Scaling limits and critical behaviour of the Φ⁴ spin model

We consider the n-componentΦ⁴ spin model on ℤ⁴, for all n ≥ 1 with small coupling constant.We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent(Formala presented)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.