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 .

We consider the n-component |ϕ| 4 lattice spin model (n ≥ 1) and the weakly self-avoiding walk (n = 0) on Z d , in dimensions d = 1, 2, 3. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance r as r −(d+α) with α ∈ (0, 2). The upper critical dimension is d c = 2α. For ǫ > 0, and α = 1 2 (d+ǫ), the dimension d = d c −ǫ is below the upper critical dimension. For small ǫ, weak coupling, and all integers n ≥ 0, we prove that the two-point function at the critical point decays with distance as r −(d−α) . This "sticking" of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion. Introduction and main resultBroadly speaking, the mathematical understanding of critical phenomena for spin systems has progressed in dimension d = 2, where exact solutions and SLE are important tools; in dimensions d > 4, where infrared bounds and the lace expansion are useful; and in dimension d = 4, where renormalisation group (RG) methods have been applied. The physically most important case of d = 3 is more difficult, and mathematical methods are scarce.In the physics literature, the ǫ-expansion was introduced to study non-integer dimensions slightly below d = 4. An alternate approach is to consider long-range models, which change the upper critical dimension from d c = 4 to a lower value d c = 2α with α ∈ (0, 2). By choosing d = 1, 2, 3 and α = 1 2 (d + ǫ) with small ǫ, it is possible to study integer dimension d which is slightly below the upper critical dimension 2α = d + ǫ. In this paper, we consider n-component spins and the weakly self-avoiding walk in this long-range context, and prove that the critical twopoint function has mean-field decay r −(d−α) also below the upper critical dimension. Our method involves a RG analysis in the vicinity of a non-Gaussian fixed point.

We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on Z 4 , for sufficiently small attraction. We prove that the susceptibility and correlation length of order p (for any p > 0) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of |x| −2 . This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric representation of the two-point function, and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self-avoiding walk, without self-attraction. The model and main resultThe self-avoiding walk is a basic model for a linear polymer chain in a good solution. The repulsive self-avoidance constraint models the excluded volume effect of the polymer. In a poor solution, the polymer tends to avoid contact with the solution by instead making contact with itself. This is modelled by a self-attraction favouring nearest-neighbour contacts. The self-avoiding walk is already a notoriously difficult problem, and the combination of these two competing tendencies creates additional difficulties and an interesting phase diagram.In this paper, we consider a continuous-time version of the weakly self-avoiding walk with nearest-neighbour contact self-attraction on Z 4 . When both the self-avoidance and self-attraction are sufficiently weak, we prove that the susceptibility and finite-order correlation length have logarithmic corrections to mean field scaling with exponents 1 4 and 1 8 for the logarithm, respectively, and that the critical two-point function is asymptotic to a multiple of |x| −2 as |x| → ∞.

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