Renormalization Group Transformations Near the Critical Point: Some Rigorous Results

TL;DR

This paper provides rigorous results on the behavior of renormalization group transformations near the critical point for Ising-type lattice systems, revealing a band structure in the linearization of the RG map.

Contribution

It extends previous work by showing that the RG linearization matrix exhibits a band structure, offering new bounds and insights into the RG behavior near criticality.

Findings

Establishes a band structure on the matrix of partial derivatives of the renormalized interaction.

Provides an upper bound for the RG linearization near the critical point.

Builds on Dobrushin uniqueness and polymer expansion techniques for rigorous analysis.

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': exp(-H'(\sigma'))=\sum_\sigma T(\sigma, \sigma')exp(-H(\sigma)), where T(\sigma, \sigma') denotes a specific RG probability kernel, \sum_\sigma' T(\sigma, \sigma')=1, for every configuration \sigma. 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.