Critical Two-Point Function for Long-Range O(n) Models Below the Upper Critical Dimension

We consider the n-component lattice spin model () and the weakly self-avoiding walk () on , in dimensions . We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance r as with . The upper critical dimension is . For , and , the dimension is below the upper critical dimension. For small , weak coupling, and all integers , we prove that the two-point function at the critical point decays with distance as . 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.