Scaling Limits and Critical Behaviour of the $$4$$ 4 -Dimensional $$n$$ n -Component $$|\varphi |^4$$ | φ | 4 Spin Model

Abstract

We consider the $$n$$ -component $$|\varphi |^4$$ spin model on $${\mathbb {Z}}^4$$ , for all $$n \ge 1$$ , with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent $$\frac{n+2}{n+8}$$ for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for $$n =1,2,3$$ ; double logarithmic scaling for $$n=4$$ ; and is bounded when $$n>4$$ . In addition, for the model defined on the $$4$$ -dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the $$|\varphi |^4$$ model.