Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for ${d\ge d_{\rm c}}$

Abstract

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 (Chen and Sakai in Ann Probab 43:639–681, 2015), 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 ${G_{p_{\rm c}}(x)}$ decays as ${|x|^{\alpha\wedge2-d}}$ above the upper-critical dimension ${d_{\rm c}\equiv(\alpha\wedge2)m}$ , 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 ${G_{p_{\rm c}}(x)}$ for the marginal case α = 2 decays as ${|x|^{2-d}/\log|x|}$ whenever d ≥ d_c (with a large spread-out parameter L). This solves the conjecture in Chen and Sakai (2015), extended all the way down to d = d_c, and confirms a part of predictions in physics (Brezin et al. in J Stat Phys 157:855–868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.