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

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Abstract

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

Original languageEnglish
Pages (from-to)609-627
Number of pages19
JournalJournal of Statistical Physics
Volume127
Issue number3
DOIs
Publication statusPublished - 2007 May
Externally publishedYes

Fingerprint

gaskets
Renormalization Group
Fixed point
Non-negative
Path
Change of Variables
Coefficient
coefficients
Invariant Set
Modulo
polynomials
Polynomial
Generalization

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

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title = "The fixed point of a generalization of the renormalization group maps for self-avoiding paths on gaskets",
abstract = "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",
keywords = "Fixed point uniqueness, Renormalization group, Self-avoiding paths, Sierpinski gasket",
author = "Tetsuya Hattori",
year = "2007",
month = "5",
doi = "10.1007/s10955-007-9283-3",
language = "English",
volume = "127",
pages = "609--627",
journal = "Journal of Statistical Physics",
issn = "0022-4715",
publisher = "Springer New York",
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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

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