Abstract
We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2Δ 4 −γ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2Δ 4 −γ. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios <R 2 g >/<R 2 e >=0.14026±0.00007, <R 2 m >/<R 2 e >=0.43961±0.00034, and Ψ*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponentΔ 1=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy forΔ 1. Earlier Monte Carlo estimates ofv, which were ≈0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios <R 2 g >/<R 2 e >=0.1599±0.0002 and Ψ*=0.2471±0.0003; since Ψ*>0, hyperscaling holds. The approach to Ψ* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2Δ 4 −γ for two-dimensional SAWs.
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References
N. Madras and G. Slade,The Self-Avoiding Walk (Birkhäuser, Boston, 1993).
P. G. de Gennes,Phys. Lett. A 38:339 (1972).
J. des Cloizeaux,J. Phys. (Paris)36:281 (1975).
M. Daoud, J. P. Cotton, B. Farnoux, G. Jannink, G. Sarma, H. Benoit, R. Duplessix, C. Picot, and P. G. de Gennes,Macromolecules 8:804 (1975).
V. J. Emery,Phys. Rev. B 11:239 (1975).
C. Aragão de Carvalho, S. Caracciolo, and J. Fröhlich,Nucl. Phys. B 215[FS7]:209 (1983).
R. Fernández, J. Fröhlich, and A. D. Sokal,Random Walks, Critical Phenomena, and Trivality in Quantum Field Theory (Springer-Verlag, Berlin, 1992).
M. Lal,Mol. Phys. 17:57 (1969).
B. MacDonald, N. Jan, D. L. Hunter, and M. O. Steinitz,J. Phys. A: Math. Gen. 18:2627 (1985).
N. Madras and A. D. Sokal,J. Stat. Phys. 50:109 (1988).
K. F. Freed,Renormalization Group Theory of Macromolecules (Wiley, New York, 1987).
J. des Cloizeaux and G. Jannink,Polymers in Solution: Their Modelling and Structure (Oxford University Press, Oxford, 1990).
B. G. Nickel,Macromolecules 24:1358 (1991).
A. D. Sokal,Europhys. Lett. 27:661 (1994).
A. D. Sokal, Fundamental problems in the static scaling behavior of high-molecularweight polymers in dilute solution I. Critique of two-parameter theories, In preparation.
B. Widom,J. Chem. Phys. 43:3892 (1965).
M. E. Fisher,Rep. Prog. Phys. 30:615 (1967).
G. Stell,J. Chem. Phys. 51:2037 (1969).
N. S. Snider,J. Chem. Phys. 54:4587 (1971).
B. Widom,Physica 73:107 (1974).
M. E. Fisher, inCollective Properties of Physical Systems, B. Lundqvist and S. Lundqvist, eds. (Academic Press, New York, 1974).
C. K. Hall,J. Stat. Phys. 13:157 (1975).
T. Hara and G. Slade,Commun. Math. Phys. 147:101 (1992).
T. Hara and G. Slade,Rev. Math. Phys. 4:235 (1992).
T. Hara and G. Slade,Commun. Math. Phys. 128:333 (1990).
T. Hara,Prob. Theory Related Fields 86:337 (1990).
W. J. Camp, D. M. Saul, J. P. Van Dyke, and M. Wortis,Phys. Rev. B 14: 3990 (1976).
G. A. Baker Jr.,Phys. Rev. B 15:1552 (1977).
B. G. Nickel and B. Sharpe,J. Phys. A 12:1819 (1979).
J. J. Rehr,J. Phys. A 12:L179 (1979).
J. Zinn-Justin,J. Phys. (Paris)40:969 (1979).
B. Nickel,Physica 106A:48 (1981).
B. G. Nickel, inPhase Transitions, M. Lévy, J.-C. LeGuillou, and J. Zinn-Justin, eds. (Plenum Press, New York, 1982).
R. Roskies,Phys. Rev. B 23:6037 (1981).
R. Z. Roskies,Phys. Rev. B 24:5305 (1981).
J. Zinn-Justin,J. Phys. (Paris)42:783 (1981).
G. A. Baker jr. and J. M. Kincaid,J. Stat. Phys. 24:469 (1981).
J.-H. Chen, M. E. Fisher, and B. G. Nickel,Phys. Rev. Lett. 48:630 (1982).
J. Adler, M. Moshe, and V. Privman,Phys. Rev. B 26:3958 (1982).
M. Ferer and M. J. Velgakis,Phys. Rev. B 27:2839 (1983).
M. E. Fisher and J.-H. Chen,J. Phys. (Paris)46:1645 (1985).
A. J. Guttmann,Phys. Rev. B 33:5089 (1986).
R. Schrader and E. Tränkle,J. Stat. Phys. 25:269 (1981).
B. A. Freedman and G. A. Baker Jr.,J. Phys. A 15:L715 (1982).
M. N. Barber, R. B. Pearson, D. Toussaint, and J. L. Richardson,Phys. Rev. B 32: 1720 (1985).
K. Binder, M. Nauenberg, V. Privman, and A. P. Young,Phys. Rev. B 31:1498 (1985).
A. Hoogland, A. Compagner, and H. W. J. Blöte,Physica A 132:593 (1985).
G. A. Baker Jr. and N. Kawashima, Renormalized coupling constant for the three dimensional Ising model, Los Alamos preprint (1994).
M. Aizenman,Commun. Math. Phys. 86:1 (1982).
J. Glimm and A. Jaffe,Ann. Inst. Henri Poincaré A 22:97 (1975).
R. Schrader,Phys. Rev. B 14:172 (1976).
R. Schrader,Commun. Math. Phys. 49:131 (1976).
A. D. Sokal,Ann. Inst. Henri Poincaré A 37:317 (1982).
G. A. Baker Jr., inPhase Transitions and Critical Phenomena, Vol. 9, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1984).
M. E. Fisher, inRenormalization Group in Critical Phenomena and Quantum Fields, J. D. Gunton and M. S. Green, eds. (Temple University, Philadelphia, 1974).
F. J. Wegner and E. K. Riedel,Phys. Rev. B 7:248 (1973).
S.-K. Ma,Modern Theory of Critical Phenomena (Benjamin, Reading, Massachusetts, 1976).
D. J. Amit and L. Peliti,Ann. Phys. 140:207 (1982).
M. E. Fisher, inCritical Phenomena (Stellenbosh 1982), F. J. W. Hahne, ed. (Springer-Verlag, Berlin, 1983), pp. 1–139.
A. C. D. van Enter, R. Fernández, and A. D. Sokal,J. Stat. Phys. 72:879 (1993).
K. Gawedzki and A. Kupiainen,Commun. Math. Phys. 99:197 (1985).
K. Gawedzki and A. Kupiainen,Nucl. Phys. B 257[FS14]:474 (1985).
K. Gawedzki and A. Kupiainen,Commun. Math. Phys. 102:1 (1985).
J. Feldman, J. Magnen, V. Rivasseau, and R. Sénéor,Commun. Math. Phys. 103: 67 (1986).
K. Gawedzki and A. Kupiainen,Commun. Math. Phys. 89:191 (1983).
K. Gawedzki and A. Kupiainen,J. Stat. Phys. 35:267 (1984).
K. Gawedzki and A. Kupiainen,Commun. Math. Phys. 106:533 (1986).
G. Felder,Commun. Math. Phys. 102:139 (1985).
T. Niemeijer and J. M. J. van Leuuwen, inPhase Transitions and Critical Phenomena, Vol. 6, C. Domb and M. S. Green, eds. (Academic Press, New York, 1976).
R. H. Swendsen, inPhase Transitions, M. Lévy, J.-C. LeGuillou, and J. Zinn-Justin, eds. (Plenum Press, New York, 1982).
E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, inPhase Transitions and Critical Phenomena, Vol. 6, C. Domb and M. S. Green, eds. (Academic Press, New York, 1976).
D. S. Gaunt and A. J. Guttmann, inPhase Transitions and Critical Phenomena, Vol. 3, C. Domb and A. J. Green, eds. (Academic Press, New York, 1974).
A. J. Guttmann, inPhase Transitions and Critical Phenomena, Vol. 13, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1989).
J. Adler, M. Moshe, and V. Privman, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Israel Physical Society, 1983).
V. Privman,J. Phys. A. 16:3097 (1983).
M. N. Barber, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1983).
J. L. Cardy, ed.,Finite-Size Scaling (North-Holland, Amsterdam, 1988).
V. Privman, ed.,Finite Size Scaling and Numerical Simulation of Statistical Systems (World Scientific, Singapore, 1990).
A. D. Sokal, Monte Carlo methods in statistical mechanics: Foundations and new algorithms, Cours de Troisième Cycle de la Physique en Suisse Romande, Lausanne (June 1989).
A. D. Sokal,Nucl. Phys. B (Proc. Suppl.) 20:55 (1991).
A. D. Sokal, inQuantum Fields on the Computer, M. Creutz, ed. (World Scientific, Singapore, 1992).
S. Caracciolo, R. G. Edwards, S. J. Ferreira, A. Pelissetto, and A. D. Sokal, Extrapolating Monte Carlo simulations to infinite volume: finite-size scaling at ζ/L≫1,Phys. Rev. Lett., to appear.
A. D. Sokal, inMonte Carlo and Molecular Dynamics Simulations in Polymer Science, K. Binder, ed. (Oxford University Press, Oxford, 1995).
A. Baumgärtner and K. Binder,J. Chem. Phys. 71:2541 (1979).
S. F. Edwards,Proc. Phys. Soc. Lond. 85:613 (1965).
S. R. S. Varadhan, Appendix to K. Symanzik inLocal Quantum Theory, R. Jost, ed. (Academic Press, New York, 1969).
J. Westwater,Commun. Math. Phys. 72:131 (1980).
J. Westwater,Commun. Math. Phys. 84:459 (1982).
A. Bovier, G. Felder, and J. Fröhlich,Nucl. Phys. B 230[FS10]:119 (1984).
V. Privman, P. C. Hohenberg, and A. Aharony, inPhase Transitions and Critical Phenomena, Vol. 14, C. Domb and J. L. Lebowitz, eds. (Academic Press, San Diego, 1991).
M. Muthukumar and B. G. Nickel,J. Chem. Phys. 80:5839 (1984).
J. des Cloizeaux, R. Conte, and G. Jannink,J. Phys. Lett. (Paris)46:L-595 (1985).
M. Muthukumar and B. G. Nickel,J. Chem. Phys. 86:460 (1987).
A. J. Barrett and B. G. Nickel, Private communication.
H. Fujita and T. Norisuye,Macromolecules 18:1637 (1985).
K. Huber and W. H. Stockmayer,Macromolecules 20:1400 (1987).
H. Fujita,Macromolecules 21:179 (1988).
H. Fujita,Polymer Solutions (Elsevier, Amsterdam, 1990).
A. J. Liu and M. E. Fisher,J. Stat. Phys. 58:431 (1990).
T. Hara, G. Slade, and A. D. Sokal,J. Stat. Phys. 72:479 (1993).
J. L. Cardy and A. J. Guttmann,J. Phys. A 26:2485 (1993).
G. Slade,Commun. Math. Phys. 110:661 (1987).
G. Slade,Ann. Prob. 17:91 (1989).
G. Slade,J. Phys. A: Math. Gen. 21:L417 (1988).
G. E. Uhlenbeck and G. W. Ford, inStudies in Statistical Mechanics, Vol. I, J. de Boer and G. E. Uhlenbeck, eds. (North-Holland, Amsterdam, 1962).
J. des Cloizeaux, Private communication cited in E. Brézin, inOrder and Fluctuation in Equilibrium and Nonequilibrium Statistical Mechanics, G. Nichols, G. Dewel, and J. W. Turner, eds. (Wiley-Interscience, New York, 1981).
J. C. LeGuillou and J. Zinn-Justin,Phys. Rev. B 21:3976 (1980).
J. C. LeGuillou and J. Zinn-Justin,J. Phys. Lett. 46:L-137 (1985).
J. C. LeGuillou and J. Zinn-Justin,J. Phys. (Paris)50:1365 (1989).
D. B. Murray and B. G. Nickel, Revised estimates for critical exponents for the continuumn-vector model in 3 dimensions, University of Guelph preprint (1991).
B. Nienhuis,Phys. Rev. Lett. 49:1062 (1982).
B. Nienhuis,J. Stat. Phys. 34:731 (1984).
D. S. McKenzie and C. Domb,Proc. Phys. Soc. Lond. 92:632 (1967).
F. J. Wegner,Phys. Rev. B 5:4529 (1972).
H. E. Staneley,Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 1971).
N. Madras, A. Orlitsky, and L. A. Shepp,J. Stat. Phys. 58:159 (1990).
G. Zifferer,Macromolecules 23:3166 (1990).
N. Eizenberg and J. Klafter,J. Chem. Phys. 99:3976 (1993).
D. E. Knuth,The Art of Computer Programming, Vol. 3 (Addison-Wesley, Reading, Massachusetts, 1973), Section 6.4.
T. H. Cormen, C. E. Leiserson, and R. L. Rivest,Introduction to Algorithms (MIT Press/ McGraw-Hill, Cambridge, Massachusetts/New York, 1990), Chapter 12.
K. Suzuki,Bull. Chem. Soc. Japan 41:538 (1968).
G. Zifferer,Mol. Simul. 6:103 (1991).
S. Redner and P. J. Reynolds,J. Phys. A 14:2679 (1981).
A. Berretti and A. D. Sokal,J. Stat. Phys. 40:483 (1985).
A. J. Barrett,Macromolecules 18:196 (1985), Section 3.
R. M. Karp and M. Luby, In24th IEEE Symposium on Foundations of Computer Science (IEEE, New York, 1983), pp. 56–64.
R. M. Karp, M. Luby, and N. Madras,J. Algorithms 10:429 (1989).
S. D. Silvey,Statistical Inference (Chapman and Hall, London, 1975).
S. Caracciolo, A. J. Guttmann, B. Li, A. Pelissetto, and A. D. Sokal, Correction-to-scaling exponents for two-dimensional self-avoiding walks, in preparation.
D. C. Rapaport,J. Phys. A 18:L39 (1985).
S. Caracciolo, A. Pelissetto, and A. D. Sokal,J. Phys. A 23:L969 (1990).
A. J. Barrett, M. Mansfield, and B. C. Benesch,Macromolecules 24:1615 (1991).
J. L. Cardy and H. Saleur,J. Phys. A 22:L601 (1989).
A. J. Guttmann, S. Merrilees, and A. D. Sokal, Unpublished (1985).
D. C. Rapaport,J. Phys. A 18:113 (1985).
J. Dayantis and J.-F. Palierne,J. Chem. Phys. 95:6088 (1991).
L. A. Johnson, A. Monge, and R. A. Friesner,J. Chem. Phys. 97:9355 (1992).
F. Shanes and B. G. Nickel, Calculation of the radius of gyration for a linear flexible polymer chain with excluded volume interaction,J. Chem. Phys., to appear.
J. Dayantis and J.-F. Palierne,Phys. Rev. B 49:3217 (1994).
A. J. Guttmann,J. Phys. A 22:2807 (1989).
B. G. Nickel,Physica A 177:189 (1991).
K. E. Newman and E. K. Riedel,Phys. Rev. B 30:6615 (1984).
A. D. Sokal, Fundamental problems in the static scaling behavior of high-molecularweight polymers in dilute solution II. Critical review of the experimental literature, In preparation.
A. Yamamoto, M. Fujii, G. Tanaka, and H. Yamakawa,Polymer J. 2:799 (1971).
M. Fukuda, M. Fukutomi, Y. Kato, and T. Hashimoto,J. Polymer Sci.: Polymer Phys. Ed. 12:871 (1974).
Y. Miyaki, Y. Einaga, and H. Fujita,Macromolecules 11:1180 (1978).
J.-C. Leguillou and J. Zinn-Justin,Phys. Rev. Lett. 39:95 (1977).
J. P. Cotton,J. Phys. Lett. (Paris)41:L-231 (1980).
H. Utiyama, S. Utsumi, Y. Tsunashima, and M. Kurata,Macromolecules 11:506 (1978).
B. Appelt and C. Meyerhoff,Macromolecules 13:657 (1980).
H. R. Haller, C. Destor, and D. S. Cannell,Rev. Sci. Instrum. 54:973 (1983).
B. Chu, R. Xu, T. Maeda, and H. S. Dhadwal,Rev. Sci. Instrum. 59:716 (1988).
K. B. Strawbridge, F. R. Hallett, and J. Watton,Can. J. Appl. Spectrosc. 36:53 (1991).
A. D. Sokal, Optimal statistical analysis of static light-scattering data from dilute polymer solutions, In preparation.
M. Benhamou and G. Mahoux,J. Phys. Lett. (Paris)46:L-689 (1985).
C. Domb and F. T. Hioe,J. Chem. Phys. 51:1915 (1969).
M. van Prooyen and B. G. Nickel, The second virial coefficient for self-avoiding walks on a lattice, In preparation.
P. J. Flory,Principles of Polymer Chemistry (Cornell University Press, Ithaca, New York, 1953).
H. Yamakawa,Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
P. G. DeGennes,Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, New York, 1979).
C. Domb and A. J. Barrett,Polymer 17:179 (1976).
J. F. Douglas and K. F. Freed,Macromolecules 18:201 (1985).
J. F. Douglas and K. F. Freed,J. Phys. Chem. 88:6613 (1984).
Z. Y. Chen and J. Noolandi,J. Chem. Phys. 96:1540 (1992).
Z. Y. Chen and J. Noolandi,Macromolecules 25:4978 (1992).
B. Krüger and L. Schäfer,J. Phys. I (Paris)4:757 (1994).
L. Schäfer,Phys. Rev. E 50:3517 (1994).
C. Bagnuls and C. Bervillier,Phys. Rev. B 41:402 (1990).
K. G. Wilson and J. Kogut,Phys. Rev. 12C:75 (1974).
K. Gawędzki and A. Kupiainen, inCritical Phenomena, Random Systems, Gauge Theories [Les Houches 1984], Part I, K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986), pp. 185–293.
J. Polchinski,Nucl. Phys. B 231:269 (1984).
J. Hughes and J. Liu,Nucl. Phys. B 307:183 (1988).
S. Weinberg,Phys. Rev. D 8:3497 (1973).
J. C. Collins and A. J. Macfarlane,Phys. Rev. D 10:1201 (1974).
S. W. MacDowell,Phys. Rev. D 12:1089 (1975).
B. Duplantier,J. Phys. (Paris)43:991 (1982).
B. Duplantier,J. Chem. Phys. 86:4233 (1987).
B. Duplantier,Phys. Rev. A 38:3647 (1988).
C. Domb and G. S. Joyce,J. Phys. C 5:956 (1972).
B. Duplantier and H. Saleur,Phys. Rev. Lett. 59:539 (1987).
S. Caracciolo, G. Ferraro, A. Pelissetto, and A. D. Sokal, Work in progress.
E. Orlandini, M. C. Tesi, and S. G. Whittington, Private communication.
G. Tanaka and K. Šolc,Macromolecules 15:791 (1982).
J. des Cloizeaux,J. Phys. (Paris)42:635 (1981).
J. F. Douglas and K. F. Freed,Macromolecules 17:1854 (1984).
S. Sternberg,Am. J. Math. 79:809 (1957);80:623 (1958);81:578 (1959).
V. I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed. (Springer-Verlag, Berlin, 1988), Chapter 5.
R. de la Llave, Invariant manifolds associated to non-resonant spectral subspaces, University of Texas preprint (1994).
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Li, B., Madras, N. & Sokal, A.D. Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks. J Stat Phys 80, 661–754 (1995). https://doi.org/10.1007/BF02178552
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DOI: https://doi.org/10.1007/BF02178552