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

Abstract

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 $\varepsilon >0$, and $\alpha = \frac{1}{2} (d+\varepsilon )$, the dimension $d=d_c-\varepsilon$ is below the upper critical dimension. For small $\varepsilon$, 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.