Abstract
We compute by direct Monte Carlo simulation the main critical exponentsα, γ,Δ 4, andv and the effective coordination numberμ for the self-avoiding random walk in three dimensions on a cubic lattice. We find both hyperscaling relationsdv=2−α anddv− 2Δ 4+γ=0 satisfied ind = 3.
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J. Ambjørn, Ph. de Forcrand, F. Koukiou, and D. Petritis,Monte Carlo simulations of triangulated random surfaces, Preprint, Lausanne University (1987).
M. Aizenman,Commun. Math. Phys. 86:1–48 (1982).
M. Aizenman,Commun. Math. Phys. 97:91–110 (1985).
C. Aragão de Carvalho, S. Caracciolo, and J. Fröhlich,Nucl. Phys. B 215:209–248 (1983).
G. A. Baker, Jr.,Phys. Rev. B 15:1552–1559 (1975).
B. Baumann,Noncanonical path and surface simulation, DESY Preprint 86-134 (1986).
A. Berretti and A. D. Sokal,J. Stat. Phys. 40:483–531 (1985).
D. C. Brydges, J. Fröhlich, and T. Spencer,Commun. Math. Phys. 83:123–150 (1982).
D. C. Brydges, J. Fröhlich, and A. D. Sokal,Commun. Math. Phys. 91:141–186 (1983).
D. C. Brydges and T. Spencer,Commun. Math. Phys. 97:125–148 (1985).
Des Cloizeaux, private communivation, cited by E. Brézin, inOrder and Fluctuations in Equilibrium and Noneequilibrium Statistical Mechanics (17th Solvay Conference 1978), G. Nicolis, G. Dewel, and J. W. Turner, eds. (Wiley-Interscience, New York, 1981).
B. Derrida,J. Phys. A: Math. Gen. 14:L5-L9 (1981).
M. E. Fisher and Jing-Huei Chen,J. Phys. (Paris) 46:1645–1654 (1985).
Ph. de Forcrand, F. Koukiou, and D. Petritis,J. Stat. Phys. 45:459–470 (1986).
Ph. de Forcrand, F. Koukiou, and D. Petritis,Phys. Lett. B 189:341–342 (1987).
P. G. de Gennes,Phys. Lett. 38A:339–340 (1972);Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, New York, 1979).
A. J. Guttmann,J. Phys. A: Math. Gen. 17:455–468 (1980).
A. J. Guttmann,Phys. Rev. B 33:5089–5092 (1986).
A. J. Guttmann,On the critical behaviour of self-avoiding walks, University of Melbourne preprint (1986).
G. F. Lawler,Duke Math. J. 47:655–693 (1980).
G. F. Lawler,Commun. Math. Phys. 86:539–554 (1982).
G. F. Lawler,Commun. Math. Phys. 97:583–594 (1985).
J. C. LeGuillou and J. Zinn-Justin,Phys. Rev. B 21:3976–3998 (1980).
B. Nienhuis,Phys. Rev. Lett. 49:1062–1064 (1982).
B. G. Nickel and B. Sharpe,J. Phys. A: Math. Gen. 12:1819–1834 (1979).
G. Parisi and N. Sourlas,J. Phys. Lett. (Paris) 41:L403-L406 (1980).
J. J. Rehr,J. Phys. A: Math. Gen. 12:L179-L183 (1979).
D. C. Rapaport,J. Phys. A: Math. Gen. 18:113–126 (1985).
A. D. Sokal, unpublished result.
G. Slade, The diffusion of self-avoiding walk in high dimensions, University of Virginia preprint (1986).
S. R. S. Varadhan, Appendix to the course by K. Symanzik,Euclidean Quantum Field Theory (Academic Press, New York, 1969).
M. J. Westwater,Commun. Math. Phys. 72:131–174 (1980).
J. Zinn-Justin,J. Phys. (Paris) 40:969–975 (1979).
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de Forcrand, P., Koukiou, F. & Petritis, D. Critical exponents for the self-avoiding random walk in three dimensions. J Stat Phys 49, 223–234 (1987). https://doi.org/10.1007/BF01009959
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DOI: https://doi.org/10.1007/BF01009959