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Critical two-point function for long-range \(O(n)\) models below the upper critical dimension. (English) Zbl 1387.82012
Summary: We consider the \(n\)-component \(|\phi |^4\) lattice spin model (\(n \geq 1\)) 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 \(\alpha = \frac{1}{2} (d+\epsilon )\), the dimension \(d=d_c-\epsilon \) is below the upper critical dimension. For small \(\epsilon \), weak coupling, and all integers \(n \geq 0\), 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.

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B28 Renormalization group methods in equilibrium statistical mechanics
82B27 Critical phenomena in equilibrium statistical mechanics
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