### Abstract

Let W(x,y) = a x^{3} + b x^{4} + f_{5} x ^{5} + f_{6} x^{6} + (3 a x^{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 = ∂W/∂x, Y = ∂W/∂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) ∈ ℝ _{+}^{2}

Original language | English |
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Pages (from-to) | 609-627 |

Number of pages | 19 |

Journal | Journal of Statistical Physics |

Volume | 127 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2007 May 1 |

Externally published | Yes |

### Keywords

- Fixed point uniqueness
- Renormalization group
- Self-avoiding paths
- Sierpinski gasket

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics