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We consider the $n$-component $|\varphi|^4$ lattice spin model ($n \ge 1$)and the weakly self-avoiding walk ($n=0$) on $\mathbb{Z}^d$, in dimensions$d=1,2,3$. We study long-range models based on the fractional Laplacian, withspin-spin interactions or walk step probabilities decaying with distance $r$ as$r^{-(d+\alpha)}$ with $\alpha \in (0,2)$. The upper critical dimension is$d_c=2\alpha$. For $\epsilon >0$, and $\alpha = \frac 12 (d+\epsilon)$, thedimension $d=d_c-\epsilon$ is below the upper critical dimension. For small$\epsilon$, weak coupling, and all integers $n \ge 0$, we prove that thetwo-point function at the critical point decays with distance as$r^{-(d-\alpha)}$. This "sticking" of the critical exponent at its mean-fieldvalue was first predicted in the physics literature in 1972. Our proof is basedon a rigorous renormalisation group method. The treatment of observablesdiffers from that used in recent work on the nearest-neighbour 4-dimensionalcase, via our use of a cluster expansion.

Critical two-point function for long-range $O(n)$ models below the upper critical dimension