Abstracts
Let W(x,y) = ax 3+ bx 4+ f 5 x 5+ f 6 x 6+ (3 ax 2)2 y+ g 5 x 5 y + h 3 x 3 y 2 + h 4 x 4 y 2 + n 3 x 3 y 3+a 24 x 2 y 4+a 05 y 5+a 15 xy 5+a 06 y 6, and X = \( X={\frac{\partial{W}}{\partial{x}}}\), \( Y={\frac{\partial{W}}{\partial{y}}}\), where the coefficients are non-negative constants, with a > 0, such that X 2(x,x 2)−Y(x,x 2) is a polynomial of x with non-negative coefficients.
Examples of the 2 dimensional map Φ: (x,y)↦ (X(x,y),Y(x,y)) satisfying the conditions are the renormalization group (RG) maps (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets.
We prove that there exists a unique fixed point (x f ,y f ) of Φ in the invariant set \(\{(x,y)\in{\mathbb R}^2\mid x^2 \geqq y\} \setminus\backslash \{0\}\).
Similar content being viewed by others
References
D. Dhar, Self-avoiding random walks: Some exactly soluble cases. J. Math. Phys. 19: 5–11 (1978).
K. Hattori, Exact Hausdorff dimension of self-avoiding processes on the multi-dimensional Sierpiński gasket. J. Math. Sci. Uni. Tokyo 7: 57–98 (2000).
K. Hattori and T. Hattori, Self-avoiding process on the Sierpiński gasket. Probab. Theory Rel. Fields 88: 405–428 (1991).
K. Hattori, T. Hattori, and S. Kusuoka, Self-avoiding paths on the pre-Sierpiński gasket. Probab. Theory Rel. Fields 84: 1–26 (1990).
K. Hattori, T. Hattori, and S. Kusuoka, Self-avoiding paths on the three dimensional Sierpiński gasket. Publications RIMS 29: 455–509 (1993).
T. Hattori, Random Walk and Renormalization Group—An Introduction to Mathematical Physics, (Kyoritsu Publishing, 2004) [in Japanese].
T. Hattori and S. Kusuoka, The exponent for mean square displacement of self-avoiding random walk on Sierpiński gasket. Probab. Theory Rel. Fields 93: 273–284 (1992).
T. Hattori and T. Tsuda, Renormalization group analysis of the self-avoiding paths on the d-dimensional Sierpi’nski gaskets. J. Stat. Phys. 109: 39–66 (2002).
S. Kumar, Y. Singh, and Y. P. Joshi, Critical exponents of self-avoiding walks on a family of truncated $n$-simplex lattices. J. Phys. A 23: 2987–3002 (1990).
R. Rammal, G. Toulouse, and J. Vannimenus, Self-avoiding walks on fractal spaces: Exact results and Flory approximation. Journal de Physique 45: 389–394 (1984).
Author information
Authors and Affiliations
Corresponding author
Additional information
2000 Mathematics Subject Classification Numbers: 82B28; 60G99; 81T17; 82C41.
Rights and permissions
About this article
Cite this article
Hattori, T. The Fixed Point of a Generalization of the Renormalization Group Maps for Self-Avoiding Paths on Gaskets. J Stat Phys 127, 609–627 (2007). https://doi.org/10.1007/s10955-007-9283-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-007-9283-3