Absence of renormalization group pathologies near the critical temperature. Two examples

Karl Haller, Thomas G Kennedy

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We consider real-space renormalization group transformations for Ising-type systems which are formally defined by exp[ - H′(σ′)] = ∑ T(σ, σ′) exp[ - H(σ)] where T(σ, σ′) is a probability kernel, i.e., ∑σ′, T(σ, σ′) = 1, for every configuration σ. For each choice of the block spin configuration σ′, let μσ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional to T(σ, σ′) exp[ - H(σ)]. We give a condition which is sufficient to imply that the renormalized Hamiltonian H′ is defined. Roughly speaking, the condition is that the collection of measures μσ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacing b = 2 on the square lattice with β < 1.36βc and the Kadanoff transformation with parameter p on the triangular lattice in a subset of the β, p plane that includes values of β greater than βc.

Original languageEnglish (US)
Pages (from-to)607-637
Number of pages31
JournalJournal of Statistical Physics
Volume85
Issue number5-6
StatePublished - Dec 1996

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pathology
Critical Temperature
Renormalization Group
critical temperature
Configuration
configurations
Decimation
Triangular Lattice
Type Systems
Square Lattice
Ising
set theory
Spacing
iteration
theorems
Directly proportional
spacing
Sufficient
kernel
Iteration

Keywords

  • Completely analytic potentials
  • Dobrushin uniqueness theorem
  • Ising model
  • Renormalization group pathologies

ASJC Scopus subject areas

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

Cite this

Absence of renormalization group pathologies near the critical temperature. Two examples. / Haller, Karl; Kennedy, Thomas G.

In: Journal of Statistical Physics, Vol. 85, No. 5-6, 12.1996, p. 607-637.

Research output: Contribution to journalArticle

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