We prove that the susceptibility of the continuous-time weakly self-avoiding walk on Z d , in the critical dimension d = 4, has a logarithmic correction to mean-field scaling behaviour as the critical point is approached, with exponent 1 4 for the logarithm. The susceptibility has been well understood previously for dimensions d ≥ 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof begins by rewriting the walk twopoint function as the two-point function of a supersymmetric field theory. The field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers. By providing a setting where the methods of the companion papers are applied together, the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner.

We prove |x| −2 decay of the critical two-point function for the continuous-time weakly self-avoiding walk on Z d , in the upper critical dimension d = 4. This is a statement that the critical exponent η exists and is equal to zero. Results of this nature have been proved previously for dimensions d ≥ 5 using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.1 Main result IntroductionThe critical behaviour of the self-avoiding walk depends on the spatial dimension d. The upper critical dimension is 4, and for d ≥ 5 the lace expansion has been used to prove that the asymptotic behaviour is Gaussian [20,[29][30][31]43]. In particular, for the strictly self-avoiding walk in dimensions d ≥ 5, the critical two-point function has |x| −(d−2+η) decay with critical exponent η = 0, both for spread-out walks [21,30] and for the nearest-neighbour walk [29]. For d = 3, the problem remains completely unsolved from a mathematical point of view, but numerical and other evidence provides convincing evidence that the behaviour is not Gaussian. In particular, numerical values of the critical exponents γ and ν [22,42], together with Fisher's relation γ = (2 − η)ν, indicate that the critical two-point function has approximate decay |x| −1.031 for d = 3. For d = 2, the critical two-point function is predicted to decay as |x| −5/24 [40], and recent work suggests that the scaling *

This paper is the third in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. In this paper, we motivate and present a general approach towards second-order perturbative renormalisation, and apply it to a specific supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on Z d . Our focus is on the critical dimension d = 4. The results include the derivation of the perturbative flow of the coupling constants, with accompanying estimates on the coefficients in the flow. These are essential results for subsequent application to the 4-dimensional weakly self-avoiding walk, including a proof of existence of logarithmic corrections to their critical scaling. With minor modifications, our results also apply to the 4-dimensional n-component |ϕ| 4 spin model.

We consider random d-regular graphs on N vertices, with degree d at least (log N ) 4 . We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by Wigner's semicircle law, down to the optimal scale given by the typical eigenvalue spacing (up to a logarithmic correction). Aside from well-known consequences for the local eigenvalue distribution, this result implies the complete (isotropic) delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity.