Long range correlation inequalities for massless Euclidean fields

Abstract

In this paper, new correlation inequalities are obtained for massless Euclidean fields on the $d$ dimensional integer lattice. Some of the inequalities have been obtained previously, in the case where the Lagrangian is a very small perturbation of a quadratic, using the renormalization group method. The results of the present paper apply provided the Lagrangian is uniformly convex. They therefore hold for the Coulomb dipole gas in which particle density can be of order $1$. The approach of the present paper is based on the methodology of Naddaf–Spencer, which relates second moment correlation functions for the Euclidean field to expectations of Green’s functions for parabolic PDE with random coefficients.