Scale limit and low momentum behavior of Euclidean fields. II

Abstract

The problem of the limit behavior of a rescaled random field εαφ (x/ε) as ε→0 is considered as an analog for random fields of the classical limit theorems for sums of random variables. If the random field φ does not depend on the scale ε, the limit is Gaussian in Euclidean field theory with a mass gap. If φ itself is a function of ε (giving a double sequence scheme), any infinitely divisible distribution can appear as the limit. In this formulation the limit of ε−dφ (x/ε) as ε→0 is discussed. It is proved that the limit field (if it exists) is a random field with values independent at every point. A correspondence between this scale limit and low momentum behavior is revealed. It is shown also that the scale limit may be considered as a limit of field theory when m→∞ and the coupling constant g→∞ in superrenormalizable theories and m→∞, g→0 for nonrenormalizable interactions. The scale limit of some ultraviolet regularized Euclidean fields is calculated. As an application some functional integrals are evaluated in the Ginzburg–Landau model in the low momentum limit.