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