Journal of Mathematics and Statistics

Convergence of Renormalization Group Transformations of Gibbs Random Field

Farida Kachapova and Ilias Kachapov

DOI : 10.3844/jmssp.2016.135.151

Journal of Mathematics and Statistics

Volume 12, Issue 3

Pages 135-151


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.


© 2016 Farida Kachapova and Ilias Kachapov. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.