메뉴 건너뛰기




Volumn 485, Issue 3, 1997, Pages 551-582

On the number of incipient spanning clusters

Author keywords

Critical behavior; Hyperscaling; Incipient infinite cluster; Incipient spanning clusters; Percolation; Scaling limit

Indexed keywords


EID: 0031561930     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0550-3213(96)00626-8     Document Type: Article
Times cited : (137)

References (55)
  • 1
    • 0002656525 scopus 로고
    • Shapes, surfaces, and interfaces in percolation clusters
    • ed. M. Daoud and N. Boccara Springer, Berlin
    • [1] A. Coniglio, Shapes, surfaces, and interfaces in percolation clusters, in Proc. Les Houches Conf. on Physics of finely divided matter ed. M. Daoud and N. Boccara (Springer, Berlin, 1985).
    • (1985) Proc. Les Houches Conf. on Physics of Finely Divided Matter
    • Coniglio, A.1
  • 2
    • 0010851173 scopus 로고
    • Fractal and multifractal approaches to percolation: Some exact and not-so-exact results
    • ed. H. Kesten Springer, Berlin
    • [2] H.E. Stanley, Fractal and multifractal approaches to percolation: some exact and not-so-exact results, in Percolation Theory and Ergodic Theory of Infinite Particle Systems, ed. H. Kesten (Springer, Berlin, 1987).
    • (1987) Percolation Theory and Ergodic Theory of Infinite Particle Systems
    • Stanley, H.E.1
  • 4
    • 0000718295 scopus 로고
    • Fractals in physics: Squig clusters, diffusions, fractal measures, and unicity of fractal dimensionality
    • [4] B. Mandelbrot, Fractals in physics: Squig clusters, diffusions, fractal measures, and unicity of fractal dimensionality, J. Stat. Phys. 34 (1984) 895.
    • (1984) J. Stat. Phys. , vol.34 , pp. 895
    • Mandelbrot, B.1
  • 5
    • 51649183278 scopus 로고
    • Perspectives from the theory of phase transitions
    • [5] G. Toulouse, Perspectives from the theory of phase transitions, Nuovo Cimento B 23 (1974) 234.
    • (1974) Nuovo Cimento B , vol.23 , pp. 234
    • Toulouse, G.1
  • 6
  • 7
    • 34249952953 scopus 로고
    • Mean-field critical behavior for percolation in high dimensions
    • [7] T. Hara and G. Slade, Mean-field critical behavior for percolation in high dimensions, Commun. Math. Phys. 128 (1990) 333.
    • (1990) Commun. Math. Phys. , vol.128 , pp. 333
    • Hara, T.1    Slade, G.2
  • 8
    • 0010851861 scopus 로고
    • The lace expansion and the upper critical dimension for percolation
    • Amer. Math. Soc.
    • [8] G. Slade, The lace expansion and the upper critical dimension for percolation, in Mathematics of Random Media, Lectures in Applied Mathematics, vol. 27, Amer. Math. Soc., 1991.
    • (1991) Mathematics of Random Media, Lectures in Applied Mathematics , vol.27
    • Slade, G.1
  • 9
    • 0000035292 scopus 로고
    • Scaling at the percolation threshold above six dimensions
    • [9] A. Aharony, Y. Gefen, and A. Kapitulnik, Scaling at the percolation threshold above six dimensions, J. Phys. A 17 (1984) L197.
    • (1984) J. Phys. A , vol.17
    • Aharony, A.1    Gefen, Y.2    Kapitulnik, A.3
  • 10
    • 0000389586 scopus 로고
    • Branched polymer approach to the structure of lattice animals and percolation clusters
    • [10] S. Alexander, G. Grest, H. Nakanishi and T. Witten, Jr., Branched polymer approach to the structure of lattice animals and percolation clusters, J. Phys. A 17 (1984) L185.
    • (1984) J. Phys. A , vol.17
    • Alexander, S.1    Grest, G.2    Nakanishi, H.3    Witten T., Jr.4
  • 11
    • 0001639861 scopus 로고
    • Tree graph inequalities and critical behavior in percolation models
    • [11] M. Aizenman and C. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys. 36 (1984) 107.
    • (1984) J. Stat. Phys. , vol.36 , pp. 107
    • Aizenman, M.1    Newman, C.2
  • 13
    • 34250275869 scopus 로고
    • A note on percolation
    • [13] L. Russo, A note on percolation, Z. Wahr. 43 (1978) 39.
    • (1978) Z. Wahr. , vol.43 , pp. 39
    • Russo, L.1
  • 14
    • 77957093986 scopus 로고
    • Percolation probabilities on the square lattice
    • ed. B. Bollobás, North-Holland, Amsterdam
    • [14] P. Seymour and D. Welsh, Percolation probabilities on the square lattice, in Advances in Graph Theory, Annals of Discrete Mathematics ed. B. Bollobás, vol. 3 (North-Holland, Amsterdam, 1978).
    • (1978) Advances in Graph Theory, Annals of Discrete Mathematics , vol.3
    • Seymour, P.1    Welsh, D.2
  • 15
    • 0002790841 scopus 로고    scopus 로고
    • Universal scaling functions for numbers of percolating clusters on planar lattices
    • [15] C.-K. Hu and C.-Y. Lin, Universal scaling functions for numbers of percolating clusters on planar lattices, Phys. Rev. Lett. 77 ( 1996) 8.
    • (1996) Phys. Rev. Lett. , vol.77 , pp. 8
    • Hu, C.-K.1    Lin, C.-Y.2
  • 16
    • 0010788259 scopus 로고
    • Multiplicity of infinite clusters in percolation above six dimensions
    • [16] L. de Arcangelis, Multiplicity of infinite clusters in percolation above six dimensions, J. Phys. A 20 (1987) 3057.
    • (1987) J. Phys. A , vol.20 , pp. 3057
    • De Arcangelis, L.1
  • 17
    • 0007122530 scopus 로고
    • The geometry of critical percolation and conformal invariance
    • ed. Hao Bai-lin World Scientific, Singapore
    • [17] M. Aizenman, The geometry of critical percolation and conformal invariance, STATPHYS 19, Proceedings Xiamen 1995, ed. Hao Bai-lin (World Scientific, Singapore, 1995).
    • (1995) STATPHYS 19, Proceedings Xiamen 1995
    • Aizenman, M.1
  • 18
    • 0001227786 scopus 로고
    • Infinite clusters in percolation models
    • [18] C. Newman and L. Schulman, Infinite clusters in percolation models, J. Stat. Phys. 26 (1981) 613.
    • (1981) J. Stat. Phys. , vol.26 , pp. 613
    • Newman, C.1    Schulman, L.2
  • 19
    • 0001607449 scopus 로고
    • Uniqueness of the infinite cluster and continuity of connectivity functions for short- and long-range percolation
    • [19] M. Aizenman, H. Kesten, and C.M. Newman, Uniqueness of the infinite cluster and continuity of connectivity functions for short-and long-range percolation, Commun. Math. Phys. 111 (1987) 505.
    • (1987) Commun. Math. Phys. , vol.111 , pp. 505
    • Aizenman, M.1    Kesten, H.2    Newman, C.M.3
  • 20
    • 0000393829 scopus 로고
    • Density and uniqueness in percolation
    • [20] R.M. Burton and M. Keane, Density and uniqueness in percolation, Commun. Math. Phys. 121 (1989) 501.
    • (1989) Commun. Math. Phys. , vol.121 , pp. 501
    • Burton, R.M.1    Keane, M.2
  • 21
    • 0001710365 scopus 로고
    • The incipient infinite cluster in two-dimensional percolation
    • [21] H. Kesten, The incipient infinite cluster in two-dimensional percolation, Prob. Th. Rel. Fields 73 (1986) 369.
    • (1986) Prob. Th. Rel. Fields , vol.73 , pp. 369
    • Kesten, H.1
  • 22
    • 30544439717 scopus 로고
    • Cluster shapes at percolation thresholds: An effective cluster dimension and its connection with critical-point exponents
    • [22] H.E. Stanley, Cluster shapes at percolation thresholds: An effective cluster dimension and its connection with critical-point exponents, J. Phys. A 10 (1977) L211.
    • (1977) J. Phys. A , vol.10
    • Stanley, H.E.1
  • 23
    • 0345858335 scopus 로고
    • Cluster structure near the percolation threshold
    • [23] A. Coniglio, Cluster structure near the percolation threshold, J. Phys. A 15 (1982) 3829.
    • (1982) J. Phys. A , vol.15 , pp. 3829
    • Coniglio, A.1
  • 25
    • 0000104819 scopus 로고
    • On the critical percolation probabilities
    • [25] L. Russo, On the critical percolation probabilities, Z. Wahr. 56 (1981) 229.
    • (1981) Z. Wahr. , vol.56 , pp. 229
    • Russo, L.1
  • 26
    • 0002697969 scopus 로고
    • The critical probability of the bond percolation on the square lattice equals 1/2
    • [26] H. Kesten, The critical probability of the bond percolation on the square lattice equals 1/2, Commun. Math. Phys. 74 (1980) 41.
    • (1980) Commun. Math. Phys. , vol.74 , pp. 41
    • Kesten, H.1
  • 27
    • 0001755621 scopus 로고
    • Scaling relations for percolation
    • [27] H. Kesten, Scaling relations for percolation, Commun. Math. Phys. 109 (1987) 109.
    • (1987) Commun. Math. Phys. , vol.109 , pp. 109
    • Kesten, H.1
  • 28
    • 0002190594 scopus 로고
    • On a sharp transition from Area Law to Perimeter Law in a system of random surfaces
    • [28] M. Aizenman, J. Chayes, L. Chayes, J. Fröhlich and L. Russo, On a sharp transition from Area Law to Perimeter Law in a system of random surfaces, Commun. Math. Phys. 92 (1983) 19.
    • (1983) Commun. Math. Phys. , vol.92 , pp. 19
    • Aizenman, M.1    Chayes, J.2    Chayes, L.3    Fröhlich, J.4    Russo, L.5
  • 29
    • 85033088704 scopus 로고    scopus 로고
    • The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees
    • [29] K. Alexander, The RSW theorem for continuum percolation and the CLT for Euclidean minimal spanning trees, to appear in Ann. Probab.
    • Ann. Probab.
    • Alexander, K.1
  • 30
    • 0030501959 scopus 로고    scopus 로고
    • Non-uniqueness of spanning clusters in 2 to 5 dimensions
    • [30] P. Sen, Non-uniqueness of spanning clusters in 2 to 5 dimensions, Int. J. Mod. Phys. C 7 (1996) 603.
    • (1996) Int. J. Mod. Phys. C , vol.7 , pp. 603
    • Sen, P.1
  • 31
    • 0022116584 scopus 로고
    • Inequalities with applications to percolation and reliability
    • [31] J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability, J. Appl. Prob. 22 (1985) 556.
    • (1985) J. Appl. Prob. , vol.22 , pp. 556
    • Van Den Berg, J.1    Kesten, H.2
  • 32
    • 0000898051 scopus 로고
    • Critical percolation in finite geometries
    • [32] J. Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992) L201.
    • (1992) J. Phys. A , vol.25
    • Cardy, J.1
  • 33
    • 0001851446 scopus 로고
    • Coulomb gas formulation of two-dimensional phase transitions
    • (C. Domb and J. Lebowitz, eds.), Academic Press
    • [33] B. Nienhuis, Coulomb gas formulation of two-dimensional phase transitions, in Phase Transitions and Critical Phenomena (C. Domb and J. Lebowitz, eds.), Academic Press, 1987.
    • (1987) Phase Transitions and Critical Phenomena
    • Nienhuis, B.1
  • 34
    • 0003332552 scopus 로고
    • Conformal invariance and statistical mechanics
    • ed. E. Brézin and J. Zinn-Justin North-Holland, Amsterdam
    • [34] J. Cardy, Conformal invariance and statistical mechanics, in Fields, Strings and Critical Phenomena, ed. E. Brézin and J. Zinn-Justin (North-Holland, Amsterdam, 1990).
    • (1990) Fields, Strings and Critical Phenomena
    • Cardy, J.1
  • 35
    • 0003002413 scopus 로고
    • Conformal invariance in two-dimensional percolation
    • [35] R. Langlands, P. Pouiliot, and Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. AMS 30 (1994) 1.
    • (1994) Bull. AMS , vol.30 , pp. 1
    • Langlands, R.1    Pouiliot, P.2    Saint-Aubin, Y.3
  • 37
    • 0000699765 scopus 로고
    • Percolation critical exponents under the triangle condition
    • [37] D. Barsky and M. Aizenman, Percolation critical exponents under the triangle condition, Ann. Prob. 19 (1991) 1520.
    • (1991) Ann. Prob. , vol.19 , pp. 1520
    • Barsky, D.1    Aizenman, M.2
  • 39
    • 0000492120 scopus 로고
    • Finite size effects in phase transitions
    • [39] E. Brézin and J. Zinn-Justin, Finite size effects in phase transitions, Nucl. Phys. B [FS] 14 (1985) 867.
    • (1985) Nucl. Phys. B [FS] , vol.14 , pp. 867
    • Brézin, E.1    Zinn-Justin, J.2
  • 40
    • 0002446794 scopus 로고
    • A lower bound for the critical probability in a certain percolation process
    • [40] T. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Camb. Phil. Soc. 56 (1960) 13.
    • (1960) Proc. Camb. Phil. Soc. , vol.56 , pp. 13
    • Harris, T.1
  • 43
    • 85033091565 scopus 로고    scopus 로고
    • private communication
    • [43] D. Staufffer, private communication.
    • Staufffer, D.1
  • 44
    • 36149021418 scopus 로고
    • Two-dimensional Ising model as a soluble problem of many fermions
    • [44] T. Schultz, D. Mattis, and E. Lieb, Two-dimensional Ising model as a soluble problem of many fermions, Rev. Mod. Phys. 36 (1964) 856.
    • (1964) Rev. Mod. Phys. , vol.36 , pp. 856
    • Schultz, T.1    Mattis, D.2    Lieb, E.3
  • 45
    • 0001288503 scopus 로고
    • Inhomogeneous percolation problems and incipient infinite cluster
    • [45] J. Chayes, L. Chayes, and R. Durrett, Inhomogeneous percolation problems and incipient infinite cluster, J. Phys A: Math. Gen. 20 (1987) 1521.
    • (1987) J. Phys A: Math. Gen. , vol.20 , pp. 1521
    • Chayes, J.1    Chayes, L.2    Durrett, R.3
  • 47
    • 21344477318 scopus 로고
    • Tree-based models for random distribution of mass
    • [47] D. Aldous, Tree-based models for random distribution of mass, J. Stat. Phys. 73 (1993) 625.
    • (1993) J. Stat. Phys. , vol.73 , pp. 625
    • Aldous, D.1
  • 48
    • 0041855748 scopus 로고
    • On the random cluster model. I. Introduction and relation to other models
    • [48] C. Fortuin and P. Kasteleyn, On the random cluster model. I. introduction and relation to other models, Physica 57 (1972) 536.
    • (1972) Physica , vol.57 , pp. 536
    • Fortuin, C.1    Kasteleyn, P.2
  • 50
    • 0030597265 scopus 로고    scopus 로고
    • Renormalized coupling constant in the Ising model
    • [50] G. Baker, Renormalized coupling constant in the Ising model, 1996 preprint, submitted to J. Phys. A.
    • (1996) J. Phys. A.
    • Baker, G.1
  • 51
    • 84976179196 scopus 로고
    • Percolation processes II. The connective constant
    • [51] J. Hammersley, Percolation processes II. the connective constant, Proc. Camb. Phil. Soc., 53 (1957) 642.
    • (1957) Proc. Camb. Phil. Soc. , vol.53 , pp. 642
    • Hammersley, J.1
  • 52
    • 0000280765 scopus 로고
    • Percolation in half-spaces; equality of critical densities and continuity of the percolation probability
    • [52] D. Barsky, G. Grimmett and C. Newman, Percolation in half-spaces; equality of critical densities and continuity of the percolation probability, Prob. Th. Rel. Fields 90 (1991) 111.
    • (1991) Prob. Th. Rel. Fields , vol.90 , pp. 111
    • Barsky, D.1    Grimmett, G.2    Newman, C.3
  • 53
    • 0001626028 scopus 로고
    • The supercritical phase of percolation is well behaved
    • [53] G. Grimmett and J. Marstrand, The supercritical phase of percolation is well behaved, Proc. R. Soc. Lond. Ser. A 430 (1990) 439.
    • (1990) Proc. R. Soc. Lond. Ser. A , vol.430 , pp. 439
    • Grimmett, G.1    Marstrand, J.2
  • 54
    • 0000136970 scopus 로고
    • Sharpness of the phase transition in percolation models
    • [54] M. Aizenman and D. Barsky, Sharpness of the phase transition in percolation models, Commun. Math. Phys. 108 (1987) 489.
    • (1987) Commun. Math. Phys. , vol.108 , pp. 489
    • Aizenman, M.1    Barsky, D.2
  • 55
    • 0002637143 scopus 로고
    • Coincidence of critical points for Poisson percolation models
    • (547 in translation)
    • [55] M. Menshikov and A. Sidorenko, Coincidence of critical points for Poisson percolation models, Th. Prob. Appl. (in Russian) 32 (1987) 603 (547 in translation).
    • (1987) Th. Prob. Appl. (In Russian) , vol.32 , pp. 603
    • Menshikov, M.1    Sidorenko, A.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.