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The Self-Avoiding-Walk and Percolation Critical Points in High Dimensions

Published online by Cambridge University Press:  12 September 2008

Takashi Hara  and
Gordon Slade
Takashi Hara
Affiliation:
Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 OEH, UK
Gordon Slade
Affiliation:
Isaac Newton Institute for Mathematical Sciences, 20 Clarkson Road, Cambridge CB3 OEH, UK
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Abstract

We prove the existence of an asymptotic expansion in the inverse dimension, to all orders, for the connective constant for self-avoiding walks on ℤd. For the critical point, defined as the reciprocal of the connective constant, the coefficients of the expansion are computed through order d−6, with a rigorous error bound of order d−7 Our method for computing terms in the expansion also applies to percolation, and for nearest-neighbour independent Bernoulli bond percolation on ℤd gives the 1/d-expansion for the critical point through order d−3, with a rigorous error bound of order d−4 The method uses the lace expansion.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 3 , September 1995 , pp. 197 - 215
Copyright
Copyright © Cambridge University Press 1995

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References

[1]Alm, S. E. (1993) Upper bounds for the connective constant of self-avoiding walks. Combinatorics, Probability and Computing 2 115–136.CrossRefGoogle Scholar
[2]Bollobás, B. and Kohayakawa, Y. (1994) Percolation in high dimensions. Euro. J. Combinatorics 15 113–125.CrossRefGoogle Scholar
[3]Brydges, D. C. and Spencer, T. (1985) Self-avoiding walk in 5 or more dimensions. Commun. Math. Phys. 97 125–148.CrossRefGoogle Scholar
[4]Conway, A. R. and Guttmann, A. J. (1993) Lower bound on the connective constant for square lattice self-avoiding walks. J. Phys. A: Math. Gen. 26 3719–3724.CrossRefGoogle Scholar
[5]Fisher, M. E. and Gaunt, D. S. (1964) Ising model and self-avoiding walks on hypercubical lattices and ‘high-density’ expansions. Phys. Rev. 133 A224–A239.CrossRefGoogle Scholar
[6]Fisher, M. E. and Singh, R. R. P. (1990) Critical points, large-dimensionality expansions, and the Ising spin glass. In Grimmett, G. R. and Welsh, D. J. A., eds., Disorder in Physical Systems. Clarendon Press.Google Scholar
[7]Gaunt, D. S. (1986) 1/d expansions for critical amplitudes. J. Phys. A: Math. Gen. 19 L149–L153.CrossRefGoogle Scholar
[8]Gaunt, D. S. and Ruskin, H. (1978) Bond percolation processes in d dimensions. J. Phys. A: Math. Gen. 11 1369–1380.CrossRefGoogle Scholar
[9]Gaunt, D. S., Sykes, M. F. and Ruskin, H. (1976) Percolation processes in d-dimensions. J. Phys. A: Math. Gen. 9 1899–1911.CrossRefGoogle Scholar
[10]Gerber, P. R. and Fisher, M. E. (1974) Critical temperatures of classical n-vector models on hypercubic lattices. Phys. Rev. B 10 4697–4703.CrossRefGoogle Scholar
[11]Gordon, D. M. (1991) Percolation in high dimensions. J. Lond. Math. Soc. (2) 44 373–384.CrossRefGoogle Scholar
[12]Grimmett, G. (1989) Percolation. Springer-Verlag.CrossRefGoogle Scholar
[13]Guttmann, A. J. (1987) On the critical behaviour of self-avoiding walks. J. Phys. A: Math. Gen. 20 1839–1854.CrossRefGoogle Scholar
[14]Hammersley, J. M. and Morton, K. W. (1954) Poor man's Monte Carlo. J. Roy. Stat. Soc. B 16 23–38.Google Scholar
[15]Hara, T. and Slade, G. (1990) Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 333–391.CrossRefGoogle Scholar
[16]Hara, T. and Slade, G. (1990) The lace expansion for self-avoiding walk in five or more dimensions. Reviews in Math. Phys. 4 235–327.CrossRefGoogle Scholar
[17]Hara, T. and Slade, G. (1992) Self-avoiding walk in five or more dimensions. I. The critical behaviour. Comm. Math. Phys. 147 101–136.CrossRefGoogle Scholar
[18]Hara, T. and Slade, G. (1993) Unpublished note.Google Scholar
[19]Hara, T., Slade, G. and Sokal, A. D. (1993) New lower bounds on the self-avoiding-walk connective constant. J. Stat. Phys. 72 479–517. Erratum. J. Stat. Phys. 78 1187–1188 (1995).CrossRefGoogle Scholar
[20]Kesten, H. (1964) On the number of self-avoiding walks. II. J. Math. Phys. 5 1128–1137.CrossRefGoogle Scholar
[21]Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals ½. Comm. Math. Phys. 74 41–59.CrossRefGoogle Scholar
[22]Kesten, H. (1990) Asymptotics in high dimensions for percolation. In Grimmett, G. R. and Welsh, D. J. A. eds., Disorder in Physical Systems. Clarendon Press.Google Scholar
[23]Madras, N. and Slade, G. (1993) The Self-Avoiding Walk. Birkhäuser.Google Scholar
[24]Nemirovsky, A. M., Freed, K. F., Ishinabe, T. and Douglas, J. F. (1992) End-to-end distance of a single self-interacting self-avoiding polymer chain: d −1 expansion. Phys. Lett. A 162 469–474.CrossRefGoogle Scholar
[25]Nemirovsky, A. M., Freed, K. F., Ishinabe, T. and Douglas, J. F. (1992) Marriage of exact enumeration and 1/d expansion methods: Lattice model of dilute polymers. J. Stat. Phys. 67 1083–1108.CrossRefGoogle Scholar
[26]Nguyen, B. G. and Yang, W.-S. (1991) Triangle condition for oriented percolation in high dimensions. Ann. Probab. 21 1809–1844.Google Scholar
[27]Slade, G. (1987) The diffusion of self-avoiding random walk in high dimensions. Comm. Math. Phys. 110 661–683.CrossRefGoogle Scholar