Research Article Open Access

Convergence of Renormalization Group Transformations of Gibbs Random Field

Farida Kachapova1 and Ilias Kachapov2
  • 1 Auckland University of Technology, New Zealand
  • 2 University of Auckland, New Zealand

Abstract

Statistical mechanics describes interaction between particles of a physical system. Particle properties of the system can be modelled with a random field on a lattice and studied at different distance scales using renormalization group transformation. Here we consider a thermodynamic limit of a lattice model with weak interaction and we use semi-invariants to prove that random fields transformed by renormalization group converge in distribution to an independent field with Gaussian distribution as the distance scale infinitely increases; it is a generalization of the central limit theorem to weakly dependent fields on a lattice.

Journal of Mathematics and Statistics
Volume 12 No. 3, 2016, 135-151

DOI: https://doi.org/10.3844/jmssp.2016.135.151

Submitted On: 13 March 2016 Published On: 27 July 2016

How to Cite: Kachapova, F. & Kachapov, I. (2016). Convergence of Renormalization Group Transformations of Gibbs Random Field. Journal of Mathematics and Statistics, 12(3), 135-151. https://doi.org/10.3844/jmssp.2016.135.151

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Keywords

  • Gibbs Measure
  • Renormalization Group
  • Semi-Invariant
  • Thermodymanic Limit
  • Weak Dependence