### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**The fixed point of a generalization of the renormalization group maps for self-avoiding paths on gaskets.** / Hattori, Tetsuya.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The fixed point of a generalization of the renormalization group maps for self-avoiding paths on gaskets

AU - Hattori, Tetsuya

PY - 2007/5

Y1 - 2007/5

N2 - Let W(x,y) = a x3 + b x4 + f5 x 5 + f6 x6 + (3 a x2)2 y + g5 x5 y + h3 x3 y2 + h4 x4 y2 + n3 x3 y 3 + a24 x2 y4 + a05 y5 + a15 xy5 + a06 y6, and X = ∂W/∂x, Y = ∂W/∂y, where the coefficients are non-negative constants, with a > 0, such that X2(x, x2) - Y(x, x2) 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 (xf, yf) of Φ in the invariant set {(x, y) ∈ ℝ +2

AB - Let W(x,y) = a x3 + b x4 + f5 x 5 + f6 x6 + (3 a x2)2 y + g5 x5 y + h3 x3 y2 + h4 x4 y2 + n3 x3 y 3 + a24 x2 y4 + a05 y5 + a15 xy5 + a06 y6, and X = ∂W/∂x, Y = ∂W/∂y, where the coefficients are non-negative constants, with a > 0, such that X2(x, x2) - Y(x, x2) 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 (xf, yf) of Φ in the invariant set {(x, y) ∈ ℝ +2

KW - Fixed point uniqueness

KW - Renormalization group

KW - Self-avoiding paths

KW - Sierpinski gasket

UR - http://www.scopus.com/inward/record.url?scp=34247253248&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34247253248&partnerID=8YFLogxK

U2 - 10.1007/s10955-007-9283-3

DO - 10.1007/s10955-007-9283-3

M3 - Article

AN - SCOPUS:34247253248

VL - 127

SP - 609

EP - 627

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -