Abstract
“Normal” and indefinitely-growing (IG) self-avoiding walks (SAWs) are exactly enumerated on several deterministic fractals (the Manderbrot-Given curve with and without dangling bonds, and the 3-simplex). On then th fractal generation, of linear sizeL, the average number of steps behaves asymptotically as 〈N〉=AL D saw+B. In contrast to SAWs on regular lattices, on these factals IGSAWs and “normal” SAWs have the same fractal dimensionD saw. However, they have different amplitudes (A) and correction terms (B).
Similar content being viewed by others
References
P. G. de Gennes,Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, New York, 1979).
A. B. Harris,Z. Phys. B 49:347 (1983).
D. Stauffer and A. Aharony,Introduction to Percolation Theory (Taylor & Francis, London, 1992). Revised 2nd edition, 1994.
A. Coniglio, N. Jan, I. Majid, and H. E. Stanley,Phys. Rev. B 35:3617 (1987).
R. M. Ziff,Physica D 38:377 (1989).
D. Dhar,J. Math. Phys. 19(1):5 (1978).
R. Rammal, G. Toulouse, and J. Vannimenus,J. Phys. (Paris)45:389 (1984).
B. Shapiro,J. Phys. C: Solid State Phys. 11:2829 (1978).
K. Kremer and J. W. Lyklema,Phys. Rev. Lett. 54:267 (1985).
K. Kremer and J. W. Lyklema,J. Phys. A 18:1515 (1985).
S. Redner and P. J. Reynolds,J. Phys. A 14:2679 (1981).
B. Berg and D. Forster,Phys. Lett. 106B:323 (1981).
C. A. de Carvalho and S. Caracciolo,J. Phys. (Paris)44:323 (1983).
B. B. Mandelbrot and J. A. Given:Phys. Rev. Lett. 52:1853 (1984).
H. E. Stanley,J. Phys. A 10:L211 (1977).
Y. Gefen, A. Aharony, B. B. Mandelbrot, and S. Kirkpatrick,Phys. Rev. Lett. 47: 1771 (1981).
P. Le Doussal and J. Machta,J. Stat. Phys. 64:541 (1991).
J. Vannimenus,Physica D 38:351 (1989).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shussman, Y., Aharony, A. Different types of self-avoiding walks on deterministic fractals. J Stat Phys 77, 545–563 (1994). https://doi.org/10.1007/BF02179449
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179449