A cluster expansion approach to renormalization group transformations

TL;DR

This paper rigorously defines the renormalization group map for classical Ising systems at high temperature using cluster expansion, demonstrating properties of the linearization of the RG transformation.

Contribution

It introduces a rigorous cluster expansion approach to analyze the RG map and its derivatives for lattice systems, establishing convergence and approximate band properties.

Findings

Existence of well-defined derivatives of the RG map.

Convergence of the cluster expansion for the RG derivatives.

Upper bounds for the RG linearization matrix.

Abstract

The renormalization group (RG) approach is largely responsible for the considerable success which has been achieved in developing a quantitative theory of phase transitions. This work treats the rigorous definition of the RG map for classical Ising-type lattice systems in the infinite volume limit at high temperature. A cluster expansion is used to justify the existence of the partial derivatives of the renormalized interaction with respect to the original interaction. This expansion is derived from the formal expressions, but it is itself well-defined and convergent. Suppose in addition that the original interaction is finite-range and translation-invariant. We will show that the matrix of partial derivatives in this case displays an approximate band property. This in turn gives an upper bound for the RG linearization.