  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, with
spin-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 $\alpha = \frac 12 (d+\epsilon)$, the
dimension $d=d_c-\epsilon$ is below the upper critical dimension. For small
$\epsilon$, weak coupling, and all integers $n \ge 0$, we prove that the
two-point function at the critical point decays with distance as
$r^{-(d-\alpha)}$. 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.
