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Critical two-point function for long-range models with power-law couplings: the marginal case for \({d\ge d_{\mathrm{c}}}\). (English) Zbl 1431.82016
Summary: Consider the long-range models on \({\mathbb{Z}^d}\) of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as \({|x|^{-d-\alpha}}\) for some \({\alpha > 0} \). In the previous work [the authors, Ann. Probab. 43, No. 2, 639–681 (2015; Zbl 1342.60162)], we have shown in a unified fashion for all \({\alpha\ne2}\) that, assuming a bound on the “derivative” of the \({n}\)-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function \({G_{p_{\mathrm {c}}}(x)}\) decays as \({|x|^{\alpha\wedge2-d}}\) above the upper-critical dimension \({d_{\mathrm {c}}\equiv(\alpha\wedge2)m}\), where \(m = 2\) for self-avoiding walk and the Ising model and \(m = 3\) for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the \(n\)-step distribution, that \({G_{p_{\mathrm {c}}}(x)}\) for the marginal case \(\alpha = 2\) decays as \({|x|^{2-d}/\log|x|}\) whenever \(d \geq d_c\) (with a large spread-out parameter \(L)\). This solves the conjecture in [the authors, loc. cit.], extended all the way down to \(d = d_c\), and confirms a part of predictions in physics [E. Brezin et al., J. Stat. Phys. 157, No. 4–5, 855–868 (2014; Zbl 1318.82017)]. The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.
MSC:
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B43 Percolation
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B27 Critical phenomena in equilibrium statistical mechanics
82D40 Statistical mechanics of magnetic materials
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