Journal of Statistical Physics

, Volume 127, Issue 3, pp 609–627 | Cite as

The Fixed Point of a Generalization of the Renormalization Group Maps for Self-Avoiding Paths on Gaskets

  • Tetsuya Hattori


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 \(\{(x,y)\in{\mathbb R}^2\mid x^2 \geqq y\} \setminus\backslash \{0\}\).


renormalization group fixed point uniqueness self-avoiding paths Sierpinski gasket 


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Mathematical Institute, Graduate School of ScienceTohoku UniversitySendaiJapan

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