Renormalization group transformations near the critical point: Some rigorous results

Abstract

We consider renormalization group (RG) transformations for classical Ising-type lattice spin systems in the infinite-volume limit. Formally, the RG maps a Hamiltonian H into a renormalized Hamiltonian H′, \documentclass[12pt]{minimal}\begin{document}$\exp (-H^{\prime }(\sigma ^{\prime }))=\sum _{\sigma }T(\sigma , \sigma ^{\prime })\break\exp (-H(\sigma )),$\end{document}exp(−H′(σ′))=∑σT(σ,σ′)exp(−H(σ)), where T(σ, σ′) denotes a specific RG probability kernel, \documentclass[12pt]{minimal}\begin{document}$\sum _{\sigma ^{\prime }}T(\sigma , \sigma ^{\prime })=1$\end{document}∑σ′T(σ,σ′)=1, for every configuration σ. With the help of the Dobrushin uniqueness condition and standard results on the polymer expansion, Haller and Kennedy gave a sufficient condition for the existence of the renormalized Hamiltonian in a neighborhood of the critical point. By a more complicated but reasonably straightforward application of the cluster expansion machinery, the present investigation shows that their condition would further imply a band structure on the matrix of partial derivatives of the renormalized interaction with respect to the original interaction. This in turn gives an upper bound for the RG linearization.