We extend and apply a rigorous renormalisation group method to study critical correlation functions, on the 4-dimensional lattice Z 4 , for the weakly coupled n-component |ϕ| 4 spin model for all n ≥ 1, and for the continuous-time weakly self-avoiding walk. For the |ϕ| 4 model, we prove that the critical two-point function has |x| −2 (Gaussian) decay asymptotically, for n ≥ 1. We also determine the asymptotic decay of the critical correlations of the squares of components of ϕ, including the logarithmic corrections to Gaussian scaling, for n ≥ 1. The above extends previously known results for n = 1 to all n ≥ 1, and also observes new phenomena for n > 1, all with a new method of proof. For the continuous-time weakly selfavoiding walk, we determine the decay of the critical generating function for the "watermelon" network consisting of p weakly mutually-and self-avoiding walks, for all p ≥ 1, including the logarithmic corrections. This extends a previously known result for p = 1, for which there is no logarithmic correction, to a much more general setting. In addition, for both models, we study the approach to the critical point and prove existence of logarithmic corrections to scaling for certain correlation functions. Our method gives a rigorous analysis of the weakly self-avoiding walk as the n = 0 case of the |ϕ| 4 model, and provides a unified treatment of both models, and of all the above results.

We study the 4-dimensional n-component |ϕ| 4 spin model for all integers n ≥ 1, and the 4-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case n = 0 interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order p, and prove the existence of a logarithmic correction to mean-field scaling, with power 1 2 n+2 n+8 , for all n ≥ 0 and p > 0. The proof is based on an improvement of a rigorous renormalisation group method developed previously.1 2 n+2 n+8 . The independence of p in the exponents exemplifies the conventional wisdom that in critical phenomena all naturally defined length scales should exhibit the same asymptotic behaviour. The correlation length ξ is predicted to diverge in the same manner, but our method would require further development to prove this. Definitions of the modelsBefore defining the models, we establish some notation. Let L > 1 be an integer (which we will need to fix large). Consider the sequence Λ = Λ N = Z d /(L N Z d ) of discrete d-dimensional tori of side lengths L N , with N → ∞ corresponding to the infinite volume limit Λ N ↑ Z d . Throughout the paper, we only consider d = 4, but we sometimes write d instead of 4 to emphasise the role of dimension. For any of the 2d unit vectors e ∈ Z d , we define the discrete gradient of a function f : Λ N → R by ∇ e f x = f x+e − f x , and the discrete Laplacian by ∆ = − 1 2 e∈Z d :|e|=1 ∇ −e ∇ e . The gradient and Laplacian operators act component-wise on vector-valued functions. We also use the discrete Laplacian ∆ Z d on Z d , and the continuous Laplacian ∆ R d on R d .

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