TY - JOUR
T1 - The fixed point of a generalization of the renormalization group maps for self-avoiding paths on gaskets
AU - Hattori, Tetsuya
N1 - Funding Information:
The research is supported in part by a Grant-in-Aid for Scientific Research (B) 17340022 from the Ministry of Education, Culture, Sports, Science and Technology.
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 | x2 ≧ y} \ {0}.
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 | x2 ≧ y} \ {0}.
KW - Fixed point uniqueness
KW - Renormalization group
KW - Self-avoiding paths
KW - Sierpinski gasket
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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 -