A mass zero cluster expansion. Part 1: The Expansion

Abstract

A cluster expansion is developed and applied to study the perturbation λ(Δφ) 4 of the massless lattice field φ in dimension 3. The method is loosely inspired by the work of Gawedzki and Kupiainen on block spin techniques for the system. The cluster expansion is given in terms of expansion coefficients for the field as a sum of certain special block spin functions. These functions are chosen with a large number of moments zero, so that the interaction couples spatially separated functions with an interaction falling off as a high inverse power of the separation distance. The present techniques, with some technical development, should work for broad classes of other models, including the lattice dipole gas and the model. Models λ(, Δ , ; φ ) 2 s , α>1/2, are essentially included in the present work.