N-cluster correlations in four- and five-dimensional percolation

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 L^d−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.

Keywords

critical exponents / percolation / logarithmic conformal field theory / Monte Carlo algorithm

References

[1] S. R. Broadbent and J. M. Hammersley, Percolation processes, Proc. Camb. Philos. Soc. 53(3), 629 (1957)

[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

[4] B. Bollobás and O. Riordan, Percolation, Cambridge: Cambridge University Press, 2006

[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)

[7] F. Y. Wu, The Potts model, Rev. Mod. Phys. 54(1), 235 (1982)

[8] H. A. Kramers and G. H. Wannier, Statistics of the twodimensional ferromagnet (Part I), Phys. Rev. 60(3), 252 (1941)

[9] E. H. Lieb, Exact solution of the problem of the entropy of two-dimensional ice, Phys. Rev. Lett. 18(17), 692 (1967)

[10] R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys. 70(1), 193 (1972)

[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)

[12] D. Friedan, Z. Qiu, and S. Shenker, Conformal invariance, unitarity, and critical exponents in two dimensions, Phys. Rev. Lett. 52(18), 1575 (1984)

[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)

[16] S. Smirnov and W. Werner, Critical exponents for twodimensional percolation, Math. Res. Lett. 8(6), 729 (2001)

[17] A. Aharony, Y. Gefen, and A. Kapitulnik, Scaling at the percolation threshold above six dimension, J. Phys. A 17, L197 (1984)

[18] T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128(2), 333 (1990)

[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)

[20] J. A. Gracey, Four loop renormalization of ϕ³ theory in six dimensions, Phys. Rev. D 92(2), 025012 (2015)

[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)

[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)

[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)

[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)

[26] M. Aizenman, B. Duplantier, and A. Aharony, Pathcrossing exponents and the external perimeter in 2D percolation, Phys. Rev. Lett. 83(7), 1359 (1999)

[27] R. Vasseur, J. L. Jacobsen, and H. Saleur, Logarithmic observables in critical percolation, J. Stat. Mech.: Theory Exp. 07, L07001 (2012)

[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)

[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)

[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)

[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)

[32] P. Mathieu and D. Ridout, From percolation to logarithmic conformal field theory, Phys. Lett. B 657(1–3), 120 (2007)

[33] R. Vasseur, J. L. Jacobsen, and H. Saleur, Indecomposability parameters in chiral logarithmic conformal field theory, Nucl. Phys. B 851(2), 314 (2011)

[34] V. Gurarie and A. W. W. Ludwig, Conformal algebras of two-dimensional disordered systems, J. Phys. Math. Gen. 35(27), L377 (2002)

[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)

[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)

[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)

[38] B. A. Galler and M. J. Fisher, An improved equivalence algorithm, Commun. ACM 7(5), 301 (1964)

[39] R. E. Tarjan and J. Van Leeuwen, Worst-case analysis of set union algorithms, J. Assoc. Comput. Mach. 31(2), 245 (1984)

[40] R. E. Tarjan, A class of algorithms which require nonlinear time to maintain disjoint sets, J. Comput. Syst. Sci. 18(2), 110 (1979)

[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)

[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)

[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)

[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)

[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)

[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)

[48] Z. P. Xun and R. M. Ziff, Precise bond percolation thresholds on several four-dimensional lattices, Phys. Rev. Research 2(1), 013067 (2020)

[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)

[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)

[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)

[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)

[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)

[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)

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