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

## TL;DR

This paper rigorously proves that in long-range $O(n)$ models below the upper critical dimension, the critical two-point function decays with a mean-field exponent, confirming a prediction from physics literature using renormalisation group and cluster expansion techniques.

## Contribution

It provides the first rigorous proof of the critical two-point function decay with mean-field exponent in long-range models below the upper critical dimension.

## Key findings

- Critical two-point function decays as $r^{-(d-rac{1}{2}(d+ ext{epsilon}))}$
- Validation of physics prediction from 1972 about critical exponents
- Application of rigorous renormalisation group and cluster expansion methods

## 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 $\epsilon >0$, 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.