A cluster expansion approach to renormalization group transformations

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.Comment: 13 page

Renormalization group transformations near the critical point: Some rigorous results

“…Using cluster expansion techniques as in [22], we show that the Hypothesis also guarantees the existence of the partial derivatives of the RG transformation (Theorem 4.6). By a more careful analysis, we will then show that the partial derivative decays sub-exponentially as the distance between the set in the original lattice and the set in the image lattice gets large.…”