067 (2020)
https://doi.org/10.1103/PhysRevResearch.2.013067
49 J. Cardy, Finite-size Scaling, Vol. 2, Elsevier, 2012
50 H. E. Stanley, Cluster shapes at the percolation threshold: and effective cluster dimensionality and its connection with critical-point exponents, J. Phys. Math. Gen. 10(11), L211 (1977)
https://doi.org/10.1088/0305-4470/10/11/008
51 X. Xu, J. F. Wang, Z. Z. Zhou, T. M. Garoni, and Y. J. Deng, Geometric structure of percolation clusters, Phys. Rev. E 89(1), 012120 (2014)
https://doi.org/10.1103/PhysRevE.89.012120
52 H. G. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. Muñoz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, Measures of critical exponents in the four-dimensional site percolation, Phys. Lett. B 400(3–4), 346 (1997)
https://doi.org/10.1016/S0370-2693(97)00337-7
53 Z. J. Zhang, P. C. Hou, S. Fang, H. Hu, and Y. J. Deng, Critical exponents and universal excess cluster number of percolation in four and five dimensions, arXiv: 2004.11289 (2020)
54 N. A. M. Araújo, P. Grassberger, B. Kahng, K. J. Schrenk, and R. M. Ziff, Recent advances and open challenges in percolation, Eur. Phys. J. Spec. Top. 223, 2307 (2014)
https://doi.org/10.1140/epjst/e2014-02266-y
55 M. Luczak and T. Luczak, The phase transition in the cluster-scaled model of a random graph, Random Structures Algorithms 28(2), 215 (2006)
https://doi.org/10.1002/rsa.20088
56 A. J. Guttmann and J. L. Jacobsen, Lattice models and integrability: A special issue in honour of F. Y. Wu, J. Phys. A Math. Theor. 45(49), 490301 (2012)
https://doi.org/10.1088/1751-8113/45/49/490301
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