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

A cluster expansion approach to renormalization group transformations