LAUSR

We consider the n-component $$|\varphi |^4$$|φ|4 lattice spin model ($$n \ge 1$$n≥1) and the weakly self-avoiding walk ($$n=0$$n=0) on $$\mathbb Z^d$$Zd, in dimensions $$d=1,2,3$$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 $$r^{-(d+\alpha )}$$r-(d+α) with $$\alpha \in (0,2)$$α∈(0,2). The upper critical dimension is $$d_c=2\alpha $$dc=2α. For $$\varepsilon >0$$ε>0, and $$\alpha = \frac{1}{2} (d+\varepsilon )$$α=12(d+ε), the dimension $$d=d_c-\varepsilon $$d=dc-ε is below the upper critical dimension. For small $$\varepsilon $$ε, weak coupling, and all integers $$n \ge 0$$n≥0, we prove that the two-point function at the critical point decays with distance as $$r^{-(d-\alpha )}$$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.