A mass zero cluster expansion

AbstractA 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 $$\lambda (\mathop \nabla \limits^ \to \phi )^4 $$ 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 $$\lambda (\mathop \nabla \limits^ \to \phi )^4 $$ model. Models λ(,Δ,;φ)2s, α>1/2, are essentially included in the present work.