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

DOI10.1007/S10955-017-1904-XzbMATH Open1387.82012arXiv1705.08540OpenAlexW3101029568MaRDI QIDQ1703078

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Publication date: 1 March 2018

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Abstract: We consider the $n$-component $|varphi{|}^4$ lattice spin model ($nge1$) 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 $alphain(0,2)$. The upper critical dimension is $d_c=2alpha$. For $epsilon>0$, and $alpha=frac12(d+epsilon)$, the dimension $d=d_c-epsilon$ is below the upper critical dimension. For small $epsilon$, weak coupling, and all integers $nge0$, 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.

Full work available at URL: https://arxiv.org/abs/1705.08540

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